# NAME

PDL::LinearAlgebra::Real - PDL interface to the real lapack linear algebra programming library

# SYNOPSIS

```
use PDL::LinearAlgebra::Real;
$a = random (100,100);
$s = zeroes(100);
$u = zeroes(100,100);
$v = zeroes(100,100);
$info = 0;
$job = 0;
gesdd($a, $job, $info, $s , $u, $v);
```

# DESCRIPTION

This module provides an interface to parts of the real lapack library. These routines accept either float or double ndarrays.

# FUNCTIONS

## gtsv

` Signature: ([phys]DL(n); [phys]D(n); [phys]DU(n); [io,phys]B(n,nrhs); int [o,phys]info())`

Solves the equation

` A * X = B`

where A is an `n`

by `n`

tridiagonal matrix, by Gaussian elimination with partial pivoting, and B is an `n`

by `nrhs`

matrix.

Note that the equation `A**T*X = B`

may be solved by interchanging the order of the arguments DU and DL.

**NB** This differs from the LINPACK function `dgtsl`

in that `DL`

starts from its first element, while the LINPACK equivalent starts from its second element.

```
Arguments
=========
DL: On entry, DL must contain the (n-1) sub-diagonal elements of A.
On exit, DL is overwritten by the (n-2) elements of the
second super-diagonal of the upper triangular matrix U from
the LU factorization of A, in DL(1), ..., DL(n-2).
D: On entry, D must contain the diagonal elements of A.
On exit, D is overwritten by the n diagonal elements of U.
DU: On entry, DU must contain the (n-1) super-diagonal elements of A.
On exit, DU is overwritten by the (n-1) elements of the
first super-diagonal of the U.
B: On entry, the n by nrhs matrix of right hand side matrix B.
On exit, if info = 0, the n by nrhs solution matrix X.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, U(i,i) is exactly zero, and the solution
has not been computed. The factorization has not been
completed unless i = n.
```

```
$dl = random(float, 9);
$d = random(float, 10);
$du = random(float, 9);
$b = random(10,5);
gtsv($dl, $d, $du, $b, ($info=null));
print "X is:\n$b" unless $info;
```

gtsv ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gesvd

` Signature: ([io]A(m,n); int jobu(); int jobvt(); [o]s(minmn); [o]U(p,p); [o]VT(s,s); int [o]info())`

Computes the singular value decomposition (SVD) of a real M-by-N matrix A.

The SVD is written

` A = U * SIGMA * V'`

where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.

Note that the routine returns VT = V', not V.

```
jobu: Specifies options for computing all or part of the matrix U:
= 0: no columns of U (no left singular vectors) are
computed.
= 1: all M columns of U are returned in array U:
= 2: the first min(m,n) columns of U (the left singular
vectors) are returned in the array U;
= 3: the first min(m,n) columns of U (the left singular
vectors) are overwritten on the array A;
jobvt: Specifies options for computing all or part of the matrix
V':
= 0: no rows of V' (no right singular vectors) are
computed.
= 1: all N rows of V' are returned in the array VT;
= 2: the first min(m,n) rows of V' (the right singular
vectors) are returned in the array VT;
= 3: the first min(m,n) rows of V' (the right singular
vectors) are overwritten on the array A;
jobvt and jobu cannot both be 3.
A: On entry, the M-by-N matrix A.
On exit,
if jobu = 3, A is overwritten with the first min(m,n)
columns of U (the left singular vectors,
stored columnwise);
if jobvt = 3, A is overwritten with the first min(m,n)
rows of V' (the right singular vectors,
stored rowwise);
if jobu != 3 and jobvt != 3, the contents of A
are destroyed.
s: The singular values of A, sorted so that s(i) >= s(i+1).
U: If jobu = 1, U contains the M-by-M orthogonal matrix U;
if jobu = 3, U contains the first min(m,n) columns of U
(the left singular vectors, stored columnwise);
if jobu = 0 or 3, U is not referenced.
Min size = [1,1].
VT: If jobvt = 1, VT contains the N-by-N orthogonal matrix
V';
if jobvt = 2, VT contains the first min(m,n) rows of
V' (the right singular vectors, stored rowwise);
if jobvt = 0 or 3, VT is not referenced.
Min size = [1,1].
info: = 0: successful exit.
< 0: if info = -i, the i-th argument had an illegal value.
> 0: if bdsqr did not converge, info specifies how many
superdiagonals of an intermediate bidiagonal form B
did not converge to zero.
```

```
$a = random (float, 100,100);
$s = zeroes(float, 100);
$u = zeroes(float, 100,100);
$vt = zeroes(float, 100,100);
$info = pdl(long, 0);
gesvd($a, 2, 2, $s , $u, $vt, $info);
```

gesvd ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gesdd

` Signature: ([io]A(m,n); int jobz(); [o]s(minmn); [o]U(p,p); [o]VT(s,s); int [o]info(); int [t]iwork(iworkn))`

Computes the singular value decomposition (SVD) of a real M-by-N matrix A.

This routine use the Coppen's divide and conquer algorithm. It is much faster than the simple driver for large matrices, but uses more workspace.

```
jobz: Specifies options for computing all or part of matrix:
= 0: no columns of U or rows of V' are computed;
= 1: all M columns of U and all N rows of V' are
returned in the arrays U and VT;
= 2: the first min(M,N) columns of U and the first
min(M,N) rows of V' are returned in the arrays U
and VT;
= 3: If M >= N, the first N columns of U are overwritten
on the array A and all rows of V' are returned in
the array VT;
otherwise, all columns of U are returned in the
array U and the first M rows of V' are overwritten
on the array A.
A: On entry, the M-by-N matrix A.
On exit,
if jobz = 3, A is overwritten with the first N columns
of U (the left singular vectors, stored
columnwise) if M >= N;
A is overwritten with the first M rows
of V' (the right singular vectors, stored
rowwise) otherwise.
if jobz != 3, the contents of A are destroyed.
s: The singular values of A, sorted so that s(i) >= s(i+1).
U: If jobz = 1 or jobz = 3 and M < N, U contains the M-by-M
orthogonal matrix U;
if jobz = 2, U contains the first min(M,N) columns of U
(the left singular vectors, stored columnwise);
if jobz = 3 and M >= N, or jobz = 0, U is not referenced.
Min size = [1,1].
VT: If jobz = 1 or jobz = 3 and M >= N, VT contains the
N-by-N orthogonal matrix V';
if jobz = 2, VT contains the first min(M,N) rows of
V' (the right singular vectors, stored rowwise);
if jobz = 3 and M < N, or jobz = 0, VT is not referenced.
Min size = [1,1].
info: = 0: successful exit.
< 0: if info = -i, the i-th argument had an illegal value.
> 0: bdsdc did not converge, updating process failed.
```

```
$lines = 50;
$columns = 100;
$a = random (float, $lines, $columns);
$min = $lines < $columns ? $lines : $columns;
$s = zeroes(float, $min);
$u = zeroes(float, $lines, $lines);
$vt = zeroes(float, $columns, $columns);
$info = long (0);
gesdd($a, 1, $s , $u, $vt, $info);
```

gesdd ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ggsvd

` Signature: ([io]A(m,n); int jobu(); int jobv(); int jobq(); [io]B(p,n); int [o]k(); int [o]l();[o]alpha(n);[o]beta(n); [o]U(q,q); [o]V(r,r); [o]Q(s,s); int [o]iwork(n); int [o]info())`

Computes the generalized singular value decomposition (GSVD) of an M-by-N real matrix A and P-by-N real matrix B:

```
U'*A*Q = D1*( 0 R ), V'*B*Q = D2*( 0 R )
where U, V and Q are orthogonal matrices, and Z' is the transpose
of Z.
```

Let K+L = the effective numerical rank of the matrix (A',B')', then R is a K+L-by-K+L nonsingular upper triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) "diagonal" matrices and of the following structures, respectively:

```
If M-K-L >= 0,
K L
D1 = K ( I 0 )
L ( 0 C )
M-K-L ( 0 0 )
K L
D2 = L ( 0 S )
P-L ( 0 0 )
N-K-L K L
( 0 R ) = K ( 0 R11 R12 )
L ( 0 0 R22 )
where
C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
S = diag( BETA(K+1), ... , BETA(K+L) ),
C**2 + S**2 = I.
R is stored in A(1:K+L,N-K-L+1:N) on exit.
If M-K-L < 0,
K M-K K+L-M
D1 = K ( I 0 0 )
M-K ( 0 C 0 )
K M-K K+L-M
D2 = M-K ( 0 S 0 )
K+L-M ( 0 0 I )
P-L ( 0 0 0 )
N-K-L K M-K K+L-M
( 0 R ) = K ( 0 R11 R12 R13 )
M-K ( 0 0 R22 R23 )
K+L-M ( 0 0 0 R33 )
where
C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1), ... , BETA(M) ),
C**2 + S**2 = I.
(R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N), and R33 is stored
( 0 R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.
```

The routine computes C, S, R, and optionally the orthogonal transformation matrices U, V and Q.

In particular, if B is an N-by-N nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B):

` A*inv(B) = U*(D1*inv(D2))*V'.`

If ( A',B')' has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:

` A'*A x = lambda* B'*B x.`

In some literature, the GSVD of A and B is presented in the form

```
U'*A*X = ( 0 D1 ), V'*B*X = ( 0 D2 )
where U and V are orthogonal and X is nonsingular, D1 and D2 are "diagonal".
```

The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as

```
X = Q*( I 0 )
( 0 inv(R) ).
Arguments
=========
jobu: = 0: U is not computed.
= 1: Orthogonal matrix U is computed;
jobv: = 0: V is not computed.
= 1: Orthogonal matrix V is computed;
jobq: = 0: Q is not computed.
= 1: Orthogonal matrix Q is computed;
k:
l: On exit, k and l specify the dimension of the subblocks
described in the Purpose section.
k + l = effective numerical rank of (A',B')'.
A: On entry, the M-by-N matrix A.
On exit, A contains the triangular matrix R, or part of R.
B: On entry, the P-by-N matrix B.
On exit, B contains the triangular matrix R if M-k-l < 0.
alpha:
beta: On exit, alpha and beta contain the generalized singular
value pairs of A and B;
alpha(1:k) = 1,
beta(1:k) = 0,
and if M-k-l >= 0,
alpha(k+1:k+l) = C,
beta(k+1:k+l) = S,
or if M-k-l < 0,
alpha(k+1:M)=C, alpha(M+1:k+l)=0
beta(k+1:M) =S, beta(M+1:k+l) =1
and
alpha(k+l+1:N) = 0
beta(k+l+1:N) = 0
U: If jobu = 1, U contains the M-by-M orthogonal matrix U.
If jobu = 0, U is not referenced.
Need a minimum array of (1,1) if jobu = 0;
V: If jobv = 1, V contains the P-by-P orthogonal matrix V.
If jobv = 0, V is not referenced.
Need a minimum array of (1,1) if jobv = 0;
Q: If jobq = 1, Q contains the N-by-N orthogonal matrix Q.
If jobq = 0, Q is not referenced.
Need a minimum array of (1,1) if jobq = 0;
iwork: On exit, iwork stores the sorting information. More
precisely, the following loop will sort alpha
for I = k+1, min(M,k+l)
swap alpha(I) and alpha(iwork(I))
endfor
such that alpha(1) >= alpha(2) >= ... >= alpha(N).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
> 0: if info = 1, the Jacobi-type procedure failed to
converge. For further details, see subroutine tgsja.
```

```
$k = null;
$l = null;
$A = random(5,6);
$B = random(7,6);
$alpha = zeroes(6);
$beta = zeroes(6);
$U = zeroes(5,5);
$V = zeroes(7,7);
$Q = zeroes(6,6);
$iwork = zeroes(long, 6);
$info = null;
ggsvd($A,1,1,1,$B,$k,$l,$alpha, $beta,$U, $V, $Q, $iwork,$info);
```

ggsvd ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## geev

` Signature: (A(n,n); int jobvl(); int jobvr(); [o]wr(n); [o]wi(n); [o]vl(m,m); [o]vr(p,p); int [o]info())`

Computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

The right eigenvector v(j) of A satisfies: A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue.

The left eigenvector u(j) of A satisfies: u(j)**H * A = lambda(j) * u(j)**H where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.

```
Arguments
=========
jobvl: = 0: left eigenvectors of A are not computed;
= 1: left eigenvectors of A are computed.
jobvr: = 0: right eigenvectors of A are not computed;
= 1: right eigenvectors of A are computed.
A: A is overwritten.
wr:
wi: wr and wi contain the real and imaginary parts,
respectively, of the computed eigenvalues. Complex
conjugate pairs of eigenvalues appear consecutively
with the eigenvalue having the positive imaginary part
first.
vl: If jobvl = 1, the left eigenvectors u(j) are stored one
after another in the columns of vl, in the same order
as their eigenvalues else vl is not referenced.
If the j-th eigenvalue is real, then u(j) = vl(:,j),
the j-th column of vl.
If the j-th and (j+1)-st eigenvalues form a complex
conjugate pair, then u(j) = vl(:,j) + i*vl(:,j+1) and
u(j+1) = vl(:,j) - i*vl(:,j+1).
Min size = [1].
vr: If jobvr = 1, the right eigenvectors v(j) are stored one
after another in the columns of vr, in the same order
as their eigenvalues else vr is not referenced.
If the j-th eigenvalue is real, then v(j) = vr(:,j),
the j-th column of vr.
If the j-th and (j+1)-st eigenvalues form a complex
conjugate pair, then v(j) = vr(:,j) + i*vr(:,j+1) and
v(j+1) = vr(:,j) - i*vr(:,j+1).
Min size = [1].
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
> 0: if info = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors have been computed;
elements i+1:N of wr and wi contain eigenvalues which
have converged.
```

```
$a = random (5, 5);
$wr = zeroes(5);
$wi = zeroes($wr);
$vl = zeroes($a);
$vr = zeroes($a);
$info = null;
geev($a, 1, 1, $wr, $wi, $vl, $vr, $info);
```

geev ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## geevx

` Signature: ([io]A(n,n); int jobvl(); int jobvr(); int balance(); int sense(); [o]wr(n); [o]wi(n); [o]vl(m,m); [o]vr(p,p); int [o]ilo(); int [o]ihi(); [o]scale(n); [o]abnrm(); [o]rconde(q); [o]rcondv(r); int [o]info(); int [t]iwork(iworkn))`

Computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ilo, ihi, scale, and abnrm), reciprocal condition numbers for the eigenvalues (rconde), and reciprocal condition numbers for the right eigenvectors (rcondv).

The right eigenvector v(j) of A satisfies:

```
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
```

The left eigenvector u(j) of A satisfies:

```
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).
```

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.

Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation D * A * D**(-1), where D is a diagonal matrix, to make its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.10.2 of the LAPACK Users' Guide.

```
Arguments
=========
balance:
Indicates how the input matrix should be diagonally scaled
and/or permuted to improve the conditioning of its
eigenvalues.
= 0: Do not diagonally scale or permute;
= 1: Perform permutations to make the matrix more nearly
upper triangular. Do not diagonally scale;
= 2: Diagonally scale the matrix, i.e. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute;
= 3: Both diagonally scale and permute A.
Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.
jobvl: = 0: left eigenvectors of A are not computed;
= 1: left eigenvectors of A are computed.
If sense = 1 or 3, jobvl must = 1.
jobvr; = 0: right eigenvectors of A are not computed;
= 1: right eigenvectors of A are computed.
If sense = 1 or 3, jobvr must = 1.
sense: Determines which reciprocal condition numbers are computed.
= 0: None are computed;
= 1: Computed for eigenvalues only;
= 2: Computed for right eigenvectors only;
= 3: Computed for eigenvalues and right eigenvectors.
If sense = 1 or 3, both left and right eigenvectors
must also be computed (jobvl = 1 and jobvr = 1).
A: The N-by-N matrix.
It is overwritten. If jobvl = 1 or
jobvr = 1, A contains the real Schur form of the balanced
version of the input matrix A.
wr
wi: wr and wi contain the real and imaginary parts,
respectively, of the computed eigenvalues. Complex
conjugate pairs of eigenvalues will appear consecutively
with the eigenvalue having the positive imaginary part
first.
vl: If jobvl = 1, the left eigenvectors u(j) are stored one
after another in the columns of vl, in the same order
as their eigenvalues else vl is not referenced.
If the j-th eigenvalue is real, then u(j) = vl(:,j),
the j-th column of vl.
If the j-th and (j+1)-st eigenvalues form a complex
conjugate pair, then u(j) = vl(:,j) + i*vl(:,j+1) and
u(j+1) = vl(:,j) - i*vl(:,j+1).
Min size = [1].
vr: If jobvr = 1, the right eigenvectors v(j) are stored one
after another in the columns of vr, in the same order
as their eigenvalues else vr is not referenced.
If the j-th eigenvalue is real, then v(j) = vr(:,j),
the j-th column of vr.
If the j-th and (j+1)-st eigenvalues form a complex
conjugate pair, then v(j) = vr(:,j) + i*vr(:,j+1) and
v(j+1) = vr(:,j) - i*vr(:,j+1).
Min size = [1].
ilo,ihi:Integer values determined when A was
balanced. The balanced A(i,j) = 0 if I > J and
J = 1,...,ilo-1 or I = ihi+1,...,N.
scale: Details of the permutations and scaling factors applied
when balancing A. If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then
scale(J) = P(J), for J = 1,...,ilo-1
= D(J), for J = ilo,...,ihi
= P(J) for J = ihi+1,...,N.
The order in which the interchanges are made is N to ihi+1,
then 1 to ilo-1.
abnrm: The one-norm of the balanced matrix (the maximum
of the sum of absolute values of elements of any column).
rconde: rconde(j) is the reciprocal condition number of the j-th
eigenvalue.
rcondv: rcondv(j) is the reciprocal condition number of the j-th
right eigenvector.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
> 0: if info = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors or condition numbers
have been computed; elements 1:ilo-1 and i+1:N of wr
and wi contain eigenvalues which have converged.
```

```
$a = random (5,5);
$wr = zeroes(5);
$wi = zeroes(5);
$vl = zeroes(5,5);
$vr = zeroes(5,5);
$ilo = null;
$ihi = null;
$scale = zeroes(5);
$abnrm = null;
$rconde = zeroes(5);
$rcondv = zeroes(5);
$info = null;
geevx($a, 1,1,3,3,$wr, $wi, $vl, $vr, $ilo, $ihi, $scale, $abnrm,$rconde, $rcondv, $info);
```

geevx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ggev

` Signature: (A(n,n); int [phys]jobvl();int [phys]jobvr();B(n,n);[o]alphar(n);[o]alphai(n);[o]beta(n);[o]VL(m,m);[o]VR(p,p);int [o]info())`

Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero.

The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

` A * v(j) = lambda(j) * B * v(j).`

The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

```
u(j)**H * A = lambda(j) * u(j)**H * B .
where u(j)**H is the conjugate-transpose of u(j).
Arguments
=========
jobvl: = 0: do not compute the left generalized eigenvectors;
= 1: compute the left generalized eigenvectors.
jobvr: = 0: do not compute the right generalized eigenvectors;
= 1: compute the right generalized eigenvectors.
A: On entry, the matrix A in the pair (A,B).
On exit, A has been overwritten.
B: On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten.
alphar:
alphai:
beta: On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,...,N, will
be the generalized eigenvalues. If alphai(j) is zero, then
the j-th eigenvalue is real; if positive, then the j-th and
(j+1)-st eigenvalues are a complex conjugate pair, with
alphai(j+1) negative.
Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j)
may easily over- or underflow, and beta(j) may even be zero.
Thus, the user should avoid naively computing the ratio
alpha/beta. However, alphar and alphai will be always less
than and usually comparable with norm(A) in magnitude, and
beta always less than and usually comparable with norm(B).
VL: If jobvl = 1, the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
u(j) = VL(:,j), the j-th column of VL. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
Each eigenvector will be scaled so the largest component have
abs(real part)+abs(imag. part)=1.
Not referenced if jobvl = 0.
VR: If jobvr = 1, the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
v(j) = VR(:,j), the j-th column of VR. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
Each eigenvector will be scaled so the largest component have
abs(real part)+abs(imag. part)=1.
Not referenced if jobvr = 0.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
= 1,...,N:
The QZ iteration failed. No eigenvectors have been
calculated, but alphar(j), alphai(j), and beta(j)
should be correct for j=info+1,...,N.
> N: =N+1: other than QZ iteration failed in hgeqz.
=N+2: error return from tgevc.
```

```
$a = random(5,5);
$b = random(5,5);
$alphar = zeroes(5);
$alphai = zeroes(5);
$beta = zeroes(5);
$vl = zeroes(5,5);
$vr = zeroes(5,5);
ggev($a, 1, 1, $b, $alphar, $alphai, $beta, $vl, $vr, ($info=null));
```

ggev ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ggevx

` Signature: ([io,phys]A(n,n);int balanc();int jobvl();int jobvr();int sense();[io,phys]B(n,n);[o]alphar(n);[o]alphai(n);[o]beta(n);[o]VL(m,m);[o]VR(p,p);int [o]ilo();int [o]ihi();[o]lscale(n);[o]rscale(n);[o]abnrm();[o]bbnrm();[o]rconde(r);[o]rcondv(s);int [o]info(); int [t]bwork(bworkn); int [t]iwork(iworkn))`

Computes for a pair of N-by-N real nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors.

Optionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ilo, ihi, lscale, rscale, abnrm, and bbnrm), reciprocal condition numbers for the eigenvalues (rconde), and reciprocal condition numbers for the right eigenvectors (rcondv).

A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A - lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero.

The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

` A * v(j) = lambda(j) * B * v(j) .`

The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies

```
u(j)**H * A = lambda(j) * u(j)**H * B.
where u(j)**H is the conjugate-transpose of u(j).
```

Further Details ===============

Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, second, applying diagonal similarity transformation to the rows and columns to make the rows and columns as close in norm as possible. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide.

An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is

` chord(w, lambda) <= EPS * norm(abnrm, bbnrm) / rconde(I)`

An approximate error bound for the angle between the i-th computed eigenvector vl(i) or vr(i) is given by

` EPS * norm(abnrm, bbnrm) / DIF(i).`

For further explanation of the reciprocal condition numbers rconde and rcondv, see section 4.11 of LAPACK User's Guide.

```
Arguments
=========
balanc: Specifies the balance option to be performed.
= 0: do not diagonally scale or permute;
= 1: permute only;
= 2: scale only;
= 3: both permute and scale.
Computed reciprocal condition numbers will be for the
matrices after permuting and/or balancing. Permuting does
not change condition numbers (in exact arithmetic), but
balancing does.
jobvl: = 0: do not compute the left generalized eigenvectors;
= 1: compute the left generalized eigenvectors.
jobvr: = 0: do not compute the right generalized eigenvectors;
= 1: compute the right generalized eigenvectors.
sense: Determines which reciprocal condition numbers are computed.
= 0: none are computed;
= 1: computed for eigenvalues only;
= 2: computed for eigenvectors only;
= 3: computed for eigenvalues and eigenvectors.
A: On entry, the matrix A in the pair (A,B).
On exit, A has been overwritten. If jobvl=1 or jobvr=1
or both, then A contains the first part of the real Schur
form of the "balanced" versions of the input A and B.
B: On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten. If jobvl=1 or jobvr=1
or both, then B contains the second part of the real Schur
form of the "balanced" versions of the input A and B.
alphar:
alphai:
beta: On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,...,N, will
be the generalized eigenvalues. If alphai(j) is zero, then
the j-th eigenvalue is real; if positive, then the j-th and
(j+1)-st eigenvalues are a complex conjugate pair, with
alphai(j+1) negative.
Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j)
may easily over- or underflow, and beta(j) may even be zero.
Thus, the user should avoid naively computing the ratio
ALPHA/beta. However, alphar and alphai will be always less
than and usually comparable with norm(A) in magnitude, and
beta always less than and usually comparable with norm(B).
vl: If jobvl = 1, the left eigenvectors u(j) are stored one
after another in the columns of vl, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
u(j) = vl(:,j), the j-th column of vl. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
u(j) = vl(:,j)+i*vl(:,j+1) and u(j+1) = vl(:,j)-i*vl(:,j+1).
Each eigenvector will be scaled so the largest component have
abs(real part) + abs(imag. part) = 1.
Not referenced if jobvl = 0.
vr: If jobvr = 1, the right eigenvectors v(j) are stored one
after another in the columns of vr, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
v(j) = vr(:,j), the j-th column of vr. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
v(j) = vr(:,j)+i*vr(:,j+1) and v(j+1) = vr(:,j)-i*vr(:,j+1).
Each eigenvector will be scaled so the largest component have
abs(real part) + abs(imag. part) = 1.
Not referenced if jobvr = 0.
ilo,ihi:ilo and ihi are integer values such that on exit
A(i,j) = 0 and B(i,j) = 0 if i > j and
j = 1,...,ilo-1 or i = ihi+1,...,N.
If balanc = 0 or 2, ilo = 1 and ihi = N.
lscale: Details of the permutations and scaling factors applied
to the left side of A and B. If PL(j) is the index of the
row interchanged with row j, and DL(j) is the scaling
factor applied to row j, then
lscale(j) = PL(j) for j = 1,...,ilo-1
= DL(j) for j = ilo,...,ihi
= PL(j) for j = ihi+1,...,N.
The order in which the interchanges are made is N to ihi+1,
then 1 to ilo-1.
rscale: Details of the permutations and scaling factors applied
to the right side of A and B. If PR(j) is the index of the
column interchanged with column j, and DR(j) is the scaling
factor applied to column j, then
rscale(j) = PR(j) for j = 1,...,ilo-1
= DR(j) for j = ilo,...,ihi
= PR(j) for j = ihi+1,...,N
The order in which the interchanges are made is N to ihi+1,
then 1 to ilo-1.
abnrm: The one-norm of the balanced matrix A.
bbnrm: The one-norm of the balanced matrix B.
rconde: If sense = 1 or 3, the reciprocal condition numbers of
the selected eigenvalues, stored in consecutive elements of
the array. For a complex conjugate pair of eigenvalues two
consecutive elements of rconde are set to the same value.
Thus rconde(j), rcondv(j), and the j-th columns of vl and vr
all correspond to the same eigenpair (but not in general the
j-th eigenpair, unless all eigenpairs are selected).
If sense = 2, rconde is not referenced.
rcondv: If sense = 2 or 3, the estimated reciprocal condition
numbers of the selected eigenvectors, stored in consecutive
elements of the array. For a complex eigenvector two
consecutive elements of rcondv are set to the same value. If
the eigenvalues cannot be reordered to compute rcondv(j),
rcondv(j) is set to 0; this can only occur when the true
value would be very small anyway.
If sense = 1, rcondv is not referenced.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
= 1,...,N:
The QZ iteration failed. No eigenvectors have been
calculated, but alphar(j), alphai(j), and beta(j)
should be correct for j=info+1,...,N.
> N: =N+1: other than QZ iteration failed in hgeqz.
=N+2: error return from tgevc.
```

```
$a = random(5,5);
$b = random(5,5);
$alphar = zeroes(5);
$alphai = zeroes(5);
$beta = zeroes(5);
$vl = zeroes(5,5);
$vr = zeroes(5,5);
$lscale = zeroes(5);
$rscale = zeroes(5);
$ilo = null;
$ihi = null;
$abnrm = null;
$bbnrm = null;
$rconde = zeroes(5);
$rcondv = zeroes(5);
ggevx($a, 3, 1, 1, 3, $b, $alphar, $alphai, $beta, $vl, $vr,
$ilo, $ihi, $lscale, $rscale, $abnrm, $bbnrm, $rconde,$rcondv,($info=null));
```

ggevx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gees

` Signature: ([io]A(n,n); int jobvs(); int sort(); [o]wr(n); [o]wi(n); [o]vs(p,p); int [o]sdim(); int [o]info(); int [t]bwork(bworkn); SV* select_func)`

Computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*Z'.

Optionally, it also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left. The leading columns of Z then form an orthonormal basis for the invariant subspace corresponding to the selected eigenvalues.

A matrix is in real Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the form

```
[ a b ]
[ c a ]
where b*c < 0.
```

The eigenvalues of such a block are a +- sqrt(bc).

```
Arguments
=========
jobvs: = 0: Schur vectors are not computed;
= 1: Schur vectors are computed.
sort: Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 0: Eigenvalues are not ordered;
= 1: Eigenvalues are ordered (see select_func).
select_func:
If sort = 1, select_func is used to select eigenvalues to sort
to the top left of the Schur form.
If sort = 0, select_func is not referenced.
An eigenvalue wr(j)+sqrt(-1)*wi(j) is selected if
select_func(SCALAR(wr(j)), SCALAR(wi(j))) is true; i.e.,
if either one of a complex conjugate pair of eigenvalues
is selected, then both complex eigenvalues are selected.
Note that a selected complex eigenvalue may no longer
satisfy select_func(wr(j),wi(j)) = 1 after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned); in this
case info is set to N+2 (see info below).
A: The N-by-N matrix A.
On exit, A has been overwritten by its real Schur form T.
sdim: If sort = 0, sdim = 0.
If sort = 1, sdim = number of eigenvalues (after sorting)
for which select_func is true. (Complex conjugate
pairs for which select_func is true for either
eigenvalue count as 2.)
wr:
wi: wr and wi contain the real and imaginary parts,
respectively, of the computed eigenvalues in the same order
that they appear on the diagonal of the output Schur form T.
Complex conjugate pairs of eigenvalues will appear
consecutively with the eigenvalue having the positive
imaginary part first.
vs: If jobvs = 1, vs contains the orthogonal matrix Z of Schur
vectors else vs is not referenced.
info = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
> 0: if info = i, and i is
<= N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ILO-1 and i+1:N of wr and wi
contain those eigenvalues which have converged; if
jobvs = 1, vs contains the matrix which reduces A
to its partially converged Schur form.
= N+1: the eigenvalues could not be reordered because some
eigenvalues were too close to separate (the problem
is very ill-conditioned);
= N+2: after reordering, roundoff changed values of some
complex eigenvalues so that leading eigenvalues in
the Schur form no longer satisfy select_func = 1 This
could also be caused by underflow due to scaling.
```

```
sub select_function{
my ($a, $b ) = @_;
# Stable "continuous time" eigenspace
return $a < 0 ? 1 : 0;
}
$A = random (5,5);
$wr= zeroes(5);
$wi = zeroes(5);
$vs = zeroes(5,5);
$sdim = null;
$info = null;
gees($A, 1,1, $wr, $wi, $vs, $sdim, $info,\&select_function);
```

gees ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## geesx

` Signature: ([io]A(n,n); int jobvs(); int sort(); int sense(); [o]wr(n); [o]wi(n); [o]vs(p,p); int [o]sdim(); [o]rconde();[o]rcondv(); int [o]info(); int [t]bwork(bworkn); SV* select_func)`

Computes for an N-by-N real nonsymmetric matrix A, the eigenvalues, the real Schur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*Z'.

Optionally, it also orders the eigenvalues on the diagonal of the real Schur form so that selected eigenvalues are at the top left; computes a reciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace corresponding to the selected eigenvalues (rcondv). The leading columns of Z form an orthonormal basis for this invariant subspace.

For further explanation of the reciprocal condition numbers rconde and rcondv, see Section 4.10 of the LAPACK Users' Guide (where these quantities are called s and sep respectively).

A real matrix is in real Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the form

```
[ a b ]
[ c a ]
where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).
Arguments
=========
jobvs: = 0: Schur vectors are not computed;
= 1: Schur vectors are computed.
sort: Specifies whether or not to order the eigenvalues on the
diagonal of the Schur form.
= 0: Eigenvalues are not ordered;
= 1: Eigenvalues are ordered (see select_func).
select_func:
If sort = 1, select_func is used to select eigenvalues to sort
to the top left of the Schur form else select_func is not referenced.
An eigenvalue wr(j)+sqrt(-1)*wi(j) is selected if
select_func(wr(j),wi(j)) is true; i.e., if either one of a
complex conjugate pair of eigenvalues is selected, then both
are. Note that a selected complex eigenvalue may no longer
satisfy select_func(wr(j),wi(j)) = 1 after ordering, since
ordering may change the value of complex eigenvalues
(especially if the eigenvalue is ill-conditioned); in this
case info may be set to N+3 (see info below).
sense: Determines which reciprocal condition numbers are computed.
= 0: None are computed;
= 1: Computed for average of selected eigenvalues only;
= 2: Computed for selected right invariant subspace only;
= 3: Computed for both.
If sense = 1, 2 or 3, sort must equal 1.
A: On entry, the N-by-N matrix A.
On exit, A is overwritten by its real Schur form T.
sdim: If sort = 0, sdim = 0.
If sort = 1, sdim = number of eigenvalues (after sorting)
for which select_func is 1. (Complex conjugate
pairs for which select_func is 1 for either
eigenvalue count as 2.)
wr:
wi: wr and wi contain the real and imaginary parts, respectively,
of the computed eigenvalues, in the same order that they
appear on the diagonal of the output Schur form T. Complex
conjugate pairs of eigenvalues appear consecutively with the
eigenvalue having the positive imaginary part first.
vs If jobvs = 1, vs contains the orthogonal matrix Z of Schur
vectors else vs is not referenced.
rconde: If sense = 1 or 3, rconde contains the reciprocal
condition number for the average of the selected eigenvalues.
Not referenced if sense = 0 or 2.
rcondv: If sense = 2 or 3, rcondv contains the reciprocal
condition number for the selected right invariant subspace.
Not referenced if sense = 0 or 1.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
> 0: if info = i, and i is
<= N: the QR algorithm failed to compute all the
eigenvalues; elements 1:ilo-1 and i+1:N of wr and wi
contain those eigenvalues which have converged; if
jobvs = 1, vs contains the transformation which
reduces A to its partially converged Schur form.
= N+1: the eigenvalues could not be reordered because some
eigenvalues were too close to separate (the problem
is very ill-conditioned);
= N+2: after reordering, roundoff changed values of some
complex eigenvalues so that leading eigenvalues in
the Schur form no longer satisfy select_func=1 This
could also be caused by underflow due to scaling.
```

```
sub select_function{
my ($a, $b) = @_;
# Stable "discrete time" eigenspace
return sqrt($a**2 + $b**2) < 1 ? 1 : 0;
}
$A = random (5,5);
$wr= zeroes(5);
$wi = zeroes(5);
$vs = zeroes(5,5);
$sdim = null;
$rconde = null;
$rcondv = null;
$info = null;
geesx($A, 1,1, 3, $wr, $wi, $vs, $sdim, $rconde, $rcondv, $info, \&select_function);
```

geesx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gges

` Signature: ([io]A(n,n); int jobvsl();int jobvsr();int sort();[io]B(n,n);[o]alphar(n);[o]alphai(n);[o]beta(n);[o]VSL(m,m);[o]VSR(p,p);int [o]sdim();int [o]info(); int [t]bwork(bworkn); SV* select_func)`

Computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the generalized real Schur form (S,T), optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization

` (A,B) = ( (VSL)*S*(VSR)', (VSL)*T*(VSR)' )`

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T.The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the driver ggev instead, which is faster.)

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or both being zero.

A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be "standardized" by making the corresponding elements of T have the form:

```
[ a 0 ]
[ 0 b ]
```

and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues.

```
Arguments
=========
jobvsl: = 0: do not compute the left Schur vectors;
= 1: compute the left Schur vectors.
jobvsr: = 0: do not compute the right Schur vectors;
= 1: compute the right Schur vectors.
sort: Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 0: Eigenvalues are not ordered;
= 1: Eigenvalues are ordered (see delztg);
delztg: If sort = 0, delztg is not referenced.
If sort = 1, delztg is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue (alphar(j)+alphai(j))/beta(j) is selected if
delztg(alphar(j),alphai(j),beta(j)) is true; i.e. if either
one of a complex conjugate pair of eigenvalues is selected,
then both complex eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex
eigenvalue may no longer satisfy delztg(alphar(j),alphai(j),
beta(j)) = 1 after ordering. info is to be set to N+2
in this case.
A: On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
B: On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
sdim: If sort = 0, sdim = 0.
If sort = 1, sdim = number of eigenvalues (after sorting)
for which delztg is true. (Complex conjugate pairs for which
delztg is true for either eigenvalue count as 2.)
alphar:
alphai:
beta: On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,...,N, will
be the generalized eigenvalues. alphar(j) + alphai(j)*i,
and beta(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
If alphai(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with alphai(j+1) negative.
Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j)
may easily over- or underflow, and beta(j) may even be zero.
Thus, the user should avoid naively computing the ratio.
However, alphar and alphai will be always less than and
usually comparable with norm(A) in magnitude, and beta always
less than and usually comparable with norm(B).
VSL: If jobvsl = 1, VSL will contain the left Schur vectors.
Not referenced if jobvsl = 0.
The leading dimension must always be >=1.
VSR: If jobvsr = 1, VSR will contain the right Schur vectors.
Not referenced if jobvsr = 0.
The leading dimension must always be >=1.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
= 1,...,N:
The QZ iteration failed. (A,B) are not in Schur
form, but alphar(j), alphai(j), and beta(j) should
be correct for j=info+1,...,N.
> N: =N+1: other than QZ iteration failed in hgeqz.
=N+2: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Generalized Schur form no
longer satisfy delztg=1 This could also
be caused due to scaling.
=N+3: reordering failed in tgsen.
```

```
sub my_select{
my ($zr, $zi, $d) = @_;
# stable generalized eigenvalues for continuous time
return ( ($zr < 0 && $d > 0 ) || ($zr > 0 && $d < 0) ) ? 1 : 0;
}
$a = random(5,5);
$b = random(5,5);
$sdim = null;
$alphar = zeroes(5);
$alphai = zeroes(5);
$beta = zeroes(5);
$vsl = zeroes(5,5);
$vsr = zeroes(5,5);
gges($a, 1, 1, 1, $b, $alphar, $alphai, $beta, $vsl, $vsr, $sdim,($info=null), \&my_select);
```

gges ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ggesx

` Signature: ([io]A(n,n); int jobvsl();int jobvsr();int sort();int sense();[io]B(n,n);[o]alphar(n);[o]alphai(n);[o]beta(n);[o]VSL(m,m);[o]VSR(p,p);int [o]sdim();[o]rconde(q=2);[o]rcondv(q=2);int [o]info(); int [t]bwork(bworkn); int [t]iwork(iworkn); SV* select_func)`

Computes for a pair of N-by-N real nonsymmetric matrices (A,B), the generalized eigenvalues, the real Schur form (S,T), and, optionally, the left and/or right matrices of Schur vectors (VSL and VSR). This gives the generalized Schur factorization

` (A,B) = ( (VSL) S (VSR)', (VSL) T (VSR)' )`

Optionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T; computes a reciprocal condition number for the average of the selected eigenvalues (RCONDE); and computes a reciprocal condition number for the right and left deflating subspaces corresponding to the selected eigenvalues (RCONDV). The leading columns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces (deflating subspaces).

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A - w*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0 or for both being zero.

A pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be "standardized" by making the corresponding elements of T have the form:

```
[ a 0 ]
[ 0 b ]
```

and the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized eigenvalues.

Further details ===============

An approximate (asymptotic) bound on the average absolute error of the selected eigenvalues is

` EPS * norm((A, B)) / RCONDE( 1 ).`

An approximate (asymptotic) bound on the maximum angular error in the computed deflating subspaces is

` EPS * norm((A, B)) / RCONDV( 2 ).`

See LAPACK User's Guide, section 4.11 for more information.

```
Arguments
=========
jobvsl: = 0: do not compute the left Schur vectors;
= 1: compute the left Schur vectors.
jobvsr: = 0: do not compute the right Schur vectors;
= 1: compute the right Schur vectors.
sort: Specifies whether or not to order the eigenvalues on the
diagonal of the generalized Schur form.
= 0: Eigenvalues are not ordered;
= 1: Eigenvalues are ordered (see delztg);
delztg: If sort = 0, delztg is not referenced.
If sort = 1, delztg is used to select eigenvalues to sort
to the top left of the Schur form.
An eigenvalue (alphar(j)+alphai(j))/beta(j) is selected if
delztg(alphar(j),alphai(j),beta(j)) is true; i.e. if either
one of a complex conjugate pair of eigenvalues is selected,
then both complex eigenvalues are selected.
Note that in the ill-conditioned case, a selected complex
eigenvalue may no longer satisfy delztg(alphar(j),alphai(j),
beta(j)) = 1 after ordering. info is to be set to N+2
in this case.
sense: Determines which reciprocal condition numbers are computed.
= 0 : None are computed;
= 1 : Computed for average of selected eigenvalues only;
= 2 : Computed for selected deflating subspaces only;
= 3 : Computed for both.
If sense = 1, 2, or 3, sort must equal 1.
A: On entry, the first of the pair of matrices.
On exit, A has been overwritten by its generalized Schur
form S.
B: On entry, the second of the pair of matrices.
On exit, B has been overwritten by its generalized Schur
form T.
sdim: If sort = 0, sdim = 0.
If sort = 1, sdim = number of eigenvalues (after sorting)
for which delztg is true. (Complex conjugate pairs for which
delztg is true for either eigenvalue count as 2.)
alphar:
alphai:
beta: On exit, (alphar(j) + alphai(j)*i)/beta(j), j=1,...,N, will
be the generalized eigenvalues. alphar(j) + alphai(j)*i,
and beta(j),j=1,...,N are the diagonals of the complex Schur
form (S,T) that would result if the 2-by-2 diagonal blocks of
the real Schur form of (A,B) were further reduced to
triangular form using 2-by-2 complex unitary transformations.
If alphai(j) is zero, then the j-th eigenvalue is real; if
positive, then the j-th and (j+1)-st eigenvalues are a
complex conjugate pair, with alphai(j+1) negative.
Note: the quotients alphar(j)/beta(j) and alphai(j)/beta(j)
may easily over- or underflow, and beta(j) may even be zero.
Thus, the user should avoid naively computing the ratio.
However, alphar and alphai will be always less than and
usually comparable with norm(A) in magnitude, and beta always
less than and usually comparable with norm(B).
VSL: If jobvsl = 1, VSL will contain the left Schur vectors.
Not referenced if jobvsl = 0.
The leading dimension must always be >=1.
VSR: If jobvsr = 1, VSR will contain the right Schur vectors.
Not referenced if jobvsr = 0.
The leading dimension must always be >=1.
rconde: If sense = 1 or 3, rconde(1) and rconde(2) contain the
reciprocal condition numbers for the average of the selected
eigenvalues.
Not referenced if sense = 0 or 2.
rcondv: If sense = 2 or 3, rcondv(1) and rcondv(2) contain the
reciprocal condition numbers for the selected deflating
subspaces.
Not referenced if sense = 0 or 1.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
= 1,...,N:
The QZ iteration failed. (A,B) are not in Schur
form, but alphar(j), alphai(j), and beta(j) should
be correct for j=info+1,...,N.
> N: =N+1: other than QZ iteration failed in hgeqz.
=N+2: after reordering, roundoff changed values of
some complex eigenvalues so that leading
eigenvalues in the Generalized Schur form no
longer satisfy delztg=1 This could also
be caused due to scaling.
=N+3: reordering failed in tgsen.
```

```
sub my_select{
my ($zr, $zi, $d) = @_;
# Eigenvalue : (ZR/D) + sqrt(-1)*(ZI/D)
# stable generalized eigenvalues for discrete time
return (sqrt($zr**2 + $zi**2) < abs($d) ) ? 1 : 0;
}
$a = random(5,5);
$b = random(5,5);
$sdim = null;
$alphar = zeroes(5);
$alphai = zeroes(5);
$beta = zeroes(5);
$vsl = zeroes(5,5);
$vsr = zeroes(5,5);
$rconde = zeroes(2);
$rcondv = zeroes(2);
ggesx($a, 1, 1, 1, 3,$b, $alphar, $alphai, $beta, $vsl, $vsr, $sdim, $rconde, $rcondv, ($info=null), \&my_select);
```

ggesx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## syev

` Signature: ([io,phys]A(n,n); int jobz(); int uplo(); [o,phys]w(n); int [o,phys]info())`

Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A.

```
Arguments
=========
jobz: = 0: Compute eigenvalues only;
= 1: Compute eigenvalues and eigenvectors.
uplo = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If uplo = 1,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if jobz = 1, then if info = 0, A contains the
orthonormal eigenvectors of the matrix A.
If jobz = 0, then on exit the lower triangle (if uplo=1)
or the upper triangle (if uplo=0) of A, including the
diagonal, is destroyed.
w: If info = 0, the eigenvalues in ascending order.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, the algorithm failed to converge; i
off-diagonal elements of an intermediate tridiagonal
form did not converge to zero.
```

```
# Assume $a is symmetric ;)
$a = random (5,5);
syev($a, 1,1, (my $w = zeroes(5)), (my $info=null));
```

syev ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## syevd

` Signature: ([io,phys]A(n,n); int jobz(); int uplo(); [o,phys]w(n); int [o,phys]info())`

Computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. If eigenvectors are desired, it uses a divide and conquer algorithm.

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

Because of large use of BLAS of level 3, syevd needs N**2 more workspace than syevx.

```
Arguments
=========
jobz: = 0: Compute eigenvalues only;
= 1: Compute eigenvalues and eigenvectors.
uplo = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If uplo = 1,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if jobz = 1, then if info = 0, A contains the
orthonormal eigenvectors of the matrix A.
If jobz = 0, then on exit the lower triangle (if uplo=1)
or the upper triangle (if uplo=0) of A, including the
diagonal, is destroyed.
w: If info = 0, the eigenvalues in ascending order.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, the algorithm failed to converge; i
off-diagonal elements of an intermediate tridiagonal
form did not converge to zero.
```

```
# Assume $a is symmetric ;)
$a = random (5,5);
syevd($a, 1,1, (my $w = zeroes(5)), (my $info=null));
```

syevd ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## syevx

` Signature: (A(n,n); int jobz(); int range(); int uplo(); vl(); vu(); int il(); int iu(); abstol(); int [o]m(); [o]w(n); [o]z(p,p);int [o]ifail(n); int [o]info(); int [t]iwork(iworkn))`

Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

```
Arguments
=========
jobz: = 0: Compute eigenvalues only;
= 1: Compute eigenvalues and eigenvectors.
range: = 0: all eigenvalues will be found.
= 1: all eigenvalues in the half-open interval (vl,vu]
will be found.
= 1: the il-th through iu-th eigenvalues will be found.
uplo = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If uplo = 1,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if uplo=1) or the upper
triangle (if uplo=0) of A, including the diagonal, is
destroyed.
vl:
vu: If range=1, the lower and upper bounds of the interval to
be searched for eigenvalues. vl < vu.
Not referenced if range = 0 or 2.
il:
iu: If range=2, the indices (in ascending order) of the
smallest and largest eigenvalues to be returned.
1 <= il <= iu <= N, if N > 0; il = 1 and iu = 0 if N = 0.
Not referenced if range = 0 or 1.
abstol: The absolute error tolerance for the eigenvalues.
An approximate eigenvalue is accepted as converged
when it is determined to lie in an interval [a,b]
of width less than or equal to
abstol + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If abstol is less than
or equal to zero, then EPS*|T| will be used in its place,
where |T| is the 1-norm of the tridiagonal matrix obtained
by reducing A to tridiagonal form.
Eigenvalues will be computed most accurately when abstol is
set to twice the underflow threshold 2*lamch(1), not zero.
If this routine returns with info>0, indicating that some
eigenvectors did not converge, try setting abstol to
2*lamch(1).
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and
Kahan, LAPACK Working Note #3.
m: The total number of eigenvalues found. 0 <= m <= N.
If range = 0, m = N, and if range = 2, m = iu-il+1.
w: On normal exit, the first M elements contain the selected
eigenvalues in ascending order.
z: If jobz = 1, then if info = 0, the first m columns of z
contain the orthonormal eigenvectors of the matrix A
corresponding to the selected eigenvalues, with the i-th
column of z holding the eigenvector associated with w(i).
If an eigenvector fails to converge, then that column of z
contains the latest approximation to the eigenvector, and the
index of the eigenvector is returned in ifail.
If jobz = 0, then z is not referenced.
Note: the user must ensure that at least max(1,m) columns are
supplied in the array z; if range = 1, the exact value of m
is not known in advance and an upper bound must be used.
ifail: If jobz = 1, then if info = 0, the first m elements of
ifail are zero. If info > 0, then ifail contains the
indices of the eigenvectors that failed to converge.
If jobz = 0, then ifail is not referenced.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, then i eigenvectors failed to converge.
Their indices are stored in array ifail.
```

```
# Assume $a is symmetric ;)
$a = random (5,5);
$unfl = lamch(1);
$ovfl = lamch(9);
labad($unfl, $ovfl);
$abstol = $unfl + $unfl;
$m = null;
$info = null;
$ifail = zeroes(5);
$w = zeroes(5);
$z = zeroes(5,5);
syevx($a, 1,0,1,0,0,0,0,$abstol, $m, $w, $z ,$ifail, $info);
```

syevx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## syevr

` Signature: ([phys]A(n,n); int jobz(); int range(); int uplo(); [phys]vl(); [phys]vu(); int [phys]il(); int [phys]iu();[phys]abstol();int [o,phys]m();[o,phys]w(n); [o,phys]z(p,q);int [o,phys]isuppz(r); int [o,phys]info())`

Computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

Whenever possible, syevr calls stegr to compute the eigenspectrum using Relatively Robust Representations. stegr computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various "good" L D L^T representations (also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as follows. For the i-th unreduced block of T,

```
(a) Compute T - sigma_i = L_i D_i L_i^T, such that L_i D_i L_i^T
is a relatively robust representation,
(b) Compute the eigenvalues, lambda_j, of L_i D_i L_i^T to high
relative accuracy by the dqds algorithm,
(c) If there is a cluster of close eigenvalues, "choose" sigma_i
close to the cluster, and go to step (a),
(d) Given the approximate eigenvalue lambda_j of L_i D_i L_i^T,
compute the corresponding eigenvector by forming a
rank-revealing twisted factorization.
```

The desired accuracy of the output can be specified by the input parameter abstol.

For more details, see "A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem", by Inderjit Dhillon, Computer Science Division Technical Report No. UCB//CSD-97-971, UC Berkeley, May 1997.

Note 1 : syevr calls stegr when the full spectrum is requested on machines which conform to the ieee-754 floating point standard. syevr calls stebz and stein on non-ieee machines and when partial spectrum requests are made.

Normal execution of stegr may create NaNs and infinities and hence may abort due to a floating point exception in environments which do not handle NaNs and infinities in the ieee standard default manner.

```
Arguments
=========
jobz: = 0: Compute eigenvalues only;
= 1: Compute eigenvalues and eigenvectors.
range: = 0: all eigenvalues will be found.
= 1: all eigenvalues in the half-open interval (vl,vu]
will be found.
= 2: the il-th through iu-th eigenvalues will be found.
********* For range = 1 or 2 and iu - il < N - 1, stebz and
********* stein are called
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If uplo = 1,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if uplo=1) or the upper
triangle (if uplo=0) of A, including the diagonal, is
destroyed.
vl:
vu: If range=1, the lower and upper bounds of the interval to
be searched for eigenvalues. vl < vu.
Not referenced if range = 0 or 2.
il:
iu: If range=2, the indices (in ascending order) of the
smallest and largest eigenvalues to be returned.
1 <= il <= iu <= N, if N > 0; il = 1 and iu = 0 if N = 0.
Not referenced if range = 0 or 1.
abstol: The absolute error tolerance for the eigenvalues.
An approximate eigenvalue is accepted as converged
when it is determined to lie in an interval [a,b]
of width less than or equal to
abstol + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If abstol is less than
or equal to zero, then EPS*|T| will be used in its place,
where |T| is the 1-norm of the tridiagonal matrix obtained
by reducing A to tridiagonal form.
See "Computing Small Singular Values of Bidiagonal Matrices
with Guaranteed High Relative Accuracy," by Demmel and
Kahan, LAPACK Working Note #3.
If high relative accuracy is important, set abstol to
lamch(1). Doing so will guarantee that
eigenvalues are computed to high relative accuracy when
possible in future releases. The current code does not
make any guarantees about high relative accuracy, but
future releases will. See J. Barlow and J. Demmel,
"Computing Accurate Eigensystems of Scaled Diagonally
Dominant Matrices", LAPACK Working Note #7, for a discussion
of which matrices define their eigenvalues to high relative
accuracy.
m: The total number of eigenvalues found. 0 <= m <= N.
If range = 0, m = N, and if range = 2, m = iu-il+1.
w: The first m elements contain the selected eigenvalues in
ascending order.
z: If jobz = 1, then if info = 0, the first m columns of z
contain the orthonormal eigenvectors of the matrix A
corresponding to the selected eigenvalues, with the i-th
column of z holding the eigenvector associated with w(i).
If jobz = 0, then z is not referenced.
Note: the user must ensure that at least max(1,m) columns are
supplied in the array z; if range = 1, the exact value of m
is not known in advance and an upper bound must be used.
isuppz: array of int, dimension ( 2*max(1,m) )
The support of the eigenvectors in z, i.e., the indices
indicating the nonzero elements in z. The i-th eigenvector
is nonzero only in elements isuppz( 2*i-1 ) through
isuppz( 2*i ).
********* Implemented only for range = 0 or 2 and iu - il = N - 1
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: Internal error
```

```
# Assume $a is symmetric ;)
$a = random (5,5);
$unfl = lamch(1);
$ovfl = lamch(9);
labad($unfl, $ovfl);
$abstol = $unfl + $unfl;
$m = null;
$info = null;
$isuppz = zeroes(10);
$w = zeroes(5);
$z = zeroes(5,5);
syevr($a, 1,0,1,0,0,0,0,$abstol, $m, $w, $z ,$isuppz, $info);
```

syevr ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sygv

` Signature: ([io,phys]A(n,n);int [phys]itype();int jobz(); int uplo();[io,phys]B(n,n);[o,phys]w(n); int [o,phys]info())`

Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric and B is also positive definite.

```
Arguments
=========
itype: Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
jobz: = 0: Compute eigenvalues only;
= 1: Compute eigenvalues and eigenvectors.
uplo: = 0: Upper triangles of A and B are stored;
= 1: Lower triangles of A and B are stored.
A: On entry, the symmetric matrix A. If uplo = 0, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If uplo = 1,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if jobz = 1, then if info = 0, A contains the
matrix Z of eigenvectors. The eigenvectors are normalized
as follows:
if itype = 1 or 2, Z'*B*Z = I;
if itype = 3, Z'*inv(B)*Z = I.
If jobz = 0, then on exit the upper triangle (if uplo=0)
or the lower triangle (if uplo=1) of A, including the
diagonal, is destroyed.
B: On entry, the symmetric positive definite matrix B.
If uplo = 0, the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If uplo = 1, the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.
On exit, if info <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U'*U or B = L*L'.
W: If info = 0, the eigenvalues in ascending order.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: potrf or syev returned an error code:
<= N: if info = i, syev failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N: if info = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
```

```
# Assume $a is symmetric ;)
$a = random (5,5);
# Assume $a is symmetric and positive definite ;)
$b = random (5,5);
sygv($a, 1,1, 0, $b, (my $w = zeroes(5)), (my $info=null));
```

sygv ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sygvd

` Signature: ([io,phys]A(n,n);int [phys]itype();int jobz(); int uplo();[io,phys]B(n,n);[o,phys]w(n); int [o,phys]info())`

Computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric and B is also positive definite.

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

```
Arguments
=========
itype: Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
jobz: = 0: Compute eigenvalues only;
= 1: Compute eigenvalues and eigenvectors.
uplo: = 0: Upper triangles of A and B are stored;
= 1: Lower triangles of A and B are stored.
A: On entry, the symmetric matrix A. If uplo = 0, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If uplo = 1,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, if jobz = 1, then if info = 0, A contains the
matrix Z of eigenvectors. The eigenvectors are normalized
as follows:
if itype = 1 or 2, Z'*B*Z = I;
if itype = 3, Z'*inv(B)*Z = I.
If jobz = 0, then on exit the upper triangle (if uplo=0)
or the lower triangle (if uplo=1) of A, including the
diagonal, is destroyed.
B: On entry, the symmetric positive definite matrix B.
If uplo = 0, the leading N-by-N upper triangular part of B
contains the upper triangular part of the matrix B.
If uplo = 1, the leading N-by-N lower triangular part of B
contains the lower triangular part of the matrix B.
On exit, if info <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U'*U or B = L*L'.
W: If info = 0, the eigenvalues in ascending order.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: potrf or syev returned an error code:
<= N: if info = i, syevd failed to converge;
i off-diagonal elements of an intermediate
tridiagonal form did not converge to zero;
> N: if info = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
```

```
# Assume $a is symmetric ;)
$a = random (5,5);
# Assume $b is symmetric positive definite ;)
$b = random (5,5);
sygvd($a, 1,1, 0, $b, (my $w = zeroes(5)), (my $info=null));
```

sygvd ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sygvx

```
Signature: ([io]A(n,n); int itype(); int jobz(); int range();
int uplo(); [io]B(n,n); vl(); vu(); int il(); int iu(); abstol();
int [o]m(); [o]w(n); [o]Z(p,p); int [o]ifail(n); int [o]info();
int [t]iwork(iworkn);
)
```

Computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite eigenproblem, of the form A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are assumed to be symmetric and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues.

```
Arguments
=========
itype: Specifies the problem type to be solved:
= 1: A*x = (lambda)*B*x
= 2: A*B*x = (lambda)*x
= 3: B*A*x = (lambda)*x
jobz: = 0: Compute eigenvalues only;
= 1: Compute eigenvalues and eigenvectors.
range: = 0: all eigenvalues will be found.
= 1: all eigenvalues in the half-open interval (vl,vu]
will be found.
= 2: the il-th through iu-th eigenvalues will be found.
uplo: = 0: Upper triangle of A and B are stored;
= 1: Lower triangle of A and B are stored.
A: On entry, the symmetric matrix A. If uplo = 0, the
leading N-by-N upper triangular part of A contains the
upper triangular part of the matrix A. If uplo = 1,
the leading N-by-N lower triangular part of A contains
the lower triangular part of the matrix A.
On exit, the lower triangle (if uplo=1) or the upper
triangle (if uplo=0) of A, including the diagonal, is
destroyed.
B: On entry, the symmetric matrix B. If uplo = 0, the
leading N-by-N upper triangular part of B contains the
upper triangular part of the matrix B. If uplo = 1,
the leading N-by-N lower triangular part of B contains
the lower triangular part of the matrix B.
On exit, if info <= N, the part of B containing the matrix is
overwritten by the triangular factor U or L from the Cholesky
factorization B = U'*U or B = L*L'.
vl:
vu: If range=1, the lower and upper bounds of the interval to
be searched for eigenvalues. vl < vu.
Not referenced if range = 0 or 2.
il:
iu: If range=2, the indices (in ascending order) of the
smallest and largest eigenvalues to be returned.
1 <= il <= iu <= N, if N > 0; il = 1 and iu = 0 if N = 0.
Not referenced if range = 0 or 1.
abstol: The absolute error tolerance for the eigenvalues.
An approximate eigenvalue is accepted as converged
when it is determined to lie in an interval [a,b]
of width less than or equal to
abstol + EPS * max( |a|,|b| ) ,
where EPS is the machine precision. If abstol is less than
or equal to zero, then EPS*|T| will be used in its place,
where |T| is the 1-norm of the tridiagonal matrix obtained
by reducing A to tridiagonal form.
Eigenvalues will be computed most accurately when abstol is
set to twice the underflow threshold 2*lamch(1), not zero.
If this routine returns with info>0, indicating that some
eigenvectors did not converge, try setting abstol to
2* lamch(1).
m: The total number of eigenvalues found. 0 <= m <= N.
If range = 0, m = N, and if range = 2, m = iu-il+1.
w: On normal exit, the first m elements contain the selected
eigenvalues in ascending order.
Z: If jobz = 0, then Z is not referenced.
If jobz = 1, then if info = 0, the first m columns of Z
contain the orthonormal eigenvectors of the matrix A
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with w(i).
The eigenvectors are normalized as follows:
if itype = 1 or 2, Z'*B*Z = I;
if itype = 3, Z'*inv(B)*Z = I.
If an eigenvector fails to converge, then that column of Z
contains the latest approximation to the eigenvector, and the
index of the eigenvector is returned in ifail.
Note: the user must ensure that at least max(1,m) columns are
supplied in the array Z; if range = 1, the exact value of m
is not known in advance and an upper bound must be used.
ifail: If jobz = 1, then if info = 0, the first M elements of
ifail are zero. If info > 0, then ifail contains the
indices of the eigenvectors that failed to converge.
If jobz = 0, then ifail is not referenced.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: potrf or syevx returned an error code:
<= N: if info = i, syevx failed to converge;
i eigenvectors failed to converge. Their indices
are stored in array ifail.
> N: if info = N + i, for 1 <= i <= N, then the leading
minor of order i of B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
```

```
# Assume $a is symmetric ;)
$a = random (5,5);
# Assume $b is symmetric positive definite ;)
$b = random (5,5);
$unfl = lamch(1);
$ovfl = lamch(9);
labad($unfl, $ovfl);
$abstol = $unfl + $unfl;
$m = null;
$w=zeroes(5);
$z = zeroes(5,5);
$ifail = zeroes(5);
sygvx($a, 1,1, 0,0, $b, 0, 0, 0, 0, $abstol, $m, $w, $z,$ifail,(my $info=null));
```

sygvx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gesv

` Signature: ([io,phys]A(n,n); [io,phys]B(n,m); int [o,phys]ipiv(n); int [o,phys]info())`

Computes the solution to a real system of linear equations

```
A * X = B,
where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
```

The LU decomposition with partial pivoting and row interchanges is used to factor A as

```
A = P * L * U,
where P is a permutation matrix, L is unit lower triangular, and U is
upper triangular.
```

The factored form of A is then used to solve the system of equations A * X = B.

```
Arguments
=========
A: On entry, the N-by-N coefficient matrix A.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
ipiv: The pivot indices that define the permutation matrix P;
row i of the matrix was interchanged with row ipiv(i).
B: On entry, the N-by-NRHS matrix of right hand side matrix B.
On exit, if info = 0, the N-by-NRHS solution matrix X.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, so the solution could not be computed.
```

```
$a = random (5,5);
$a = transpose($a);
$b = random (5,5);
$b = transpose($b);
gesv($a,$b, (my $ipiv=zeroes(5)),(my $info=null));
print "The solution matrix X is :". transpose($b)."\n" unless $info;
```

gesv ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gesvx

` Signature: ([io]A(n,n); int trans(); int fact(); [io]B(n,m); [io]af(n,n); int [io]ipiv(n); int [io]equed(); [o]r(p); [o]c(q); [o]X(n,m); [o]rcond(); [o]ferr(m); [o]berr(m);[o]rpvgrw();int [o]info(); [t]work(workn); int [t]iwork(n))`

Uses the LU factorization to compute the solution to a real system of linear equations

```
A * X = B,
where A is an N-by-N matrix and X and B are N-by-NRHS matrices.
```

Error bounds on the solution and a condition estimate are also provided.

The following steps are performed:

If fact = 2, real scaling factors are computed to equilibrate the system:

`trans = 0: diag(r)*A*diag(c) *inv(diag(c))*X = diag(c)*B trans = 1: (diag(r)*A*diag(c))' *inv(diag(r))*X = diag(c)*B trans = 2: (diag(r)*A*diag(c))**H *inv(diag(r))*X = diag(c)*B`

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(r)*A*diag(c) and B by diag(r)*B (if trans=0) or diag(c)*B (if trans = 1 or 2).

If fact = 1 or 2, the LU decomposition is used to factor the matrix A (after equilibration if fact = 2) as

`A = P * L * U, where P is a permutation matrix, L is a unit lower triangular matrix, and U is upper triangular.`

If some U(i,i)=0, so that U is exactly singular, then the routine returns with info = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, info = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

The system of equations is solved for X using the factored form of A.

Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

If equilibration was used, the matrix X is premultiplied by diag(c) (if trans = 0) or diag(r) (if trans = 1 or 2) so that it solves the original system before equilibration.

```
Arguments
=========
fact: Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 0: On entry, af and ipiv contain the factored form of A.
If equed is not 0, the matrix A has been
equilibrated with scaling factors given by r and c.
A, af, and ipiv are not modified.
= 1: The matrix A will be copied to af and factored.
= 2: The matrix A will be equilibrated if necessary, then
copied to af and factored.
trans: Specifies the form of the system of equations:
= 0: A * X = B (No transpose)
= 1: A' * X = B (Transpose)
= 2: A**H * X = B (Transpose)
A: On entry, the N-by-N matrix A. If fact = 0 and equed is
not 0, then A must have been equilibrated by the scaling
factors in r and/or c. A is not modified if fact = 0 or
1, or if fact = 2 and equed = 0 on exit.
On exit, if equed != 0, A is scaled as follows:
equed = 1: A := diag(r) * A
equed = 2: A := A * diag(c)
equed = 3: A := diag(r) * A * diag(c).
af: If fact = 0, then af is an input argument and on entry
contains the factors L and U from the factorization
A = P*L*U as computed by getrf. If equed != 0, then
af is the factored form of the equilibrated matrix A.
If fact = 1, then af is an output argument and on exit
returns the factors L and U from the factorization A = P*L*U
of the original matrix A.
If fact = 2, then af is an output argument and on exit
returns the factors L and U from the factorization A = P*L*U
of the equilibrated matrix A (see the description of A for
the form of the equilibrated matrix).
ipiv: If fact = 0, then ipiv is an input argument and on entry
contains the pivot indices from the factorization A = P*L*U
as computed by getrf; row i of the matrix was interchanged
with row ipiv(i).
If fact = 1, then ipiv is an output argument and on exit
contains the pivot indices from the factorization A = P*L*U
of the original matrix A.
If fact = 2, then ipiv is an output argument and on exit
contains the pivot indices from the factorization A = P*L*U
of the equilibrated matrix A.
equed: Specifies the form of equilibration that was done.
= 0: No equilibration (always true if fact = 1).
= 1: Row equilibration, i.e., A has been premultiplied by
diag(r).
= 2: Column equilibration, i.e., A has been postmultiplied
by diag(c).
= 3: Both row and column equilibration, i.e., A has been
replaced by diag(r) * A * diag(c).
equed is an input argument if fact = 0; otherwise, it is an
output argument.
r: The row scale factors for A. If equed = 1 or 3, A is
multiplied on the left by diag(r); if equed = 0 or 2, r
is not accessed. r is an input argument if fact = 0;
otherwise, r is an output argument. If fact = 0 and
equed = 1 or 3, each element of r must be positive.
c: The column scale factors for A. If equed = 2 or 3, A is
multiplied on the right by diag(c); if equed = 0 or 1, c
is not accessed. c is an input argument if fact = 0;
otherwise, c is an output argument. If fact = 0 and
equed = 2 or 3, each element of c must be positive.
B: On entry, the N-by-NRHS right hand side matrix B.
On exit,
if equed = 0, B is not modified;
if trans = 0 and equed = 1 or 3, B is overwritten by
diag(r)*B;
if trans = 1 or 2 and equed = 2 or 3, B is
overwritten by diag(c)*B.
X: If info = 0 or info = N+1, the N-by-NRHS solution matrix X
to the original system of equations. Note that A and B are
modified on exit if equed != 0, and the solution to the
equilibrated system is inv(diag(c))*X if trans = 0 and
equed = 2 or 3, or inv(diag(r))*X if trans = 1 or 2
and equed = 1 or 3.
rcond: The estimate of the reciprocal condition number of the matrix
A after equilibration (if done). If rcond is less than the
machine precision (in particular, if rcond = 0), the matrix
is singular to working precision. This condition is
indicated by a return code of info > 0.
ferr: The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), ferr(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for rcond, and is almost always a slight
overestimate of the true error.
berr: The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
rpvgrw: Contains the reciprocal pivot growth factor norm(A)/norm(U).
The "max absolute element" norm is used. If it is much less
than 1, then the stability of the LU factorization of the
(equilibrated) matrix A could be poor. This also means that
the solution X, condition estimator rcond, and forward error
bound ferr could be unreliable. If factorization fails with
0<info<=N, then it contains the reciprocal pivot growth factor
for the leading info columns of A.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, and i is
<= N: U(i,i) is exactly zero. The factorization has
been completed, but the factor U is exactly
singular, so the solution and error bounds
could not be computed. rcond = 0 is returned.
= N+1: U is nonsingular, but rcond is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of rcond would suggest.
```

```
$a= random(5,5);
$b = random(5,5);
$a = transpose($a);
$b = transpose($b);
$rcond = pdl(0);
$rpvgrw = pdl(0);
$equed = pdl(long,0);
$info = pdl(long,0);
$berr = zeroes(5);
$ipiv = zeroes(5);
$ferr = zeroes(5);
$r = zeroes(5);
$c = zeroes(5);
$X = zeroes(5,5);
$af = zeroes(5,5);
gesvx($a,0, 2, $b, $af, $ipiv, $equed, $r, $c, $X, $rcond, $ferr, $berr, $rpvgrw, $info);
print "The solution matrix X is :". transpose($X)."\n" unless $info;
```

gesvx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sysv

` Signature: ([io,phys]A(n,n); int uplo(); [io,phys]B(n,m); int [o]ipiv(n); int [o]info())`

Computes the solution to a real system of linear equations

```
A * X = B,
where A is an N-by-N symmetric matrix and X and B are N-by-NRHS
matrices.
```

The diagonal pivoting method is used to factor A as

```
A = U * D * U', if uplo = 0, or
A = L * D * L', if uplo = 1,
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
```

The factored form of A is then used to solve the system of equations A * X = B.

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if info = 0, the block diagonal matrix D and the
multipliers used to obtain the factor U or L from the
factorization A = U*D*U' or A = L*D*L' as computed by
sytrf.
ipiv: Details of the interchanges and the block structure of D, as
determined by sytrf. If ipiv(k) > 0, then rows and columns
k and ipiv(k) were interchanged, and D(k,k) is a 1-by-1
diagonal block. If uplo = 0 and ipiv(k) = ipiv(k-1) < 0,
then rows and columns k-1 and -ipiv(k) were interchanged and
D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If uplo = 1 and
ipiv(k) = ipiv(k+1) < 0, then rows and columns k+1 and
-ipiv(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2
diagonal block.
B: On entry, the N-by-NRHS right hand side matrix B.
On exit, if info = 0, the N-by-NRHS solution matrix X.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, so the solution could not be computed.
```

```
# Assume $a is symmetric ;)
$a = random (5,5);
$a = transpose($a);
$b = random(4,5);
$b = transpose($b);
sysv($a, 1, $b, (my $ipiv=zeroes(5)),(my $info=null));
print "The solution matrix X is :". transpose($b)."\n" unless $info;
```

sysv ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sysvx

` Signature: ([phys]A(n,n); int uplo(); int fact(); [phys]B(n,m); [io,phys]af(n,n); int [io,phys]ipiv(n); [o]X(n,m); [o]rcond(); [o]ferr(m); [o]berr(m); int [o]info(); int [t]iwork(n))`

Uses the diagonal pivoting factorization to compute the solution to a real system of linear equations A * X = B, where A is an N-by-N symmetric matrix and X and B are N-by-NRHS matrices.

Error bounds on the solution and a condition estimate are also provided.

The following steps are performed:

If fact = 0, the diagonal pivoting method is used to factor A. The form of the factorization is

`A = U * D * U', if uplo = 0, or A = L * D * L', if uplo = 1, where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.`

If some D(i,i)=0, so that D is exactly singular, then the routine returns with info = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, info = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

The system of equations is solved for X using the factored form of A.

Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

```
Arguments
=========
fact: Specifies whether or not the factored form of A has been
supplied on entry.
= 0: The matrix A will be copied to af and factored.
= 1: On entry, af and ipiv contain the factored form of
A. af and ipiv will not be modified.
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: The symmetric matrix A. If uplo = 0, the leading N-by-N
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If uplo = 1, the leading N-by-N lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.
af: If fact = 1, then af is an input argument and on entry
contains the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U' or A = L*D*L' as computed by sytrf.
If fact = 0, then af is an output argument and on exit
returns the block diagonal matrix D and the multipliers used
to obtain the factor U or L from the factorization
A = U*D*U' or A = L*D*L'.
ipiv: If fact = 1, then ipiv is an input argument and on entry
contains details of the interchanges and the block structure
of D, as determined by sytrf.
If ipiv(k) > 0, then rows and columns k and ipiv(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If uplo = 0 and ipiv(k) = ipiv(k-1) < 0, then rows and
columns k-1 and -ipiv(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If uplo = 1 and ipiv(k) =
ipiv(k+1) < 0, then rows and columns k+1 and -ipiv(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
If fact = 0, then ipiv is an output argument and on exit
contains details of the interchanges and the block structure
of D, as determined by sytrf.
B: The N-by-NRHS right hand side matrix B.
X: If info = 0 or info = N+1, the N-by-NRHS solution matrix X.
rcond: The estimate of the reciprocal condition number of the matrix
A. If rcond is less than the machine precision (in
particular, if rcond = 0), the matrix is singular to working
precision. This condition is indicated by a return code of
info > 0.
ferr: The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), ferr(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for rcond, and is almost always a slight
overestimate of the true error.
berr: The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, and i is
<= N: D(i,i) is exactly zero. The factorization
has been completed but the factor D is exactly
singular, so the solution and error bounds could
not be computed. rcond = 0 is returned.
= N+1: D is nonsingular, but rcond is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of rcond would suggest.
```

```
$a= random(5,5);
$b = random(10,5);
$a = transpose($a);
$b = transpose($b);
$X = zeroes($b);
$af = zeroes($a);
$ipiv = zeroes(long, 5);
$rcond = pdl(0);
$ferr = zeroes(10);
$berr = zeroes(10);
$info = pdl(long, 0);
# Assume $a is symmetric
sysvx($a, 0, 0, $b,$af, $ipiv, $X, $rcond, $ferr, $berr,$info);
print "The solution matrix X is :". transpose($X)."\n";
```

sysvx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## posv

` Signature: ([io,phys]A(n,n); int uplo(); [io,phys]B(n,m); int [o,phys]info())`

Computes the solution to a real system of linear equations

```
A * X = B,
where A is an N-by-N symmetric positive definite matrix and X and B
are N-by-NRHS matrices.
```

The Cholesky decomposition is used to factor A as

```
A = U'* U, if uplo = 0, or
A = L * L', if uplo = 1,
where U is an upper triangular matrix and L is a lower triangular
matrix.
```

The factored form of A is then used to solve the system of equations A * X = B.

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if info = 0, the factor U or L from the Cholesky
factorization A = U'*U or A = L*L'.
B: On entry, the N-by-NRHS right hand side matrix B.
On exit, if info = 0, the N-by-NRHS solution matrix X.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, the leading minor of order i of A is not
positive definite, so the factorization could not be
completed, and the solution has not been computed.
```

```
# Assume $a is symmetric positive definite ;)
$a = random (5,5);
$a = transpose($a);
$b = random(4,5);
$b = transpose($b);
posv($a, 1, $b, (my $info=null));
print "The solution matrix X is :". transpose($b)."\n" unless $info;
```

posv ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## posvx

` Signature: ([io,phys]A(n,n); int uplo(); int fact(); [io,phys]B(n,m); [io,phys]af(n,n); int [io]equed(); [o]s(p); [o,phys]X(n,m); [o,phys]rcond(); [o,phys]ferr(m); [o,phys]berr(m); int [o,phys]info(); int [t]iwork(n); [t]work(workn))`

Uses the Cholesky factorization A = U'*U or A = L*L' to compute the solution to a real system of linear equations

```
A * X = B,
where A is an N-by-N symmetric positive definite matrix and X and B
are N-by-NRHS matrices.
```

Error bounds on the solution and a condition estimate are also provided.

The following steps are performed:

If fact = 2, real scaling factors are computed to equilibrate the system:

`diag(s) * A * diag(s) * inv(diag(s)) * X = diag(s) * B`

Whether or not the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.

If fact = 1 or 2, the Cholesky decomposition is used to factor the matrix A (after equilibration if fact = 2) as

`A = U'* U, if uplo = 0, or A = L * L', if uplo = 1, where U is an upper triangular matrix and L is a lower triangular matrix.`

If the leading i-by-i principal minor is not positive definite, then the routine returns with info = i. Otherwise, the factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the condition number is less than machine precision, info = N+1 is returned as a warning, but the routine still goes on to solve for X and compute error bounds as described below.

The system of equations is solved for X using the factored form of A.

Iterative refinement is applied to improve the computed solution matrix and calculate error bounds and backward error estimates for it.

If equilibration was used, the matrix X is premultiplied by diag(s) so that it solves the original system before equilibration.

```
Arguments
=========
fact: Specifies whether or not the factored form of the matrix A is
supplied on entry, and if not, whether the matrix A should be
equilibrated before it is factored.
= 0: On entry, af contains the factored form of A.
If equed = 1, the matrix A has been equilibrated
with scaling factors given by s. A and af will not
be modified.
= 1: The matrix A will be copied to af and factored.
= 2: The matrix A will be equilibrated if necessary, then
copied to af and factored.
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A, except if fact = 0 and
equed = 1, then A must contain the equilibrated matrix
diag(s)*A*diag(s). If uplo = 0, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced. A is not modified if
fact = 0 or 1, or if fact = 2 and equed = 0 on exit.
On exit, if fact = 2 and equed = 1, A is overwritten by
diag(s)*A*diag(s).
af: If fact = 0, then af is an input argument and on entry
contains the triangular factor U or L from the Cholesky
factorization A = U'*U or A = L*L', in the same storage
format as A. If equed != 0, then af is the factored form
of the equilibrated matrix diag(s)*A*diag(s).
If fact = 1, then af is an output argument and on exit
returns the triangular factor U or L from the Cholesky
factorization A = U'*U or A = L*L' of the original
matrix A.
If fact = 2, then af is an output argument and on exit
returns the triangular factor U or L from the Cholesky
factorization A = U'*U or A = L*L' of the equilibrated
matrix A (see the description of A for the form of the
equilibrated matrix).
equed: Specifies the form of equilibration that was done.
= 0: No equilibration (always true if fact = 1).
= 1: Equilibration was done, i.e., A has been replaced by
diag(s) * A * diag(s).
equed is an input argument if fact = 0; otherwise, it is an
output argument.
s: The scale factors for A; not accessed if equed = 0. s is
an input argument if fact = 0; otherwise, s is an output
argument. If fact = 0 and equed = 1, each element of s
must be positive.
B: On entry, the N-by-NRHS right hand side matrix B.
On exit, if equed = 0, B is not modified; if equed = 1,
B is overwritten by diag(s) * B.
X: If info = 0 or info = N+1, the N-by-NRHS solution matrix X to
the original system of equations. Note that if equed = 1,
A and B are modified on exit, and the solution to the
equilibrated system is inv(diag(s))*X.
rcond: The estimate of the reciprocal condition number of the matrix
A after equilibration (if done). If rcond is less than the
machine precision (in particular, if rcond = 0), the matrix
is singular to working precision. This condition is
indicated by a return code of info > 0.
ferr: The estimated forward error bound for each solution vector
X(j) (the j-th column of the solution matrix X).
If XTRUE is the true solution corresponding to X(j), FERR(j)
is an estimated upper bound for the magnitude of the largest
element in (X(j) - XTRUE) divided by the magnitude of the
largest element in X(j). The estimate is as reliable as
the estimate for rcond, and is almost always a slight
overestimate of the true error.
berr: The componentwise relative backward error of each solution
vector X(j) (i.e., the smallest relative change in
any element of A or B that makes X(j) an exact solution).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, and i is
<= N: the leading minor of order i of A is
not positive definite, so the factorization
could not be completed, and the solution has not
been computed. rcond = 0 is returned.
= N+1: U is nonsingular, but rcond is less than machine
precision, meaning that the matrix is singular
to working precision. Nevertheless, the
solution and error bounds are computed because
there are a number of situations where the
computed solution can be more accurate than the
value of rcond would suggest.
```

```
$a= random(5,5);
$b = random(5,5);
$a = transpose($a);
$b = transpose($b);
# Assume $a is symmetric positive definite
$rcond = pdl(0);
$equed = pdl(long,0);
$info = pdl(long,0);
$berr = zeroes(5);
$ferr = zeroes(5);
$s = zeroes(5);
$X = zeroes(5,5);
$af = zeroes(5,5);
posvx($a,0,2,$b,$af, $equed, $s, $X, $rcond, $ferr, $berr,$info);
print "The solution matrix X is :". transpose($X)."\n" unless $info;
```

posvx ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gels

` Signature: ([io,phys]A(m,n); int trans(); [io,phys]B(p,q);int [o,phys]info())`

Solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A. It is assumed that A has full rank.

The following options are provided:

If trans = 0 and m >= n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A*X ||.

If trans = 0 and m < n: find the minimum norm solution of an underdetermined system A * X = B.

If trans = 1 and m >= n: find the minimum norm solution of an undetermined system A' * X = B.

If trans = 1 and m < n: find the least squares solution of an overdetermined system, i.e., solve the least squares problem minimize || B - A' * X ||.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

```
Arguments
=========
trans: = 0: the linear system involves A;
= 1: the linear system involves A'.
A: On entry, the M-by-N matrix A.
On exit,
if M >= N, A is overwritten by details of its QR
factorization as returned by geqrf;
if M < N, A is overwritten by details of its LQ
factorization as returned by gelqf.
B: On entry, the matrix B of right hand side vectors, stored
columnwise; B is M-by-NRHS if trans = 0, or N-by-NRHS
if trans = 1.
On exit, B is overwritten by the solution vectors, stored
columnwise:
if trans = 0 and m >= n, rows 1 to n of B contain the least
squares solution vectors; the residual sum of squares for the
solution in each column is given by the sum of squares of
elements N+1 to M in that column;
if trans = 0 and m < n, rows 1 to N of B contain the
minimum norm solution vectors;
if trans = 1 and m >= n, rows 1 to M of B contain the
minimum norm solution vectors;
if trans = 1 and m < n, rows 1 to M of B contain the
least squares solution vectors; the residual sum of squares
for the solution in each column is given by the sum of
squares of elements M+1 to N in that column.
The leading dimension of the array B >= max(1,M,N).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a= random(7,5);
# $b will contain X
# TODO better example with slice
$b = random(7,6);
gels($a, 1, $b, ($info = null));
```

gels ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gelsy

` Signature: ([io,phys]A(m,n); [io,phys]B(p,q); [phys]rcond(); int [io,phys]jpvt(n); int [o,phys]rank();int [o,phys]info())`

Computes the minimum-norm solution to a real linear least squares problem:

` minimize || A * X - B ||`

using a complete orthogonal factorization of A.

A is an M-by-N matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

The routine first computes a QR factorization with column pivoting:

```
A * P = Q * [ R11 R12 ]
[ 0 R22 ]
with R11 defined as the largest leading submatrix whose estimated
condition number is less than 1/rcond. The order of R11, rank,
is the effective rank of A.
```

Then, R22 is considered to be negligible, and R12 is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization:

```
A * P = Q * [ T11 0 ] * Z
[ 0 0 ]
```

The minimum-norm solution is then

```
X = P * Z' [ inv(T11)*Q1'*B ]
[ 0 ]
where Q1 consists of the first rank columns of Q.
Arguments
=========
A: On entry, the M-by-N matrix A.
On exit, A has been overwritten by details of its
complete orthogonal factorization.
B: On entry, the M-by-NRHS right hand side matrix B.
On exit, the N-by-NRHS solution matrix X.
The leading dimension of the array B >= max(1,M,N).
jpvt: On entry, if jpvt(i) != 0, the i-th column of A is permuted
to the front of AP, otherwise column i is a free column.
On exit, if jpvt(i) = k, then the i-th column of AP
was the k-th column of A.
rcond: rcond is used to determine the effective rank of A, which
is defined as the order of the largest leading triangular
submatrix R11 in the QR factorization with pivoting of A,
whose estimated condition number < 1/rcond.
rank: The effective rank of A, i.e., the order of the submatrix
R11. This is the same as the order of the submatrix T11
in the complete orthogonal factorization of A.
info: = 0: successful exit
< 0: If info = -i, the i-th argument had an illegal value.
```

```
$a= random(7,5);
# $b will contain X
# TODO better example with slice
$b = random(7,6);
$jpvt = zeroes(long, 5);
$eps = lamch(0);
#Threshold for rank estimation
$rcond = sqrt($eps) - (sqrt($eps) - $eps) / 2;
gelsy($a, $b, $rcond, $jpvt,($rank=null),($info = null));
```

gelsy ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gelss

` Signature: ([io,phys]A(m,n); [io,phys]B(p,q); [phys]rcond(); [o,phys]s(r); int [o,phys]rank();int [o,phys]info())`

Computes the minimum norm solution to a real linear least squares problem:

` Minimize 2-norm(| b - A*x |).`

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient.

Several right hand side vectors b and solution vectors x can be handled in a single call; they are stored as the columns of the M-by-NRHS right hand side matrix B and the N-by-NRHS solution matrix X.

The effective rank of A is determined by treating as zero those singular values which are less than rcond times the largest singular value.

```
Arguments
=========
A: On entry, the M-by-N matrix A.
On exit, the first min(m,n) rows of A are overwritten with
its right singular vectors, stored rowwise.
B: On entry, the M-by-NRHS right hand side matrix B.
On exit, B is overwritten by the N-by-NRHS solution
matrix X. If m >= n and rank = n, the residual
sum-of-squares for the solution in the i-th column is given
by the sum of squares of elements n+1:m in that column.
The leading dimension of the array B >= max(1,M,N).
s: The singular values of A in decreasing order.
The condition number of A in the 2-norm = s(1)/s(min(m,n)).
rcond: rcond is used to determine the effective rank of A.
Singular values s(i) <= rcond*s(1) are treated as zero.
If rcond < 0, machine precision is used instead.
rank: The effective rank of A, i.e., the number of singular values
which are greater than rcond*s(1).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
> 0: the algorithm for computing the SVD failed to converge;
if info = i, i off-diagonal elements of an intermediate
bidiagonal form did not converge to zero.
```

```
$a= random(7,5);
# $b will contain X
# TODO better example with slice
$b = random(7,6);
$eps = lamch(0);
$s =zeroes(5);
#Threshold for rank estimation
$rcond = sqrt($eps) - (sqrt($eps) - $eps) / 2;
gelss($a, $b, $rcond, $s, ($rank=null),($info = null));
```

gelss ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gelsd

` Signature: ([io]A(m,n); [io]B(p,q); rcond(); [o]s(minmn); int [o]rank();int [o]info(); int [t]iwork(iworkn))`

Computes the minimum-norm solution to a real linear least squares problem:

` minimize 2-norm(| b - A*x |)`

using the singular value decomposition (SVD) of A. A is an M-by-N matrix which may be rank-deficient.

The problem is solved in three steps:

Reduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the original problem into a "bidiagonal least squares problem" (BLS)

Solve the BLS using a divide and conquer approach.

Apply back all the Householder transformations to solve the original least squares problem.

The effective rank of A is determined by treating as zero those singular values which are less than rcond times the largest singular value.

The divide and conquer algorithm makes very mild assumptions about floating point arithmetic. It will work on machines with a guard digit in add/subtract, or on those binary machines without guard digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could conceivably fail on hexadecimal or decimal machines without guard digits, but we know of none.

```
Arguments
=========
A: On entry, the M-by-N matrix A.
On exit, A has been destroyed.
B: On entry, the M-by-NRHS right hand side matrix B.
On exit, B is overwritten by the N-by-NRHS solution
matrix X. If m >= n and rank = n, the residual
sum-of-squares for the solution in the i-th column is given
by the sum of squares of elements n+1:m in that column.
The leading dimension of the array B >= max(1,M,N).
s: The singular values of A in decreasing order.
The condition number of A in the 2-norm = s(1)/s(min(m,n)).
rcond: rcond is used to determine the effective rank of A.
Singular values s(i) <= rcond*s(1) are treated as zero.
If rcond < 0, machine precision is used instead.
rank: The effective rank of A, i.e., the number of singular values
which are greater than rcond*s(1).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
> 0: the algorithm for computing the SVD failed to converge;
if info = i, i off-diagonal elements of an intermediate
bidiagonal form did not converge to zero.
```

```
$a= random(7,5);
# $b will contain X
# TODO better example with slice
$b = random(7,6);
$eps = lamch(0);
$s =zeroes(5);
#Threshold for rank estimation
$rcond = sqrt($eps) - (sqrt($eps) - $eps) / 2;
gelsd($a, $b, $rcond, $s, ($rank=null),($info = null));
```

gelsd ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gglse

` Signature: ([phys]A(m,n); [phys]B(p,n);[io,phys]c(m);[phys]d(p);[o,phys]x(n);int [o,phys]info())`

Solves the linear equality-constrained least squares (LSE) problem:

```
minimize || c - A*x ||_2 subject to B*x = d
where A is an M-by-N matrix, B is a P-by-N matrix, c is a given
M-vector, and d is a given P-vector. It is assumed that
P <= N <= M+P, and
rank(B) = P and rank( ( A ) ) = N.
( ( B ) )
```

These conditions ensure that the LSE problem has a unique solution, which is obtained using a GRQ factorization of the matrices B and A.

```
Arguments
=========
A: On entry, the M-by-N matrix A.
On exit, A is destroyed.
B: On entry, the P-by-N matrix B.
On exit, B is destroyed.
c: On entry, c contains the right hand side vector for the
least squares part of the LSE problem.
On exit, the residual sum of squares for the solution
is given by the sum of squares of elements N-P+1 to M of
vector c.
d: On entry, d contains the right hand side vector for the
constrained equation.
On exit, d is destroyed.
x: On exit, x is the solution of the LSE problem.
info: = 0: successful exit.
< 0: if info = -i, the i-th argument had an illegal value.
```

```
$a = random(7,5);
$b = random(4,5);
$c = random(7);
$d = random(4);
$x = zeroes(5);
gglse($a, $b, $c, $d, $x, ($info=null));
```

gglse ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ggglm

` Signature: ([phys]A(n,m); [phys]B(n,p);[phys]d(n);[o,phys]x(m);[o,phys]y(p);int [o,phys]info())`

Solves a general Gauss-Markov linear model (GLM) problem:

```
minimize || y ||_2 subject to d = A*x + B*y
x
where A is an N-by-M matrix, B is an N-by-P matrix, and d is a
given N-vector. It is assumed that M <= N <= M+P, and
rank(A) = M and rank( A B ) = N.
```

Under these assumptions, the constrained equation is always consistent, and there is a unique solution x and a minimal 2-norm solution y, which is obtained using a generalized QR factorization of A and B.

In particular, if matrix B is square nonsingular, then the problem GLM is equivalent to the following weighted linear least squares problem

```
minimize || inv(B)*(d-A*x) ||_2
x
where inv(B) denotes the inverse of B.
Arguments
=========
A: On entry, the N-by-M matrix A.
On exit, A is destroyed.
B: On entry, the N-by-P matrix B.
On exit, B is destroyed.
d: On entry, d is the left hand side of the GLM equation.
On exit, d is destroyed.
x:
y: On exit, x and y are the solutions of the GLM problem.
info: = 0: successful exit.
< 0: if info = -i, the i-th argument had an illegal value.
```

```
$a = random(7,5);
$b = random(7,4);
$d = random(7);
$x = zeroes(5);
$y = zeroes(4);
ggglm($a, $b, $d, $x, $y,($info=null));
```

ggglm ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## getrf

` Signature: ([io]A(m,n); int [o]ipiv(p); int [o]info())`

Computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges.

The factorization has the form

```
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
```

This is the right-looking Level 3 BLAS version of the algorithm.

```
Arguments
=========
A: On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
ipiv: The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row ipiv(i).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
```

```
$a = random (float, 100,50);
$ipiv = zeroes(long, 50);
$info = null;
getrf($a, $ipiv, $info);
```

getrf ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## getf2

` Signature: ([io]A(m,n); int [o]ipiv(p); int [o]info())`

Computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges.

The factorization has the form

```
A = P * L * U
where P is a permutation matrix, L is lower triangular with unit
diagonal elements (lower trapezoidal if m > n), and U is upper
triangular (upper trapezoidal if m < n).
```

This is the right-looking Level 2 BLAS version of the algorithm.

```
Arguments
=========
A: On entry, the M-by-N matrix to be factored.
On exit, the factors L and U from the factorization
A = P*L*U; the unit diagonal elements of L are not stored.
ipiv: The pivot indices; for 1 <= i <= min(M,N), row i of the
matrix was interchanged with row ipiv(i).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, U(i,i) is exactly zero. The factorization
has been completed, but the factor U is exactly
singular, and division by zero will occur if it is used
to solve a system of equations.
```

```
$a = random (float, 100,50);
$ipiv = zeroes(long, 50);
$info = null;
getf2($a, $ipiv, $info);
```

getf2 ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sytrf

` Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]ipiv(n); int [o,phys]info())`

Computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is

```
A = U*D*U' or A = L*D*L'
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
```

This is the blocked version of the algorithm, calling Level 3 BLAS.

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).
ipiv: Details of the interchanges and the block structure of D.
If ipiv(k) > 0, then rows and columns k and ipiv(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If uplo = 0 and ipiv(k) = ipiv(k-1) < 0, then rows and
columns k-1 and -ipiv(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If uplo = 1 and ipiv(k) =
ipiv(k+1) < 0, then rows and columns k+1 and -ipiv(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.
Further Details
===============
If uplo = 0, then A = U*D*U', where
U = P(n)*U(n)* ... *P(k)U(k)* ...,
i.e., U is a product of terms P(k)*U(k), where k decreases from n to
1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by ipiv(k), and U(k) is a unit upper triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I v 0 ) k-s
U(k) = ( 0 I 0 ) s
( 0 0 I ) n-k
k-s s n-k
If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).
If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k),
and A(k,k), and v overwrites A(1:k-2,k-1:k).
If uplo = 1, then A = L*D*L', where
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
i.e., L is a product of terms P(k)*L(k), where k increases from 1 to
n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1
and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as
defined by ipiv(k), and L(k) is a unit lower triangular matrix, such
that if the diagonal block D(k) is of order s (s = 1 or 2), then
( I 0 0 ) k-1
L(k) = ( 0 I 0 ) s
( 0 v I ) n-k-s+1
k-1 s n-k-s+1
If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).
If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k),
and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).
```

```
$a = random(100,100);
$ipiv = zeroes(100);
$info = null;
# Assume $a is symmetric
sytrf($a, 0, $ipiv, $info);
```

sytrf ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sytf2

` Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]ipiv(n); int [o,phys]info())`

Computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is

```
A = U*D*U' or A = L*D*L'
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
1-by-1 and 2-by-2 diagonal blocks.
```

This is the unblocked version of the algorithm, calling Level 2 BLAS.

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, the block diagonal matrix D and the multipliers used
to obtain the factor U or L (see below for further details).
ipiv: Details of the interchanges and the block structure of D.
If ipiv(k) > 0, then rows and columns k and ipiv(k) were
interchanged and D(k,k) is a 1-by-1 diagonal block.
If uplo = 0 and ipiv(k) = ipiv(k-1) < 0, then rows and
columns k-1 and -ipiv(k) were interchanged and D(k-1:k,k-1:k)
is a 2-by-2 diagonal block. If uplo = 1 and ipiv(k) =
ipiv(k+1) < 0, then rows and columns k+1 and -ipiv(k) were
interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, D(i,i) is exactly zero. The factorization
has been completed, but the block diagonal matrix D is
exactly singular, and division by zero will occur if it
is used to solve a system of equations.
For further details see sytrf
```

```
$a = random(100,100);
$ipiv = zeroes(100);
$info = null;
# Assume $a is symmetric
sytf2($a, 0, $ipiv, $info);
```

sytf2 ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## potrf

` Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]info())`

Computes the Cholesky factorization of a real symmetric positive definite matrix A.

The factorization has the form

```
A = U' * U, if uplo = 0, or
A = L * L', if uplo = 1,
where U is an upper triangular matrix and L is lower triangular.
```

This is the block version of the algorithm, calling Level 3 BLAS.

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if info = 0, the factor U or L from the Cholesky
factorization A = U'*U or A = L*L'.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, the leading minor of order i is not
positive definite, and the factorization could not be
completed.
```

```
$a = random(100,100);
# Assume $a is symmetric positive definite
potrf($a, 0, ($info = null));
```

potrf ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## potf2

` Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]info())`

Computes the Cholesky factorization of a real symmetric positive definite matrix A.

The factorization has the form

```
A = U' * U, if uplo = 0, or
A = L * L', if uplo = 1,
where U is an upper triangular matrix and L is lower triangular.
```

This is the unblocked version of the algorithm, calling Level 2 BLAS.

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the symmetric matrix A. If uplo = 0, the leading
N-by-N upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit, if info = 0, the factor U or L from the Cholesky
factorization A = U'*U or A = L*L'.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, the leading minor of order i is not
positive definite, and the factorization could not be
completed.
```

```
$a = random(100,100);
# Assume $a is symmetric positive definite
potf2($a, 0, ($info = null));
```

potf2 ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## getri

` Signature: ([io,phys]A(n,n); int [phys]ipiv(n); int [o,phys]info())`

Computes the inverse of a matrix using the LU factorization computed by `getrf`

.

This method inverts U and then computes inv(A) by solving the system

```
inv(A)*L = inv(U) for inv(A).
Arguments
=========
A: On entry, the factors L and U from the factorization
A = P*L*U as computed by getrf.
On exit, if info = 0, the inverse of the original matrix A.
ipiv: The pivot indices from getrf; for 1<=i<=N, row i of the
matrix was interchanged with row ipiv(i).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, U(i,i) is exactly zero; the matrix is
singular and its inverse could not be computed.
```

```
$a = random (float, 100, 100);
$ipiv = zeroes(long, 100);
$info = null;
getrf($a, $ipiv, $info);
if ($info == 0){
getri($a, $ipiv, $info);
}
print "Inverse of \$a is :\n $a" unless $info;
```

getri ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sytri

` Signature: ([io]A(n,n); int uplo(); int ipiv(n); int [o]info(); [t]work(n))`

Computes the inverse of a real symmetric indefinite matrix A using the factorization A = U*D*U' or A = L*D*L' computed by `sytrf`

.

```
Arguments
=========
uplo: Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 0: Upper triangular, form is A = U*D*U';
= 1: Lower triangular, form is A = L*D*L'.
A: On entry, the block diagonal matrix D and the multipliers
used to obtain the factor U or L as computed by sytrf.
On exit, if info = 0, the (symmetric) inverse of the original
matrix. If uplo = 0, the upper triangular part of the
inverse is formed and the part of A below the diagonal is not
referenced; if uplo = 1 the lower triangular part of the
inverse is formed and the part of A above the diagonal is
not referenced.
ipiv: Details of the interchanges and the block structure of D
as determined by sytrf.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, D(i,i) = 0; the matrix is singular and its
inverse could not be computed.
```

```
$a = random (float, 100, 100);
# assume $a is symmetric
$ipiv = zeroes(long, 100);
sytrf($a, 0, $ipiv, ($info=null));
if ($info == 0){
sytri($a, 0, $ipiv, $info);
}
print "Inverse of \$a is :\n $a" unless $info;
```

sytri ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## potri

` Signature: ([io,phys]A(n,n); int uplo(); int [o,phys]info())`

Computes the inverse of a real symmetric positive definite matrix A using the Cholesky factorization A = U'*U or A = L*L' computed by `potrf`

.

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: On entry, the triangular factor U or L from the Cholesky
factorization A = U'*U or A = L*L', as computed by
potrf.
On exit, the upper or lower triangle of the (symmetric)
inverse of A, overwriting the input factor U or L.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, the (i,i) element of the factor U or L is
zero, and the inverse could not be computed.
```

```
$a = random (float, 100, 100);
# Assume $a is symmetric positive definite
potrf($a, 0, ($info = null));
if ($info == 0){ # Hum... is it positive definite????
potri($a, 0,$info);
}
print "Inverse of \$a is :\n $a" unless $info;
```

potri ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## trtri

` Signature: ([io,phys]A(n,n); int uplo(); int diag(); int [o,phys]info())`

Computes the inverse of a real upper or lower triangular matrix A.

This is the Level 3 BLAS version of the algorithm.

```
Arguments
=========
uplo: = 0: A is upper triangular;
= 1: A is lower triangular.
diag: = 0: A is non-unit triangular;
= 1: A is unit triangular.
A: On entry, the triangular matrix A. If uplo = 0, the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced. If diag = 1, the
diagonal elements of A are also not referenced and are
assumed to be 1.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, A(i,i) is exactly zero. The triangular
matrix is singular and its inverse can not be computed.
```

```
$a = random (float, 100, 100);
# assume $a is upper triangular
trtri($a, 1, ($info=null));
print "Inverse of \$a is :\n transpose($a)" unless $info;
```

trtri ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## trti2

` Signature: ([io,phys]A(n,n); int uplo(); int diag(); int [o,phys]info())`

Computes the inverse of a real upper or lower triangular matrix A.

This is the Level 2 BLAS version of the algorithm.

```
Arguments
=========
uplo: = 0: A is upper triangular;
= 1: A is lower triangular.
diag: = 0: A is non-unit triangular;
= 1: A is unit triangular.
A: On entry, the triangular matrix A. If uplo = 0, the
leading N-by-N upper triangular part of the array A contains
the upper triangular matrix, and the strictly lower
triangular part of A is not referenced. If uplo = 1, the
leading N-by-N lower triangular part of the array A contains
the lower triangular matrix, and the strictly upper
triangular part of A is not referenced. If diag = 1, the
diagonal elements of A are also not referenced and are
assumed to be 1.
On exit, the (triangular) inverse of the original matrix, in
the same storage format.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 100, 100);
# assume $a is upper triangular
trtri2($a, 1, ($info=null));
print "Inverse of \$a is :\n transpose($a)" unless $info;
```

trti2 ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## getrs

` Signature: ([phys]A(n,n); int trans(); [io,phys]B(n,m); int [phys]ipiv(n); int [o,phys]info())`

Solves a system of linear equations

` A * X = B or A' * X = B`

with a general N-by-N matrix A using the LU factorization computed by getrf.

```
Arguments
=========
trans: Specifies the form of the system of equations:
= 0: A * X = B (No transpose)
= 1: A'* X = B (Transpose)
A: The factors L and U from the factorization A = P*L*U
as computed by getrf.
ipiv: The pivot indices from getrf; for 1<=i<=N, row i of the
matrix was interchanged with row ipiv(i).
B: On entry, the right hand side matrix B.
On exit, the solution matrix X.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 100, 100);
$ipiv = zeroes(long, 100);
$b = random(100,50);
getrf($a, $ipiv, ($info=null));
if ($info == 0){
getrs($a, 0, $b, $ipiv, $info);
}
print "X is :\n $b" unless $info;
```

getrs ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sytrs

` Signature: ([phys]A(n,n); int uplo();[io,phys]B(n,m); int [phys]ipiv(n); int [o,phys]info())`

Solves a system of linear equations A*X = B with a real symmetric matrix A using the factorization A = U*D*U' or A = L*D*L' computed by `sytrf`

.

```
Arguments
=========
uplo: Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 0: Upper triangular, form is A = U*D*U';
= 1: Lower triangular, form is A = L*D*L'.
A: The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by sytrf.
ipiv: Details of the interchanges and the block structure of D
as determined by sytrf.
B: On entry, the right hand side matrix B.
On exit, the solution matrix X.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 100, 100);
$b = random(50,100);
$a = transpose($a);
$b = transpose($b);
# Assume $a is symmetric
sytrf($a, 0, ($ipiv=zeroes(100)), ($info=null));
if ($info == 0){
sytrs($a, 0, $b, $ipiv, $info);
}
print("X is :\n".transpose($b))unless $info;
```

sytrs ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## potrs

` Signature: ([phys]A(n,n); int uplo(); [io,phys]B(n,m); int [o,phys]info())`

Solves a system of linear equations A*X = B with a symmetric positive definite matrix A using the Cholesky factorization A = U'*U or A = L*L' computed by `potrf`

.

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: The triangular factor U or L from the Cholesky factorization
A = U'*U or A = L*L', as computed by potrf.
B: On entry, the right hand side matrix B.
On exit, the solution matrix X.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 100, 100);
$b = random(50,100);
$a = transpose($a);
$b = transpose($b);
# Assume $a is symmetric positive definite
potrf($a, 0, ($info=null));
if ($info == 0){
potrs($a, 0, $b, $info);
}
print("X is :\n".transpose($b))unless $info;
```

potrs ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## trtrs

` Signature: ([phys]A(n,n); int uplo(); int trans(); int diag();[io,phys]B(n,m); int [o,phys]info())`

Solves a triangular system of the form

```
A * X = B or A' * X = B,
where A is a triangular matrix of order N, and B is an N-by-NRHS
matrix.
```

A check is made to verify that A is nonsingular.

```
Arguments
=========
uplo: = 0: A is upper triangular;
= 1: A is lower triangular.
trans: Specifies the form of the system of equations:
= 0: A * X = B (No transpose)
= 1: A**T * X = B (Transpose)
diag: = 0: A is non-unit triangular;
= 1: A is unit triangular.
A: The triangular matrix A. If uplo = 0, the leading N-by-N
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If uplo = 1, the leading N-by-N lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If diag = 1, the diagonal elements of A are
also not referenced and are assumed to be 1.
B: On entry, the right hand side matrix B.
On exit, if info = 0, the solution matrix X.
info = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, the i-th diagonal element of A is zero,
indicating that the matrix is singular and the solutions
X have not been computed.
```

```
# Assume $a is upper triangular
$a = random (float, 100, 100);
$b = random(50,100);
$a = transpose($a);
$b = transpose($b);
$info = null;
trtrs($a, 0, 0, 0, $b, $info);
print("X is :\n".transpose($b))unless $info;
```

trtrs ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## latrs

` Signature: ([phys]A(n,n); int uplo(); int trans(); int diag(); int normin();[io,phys]x(n); [o,phys]scale();[io,phys]cnorm(n);int [o,phys]info())`

Solves one of the triangular systems

` A *x = s*b or A'*x = s*b`

with scaling to prevent overflow. Here A is an upper or lower triangular matrix, A' denotes the transpose of A, x and b are n-element vectors, and s is a scaling factor, usually less than or equal to 1, chosen so that the components of x will be less than the overflow threshold. If the unscaled problem will not cause overflow, the Level 2 BLAS routine `trsv`

is called. If the matrix A is singular (A(j,j) = 0 for some j), then s is set to 0 and a non-trivial solution to A*x = 0 is returned.

Further Details ======= =======

A rough bound on x is computed; if that is less than overflow, trsv is called, otherwise, specific code is used which checks for possible overflow or divide-by-zero at every operation.

A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower triangular is

```
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
end
```

Define bounds on the components of x after j iterations of the loop:

```
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
```

Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

```
Then for iteration j+1 we have
M(j+1) <= G(j) / | A(j+1,j+1) |
G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
<= G(j) ( 1 + cnorm(j+1) / | A(j+1,j+1) | )
where cnorm(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal.
```

Hence

```
G(j) <= G(0) product ( 1 + cnorm(i) / | A(i,i) | )
1<=i<=j
and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + cnorm(i) / |A(i,i)| )
1<=i< j
```

Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the reciprocal of the largest M(j), j=1,..,n, is larger than max(underflow, 1/overflow).

The bound on x(j) is also used to determine when a step in the columnwise method can be performed without fear of overflow. If the computed bound is greater than a large constant, x is scaled to prevent overflow, but if the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a non-trivial solution to A*x = 0 is found.

Similarly, a row-wise scheme is used to solve A'*x = b. The basic algorithm for A upper triangular is

```
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end
```

We simultaneously compute two bounds

```
G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
```

The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1. Then the bound on x(j) is

```
M(j) <= M(j-1) * ( 1 + cnorm(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + cnorm(i) ) / |A(i,i)| )
1<=i<=j
```

and we can safely call trsv if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).

```
Arguments
=========
uplo: Specifies whether the matrix A is upper or lower triangular.
= 0: Upper triangular
= 1: Lower triangular
trans: Specifies the operation applied to A.
= 0: Solve A * x = s*b (No transpose)
= 1: Solve A'* x = s*b (Transpose)
diag: Specifies whether or not the matrix A is unit triangular.
= 0: Non-unit triangular
= 1: Unit triangular
normin: Specifies whether cnorm has been set or not.
= 1: cnorm contains the column norms on entry
= 0: cnorm is not set on entry. On exit, the norms will
be computed and stored in cnorm.
A: The triangular matrix A. If uplo = 0, the leading n by n
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If uplo = 1, the leading n by n lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If diag = 1, the diagonal elements of A are
also not referenced and are assumed to be 1.
x: On entry, the right hand side b of the triangular system.
On exit, x is overwritten by the solution vector x.
scale: The scaling factor s for the triangular system
A * x = s*b or A'* x = s*b.
If scale = 0, the matrix A is singular or badly scaled, and
the vector x is an exact or approximate solution to A*x = 0.
cnorm: If normin = 0, cnorm is an output argument and cnorm(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
If normin = 1, cnorm is an input argument and cnorm(j)
contains the norm of the off-diagonal part of the j-th column
of A. If trans = 0, cnorm(j) must be greater than or equal
to the infinity-norm, and if trans = 1, cnorm(j)
must be greater than or equal to the 1-norm.
info: = 0: successful exit
< 0: if info = -k, the k-th argument had an illegal value
```

```
# Assume $a is upper triangular
$a = random (float, 100, 100);
$b = random(100);
$a = transpose($a);
$info = null;
$scale= null;
$cnorm = zeroes(100);
latrs($a, 0, 0, 0, 0,$b, $scale, $cnorm,$info);
```

latrs ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gecon

` Signature: (A(n,n); int norm(); anorm(); [o]rcond();int [o]info(); int [t]iwork(n); [t]work(workn))`

Estimates the reciprocal of the condition number of a general real matrix A, in either the 1-norm or the infinity-norm, using the LU factorization computed by `getrf`

.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as

```
rcond = 1 / ( norm(A) * norm(inv(A)) ).
Arguments
=========
norm: Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= 0: Infinity-norm.
= 1: 1-norm;
A: The factors L and U from the factorization A = P*L*U
as computed by getrf.
anorm: If norm = 0, the infinity-norm of the original matrix A.
If norm = 1, the 1-norm of the original matrix A.
rcond: The reciprocal of the condition number of the matrix A,
computed as rcond = 1/(norm(A) * norm(inv(A))).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 100, 100);
$anorm = $a->lange(1);
$ipiv = zeroes(long, 100);
$info = null;
getrf($a, $ipiv, $info);
($rcond, $info) = gecon($a, 1, $anorm) unless $info != 0;
```

gecon ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## sycon

` Signature: ([phys]A(n,n); int uplo(); int ipiv(n); [phys]anorm(); [o,phys]rcond();int [o,phys]info(); int [t]iwork(n); [t]work(workn))`

Estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric matrix A using the factorization A = U*D*U' or A = L*D*L' computed by `sytrf`

.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as rcond = 1 / (anorm * norm(inv(A))).

```
Arguments
=========
uplo: Specifies whether the details of the factorization are stored
as an upper or lower triangular matrix.
= 0: Upper triangular, form is A = U*D*U';
= 1: Lower triangular, form is A = L*D*L'.
A: The block diagonal matrix D and the multipliers used to
obtain the factor U or L as computed by sytrf.
ipiv: Details of the interchanges and the block structure of D
as determined by sytrf.
anorm: The 1-norm of the original matrix A.
rcond: The reciprocal of the condition number of the matrix A,
computed as rcond = 1/(anorm * aimvnm), where ainvnm is an
estimate of the 1-norm of inv(A) computed in this routine.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
```

```
# Assume $a is symmetric
$a = random (float, 100, 100);
$anorm = $a->lansy(1,1);
$ipiv = zeroes(long, 100);
$info = null;
sytrf($a, 1,$ipiv, $info);
($rcond, $info) = sycon($a, 1, $anorm) unless $info != 0;
```

sycon ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## pocon

` Signature: (A(n,n); int uplo(); anorm(); [o]rcond();int [o]info(); int [t]iwork(n); [t]work(workn))`

Estimates the reciprocal of the condition number (in the 1-norm) of a real symmetric positive definite matrix using the Cholesky factorization A = U'*U or A = L*L' computed by `potrf`

.

An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as rcond = 1 / (anorm * norm(inv(A))).

```
Arguments
=========
uplo: = 0: Upper triangle of A is stored;
= 1: Lower triangle of A is stored.
A: The triangular factor U or L from the Cholesky factorization
A = U'*U or A = L*L', as computed by potrf.
anorm: The 1-norm of the matrix A.
rcond: The reciprocal of the condition number of the matrix A,
computed as rcond = 1/(anorm * ainvnm), where ainvnm is an
estimate of the 1-norm of inv(A) computed in this routine.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
# Assume $a is symmetric positive definite
$a = random (float, 100, 100);
$anorm = $a->lansy(1,1);
$info = null;
potrf($a, 0, $info);
($rcond, $info) = pocon($a, 1, $anorm) unless $info != 0;
```

pocon ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## trcon

` Signature: (A(n,n); int norm();int uplo();int diag(); [o]rcond();int [o]info(); int [t]iwork(n); [t]work(workn))`

Estimates the reciprocal of the condition number of a triangular matrix A, in either the 1-norm or the infinity-norm.

The norm of A is computed and an estimate is obtained for norm(inv(A)), then the reciprocal of the condition number is computed as

```
rcond = 1 / ( norm(A) * norm(inv(A)) ).
Arguments
=========
norm: Specifies whether the 1-norm condition number or the
infinity-norm condition number is required:
= 0: Infinity-norm.
= 1: 1-norm;
uplo: = 0: A is upper triangular;
= 1: A is lower triangular.
diag: = 0: A is non-unit triangular;
= 1: A is unit triangular.
A: The triangular matrix A. If uplo = 0, the leading N-by-N
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If uplo = 1, the leading N-by-N lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If diag = 1, the diagonal elements of A are
also not referenced and are assumed to be 1.
rcond: The reciprocal of the condition number of the matrix A,
computed as rcond = 1/(norm(A) * norm(inv(A))).
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
# Assume $a is upper triangular
$a = random (float, 100, 100);
$info = null;
($rcond, $info) = trcon($a, 1, 1, 0) unless $info != 0;
```

trcon ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## geqp3

` Signature: ([io,phys]A(m,n); int [io,phys]jpvt(n); [o,phys]tau(k); int [o,phys]info())`

geqp3 computes a QR factorization using Level 3 BLAS with column pivoting of a matrix A:

` A*P = Q*R`

The matrix Q is represented as a product of elementary reflectors

` Q = H(1) H(2) . . . H(k), where k = min(m,n).`

Each H(i) has the form

```
H(i) = I - tau * v * v'
where tau is a real/complex scalar, and v is a real/complex vector
with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in tau(i).
Arguments
=========
A: On entry, the M-by-N matrix A.
On exit, the upper triangle of the array contains the
min(M,N)-by-N upper trapezoidal matrix R; the elements below
the diagonal, together with the array tau, represent the
orthogonal matrix Q as a product of min(M,N) elementary
reflectors.
jpvt: On entry, if jpvt(J)!=0, the J-th column of A is permuted
to the front of A*P (a leading column); if jpvt(J)=0,
the J-th column of A is a free column.
On exit, if jpvt(J)=K, then the J-th column of A*P was the
the K-th column of A.
tau: The scalar factors of the elementary reflectors.
info: = 0: successful exit.
< 0: if info = -i, the i-th argument had an illegal value.
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
$jpvt = zeroes(long, 50);
geqp3($a, $jpvt, $tau, $info);
```

geqp3 ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## geqrf

` Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())`

geqrf computes a QR factorization of a matrix A:

` A = Q * R`

The matrix Q is represented as a product of elementary reflectors

` Q = H(1) H(2) . . . H(k), where k = min(m,n).`

Each H(i) has the form

```
H(i) = I - tau * v * v'
where tau is a real/complex scalar, and v is a real/complex vector
with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in
A(i+1:m,i), and tau in tau(i).
Arguments
=========
A: On exit, the elements on and above the diagonal of the array
contain the min(M,N)-by-N upper trapezoidal matrix R (R is
upper triangular if m >= n); the elements below the diagonal,
with the array TAU, represent the orthogonal matrix Q as a
product of min(m,n) elementary reflectors.
tau: The scalar factors of the elementary reflectors.
info: = 0: successful exit.
< 0: if info = -i, the i-th argument had an illegal value.
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
geqrf($a, $tau, $info);
```

geqrf ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## orgqr

` Signature: ([io,phys]A(m,n); [phys]tau(k); int [o,phys]info())`

Generates an M-by-N real matrix Q with orthonormal columns, which is defined as the first N columns of a product of K elementary reflectors of order M

```
Q = H(1) H(2) . . . H(k)
as returned by geqrf or geqp3.
Arguments
=========
A: On entry, the i-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by geqrf or geqp3 in the first k columns of its array
argument A.
On exit, the M-by-N matrix Q.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by geqrf or geqp3.
info: = 0: successful exit
< 0: if info = -i, the i-th argument has an illegal value
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
geqrf($a, $tau, $info);
orgqr($a, $tau, $info) unless $info != 0;
```

orgqr ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ormqr

` Signature: ([phys]A(p,k); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())`

Overwrites the general real M-by-N matrix C with

```
side = 0 side = 1
trans = 0: Q * C C * Q
trans = 1: Q' * C C * Q'
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by geqrf or geqp3.
```

Q is of order M if `side`

= 0 and of order N if `side`

= 1.

```
Arguments
=========
side: = 0: apply Q or Q' from the Left;
= 1: apply Q or Q' from the Right.
trans: = 0: No transpose, apply Q;
= 1: Transpose, apply Q'.
A: The i-th column must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
geqrf or geqp3 in the first k columns of its array argument A.
A is modified by the routine but restored on exit.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by geqrf or geqp3.
C: On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 50, 100);
$a = transpose($a);
$info = null;
$tau = zeroes(float, 50);
geqrf($a, $tau, $info);
$c = random(70,50);
# $c will contain the result
$c->reshape(70,100);
$c = transpose($c);
ormqr($a, $tau, $c, $info);
```

ormqr ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gelqf

` Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())`

Computes an LQ factorization of a real M-by-N matrix A:

` A = L * Q.`

The matrix Q is represented as a product of elementary reflectors

` Q = H(k) . . . H(2) H(1), where k = min(m,n).`

Each H(i) has the form

```
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i,i+1:n),
and tau in tau(i).
Arguments
=========
A: On entry, the M-by-N matrix A.
On exit, the elements on and below the diagonal of the array
contain the m-by-min(m,n) lower trapezoidal matrix L (L is
lower triangular if m <= n); the elements above the diagonal,
with the array tau, represent the orthogonal matrix Q as a
product of elementary reflectors.
tau: The scalar factors of the elementary reflectors.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
gelqf($a, $tau, $info);
```

gelqf ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## orglq

` Signature: ([io,phys]A(m,n); [phys]tau(k); int [o,phys]info())`

Generates an M-by-N real matrix Q with orthonormal rows, which is defined as the first M rows of a product of K elementary reflectors of order N

```
Q = H(k) . . . H(2) H(1)
as returned by gelqf.
Arguments
=========
A: On entry, the i-th row must contain the vector which defines
the elementary reflector H(i), for i = 1,2,...,k, as returned
by gelqf in the first k rows of its array argument A.
On exit, the M-by-N matrix Q.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by gelqf.
info: = 0: successful exit
< 0: if info = -i, the i-th argument has an illegal value
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
gelqf($a, $tau, $info);
orglq($a, $tau, $info) unless $info != 0;
```

orglq ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ormlq

` Signature: ([phys]A(k,p); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())`

Overwrites the general real M-by-N matrix C with

```
side = 0 side = 1
trans = 0: Q * C C * Q
trans = 1: Q' * C C * Q'
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by gelqf.
```

Q is of order M if `side`

= 0 and of order N if `side`

= 1.

```
Arguments
=========
side: = 0: apply Q or Q' from the Left;
= 1: apply Q or Q' from the Right.
trans: = 0: No transpose, apply Q;
= 1: Transpose, apply Q'.
A: The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
gelqf in the first k rows of its array argument A.
A is modified by the routine but restored on exit.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by gelqf.
C: On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 50, 100);
$a = transpose($a);
$info = null;
$tau = zeroes(float, 50);
gelqf($a, $tau, $info);
$c = random(70,50);
# $c will contain the result
$c->reshape(70,100);
$c = transpose($c);
ormlq($a, $tau, $c, $info);
```

ormlq ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## geqlf

` Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())`

Computes a QL factorization of a real M-by-N matrix A:

` A = Q * L`

The matrix Q is represented as a product of elementary reflectors

` Q = H(k) . . . H(2) H(1), where k = min(m,n).`

Each H(i) has the form

```
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(m-k+i+1:m) = 0 and v(m-k+i) = 1; v(1:m-k+i-1) is stored on exit in
A(1:m-k+i-1,n-k+i), and tau in TAU(i).
Arguments
=========
A: On entry, the M-by-N matrix A.
On exit,
if m >= n, the lower triangle of the subarray
A(m-n+1:m,1:n) contains the N-by-N lower triangular matrix L;
if m <= n, the elements on and below the (n-m)-th
superdiagonal contain the M-by-N lower trapezoidal matrix L;
the remaining elements, with the array tau, represent the
orthogonal matrix Q as a product of elementary reflectors.
tau: The scalar factors of the elementary reflectors.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
geqlf($a, $tau, $info);
```

geqlf ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## orgql

` Signature: ([io,phys]A(m,n); [phys]tau(k); int [o,phys]info())`

Generates an M-by-N real matrix Q with orthonormal columns, which is defined as the last N columns of a product of K elementary reflectors of order M

```
Q = H(k) . . . H(2) H(1)
as returned by geqlf.
Arguments
=========
A: On entry, the (n-k+i)-th column must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by geqlf in the last k columns of its array
argument A.
On exit, the M-by-N matrix Q.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by geqlf.
info: = 0: successful exit
< 0: if info = -i, the i-th argument has an illegal value
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
geqlf($a, $tau, $info);
orgql($a, $tau, $info) unless $info != 0;
```

orgql ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ormql

` Signature: ([phys]A(p,k); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())`

Overwrites the general real M-by-N matrix C with

```
side = 0 side = 1
trans = 0: Q * C C * Q
trans = 1: Q' * C C * Q'
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(k) . . . H(2) H(1)
as returned by geqlf.
```

Q is of order M if `side`

= 0 and of order N if `side`

= 1.

```
Arguments
=========
side: = 0: apply Q or Q' from the Left;
= 1: apply Q or Q' from the Right.
trans: = 0: No transpose, apply Q;
= 1: Transpose, apply Q'.
A: The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
geqlf in the last k rows of its array argument A.
A is modified by the routine but restored on exit.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by geqlf.
C: On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 50, 100);
$a = transpose($a);
$info = null;
$tau = zeroes(float, 50);
geqlf($a, $tau, $info);
$c = random(70,50);
# $c will contain the result
$c->reshape(70,100);
$c = transpose($c);
ormql($a, $tau, $c, $info);
```

ormql ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gerqf

` Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())`

Computes an RQ factorization of a real M-by-N matrix A:

` A = R * Q.`

The matrix Q is represented as a product of elementary reflectors

` Q = H(1) H(2) . . . H(k), where k = min(m,n).`

Each H(i) has the form

```
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and tau in TAU(i).
Arguments
=========
A: On entry, the M-by-N matrix A.
On exit,
if m <= n, the upper triangle of the subarray
A(1:m,n-m+1:n) contains the M-by-M upper triangular matrix R;
if m >= n, the elements on and above the (m-n)-th subdiagonal
contain the M-by-N upper trapezoidal matrix R;
the remaining elements, with the array tau, represent the
orthogonal matrix Q as a product of min(m,n) elementary
reflectors (see Further Details).
tau: The scalar factors of the elementary reflectors.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
gerqf($a, $tau, $info);
```

gerqf ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## orgrq

` Signature: ([io,phys]A(m,n); [phys]tau(k); int [o,phys]info())`

Generates an M-by-N real matrix Q with orthonormal rows, which is defined as the last M rows of a product of K elementary reflectors of order N

```
Q = H(1) H(2) . . . H(k)
as returned by gerqf.
Arguments
=========
A: On entry, the (m-k+i)-th row must contain the vector which
defines the elementary reflector H(i), for i = 1,2,...,k, as
returned by gerqf in the last k rows of its array argument
A.
On exit, the M-by-N matrix Q.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by gerqf.
info: = 0: successful exit
< 0: if info = -i, the i-th argument has an illegal value
```

```
$a = random (float, 100, 50);
$info = null;
$tau = zeroes(float, 50);
gerqf($a, $tau, $info);
orgrq($a, $tau, $info) unless $info != 0;
```

orgrq ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ormrq

` Signature: ([phys]A(k,p); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())`

Overwrites the general real M-by-N matrix C with

```
side = 0 side = 1
trans = 0: Q * C C * Q
trans = 1: Q' * C C * Q'
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by gerqf.
```

Q is of order M if `side`

= 0 and of order N if `side`

= 1.

```
Arguments
=========
side: = 0: apply Q or Q' from the Left;
= 1: apply Q or Q' from the Right.
trans: = 0: No transpose, apply Q;
= 1: Transpose, apply Q'.
A: The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
gerqf in the last k rows of its array argument A.
A is modified by the routine but restored on exit.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by gerqf.
C: On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 50, 100);
$a = transpose($a);
$info = null;
$tau = zeroes(float, 50);
gerqf($a, $tau, $info);
$c = random(70,50);
# $c will contain the result
$c->reshape(70,100);
$c = transpose($c);
ormrq($a, $tau, $c, $info);
```

ormrq ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## tzrzf

` Signature: ([io,phys]A(m,n); [o,phys]tau(k); int [o,phys]info())`

Reduces the M-by-N ( M <= N ) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations.

The upper trapezoidal matrix A is factored as

```
A = ( R 0 ) * Z,
where Z is an N-by-N orthogonal matrix and R is an M-by-M upper
triangular matrix.
```

The factorization is obtained by Householder's method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the ( m - k + 1 )th row of A, is given in the form

```
Z( k ) = ( I 0 ),
( 0 T( k ) )
where
T( k ) = I - tau*u( k )*u( k )', u( k ) = ( 1 ),
( 0 )
( z( k ) )
```

tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of X.

The scalar tau is returned in the kth element of `tau`

and the vector u( k ) in the kth row of A, such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of A.

Z is given by

```
Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
Arguments
=========
A: On entry, the leading M-by-N upper trapezoidal part of the
array A must contain the matrix to be factorized.
On exit, the leading M-by-M upper triangular part of A
contains the upper triangular matrix R, and elements M+1 to
N of the first M rows of A, with the array tau, represent the
orthogonal matrix Z as a product of M elementary reflectors.
tau: The scalar factors of the elementary reflectors.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 50, 100);
$info = null;
$tau = zeroes(float, 50);
tzrzf($a, $tau, $info);
```

tzrzf ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## ormrz

` Signature: ([phys]A(k,p); int side(); int trans(); [phys]tau(k); [io,phys]C(m,n);int [o,phys]info())`

Overwrites the general real M-by-N matrix C with

```
side = 0 side = 1
trans = 0: Q * C C * Q
trans = 1: Q' * C C * Q'
where Q is a real orthogonal matrix defined as the product of k
elementary reflectors
Q = H(1) H(2) . . . H(k)
as returned by tzrzf.
```

Q is of order M if `side`

= 0 and of order N if `side`

= 1.

```
Arguments
=========
side: = 0: apply Q or Q' from the Left;
= 1: apply Q or Q' from the Right.
trans: = 0: No transpose, apply Q;
= 1: Transpose, apply Q'.
A: The i-th row must contain the vector which defines the
elementary reflector H(i), for i = 1,2,...,k, as returned by
tzrzf in the last k rows of its array argument A.
A is modified by the routine but restored on exit.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by tzrzf.
C: On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q'*C or C*Q' or C*Q.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (float, 50, 100);
$a = transpose($a);
$info = null;
$tau = zeroes(float, 50);
tzrzf($a, $tau, $info);
$c = random(70,50);
# $c will contain the result
$c->reshape(70,100);
$c = transpose($c);
ormrz($a, $tau, $c, $info);
```

ormrz ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gehrd

` Signature: ([io,phys]A(n,n); int [phys]ilo();int [phys]ihi();[o,phys]tau(k); int [o,phys]info())`

Reduces a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation: Q' * A * Q = H .

Further Details ===============

The matrix Q is represented as a product of (ihi-ilo) elementary reflectors

` Q = H(ilo) H(ilo+1) . . . H(ihi-1).`

Each H(i) has the form

```
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
exit in A(i+2:ihi,i), and tau in tau(i).
```

The contents of A are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:

```
on entry, on exit,
( a a a a a a a ) ( a a h h h h a )
( a a a a a a ) ( a h h h h a )
( a a a a a a ) ( h h h h h h )
( a a a a a a ) ( v2 h h h h h )
( a a a a a a ) ( v2 v3 h h h h )
( a a a a a a ) ( v2 v3 v4 h h h )
( a ) ( a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).
Arguments
=========
ilo:
ihi: It is assumed that A is already upper triangular in rows
and columns 1:ilo-1 and ihi+1:N. ilo and ihi are normally
set by a previous call to gebal; otherwise they should be
set to 1 and N respectively. See Further Details.
1 <= ilo <= ihi <= N, if N > 0; ilo=1 and ihi=0, if N=0.
A: On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiagonal of A
are overwritten with the upper Hessenberg matrix H, and the
elements below the first subdiagonal, with the array tau,
represent the orthogonal matrix Q as a product of elementary
reflectors. See Further Details.
tau: The scalar factors of the elementary reflectors (see Further
Details). Elements 1:ilo-1 and ihi:N-1 of tau are set to
zero. (dimension (N-1))
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
```

```
$a = random (50, 50);
$info = null;
$tau = zeroes(50);
gehrd($a, 1, 50, $tau, $info);
```

gehrd ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## orghr

` Signature: ([io,phys]A(n,n); int [phys]ilo();int [phys]ihi();[phys]tau(k); int [o,phys]info())`

Generates a real orthogonal matrix Q which is defined as the product of ihi-ilo elementary reflectors of order N, as returned by `gehrd`

:

```
Q = H(ilo) H(ilo+1) . . . H(ihi-1).
Arguments
=========
ilo:
ihi: ilo and ihi must have the same values as in the previous call
of gehrd. Q is equal to the unit matrix except in the
submatrix Q(ilo+1:ihi,ilo+1:ihi).
1 <= ilo <= ihi <= N, if N > 0; ilo=1 and ihi=0, if N=0.
A: On entry, the vectors which define the elementary reflectors,
as returned by gehrd.
On exit, the N-by-N orthogonal matrix Q.
tau: tau(i) must contain the scalar factor of the elementary
reflector H(i), as returned by gehrd.(dimension (N-1))
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (50, 50);
$info = null;
$tau = zeroes(50);
gehrd($a, 1, 50, $tau, $info);
orghr($a, 1, 50, $tau, $info);
```

orghr ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## hseqr

` Signature: ([io,phys]H(n,n); int job();int compz();int [phys]ilo();int [phys]ihi();[o,phys]wr(n); [o,phys]wi(n);[o,phys]Z(m,m); int [o,phys]info())`

Computes the eigenvalues of a real upper Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors.

Optionally Z may be postmultiplied into an input orthogonal matrix Q, so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.

```
Arguments
=========
job: = 0: compute eigenvalues only;
= 1: compute eigenvalues and the Schur form T.
compz: = 0: no Schur vectors are computed;
= 1: Z is initialized to the unit matrix and the matrix Z
of Schur vectors of H is returned;
= 2: Z must contain an orthogonal matrix Q on entry, and
the product Q*Z is returned.
ilo:
ihi: It is assumed that H is already upper triangular in rows
and columns 1:ilo-1 and ihi+1:N. ilo and ihi are normally
set by a previous call to gebal, and then passed to gehrd
when the matrix output by gebal is reduced to Hessenberg
form. Otherwise ilo and ihi should be set to 1 and N
respectively.
1 <= ilo <= ihi <= N, if N > 0; ilo=1 and ihi=0, if N=0.
H: On entry, the upper Hessenberg matrix H.
On exit, if job = 1, H contains the upper quasi-triangular
matrix T from the Schur decomposition (the Schur form);
2-by-2 diagonal blocks (corresponding to complex conjugate
pairs of eigenvalues) are returned in standard form, with
H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1) < 0. If job = 0,
the contents of H are unspecified on exit.
wr:
wi: The real and imaginary parts, respectively, of the computed
eigenvalues. If two eigenvalues are computed as a complex
conjugate pair, they are stored in consecutive elements of
wr and wi, say the i-th and (i+1)th, with wi(i) > 0 and
wi(i+1) < 0. If job = 1, the eigenvalues are stored in the
same order as on the diagonal of the Schur form returned in
H, with wr(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
diagonal block, wi(i) = sqrt(H(i+1,i)*H(i,i+1)) and
wi(i+1) = -wi(i).
Z: If compz = 0: Z is not referenced.
If compz = 1: on entry, Z need not be set, and on exit, Z
contains the orthogonal matrix Z of the Schur vectors of H.
If compz = 2: on entry Z must contain an N-by-N matrix Q,
which is assumed to be equal to the unit matrix except for
the submatrix Z(ilo:ihi,ilo:ihi); on exit Z contains Q*Z.
Normally Q is the orthogonal matrix generated by orghr after
the call to gehrd which formed the Hessenberg matrix H.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
> 0: if info = i, hseqr failed to compute all of the
eigenvalues in a total of 30*(ihi-ilo+1) iterations;
elements 1:ilo-1 and i+1:n of wr and wi contain those
eigenvalues which have been successfully computed.
```

```
$a = random (50, 50);
$info = null;
$tau = zeroes(50);
$z= zeroes(1,1);
gehrd($a, 1, 50, $tau, $info);
hseqr($a,0,0,1,50,($wr=null),($wi=null),$z,$info);
```

hseqr ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## trevc

` Signature: ([io]T(n,n); int side();int howmny();int select(q);[o]VL(m,m); [o]VR(p,p);int [o]m(); int [o]info(); [t]work(workn))`

Computes some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T.

The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:

```
T*x = w*x, y'*T = w*y'
where y' denotes the conjugate transpose of the vector y.
```

If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input orthogonal matrix. If T was obtained from the real-Schur factorization of an original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of right or left eigenvectors of A.

T must be in Schur canonical form (as returned by hseqr), that is, block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. Corresponding to each 2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part.

Further Details ===============

The algorithm used in this program is basically backward (forward) substitution, with scaling to make the the code robust against possible overflow.

Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|.

```
Arguments
=========
side: = 0 : compute both right and left eigenvectors;
= 1 : compute right eigenvectors only;
= 2 : compute left eigenvectors only.
howmny: = 0: compute all right and/or left eigenvectors;
= 1: compute all right and/or left eigenvectors,
and backtransform them using the input matrices
supplied in VR and/or VL;
= 2: compute selected right and/or left eigenvectors,
specified by the logical array select.
select: If howmny = 2, select specifies the eigenvectors to be
computed.
If howmny = 0 or 1, select is not referenced.
To select the real eigenvector corresponding to a real
eigenvalue w(j), select(j) must be set to TRUE. To select
the complex eigenvector corresponding to a complex conjugate
pair w(j) and w(j+1), either select(j) or select(j+1) must be
set to TRUE; then on exit select(j) is TRUE and
select(j+1) is FALSE.
T: The upper quasi-triangular matrix T in Schur canonical form.
VL: On entry, if side = 2 or 0 and howmny = 1, VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by hseqr).
On exit, if side = 2 or 0, VL contains:
if howmny = 0, the matrix Y of left eigenvectors of T;
VL has the same quasi-lower triangular form
as T'. If T(i,i) is a real eigenvalue, then
the i-th column VL(i) of VL is its
corresponding eigenvector. If T(i:i+1,i:i+1)
is a 2-by-2 block whose eigenvalues are
complex-conjugate eigenvalues of T, then
VL(i)+sqrt(-1)*VL(i+1) is the complex
eigenvector corresponding to the eigenvalue
with positive real part.
if howmny = 1, the matrix Q*Y;
if howmny = 2, the left eigenvectors of T specified by
select, stored consecutively in the columns
of VL, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
If side = 1, VL is not referenced.
VR: On entry, if side = 1 or 0 and howmny = 1, VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of Schur vectors returned by hseqr).
On exit, if side = 1 or 0, VR contains:
if howmny = 0, the matrix X of right eigenvectors of T;
VR has the same quasi-upper triangular form
as T. If T(i,i) is a real eigenvalue, then
the i-th column VR(i) of VR is its
corresponding eigenvector. If T(i:i+1,i:i+1)
is a 2-by-2 block whose eigenvalues are
complex-conjugate eigenvalues of T, then
VR(i)+sqrt(-1)*VR(i+1) is the complex
eigenvector corresponding to the eigenvalue
with positive real part.
if howmny = 1, the matrix Q*X;
if howmny = 2, the right eigenvectors of T specified by
select, stored consecutively in the columns
of VR, in the same order as their
eigenvalues.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
If side = 2, VR is not referenced.
m: The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors.
If howmny = 0 or 1, m is set to N.
Each selected real eigenvector occupies one column and each
selected complex eigenvector occupies two columns.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random (50, 50);
$info = null;
$tau = zeroes(50);
$z= zeroes(1,1);
gehrd($a, 1, 50, $tau, $info);
hseqr($a,0,0,1,50,($wr=null),($wi=null),$z,$info);
```

trevc ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## tgevc

` Signature: ([io]A(n,n); int side();int howmny();[io]B(n,n);int select(q);[o]VL(m,m); [o]VR(p,p);int [o]m(); int [o]info(); [t]work(workn))`

Computes some or all of the right and/or left generalized eigenvectors of a pair of real upper triangular matrices (A,B).

The right generalized eigenvector x and the left generalized eigenvector y of (A,B) corresponding to a generalized eigenvalue w are defined by:

```
(A - wB) * x = 0 and y**H * (A - wB) = 0
where y**H denotes the conjugate transpose of y.
```

If an eigenvalue w is determined by zero diagonal elements of both A and B, a unit vector is returned as the corresponding eigenvector.

If all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left eigenvectors of (A,B), or the products Z*X and/or Q*Y, where Z and Q are input orthogonal matrices. If (A,B) was obtained from the generalized real-Schur factorization of an original pair of matrices

` (A0,B0) = (Q*A*Z**H,Q*B*Z**H),`

then Z*X and Q*Y are the matrices of right or left eigenvectors of A.

A must be block upper triangular, with 1-by-1 and 2-by-2 diagonal blocks. Corresponding to each 2-by-2 diagonal block is a complex conjugate pair of eigenvalues and eigenvectors; only one eigenvector of the pair is computed, namely the one corresponding to the eigenvalue with positive imaginary part.

```
Arguments
=========
side: = 0 : compute both right and left eigenvectors;
= 1 : compute right eigenvectors only;
= 2 : compute left eigenvectors only.
howmny: = 0 : compute all right and/or left eigenvectors;
= 1 : compute all right and/or left eigenvectors, and
backtransform them using the input matrices supplied
in VR and/or VL;
= 2 : compute selected right and/or left eigenvectors,
specified by the logical array select.
select: If howmny=2, select specifies the eigenvectors to be
computed.
If howmny=0 or 1, select is not referenced.
To select the real eigenvector corresponding to the real
eigenvalue w(j), select(j) must be set to TRUE To select
the complex eigenvector corresponding to a complex conjugate
pair w(j) and w(j+1), either select(j) or select(j+1) must
be set to TRUE.
A: The upper quasi-triangular matrix A.
B: The upper triangular matrix B. If A has a 2-by-2 diagonal
block, then the corresponding 2-by-2 block of B must be
diagonal with positive elements.
VL: On entry, if side = 2 or 0 and howmny = 1, VL must
contain an N-by-N matrix Q (usually the orthogonal matrix Q
of left Schur vectors returned by hgqez).
On exit, if side = 2 or 0, VL contains:
if howmny = 0, the matrix Y of left eigenvectors of (A,B);
if howmny = 1, the matrix Q*Y;
if howmny = 2, the left eigenvectors of (A,B) specified by
select, stored consecutively in the columns of
VL, in the same order as their eigenvalues.
If side = 1, VL is not referenced.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part, and the second the imaginary part.
VR: On entry, if side = 1 or 0 and howmny = 1, VR must
contain an N-by-N matrix Q (usually the orthogonal matrix Z
of right Schur vectors returned by hgeqz).
On exit, if side = 1 or 0, VR contains:
if howmny = 0, the matrix X of right eigenvectors of (A,B);
if howmny = 1, the matrix Z*X;
if howmny = 2, the right eigenvectors of (A,B) specified by
select, stored consecutively in the columns of
VR, in the same order as their eigenvalues.
If side = 2, VR is not referenced.
A complex eigenvector corresponding to a complex eigenvalue
is stored in two consecutive columns, the first holding the
real part and the second the imaginary part.
M: The number of columns in the arrays VL and/or VR actually
used to store the eigenvectors. If howmny = 0 or 1, M
is set to N. Each selected real eigenvector occupies one
column and each selected complex eigenvector occupies two
columns.
info: = 0: successful exit.
< 0: if info = -i, the i-th argument had an illegal value.
> 0: the 2-by-2 block (info:info+1) does not have a complex
eigenvalue.
=for example
$a = random (50, 50);
$info = null;
$tau = zeroes(50);
$z= zeroes(1,1);
gehrd($a, 1, 50, $tau, $info);
hseqr($a,0,0,1,50,($wr=null),($wi=null),$z,$info);
```

tgevc ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gebal

` Signature: ([io,phys]A(n,n); int job(); int [o,phys]ilo();int [o,phys]ihi();[o,phys]scale(n); int [o,phys]info())`

Balances a general real matrix A. This involves, first, permuting A by a similarity transformation to isolate eigenvalues in the first 1 to ilo-1 and last ihi+1 to N elements on the diagonal; and second, applying a diagonal similarity transformation to rows and columns ilo to ihi to make the rows and columns as close in norm as possible. Both steps are optional.

Balancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues and/or eigenvectors.

Further Details ===============

The permutations consist of row and column interchanges which put the matrix in the form

```
( T1 X Y )
P A P = ( 0 B Z )
( 0 0 T2 )
where T1 and T2 are upper triangular matrices whose eigenvalues lie
along the diagonal. The column indices ilo and ihi mark the starting
and ending columns of the submatrix B. Balancing consists of applying
a diagonal similarity transformation inv(D) * B * D to make the
1-norms of each row of B and its corresponding column nearly equal.
```

The output matrix is

```
( T1 X*D Y )
( 0 inv(D)*B*D inv(D)*Z ).
( 0 0 T2 )
```

Information about the permutations P and the diagonal matrix D is returned in the vector `scale`

.

```
Arguments
=========
job: Specifies the operations to be performed on A:
= 0: none: simply set ilo = 1, ihi = N, scale(I) = 1.0
for i = 1,...,N;
= 1: permute only;
= 2: scale only;
= 3: both permute and scale.
A: On entry, the input matrix A.
On exit, A is overwritten by the balanced matrix.
If job = 0, A is not referenced.
See Further Details.
ilo:
ihi: ilo and ihi are set to integers such that on exit
A(i,j) = 0 if i > j and j = 1,...,ilo-1 or I = ihi+1,...,N.
If job = 0 or 2, ilo = 1 and ihi = N.
scale: Details of the permutations and scaling factors applied to
A. If P(j) is the index of the row and column interchanged
with row and column j and D(j) is the scaling factor
applied to row and column j, then
scale(j) = P(j) for j = 1,...,ilo-1
= D(j) for j = ilo,...,ihi
= P(j) for j = ihi+1,...,N.
The order in which the interchanges are made is N to ihi+1,
then 1 to ilo-1.
info: = 0: successful exit.
< 0: if info = -i, the i-th argument had an illegal value.
```

```
$a = random (50, 50);
$scale = zeroes(50);
$info = null;
$ilo = null;
$ihi = null;
gebal($a, $ilo, $ihi, $scale, $info);
```

gebal ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gebak

` Signature: ([io,phys]A(n,m); int job(); int side();int [phys]ilo();int [phys]ihi();[phys]scale(n); int [o,phys]info())`

gebak forms the right or left eigenvectors of a real general matrix by backward transformation on the computed eigenvectors of the balanced matrix output by gebal.

```
Arguments
=========
A: On entry, the matrix of right or left eigenvectors to be
transformed, as returned by hsein or trevc.
On exit, A is overwritten by the transformed eigenvectors.
job: Specifies the type of backward transformation required:
= 0 , do nothing, return immediately;
= 1, do backward transformation for permutation only;
= 2, do backward transformation for scaling only;
= 3, do backward transformations for both permutation and
scaling.
job must be the same as the argument job supplied to gebal.
side: = 0: V contains left eigenvectors.
= 1: V contains right eigenvectors;
ilo:
ihi: The integers ilo and ihi determined by gebal.
1 <= ilo <= ihi <= N, if N > 0; ilo=1 and ihi=0, if N=0.
Here N is the the number of rows of the matrix A.
scale: Details of the permutation and scaling factors, as returned
by gebal.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value.
```

```
$a = random (50, 50);
$scale = zeroes(50);
$info = null;
$ilo = null;
$ihi = null;
gebal($a, $ilo, $ihi, $scale, $info);
# Compute eigenvectors ($ev)
gebak($ev, $ilo, $ihi, $scale, $info);
```

gebak ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## lange

` Signature: ([phys]A(n,m); int norm(); [o]b(); [t]work(workn))`

Computes the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real matrix A.

```
Description
===========
returns the value
lange = ( max(abs(A(i,j))), norm = 0
(
( norm1(A), norm = 1
(
( normI(A), norm = 2
(
( normF(A), norm = 3
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a matrix norm.
Arguments
=========
norm: Specifies the value to be returned in lange as described
above.
A: The n by m matrix A.
```

```
$a = random (float, 100, 100);
$norm = $a->lange(1);
```

lange ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## lansy

` Signature: (A(n,n); int uplo(); int norm(); [o]b(); [t]work(workn))`

Computes the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a real symmetric matrix A.

```
Description
===========
returns the value
lansy = ( max(abs(A(i,j))), norm = 0
(
( norm1(A), norm = 1
(
( normI(A), norm = 2
(
( normF(A), norm = 3
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a matrix norm.
norm: Specifies the value to be returned in lansy as described
above.
uplo: Specifies whether the upper or lower triangular part of the
symmetric matrix A is to be referenced.
= 0: Upper triangular part of A is referenced
= 1: Lower triangular part of A is referenced
A: The symmetric matrix A. If uplo = 0, the leading n by n
upper triangular part of A contains the upper triangular part
of the matrix A, and the strictly lower triangular part of A
is not referenced. If uplo = 1, the leading n by n lower
triangular part of A contains the lower triangular part of
the matrix A, and the strictly upper triangular part of A is
not referenced.
```

```
# Assume $a is symmetric
$a = random (float, 100, 100);
$norm = $a->lansy(1, 1);
```

lansy ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## lantr

` Signature: (A(m,n);int uplo();int norm();int diag();[o]b(); [t]work(workn))`

Computes the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a trapezoidal or triangular matrix A.

```
Description
===========
returns the value
lantr = ( max(abs(A(i,j))), norm = 0
(
( norm1(A), norm = 1
(
( normI(A), norm = 2
(
( normF(A), norm = 3
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a matrix norm.
norm: Specifies the value to be returned in lantr as described
above.
uplo: Specifies whether the matrix A is upper or lower trapezoidal.
= 0: Upper triangular part of A is referenced
= 1: Lower triangular part of A is referenced
Note that A is triangular instead of trapezoidal if M = N.
diag: Specifies whether or not the matrix A has unit diagonal.
= 0: Non-unit diagonal
= 1: Unit diagonal
A: The trapezoidal matrix A (A is triangular if m = n).
If uplo = 0, the leading m by n upper trapezoidal part of
the array A contains the upper trapezoidal matrix, and the
strictly lower triangular part of A is not referenced.
If uplo = 1, the leading m by n lower trapezoidal part of
the array A contains the lower trapezoidal matrix, and the
strictly upper triangular part of A is not referenced. Note
that when diag = 1, the diagonal elements of A are not
referenced and are assumed to be one.
```

```
# Assume $a is upper triangular
$a = random (float, 100, 100);
$norm = $a->lantr(1, 1, 0);
```

lantr ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## gemm

` Signature: ([phys]A(m,n); int transa(); int transb(); [phys]B(p,q);[phys]alpha(); [phys]beta(); [io,phys]C(r,s))`

Performs one of the matrix-matrix operations

```
C := alpha*op( A )*op( B ) + beta*C,
where op( X ) is one of p( X ) = X or op( X ) = X',
alpha and beta are scalars, and A, B and C are matrices, with op( A )
an m by k matrix, op( B ) a k by n matrix and C an m by n matrix.
Parameters
==========
transa: On entry, transa specifies the form of op( A ) to be used in
the matrix multiplication as follows:
transa = 0, op( A ) = A.
transa = 1, op( A ) = A'.
transb: On entry, transb specifies the form of op( B ) to be used in
the matrix multiplication as follows:
transb = 0, op( B ) = B.
transb = 1, op( B ) = B'.
alpha: On entry, alpha specifies the scalar alpha.
A: Before entry with transa = 0, the leading m by k
part of the array A must contain the matrix A, otherwise
the leading k by m part of the array A must contain the
matrix A.
B: Before entry with transb = 0, the leading k by n
part of the array B must contain the matrix B, otherwise
the leading n by k part of the array B must contain the
matrix B.
beta: On entry, beta specifies the scalar beta. When beta is
supplied as zero then C need not be set on input.
C: Before entry, the leading m by n part of the array C must
contain the matrix C, except when beta is zero, in which
case C need not be set on entry.
On exit, the array C is overwritten by the m by n matrix
( alpha*op( A )*op( B ) + beta*C ).
```

```
$a = random(5,4);
$b = random(5,4);
$alpha = pdl(0.5);
$beta = pdl(0);
$c = zeroes(5,5);
gemm($a, 0, 1,$b, $alpha, $beta, $c);
```

gemm ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## mmult

` Signature: ([phys]A(m,n); [phys]B(p,m); [o,phys]C(p,n))`

Blas matrix multiplication based on gemm

mmult ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## crossprod

` Signature: ([phys]A(n,m); [phys]B(p,m); [o,phys]C(p,n))`

Blas matrix cross product based on gemm

crossprod ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## syrk

` Signature: ([phys]A(m,n); int uplo(); int trans(); [phys]alpha(); [phys]beta(); [io,phys]C(p,p))`

Performs one of the symmetric rank k operations

` C := alpha*A*A' + beta*C,`

or

```
C := alpha*A'*A + beta*C,
where alpha and beta are scalars, C is an n by n symmetric matrix
and A is an n by k matrix in the first case and a k by n matrix
in the second case.
Parameters
==========
uplo: On entry, uplo specifies whether the upper or lower
triangular part of the array C is to be referenced as
follows:
uplo = 0 Only the upper triangular part of C
is to be referenced.
uplo = 1 Only the lower triangular part of C
is to be referenced.
Unchanged on exit.
trans: On entry, trans specifies the operation to be performed as
follows:
trans = 0 C := alpha*A*A' + beta*C.
trans = 1 C := alpha*A'*A + beta*C.
alpha: On entry, alpha specifies the scalar alpha.
Unchanged on exit.
A: Before entry with trans = 0, the leading n by k
part of the array A must contain the matrix A, otherwise
the leading k by n part of the array A must contain the
matrix A.
beta: On entry, beta specifies the scalar beta.
C: Before entry with uplo = 0, the leading n by n
upper triangular part of the array C must contain the upper
triangular part of the symmetric matrix and the strictly
lower triangular part of C is not referenced. On exit, the
upper triangular part of the array C is overwritten by the
upper triangular part of the updated matrix.
Before entry with uplo = 1, the leading n by n
lower triangular part of the array C must contain the lower
triangular part of the symmetric matrix and the strictly
upper triangular part of C is not referenced. On exit, the
lower triangular part of the array C is overwritten by the
lower triangular part of the updated matrix.
```

```
$a = random(5,4);
$b = zeroes(5,5);
$alpha = 1;
$beta = 0;
syrk ($a, 1,0,$alpha, $beta , $b);
```

syrk ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## dot

` Signature: ([phys]a(n);[phys]b(n);[o,phys]c())`

Dot product of two vectors using Blas.

```
$a = random(5);
$b = random(5);
$c = dot($a, $b)
```

dot ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## axpy

` Signature: ([phys]a(n);[phys] alpha();[io,phys]b(m))`

Linear combination of vectors ax + b using Blas. Returns result in b.

```
$a = random(5);
$b = random(5);
axpy($a, 12, $b)
```

axpy ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## nrm2

` Signature: ([phys]a(n);[o]b())`

Euclidean norm of a vector using Blas.

```
$a = random(5);
$norm2 = norm2($a)
```

nrm2 ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## asum

` Signature: ([phys]a(n);[o]b())`

Sum of absolute values of a vector using Blas.

```
$a = random(5);
$absum = asum($a)
```

asum ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## scal

` Signature: ([io,phys]a(n);scale())`

Scale a vector by a constant using Blas.

```
$a = random(5);
$a->scal(0.5)
```

scal ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## rot

` Signature: ([io,phys]a(n);[phys]c(); [phys]s();[io,phys]b(n))`

Applies plane rotation using Blas.

```
$a = random(5);
$b = random(5);
rot($a, 0.5, 0.7, $b)
```

rot ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## rotg

` Signature: ([io,phys]a();[io,phys]b();[o,phys]c(); [o,phys]s())`

Generates plane rotation using Blas.

```
$a = sequence(4);
rotg($a(0), $a(1),$a(2),$a(3))
```

rotg ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## lasrt

` Signature: ([io,phys]d(n); int id();int [o,phys]info())`

Sort the numbers in d in increasing order (if id = 0) or in decreasing order (if id = 1 ).

Use Quick Sort, reverting to Insertion sort on arrays of size <= 20. Dimension of stack limits N to about 2**32.

```
Arguments
=========
id: = 0: sort d in increasing order;
= 1: sort d in decreasing order.
d: On entry, the array to be sorted.
On exit, d has been sorted into increasing order
(d(1) <= ... <= d(N) ) or into decreasing order
(d(1) >= ... >= d(N) ), depending on id.
info: = 0: successful exit
< 0: if info = -i, the i-th argument had an illegal value
```

```
$a = random(5);
lasrt ($a, 0, ($info = null));
```

lasrt ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## lacpy

` Signature: ([phys]A(m,n); int uplo(); [o,phys]B(p,n))`

Copies all or part of a two-dimensional matrix A to another matrix B.

```
Arguments
=========
uplo: Specifies the part of the matrix A to be copied to B.
= 0: Upper triangular part
= 1: Lower triangular part
Otherwise: All of the matrix A
A: The m by n matrix A. If uplo = 0, only the upper triangle
or trapezoid is accessed; if uplo = 1, only the lower
triangle or trapezoid is accessed.
B: On exit, B = A in the locations specified by uplo.
```

```
$a = random(5,5);
$b = zeroes($a);
lacpy ($a, 0, $b);
```

lacpy ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## laswp

` Signature: ([io,phys]A(m,n);int [phys]k1();int [phys] k2(); int [phys]ipiv(p);int [phys]inc())`

Performs a series of row interchanges on the matrix A. One row interchange is initiated for each of rows k1 through k2 of A. Doesn't use PDL indices (start = 1).

```
Arguments
=========
A: On entry, the matrix of column dimension N to which the row
interchanges will be applied.
On exit, the permuted matrix.
k1: The first element of ipiv for which a row interchange will
be done.
k2: The last element of ipiv for which a row interchange will
be done.
ipiv: The vector of pivot indices. Only the elements in positions
k1 through k2 of ipiv are accessed.
ipiv(k) = l implies rows k and l are to be interchanged.
inc: The increment between successive values of ipiv. If ipiv
is negative, the pivots are applied in reverse order.
```

```
$a = random(5,5);
# reverse row (col for PDL)
$b = pdl([5,4,3,2,1]);
$a->laswp(1,2,$b,1);
```

laswp ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## lamch

` Signature: (cmach(); [o]precision())`

Determines precision machine parameters. Works inplace.

```
Arguments
=========
cmach: Specifies the value to be returned by lamch:
= 0 LAMCH := eps
= 1 LAMCH := sfmin
= 2 LAMCH := base
= 3 LAMCH := eps*base
= 4 LAMCH := t
= 5 LAMCH := rnd
= 6 LAMCH := emin
= 7 LAMCH := rmin
= 8 LAMCH := emax
= 9 LAMCH := rmax
where
eps = relative machine precision
sfmin = safe minimum, such that 1/sfmin does not overflow
base = base of the machine
prec = eps*base
t = number of (base) digits in the mantissa
rnd = 1.0 when rounding occurs in addition, 0.0 otherwise
emin = minimum exponent before (gradual) underflow
rmin = underflow threshold - base**(emin-1)
emax = largest exponent before overflow
rmax = overflow threshold - (base**emax)*(1-eps)
```

```
$a = lamch (0);
print "EPS is $a for double\n";
```

lamch ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## labad

` Signature: ([io,phys]small(); [io,phys]large())`

Takes as input the values computed by `lamch`

for underflow and overflow, and returns the square root of each of these values if the log of large is sufficiently large. This subroutine is intended to identify machines with a large exponent range, such as the Crays, and redefine the underflow and overflow limits to be the square roots of the values computed by `lamch`

. This subroutine is needed because lamch does not compensate for poor arithmetic in the upper half of the exponent range, as is found on a Cray.

```
Arguments
=========
small: On entry, the underflow threshold as computed by lamch.
On exit, if LOG10(large) is sufficiently large, the square
root of small, otherwise unchanged.
large: On entry, the overflow threshold as computed by lamch.
On exit, if LOG10(large) is sufficiently large, the square
root of large, otherwise unchanged.
```

```
$underflow = lamch(7);
$overflow = lamch(9);
labad ($underflow, $overflow);
```

labad ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## tricpy

` Signature: (A(m,n);[o] C(m,n); int uplo)`

tricpy(PDL(A), int(uplo), PDL(C))

```
use PDL::LinearAlgebra;
$c = $a->tricpy($uplo); # explicit uplo
$c = $a->tricpy; # default upper
or
use PDL::LinearAlgebra::Real;
tricpy($a, $uplo, $c); # modify c
```

Copy triangular part to another matrix. If uplo == 0 copy upper triangular part.

tricpy does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## cplx_eigen

` Signature: (eigreval(n);eigimval(n); eigvec(n,p);int fortran();[o]cplx_val(c=2,n);[o]cplx_vec(c=2,n,p))`

Output complex eigen-values/vectors from eigen-values/vectors as computed by geev or geevx. 'fortran' means fortran storage type.

cplx_eigen does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## augment

` Signature: (x(n); y(p);[o]out(q))`

Combine two ndarrays into a single ndarray. This routine does backward and forward dataflow automatically.

augment does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## mstack

` Signature: (x(n,m);y(n,p);[o]out(n,q))`

Combine two ndarrays into a single ndarray. This routine does backward and forward dataflow automatically.

mstack does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

## charpol

` Signature: (A(n,n);[o]Y(n,n);[o]out(p))`

Compute adjoint matrix and characteristic polynomial.

charpol does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.

# AUTHOR

Copyright (C) Grégory Vanuxem 2005-2018.

This library is free software; you can redistribute it and/or modify it under the terms of the Perl Artistic License as in the file Artistic_2 in this distribution.