NAME
Math::NumSeq::Fibonacci  Fibonacci numbers
SYNOPSIS
use Math::NumSeq::Fibonacci;
my $seq = Math::NumSeq::Fibonacci>new;
my ($i, $value) = $seq>next;
DESCRIPTION
The Fibonacci numbers F(i) = F(i1) + F(i2) starting from 0,1,
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
starting i=0
FUNCTIONS
See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes.
Iterating
($i, $value) = $seq>next()

Return the next index and value in the sequence.
When
$value
exceeds the range of a Perl unsigned integer the return is aMath::BigInt
to preserve precision. $seq>seek_to_i($i)

Move the current sequence position to
$i
. The next call tonext()
will return$i
and corresponding value.
Random Access
$value = $seq>ith($i)

Return the
$i
'th Fibonacci number.For negative <$i> the sequence is extended backwards as F[i]=F[i+2]F[i+1]. The effect is the same Fibonacci numbers but negative at negative even i.
i F[i]   0 0 1 1 2 1 <+ negative at even i 3 2  4 3 <+
When
$value
exceeds the range of a Perl unsigned integer the return is aMath::BigInt
to preserve precision. $bool = $seq>pred($value)

Return true if
$value
occurs in the sequence, so is a positive Fibonacci number. $i = $seq>value_to_i_estimate($value)

Return an estimate of the i corresponding to
$value
. See "Value to i Estimate" below.
FORMULAS
Ith
Fibonacci F[i] can be calculated by a powering procedure with two squares per step. A pair of values F[k] and F[k1] are maintained and advanced according to bits of i from high to low
start k=1, F[k]=1, F[k1]=0
add = 2 # 2*(1)^k
loop
F[2k+1] = 4*F[k]^2  F[k1]^2 + add
F[2k1] = F[k]^2 + F[k1]^2
F[2k] = F[2k+1]  F[2k1]
bit = next bit of i, high to low, skip high 1 bit
if bit == 1
take F[2k+1], F[2k] as new F[k],F[k1]
add = 2 (for next loop)
else bit == 0
take F[2k], F[2k1] as new F[k],F[k1]
add = 2 (for next loop)
For the last (least significant) bit of i an optimization can be made with a single multiple for that last step, instead of two squares.
bit = least significant bit of i
if bit == 1
F[2k+1] = (2F[k]+F[k1])*(2F[k]F[k1]) + add
else
F[2k] = F[k]*(F[k]+2F[k1])
The "add" amount is 2*(1)^k which means +2 or 2 according to k odd or even, which means whether the previous bit taken from i was 1 or 0. That can be easily noted from each bit, to be used in the following loop iteration or the final step F[2k+1] formula.
For small i it's usually faster to just successively add F[k+1]=F[k]+F[k1], but when in bignums the doubling k>2k by two squares is faster than doing k many individual additions for the same thing.
Value to i Estimate
F[i] increases as a power of phi, the golden ratio. The exact value is
F[i] = (phi^i  beta^i) / (phi  beta) # exactly
phi = (1+sqrt(5))/2 = 1.618
beta = 1/phi = 0.618
Since abs(beta)<1 the beta^i term quickly becomes small. So taking a log (natural logarithm) to get i,
log(F[i]) ~= i*log(phi)  log(phibeta)
i ~= (log(F[i]) + log(phibeta)) / log(phi)
Or the same using log base 2 which can be estimated from the highest bit position of a bignum,
log2(F[i]) ~= i*log2(phi)  log2(phibeta)
i ~= (log2(F[i]) + log2(phibeta)) / log2(phi)
SEE ALSO
Math::NumSeq, Math::NumSeq::LucasNumbers, Math::NumSeq::Fibbinary, Math::NumSeq::FibonacciWord, Math::NumSeq::Pell, Math::NumSeq::Tribonacci
Math::Fibonacci, Math::Fibonacci::Phi
HOME PAGE
http://user42.tuxfamily.org/mathnumseq/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019 Kevin Ryde
MathNumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.
MathNumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
You should have received a copy of the GNU General Public License along with MathNumSeq. If not, see <http://www.gnu.org/licenses/>.