Revision history for Perl extension Math::Polynomial::Solve.

     Oct 21 2015
	- Changed tests to use Math::Utils's polynomial and compare code.
	- More POD reformatting and clarification, extend the POD docs,
	  and change some code based on aesthetics.
	- Gave up on putting unicode directly in POD (Heinrich Dorrie's
	  name seems to defeat the pod-to-html parser, which fails to
	  recognize an o with an umlaut
	- Bump the version numbers, up to CPAN.
     Oct 19 2015
	- Reformat to POD-as-function-head (becoming both documentation
	  and function comment description), as is currently done in
	  most perl modules.
	- I'm pretty firm about (most of) the :utility functions going
	  away by 2.80, but I also have it written that they're being
	  removed in two releases in some parts of the documentation,
	  which is way too restrictive. Changed all mentions of "two
	  releases" to "version 2.80".
     Oct 8 2015
	- Document functions deprecated by Math-Utils and Math-VecStat.
	- Change internal use from now-deprecated functions to functions
	  in Math-Utils and Math-VecStat (this is mostly in the Sturm
	- New function sturm_bisection(). Code comes from
	  sturm_bisection_root(), which now calls it.
	- Fix (deprecated, but in use) fltcmp() to avoid an error that
	  could result in fltcmp thinking equal values are less than.

     Jan 27 2014
	- Made documentation changes to the README file in anticipation
	  of removing set_hessenberg() and get_hessenberg(). Also made
	  version number 2.70.
	- Removed get_hessenberg() and set_hessenberg() from the code.
     May  3 2013
	- Some documentation tweaks on the use of ascending_order().
	- Code seems stable after use; time to upload to CPAN.
     Apr 19 2013
	- Add --ascending flag to the perl scripts in the eg/ directory.
	- Added files,, and to the eg/
	  directory, and added them to the MANIFEST. Again, new scripts
	  for trying individual functions.
     Apr 17 2013
	- Fix $ascending_flag checks in sturm functions with early
	  returns, or the sturm functions that use the first entry
	  in the chain for the (unreversed) polynomial.
	- Removed zyxw.t, as the existing test files now check
	  ascending_order() themselves.
     Apr 15 2013
	- Test files sturm1.t, sturm2.t, and sturm3.t also use
	  Test::More and test with the ascending_order().
	- Add ascending_order() tests to pderival.t.
	- Add ascending_order() tests to peval.t.
     Apr 12 2013
	- Get sturm0.t testing with ascending_order(), and Test::More.
     Apr 11 2013
	- Testing via division.t tests correctly again, with or without
	  the ascending_order() set.
	- Start in on the :sturm functions to recognize $ascending_flag.
     Apr 4 2013
	- Completed (maybe) adding $ascending_flag to the :utility
     Apr 1 2013
	- Start in on using $ascending_flags for the :utility functions.
	- Note ascending_order()'s use at length in the README file.
     Mar 28 2013
	- Fixed use of $ascending_flag in the :classical and :numeric
	  functions. Left to do are :sturm and :utility functions.
	- Updated test files for quadratic, cubic, quartic, and the
	  generic poly_roots() functions. They now use Test::More
	  and a new allzeroes() function for testing.
	- New test file zyxw.t for the ascending_order() checks.
     Mar 25 2013
	- Added the ascending_order() function, the $asending_flag
	  variable, and the documentation for the function. Actual
	  code comes later.
     Jan 25 2013
	- Wrote a better paragraph describing balance_matrix().
     Jan 15 2013
	- Corrected the build_companion() example (varible was a scalar
	  instead of an array).
	- Took a first stab at documenting balance_matrix().
     Jan 14 2013
	- Document existence of Fortran code.
     Jan 3 2013
	- Found Hiroshi Murakami's Fortran code on an old flash drive!
	  Added it to the references directory, under 'qralg' (he stated
	  it was freely available for this purpose in an e-mail Nick
	  Ing-Simmons forwarded to me).
	- Replaced Dick Nickalls's cubic paper with an updated one that
	  he pointed me to (it has a few corrections).
     Dec 24 2012
	- Added a reference in the documentation for balance_matrix().
     Dec 19 2012
	- Corrected a line in the documentation that stated build_companion()
	  returned a reference to an array of arrays. It used to, but now
	  that it's exported, it returns an array of arrays (no reference).
	- Fixed a brace mismatch in example code, and deleted a redundant
	  line about laguerre() in Sturm documentation.
     Dec 16 2012
	- Removed the early return if-block in build_companion(). The code
	  it was skipping is no longer in the function to be skipped, having
	  been moved to balance_matrix().
	- Changed the single-index loops in build_companion() to map operations.
     Dec 15 2012
	- Having made build_companion() and balance_matrix()
	  exportable, change their return values from array references
	  to arrays. This makes them easier to use with other modules.
	- balance_matrix() and hqr_eigen_hessenberg() now take an array
	  as a parameter, for the same reason.
	- Moved hqr_eigen_hessenberg() local variables deeper into
	  the loop in which they're used.
     Dec 12 2012
	- Split balance_matrix() off of build_companion() to let
	  build_companion() return a 'pure' companion matrix.
	- Start documenting all this.
     Dec 11 2012
	- Add build_companion() and hqr_eigen_hessenberg() to the
	  EXPORT list.
     Dec 6 2012
	- Merged in changes: eliminated a loop in build_companion() and
	  changed one of the foreach loops to a map.
	- Added and scripts to the eg/ directory.
     Sep 28 2012
	- De-Fortran'd the matrix used by build_companion() and
	  hqr_eigen_hessenberg(): it is now zero-based instead of one-based.
	  The amount of memory it saves is probably not significant, but
	  it does put the routines in line with other matrix-handling
	  modules, which may be useful down the road.
     Sep 10 2012
	- Added Newton-Raphson to the keyword list in Build.PL. Someday
	  those keywords will get used...
	- Time to upload to CPAN.
     Sep  6 2012
	- I was confused about the use of the MYMETA files. They are
	  now removed from the MANIFEST listing.
     Sep  4 2012
	- Yet more documentation improvement involving some minor
	  re-organization of the Sturm functions and a correction
	  in the poly_iteration() example.
	- Get all the files's VERSION number up to 2.63.
	- Added 'create_license => 1' to the Build.PL list of attributes.
	- Added 'dynamic_config => 0' to the Build.PL list of attributes.
	- Added LICENSE to the MANIFEST file.
	- Up to CPAN we go.
     Aug 27 2012
	- Documentation of 'newtonraphson' option for poly_tolerance()
	  accidentally left off.
	- Better documentation for newtonraphson().
	- Removed some debugging code from laguerre() that I forgot to
	  remove earlier.
     Aug 23 2012
	- sturm_bisection_roots(), which was supposed to only
	  return a polynomial's real roots, could occasionally
	  return complex roots. This was because the laguerre()
	  function could occasionally slip into the Complex
	  plane while doing its hunting. Put a "no Math::Complex"
	  line at the start of the function to prevent this.
	- Did the same for newtonraphson(), as its primary use
	  is to find real roots.
	- Made sure the documentation makes the real-roots-only
	  condition of laguerre() and newtonraphson() clear.
     Jul 18 2012
	- Changed the Test::Pod requirement in pod.t from
	  version 1.00 to 1.41. Didn't put the version
	  requirement in Build.PL because it's silly to
	  block installation because of POD testing.
	- Put "=encoding utf8" command in the first line of my
	  POD. One of my references has a non-ASCII character in
	  his name, and Pod-Simple now warns about that as of May.
	  Some searching seems to indicate that =encoding was
	  introduced in Pod-Simple during perl version 5.10; we'll
	  see whether older perls got the updated Pod-Simple.
     Jul 10 2012
	- Comment and Smart-Comment changes. Also removed the late
	  Nick Ing-Simmons's e-mail address from comments.
	- Changed the remaining C/Fortran-style down-count loop in
	  hqr_eigen_hessenberg() to a Perl-style loop using reverse.
	  Made sure edge cases were handled, and commented *why*
	  early checks are made in that loop.
	- Made variables $s and $z in hqr_eigen_hessenberg() local
	  to their loops in an attempt to make the algorithm more
	  easily read.
     Jul 5  2012
	- Changed the first C/Fortran-style down-count loop in
	  hqr_eigen_hessenberg() to a Perl-style loop using reverse.

     Jan 2  2012
	- And more tests added to derivaluate.t. Up to CPAN we go.
     Dec 23 2011
	- Bug in poly_derivaluat(). For polynomials degree 5 or higher,
	  was returning the wrong second derivative value.
	  Unfortunately, the testing of the function was only done
	  indirectly in laguerre.t, and it only had third degree
	  polynomials. Extended laguerre.t, and added derivaluate.t
	  to test files.
     Nov 17 2011
	- Perl version requirement changed to 5.8.3 (from 5.6.0). Support
	  for 5.6.0 in various perl development tools is being dropped,
	  and 5.8.3 seems to be a safe minimum, at least for the next
	  year or so. Change made in Build.PL and
	- Changed "repository" key in Build.PL from a hash to a string,
	  as seems to be currently required under the version of
	  Module::Build that I'm developing with (the hash form looks
	  to be for a later version that I cannot install yet).
     Oct 30 2011
	- A couple more changes to Build.PL (dist_author and resources).
	  Time to push this up to git and add it to CPAN.
     Sep 20 2011
	- CPAN::Meta::Spec had a lot to say about the changes to Build.PL
	  scripts. Made those changes.
     Sep 19 2011
	- The pre-September items for this version were done separate
	  from the git and github installation. Merged it all in.
     Sep 9 2011
	- Added carp messages to get_hessenberg() and set_hessenberg().
	  Now anyone still using these functions gets nagged about it.
     July 20 2011
	- Emphasize in the documentation the fact that roots are mixed
	  real and Math::Complex - there was some confusion.
     Mar 24 2011
	- Gah. EXPORT list didn't have poly_tolerance!
	- Put in a check for accidental refs in poly_derivaluate().
     Mar 23 2011
	- Test file nr.t created, and added to MANIFEST.
	- A couple more keywords added to Build.PL.
     Mar 12 2011
	- Added Newton-Raphson function newtonraphson() to the module, just
	  to provide a comparison with laguerre().
	- Added Newton-Raphson to the keyword list.
     Mar 9 2011
	- New function poly_derivaluate(), to be used internally by
	  laguerre(). Added it to the :utility EXPORT list.
	- laguerre() now uses poly_derivaluate().
	- Documented poly_derivaluate().
	- Fixed an L<> typo.
	- Improved the Sturm function documentation.
     Mar 2 2011
	- Three more tests to cubic.t (polynomial representations
	  of trisecting a 60 deg. angle; duplicating the cube; and
	  construction a septagon; all with straightedge and compass).
	- Instead of a copy of the @cases in cubic.t with negative
	  coefficients, just use poly_constmult() in the loop.
	- Did the same to quartic.t.
	- And quadratic.t.
     Feb 21 2011
	- %iteration hash was checked in poly_tolerance(). Changed
	  to checking %tolerance, which makes much more sense.
     Jan 27 2011
	- README wasn't up to date.
	- Added some more keywords in Build.PL.
	- Documentation changes to grammar, spelling, and L<>s.
     Dec 15 2010
	- Moved fltcmp() from the test directory to the module, and
	  added it to the :utility list. Added a "fltcmp" key to the
	  %tolerance list.
	- Changed all of the test files to use the module's fltcmp();
	- More documentation clean-up!
	- Released!
     Dec 14 2010
	- Created and documented poly_tolerance(), letting the user
	  set the tolerance value for laguerre().
	- Documentation clean-up.
     Dec 3 2010
	- Function laguerre() now works with a better tolerance value.
	- Added a %tolerance hash for the iterative functions. Currently
	  only laguerre() makes use of it, and I may change the way of
	  handling later.
	- Fixed poly_iteration() - iteration limits weren't getting
	- Checking the t directory, I found four or so test files that
	  weren't in the MANIFEST file. Fixed that.
     Nov 30 2010
	- New version of laguerre() now takes multiple x-values.
	- Documented the changes in poly_evaluate().
     Nov 18 2010
	- Function poly_evaluate() can now detect whether it is being passed
	  an array of x-values, or a single x-value. Return checks to see if
	  an array of y-values is wanted, or a single value.
     Nov 17 2010
	- Add function poly_iteration() to make iteration limits adjustable.
	- Add function laguerre() for use by sturm_bisection_roots().
     Nov 7 2010
	- Add function sturm_bisection_roots(). Not complete yet.
	- De-Fortran'd some loops in hqr_eigen_hessenberg(). Now uses
	  the range operator, '..'.
     Oct 22 2010
	- The Sturm functions sturm_sign_minus_inf() and sturm_sign_minus_inf()
	  had been documented as exported, but they weren't. They are now.
	- Documentation for the Sturm functions was cut short - either I had
	  accidentally chopped it, or I left off writing it and forgot to get
	  back to it. Took a stab at completing it.
	- Added function sturm_real_root_range_count().
	- Added test file sturm2.t to the t directory, and to the MANIFEST.
     Oct 19 2010
	- The keyword metadata key isn't an argument to the Module::Build
	  constructor, it's part of the meta_merge hash which *is* an
	  argument to the constructor. Fixed that.
	- Document the varsubst option in poly_option().
	- Made the variable substitution method more efficient by
	  actually removing primes from the list as they're checked.
	  Extended the prime list that gets checked.
     Oct 18 2010
	- Aaaaaaand of course I forgot to put varsubst.t in the MANIFEST.
	- Updated Module::Build on my system. Maybe get keywords in
	  META.yml now?
     Oct 18 2010
	- Got the variable substitution code in, and created test file
	  varsubst.t. Undocumented for now.
     Oct 15 2010
	- More documentation improvments, as I clear away old
	  numeric vs. classical confusion, and add documentation
	  for poly_option().
	- Read the latest Module::Build documentation, and decided
	  to try the keyword item in Build.PL.
     Oct 12 2010
	- Added to the eg directory and to the MANIFEST.
	- Was removing leading zeros in poly_nonzero_term_count(),
	  which is pretty pointless. Removed those lines of code.
     Oct 12 2010
	- And one more document change: I found a call to set_hessenberg(0)
	  where we now want poly_option(hessenberg => 0);
     Oct 8 2010
	- Added rootf.t to test directory and MANIFEST; it tests poly_roots()
	  with the option "root_function" set.
	- Changes to the poly_option() function for better
	  option handling.
	- Change test files poly0.t and polyfp0.t (which used
	  set_hessenberg($value)) to now use
	  poly_option(hessenberg => $value) instead.
     Oct 5 2010
	- Documentation change to EXPORT section. Instead of listing
	  exported functions, link to the head3 Functions sections,
	  which are now organized by tag (see Sep 28 2010).
	- Added the poly_option() function.
     Sep 28 2010
	- Documentation improvement by categorizing the functions by
	  tag and improving the Sturm sequence explanation.
     Sep 7 2010
	- Documentation updates. Time to release the module.
     Sep 2 2010
	- Added an %option variable. Currently its fields are hessenberg,
	  varsubs, and roots; hessenberg (which has replaced
	  $use_hessenberg) is in use, the other two might not even keep
	  their names the same.
	- Let epsilon() function change the calculated machine epsilon
	  value. This is probably something you want to do rarely, if at all.
     Aug 31 2010
	- Added early return to poly_sturm_chain() when it's passed a
	  constant or a linear equation (yeah, I know, but one still has to
	- A couple more tests each for sturm0.t and sturm1.t.
	- Finally used (and then fixed) sturm_sign_chain().
	- Expose some more internal functions under the :sturm tag, and
	  document them.
     Aug 26 2010
	- Added another cubic test to cubic.t and poly0.t.
     Aug 25 2010
	- CPAN testers found a failing test in sturm0.t for some machines.
	  Altered the test to make use of fltcmp(), which I hope will solve
	  what I suspect is a precision problem.
	- Made consistent the book references in the Acknowledgements section.
     Aug 23 2010
	- poly_real_roots() becomes poly_real_root_count().
	- Added test files sturm0.t and sturm1.t.
	- Many documentation additions. Of note: the documentation
	  notes the FUTURE DEPRECATION of set_hessenberg() and
	- Set $VERSION to reflect beta condition, particularly
	  the (lone) sturm function.
     Aug 14 2010
	- Added poly_real_roots().
     Jul 22 2010
	- Added the functions to create the sign rows, currently named
	  sturm_sign_minus_infinity(), sturm_sign_plus_infinity(), and
     Jul 21 2010
	- Added function poly_sturm_chain(). Started a test file for it.
	- Added a test file for the poly_division() function and of course
	  made the function available under the :utility tag.
     Jun 24 2010
	- Stefan Petrea suggested adding Sturm's sequence to the module.  After
	  looking it up, it seemed like a good idea, so I began by:
	- Added poly_derivative() to the export list and to the :utility tag.
	- Added poly_antiderivative() to the export list and to the :utility
	  tag, mostly for completeness's sake, although I may find a legitimate
	  use for it.
     Jun 22 2010
	- Added epsilon() to the export list and to the :utility tag. It
	  existed in v2.52, I just forgot to add it.
	- Formatting changes to the POD with respect to the functions'

     Jun 16 2010
	- Added export tags. You can export the analytic functions
	  linear_roots(), quadratic_roots(), cubic_roots(), and quartic_roots()
	  via the :classical tag; poly_roots(), get_hessenberg(), and
	  set_hessenbert() via the :numeric tag; and poly_evaluate() and
	  simplified_form() via the :utility tag.
	- Fixed spelling error, mostly in the comments: it's "resolvent" cubic,
	  not "resolvant".
	- Documentation upgrade, particularly for poly_roots(), which was
	  worded a little funny given the change from the default solving
	  method in version 2.50. Added documentation for the new export tags
	  and the new functions.
     Jun 15 2010
	- Strunk & White'd the possesive form of Dr. Nickalls's name.
	  (I'm not sure how Strunk & White would react to verbing their names.)
	  I had randomly dropped the 's' before or after the apostrophe.
	- Opened up the internals a bit by providing a a simplified_form()
	  function that removes leading zero coefficients and divides the
	  remaining coefficients by the coefficient of the highest power.
     Jun 14 2010
	- Added a poly_evaluate() function to return y values of the
	  polynomial at given x values. Partly done for my own testing but
	  it's too useful not to provide generally.
     Jun 1 2010
	- Jean Connelly found a quartic where $g is greater than $epsilon,
	  but $g**2 isn't. Since it's the squared value that's used in
	  Ferrari's method, we eventually wind up dividing by zero.
	  This is a floating point representation sample that slipped
	  past my tests (which up to now were all based on integers).
	  Changed the check to use $g**2, since that's what we're
	  protecting ourselves from anyway.
	- Connelly's case is 0.9216 times an integer coeffecient set. Put
	  the integer version in poly0.t, and then created the floating point
	  tests in the new file polyfp0.t by taking the poly0.t tests and
	  multiplying everything by 0.9216. I'll extend this later.
	- Added Dr. Nickalls's quartic article to the distribution. Both of
	  his articles are in the subdirectory "reference".
     Apr 19 2010
	- Bowed to convention and reversed the order of the Change entries.
	  (And moved an obviously out-of-place Change entry to its proper
	- Added Dr. Nickalls's article to the distribution, with his permission.
	- Abandoned backward compatibility with perl version 5.05, which was
	  present despite the 5.6 requirement in Build.PL (I was supporting
	  the module on a very old server). The module now requires version 5.6,
	  which admittedly isn't cutting edge but does allow more modern
	  constructs like 'our'.

2.50 Mar 7 2009
	- Function poly_roots() now uses the QR Hessenberg method by
	  default.  To return to the old version's behavior, use
	- Update tests and documentation to reflect this.
	- Update Dr. Nickalls's e-mail address and his paper's url.

2.12 Jan 28 2009
	- Add "requires" item in Build.PL to insist on a Math::Complex
	  version greater than 1.36.  This makes the Complex.txt file
	  unnecessary since I won't have to explain about the old
	  bug in the Im() function.

2.11 Feb 19 2007
	- Andy Stein sent me cubic.m, an update to Herman Bruyninckx's
	  matlab script. The script is in the eg directory.
	- Change all debugging code to use Smart::Comments.
	  Replaces not only debug print statements but also the
	  show_matrix() subroutine.
	- Found some "$var = $var op ..." code that is now
	  "$var op= ..." in hqr_eigen_hessenberg().
	- More documentation changes.
	- Found a spot where $#coefficients was used where @coefficients
	  would be less likely to cause an infinite loop.
     Sep 5 2008
	- Added some quadratic tests.

2.10 Jul 4 2006
	- Released.

2.00 Feb 18 2004
	- For some idiotic reason, I thought that I had already uploaded the
	  module to CPAN.  Time to rectify that mistake.
	- Well, as long as I'm here, reviewed the code again, and found some
	  inconsistant epsilon comparisons ('<' vs. '<=').  Changed them all
	  to '<'.
	- Double-checked the links too.  Dr. Nickalls's article keeps moving
     Apr 3 2006
	- Upgraded the test files to use Test::Simple.
	- Moved the internal structure of the module directory around
	  to match the current layout made by the current h2xs.
	- Added a Build.PL file.
	- Generated a Meta.yml file and a Makefile.PL from Build.PL
	- Added new tests to cubic.t to cover catastrophic
	  cancellation conditions.
	- Re-factored an expression in the ($dis < -$epsilon) branch
	  of cubic_roots(), in preparation for tracking down the
	  cancellation error.  This is the branch that was failing
	  the tests for certain values of the leading coefficient
	  (a negative, for one thing).  Wouldn't you know, the tests
	  passed after doing this.  I suspect that this will not be the
	  final chapter in this story.
	- Updated the link to in the documentation.
     Jun 26 2006
	- Added a test to poly1.t (x**5 + x + 1).  Nothing special
	  except that it's solvable in radicals, and there was an
	  article on Galois theory that missed that.

1.99 April 30 2003
	- More documentation changes, including updating links to Dr.
	  Nickalls's paper.
	- Added many, many more tests.
	- Added the set_ and get_hessenberg() functions.
	- Put some code that was in a text file in one of the helper
	  perl files.

1.90 March 17 2003
	Nick Ing-Simmons had been working on a perl binding to the GNU
	Scientific Library that performed, among other things, polynomial
	solving.  He graciously (indeed, FAR beyond the call of duty)
	provided a perl version the Fortran code written by Hiroshi
	Murakami, and gave me permission to use the perl version of the code
	in this module (as Hiroshi Murakami gave permission to Nick Ing-Simmons).
	For further information on his efforts, see Math::GSL::Polynomial.
	Putting in the code resulted in:
	- Some formatting changes.  Sorry, I'm a tab user.
	- Removal of some redundant code.  Math::Polynomial::Solve
	  already calculated epsilon, and the wrapper function already
	  removed leading zero coefficients.
	- Coefficients to be passed in from higher power to lower,
	  rather than the reverse.  This matches current usage in this
	- Roots to be returned as Complex number objects, rather than paired
	  real and imaginary parts.
	- An early return for the really simple case.

	- Web site address of Karl's Calculus Tutor has changed.  Document
	  changed with it.
	- Code that popped off leading zero coefficients was flawed.
	  Negative values could get popped off too since the check was
	      $coefficients[0] <= $epsilon
	  instead of
	      abs($coefficients[0]) <= $epsilon.
	- Began writing code for symbolic solutions.
	- And then I got the e-mail from Nick Ing-Simmons.

1.00 Wed Feb 14 17:51:04 2001
	- original version