Math::Cephes::Polynomial  Perl interface to the cephes math polynomial routines
use Math::Cephes::Polynomial qw(poly);
# 'poly' is a shortcut for Math::Cephes::Polynomial>new
require Math::Cephes::Fraction; # if coefficients are fractions
require Math::Cephes::Complex; # if coefficients are complex
my $a = poly([1, 2, 3]); # a(x) = 1 + 2x + 3x^2
my $b = poly([4, 5, 6, 7]; # b(x) = 4 + 5x + 6x^2 + 7x^3
my $c = $a>add($b); # c(x) = 5 + 7x + 9x^2 + 7x^3
my $cc = $c>coef;
for (my $i=0; $i<4; $i++) {
print "term $i: $cc>[$i]\n";
}
my $x = 2;
my $r = $c>eval($x);
print "At x=$x, c(x) is $r\n";
my $u1 = Math::Cephes::Complex>new(2,1);
my $u2 = Math::Cephes::Complex>new(1,3);
my $v1 = Math::Cephes::Complex>new(1,3);
my $v2 = Math::Cephes::Complex>new(2,4);
my $z1 = Math::Cephes::Polynomial>new([$u1, $u2]);
my $z2 = Math::Cephes::Polynomial>new([$v1, $v2]);
my $z3 = $z1>add($z2);
my $z3c = $z3>coef;
for (my $i=0; $i<2; $i++) {
print "term $i: real=$z3c>{r}>[$i], imag=$z3c>{i}>[$i]\n";
}
$r = $z3>eval($x);
print "At x=$x, z3(x) has real=", $r>r, " and imag=", $r>i, "\n";
my $a1 = Math::Cephes::Fraction>new(1,2);
my $a2 = Math::Cephes::Fraction>new(2,1);
my $b1 = Math::Cephes::Fraction>new(1,2);
my $b2 = Math::Cephes::Fraction>new(2,2);
my $f1 = Math::Cephes::Polynomial>new([$a1, $a2]);
my $f2 = Math::Cephes::Polynomial>new([$b1, $b2]);
my $f3 = $f1>add($f2);
my $f3c = $f3>coef;
for (my $i=0; $i<2; $i++) {
print "term $i: num=$f3c>{n}>[$i], den=$f3c>{d}>[$i]\n";
}
$r = $f3>eval($x);
print "At x=$x, f3(x) has num=", $r>n, " and den=", $r>d, "\n";
$r = $f3>eval($a1);
print "At x=", $a1>n, "/", $a1>d,
", f3(x) has num=", $r>n, " and den=", $r>d, "\n";
This module is a layer on top of the basic routines in the cephes math library
to handle polynomials. In the following, a Math::Cephes::Polynomial object is
created as
my $p = Math::Cephes::Polynomial>new($arr_ref);
where $arr_ref is a reference to an array which can consist of one of
 •
 floating point numbers, for polynomials with floating point
coefficients,
 •
 Math::Cephes::Fraction or Math::Fraction objects, for
polynomials with fractional coefficients,
 •
 Math::Cephes::Complex or Math::Complex objects, for
polynomials with complex coefficients,
The maximum degree of the polynomials handled is set by default to 256  this
can be changed by setting
$Math::Cephes::Polynomial::MAXPOL.
A copy of a
Math::Cephes::Polynomial object may be done as
my $p_copy = $p>new();
and a string representation of the polynomial may be gotten through
print $p>as_string;
The following methods are available.
 coef: get coefficients of the polynomial

SYNOPSIS:
my $c = $p>coef;
DESCRIPTION:
This returns an array reference containing the coefficients of the
polynomial.
 clr: set a polynomial identically equal to zero

SYNOPSIS:
$p>clr($n);
DESCRIPTION:
This sets the coefficients of the polynomial identically to 0, up to
$p>[$n]. If $n is omitted, all elements are set to 0.
 add: add two polynomials

SYNOPSIS:
$c = $a>add($b);
DESCRIPTION:
This sets $c equal to $a + $b.
 sub: subtract two polynomials

SYNOPSIS:
$c = $a>sub($b);
DESCRIPTION:
This sets $c equal to $a  $b.
 mul: multiply two polynomials

SYNOPSIS:
$c = $a>mul($b);
DESCRIPTION:
This sets $c equal to $a * $b.
 div: divide two polynomials

SYNOPSIS:
$c = $a>div($b);
DESCRIPTION:
This sets $c equal to $a / $b, expanded by a Taylor series. Accuracy is
approximately equal to the degree of the polynomial, with an internal
limit of about 16.
 sbt: change of variables

SYNOPSIS:
$c = $a>sbt($b);
DESCRIPTION:
If a(x) and b(x) are polynomials, then
c(x) = a(b(x))
is a polynomial found by substituting b(x) for x in a(x). This method is not
available for polynomials with complex coefficients.
 eval: evaluate a polynomial

SYNOPSIS:
$s = $a>eval($x);
DESCRIPTION:
This evaluates the polynomial at the value $x. The returned value is of the
same type as that used to represent the coefficients of the
polynomial.
 sqt: square root of a polynomial

SYNOPSIS:
$b = $a>sqt();
DESCRIPTION:
This finds the square root of a polynomial, evaluated by a Taylor expansion.
Accuracy is approximately equal to the degree of the polynomial, with an
internal limit of about 16. This method is not available for polynomials
with complex coefficients.
 sin: sine of a polynomial

SYNOPSIS:
$b = $a>sin();
DESCRIPTION:
This finds the sine of a polynomial, evaluated by a Taylor expansion.
Accuracy is approximately equal to the degree of the polynomial, with an
internal limit of about 16. This method is not available for polynomials
with complex coefficients.
 cos: cosine of a polynomial

SYNOPSIS:
$b = $a>cos();
DESCRIPTION:
This finds the cosine of a polynomial, evaluated by a Taylor expansion.
Accuracy is approximately equal to the degree of the polynomial, with an
internal limit of about 16. This method is not available for polynomials
with complex coefficients.
 atn: arctangent of the ratio of two polynomials

SYNOPSIS:
$c = $a>atn($b);
DESCRIPTION:
This finds the arctangent of the ratio $a / $b of two polynomial, evaluated
by a Taylor expansion. Accuracy is approximately equal to the degree of
the polynomial, with an internal limit of about 16. This method is not
available for polynomials with complex coefficients.
 rts: roots of a polynomial

SYNOPSIS:
my $w = Math::Cephes::Polynomial>new([2, 0, 1, 0, 1]);
my ($flag, $r) = $w>rts();
for (my $i=0; $i<4; $i++) {
print "Root $i has real=", $r>[$i]>r, " and imag=", $r>[$i]>i, "\n";
}
DESCRIPTION:
This finds the roots of a polynomial. $flag, if
nonzero, indicates a failure of some kind. $roots in
an array reference of Math::Cephes::Complex objects holding the
real and complex values of the roots found. This method is not available
for polynomials with complex coefficients.
ACCURACY:
Termination depends on evaluation of the polynomial at the trial values of
the roots. The values of multiple roots or of roots that are nearly equal
may have poor relative accuracy after the first root in the neighborhood
has been found.
Please report any to Randy Kobes <randy@theoryx5.uwinnipeg.ca>
The C code for the Cephes Math Library is Copyright 1984, 1987, 1989, 2002 by
Stephen L. Moshier, and is available at http://www.netlib.org/cephes/. Direct
inquiries to 30 Frost Street, Cambridge, MA 02140.
The perl interface is copyright 2000, 2002 by Randy Kobes. This library is free
software; you can redistribute it and/or modify it under the same terms as
Perl itself.