NAME

Math::Cephes::Polynomial - Perl interface to the cephes math polynomial routines

SYNOPSIS

  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";

DESCRIPTION

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;

Methods

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 non-zero, 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.

BUGS

Please report any to Randy Kobes <randy@theoryx5.uwinnipeg.ca>

COPYRIGHT

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.