Finds the zeros of a polynomial with real coefficients using the Jenkins-Traub three-stage algorithm.

Required Arguments

COEFF — Vector of length NDEG + 1 containing the coefficients of the polynomial in increasing order by degree.   (Input)
The polynomial is
COEFF(NDEG + 1)*Z**NDEG + COEFF(NDEG) * Z**(NDEG -1) + … + COEFF(1).

ROOT — Complex vector of length NDEG containing the zeros of the polynomial.   (Output)

Optional Arguments

NDEG — Degree of the polynomial. 1 ≤ NDEG ≤ 100    (Input)
Default: NDEG = size (COEFF,1) – 1.

FORTRAN 90 Interface

Generic:          CALL ZPORC (COEFF, ROOT [,…])

Specific:         The specific interface names are S_ZPORC and D_ZPORC.

FORTRAN 77 Interface

Single:            CALL ZPORC (NDEG, COEFF, ROOT)

Double:          The double precision name is DZPORC.

Description

Routine ZPORC computes the n zeros of the polynomial

p(z) = anzn + an-1 zn-1 + … + a1z + a0

where the coefficients ai for i = 0, 1, …, n are real and n is the degree of the polynomial.

The routine ZPORC uses the Jenkins-Traub three-stage algorithm (Jenkins and Traub 1970; Jenkins 1975). The zeros are computed one at a time for real zeros or two at a time for complex conjugate pairs. As the zeros are found, the real zero or quadratic factor is removed by polynomial deflation.

Comments

Informational errors

Type             Code

3                    1        The first several coefficients of the polynomial are equal to zero. Several of the last roots will be set to machine infinity to compensate for this problem.

3                    2        Fewer than NDEG zeros were found. The ROOT vector will contain the value for machine infinity in the locations that do not contain zeros.

Example

This example finds the zeros of the third-degree polynomial

p(z) = z3 - 3z2 + 4z -2

where z is a complex variable.

 

      USE ZPORC_INT

      USE WRCRN_INT

 

      IMPLICIT   NONE

!                                 Declare variables

      INTEGER    NDEG

      PARAMETER  (NDEG=3)

!

      REAL       COEFF(NDEG+1)

      COMPLEX    ZERO(NDEG)

!                                 Set values of COEFF

!                                 COEFF = (-2.0  4.0 -3.0  1.0)

!

      DATA COEFF/-2.0, 4.0, -3.0, 1.0/

!

      CALL ZPORC (COEFF, ZERO)

!

      CALL WRCRN ('The zeros found are', ZERO, 1, NDEG, 1)

!

      END

Output

 

              The zeros found are
             1                2                3
( 1.000, 0.000)  ( 1.000, 1.000)  ( 1.000,-1.000)

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