ZPORC
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 | Description |
|---|
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)
