ZPOCC
Finds the zeros of a polynomial with complex coefficients.
Required Arguments
COEFF — Complex 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 < 50 (Input)
Default: NDEG = size (COEFF,1) - 1.
FORTRAN 90 Interface
Generic: CALL ZPOCC (COEFF, ROOT [, …])
Specific: The specific interface names are S_ZPOCC and D_ZPOCC.
FORTRAN 77 Interface
Single: CALL ZPOCC (NDEG, COEFF, ROOT)
Double: The double precision name is DZPOCC.
Description
Routine ZPOCC computes the n zeros of the polynomial
p(z) = anzn + an-1zn-1 + … + a1z + a0
where the coefficients ai for i = 0, 1, …, n are complex and n is the degree of the polynomial.
The routine ZPOCC uses the Jenkins-Traub three-stage complex algorithm (Jenkins and Traub 1970, 1972). The zeros are computed one at a time in roughly increasing order of modulus. As each zero is found, the polynomial is deflated to one of lower degree.
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 - (3 + 6i)z2 - (8 - 12i)z + 10
where z is a complex variable.
USE ZPOCC_INT
USE WRCRN_INT
IMPLICIT NONE
! Declare variables
INTEGER NDEG
PARAMETER (NDEG=3)
!
COMPLEX COEFF(NDEG+1), ZERO(NDEG)
! Set values of COEFF
! COEFF = ( 10.0 + 0.0i )
! ( -8.0 + 12.0i )
! ( -3.0 - 6.0i )
! ( 1.0 + 0.0i )
!
DATA COEFF/(10.0,0.0), (-8.0,12.0), (-3.0,-6.0), (1.0,0.0)/
!
CALL ZPOCC (COEFF, ZERO)
!
CALL WRCRN ('The zeros found are', ZERO, 1, NDEG, 1)
!
END
Output
The zeros found are
1 2 3
( 1.000, 1.000) ( 1.000, 2.000) ( 1.000, 3.000)
