Finds the zeros of a polynomial with real coefficients using the Jenkins-Traub, three-stage algorithm.
Synopsis
#include <imsl.h>
f_complex *imsl_f_zeros_poly (int ndeg, float coef[], …, 0)
The type d_complex function is imsl_d_zeros_poly.
Required Arguments
int ndeg
(Input)
Degree of the polynomial.
float coef[]
(Input)
Array with ndeg + 1
components containing the coefficients of the polynomial in increasing order by
degree. The polynomial is coef[n] zn + coef [n − 1] zn−1 + … + coef[0], where
n = ndeg.
Return Value
A pointer to the complex array of zeros of the polynomial. To release this space, use imsl_free. If no zeros are computed, then NULL is returned.
Synopsis with Optional Arguments
#include <imsl.h>
f_complex *imsl_f_zeros_poly (int ndeg, float coef[],
IMSL_RETURN_USER, f_complex root[],
0)
Optional Arguments
IMSL_RETURN_USER, f_complex root[]
(Output)
Array with ndeg components
containing the zeros of the polynomial.
Description
The function imsl_f_zeros_poly computes the n zeros of the polynomial
where the coefficients ai for i = 0, 1, …, n are real and n is the degree of the polynomial.
The function imsl_f_zeros_poly 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 a complex conjugate pair. As the zeros are found, the real zero, or quadratic factor, is removed by polynomial deflation.
Examples
Example 1
This example finds the zeros of the third-degree polynomial
p(z) = z3 − 3z2 + 4z – 2
where z is a complex variable.
#include <imsl.h>
#define NDEG 3
int main()
{
f_complex *zeros;
static float coeff[NDEG + 1] = {-2.0, 4.0, -3.0, 1.0};
zeros = imsl_f_zeros_poly(NDEG, coeff, 0);
imsl_c_write_matrix ("The complex zeros found are", 1, 3,
zeros, 0);
}
Output
The complex zeros found are
1 2 3
( 1, 0) ( 1, 1) ( 1, -1)
Example 2
The same problem is solved with the return option.
#include <imsl.h>
#define NDEG 3
int main()
{
f_complex zeros[3];
static float coeff[NDEG + 1] = {-2.0, 4.0, -3.0, 1.0};
imsl_f_zeros_poly(NDEG, coeff,
IMSL_RETURN_USER, zeros, 0);
imsl_c_write_matrix ("The complex zeros found are", 1, 3,
zeros, 0);
}
Output
The complex zeros found are
1 2 3
( 1, 0) ( 1, 1) ( 1, -1)
Warning Errors
IMSL_ZERO_COEFF 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.
IMSL_FEWER_ZEROS_FOUND Fewer than ndeg zeros were found. The root vector will contain the value for machine infinity in the locations that do not contain zeros.
