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

Synopsis

#include <imsl.h>

f_complex *imsl_c_zeros_poly (int ndeg, f_complex coef[], ¼, 0)

The type d_complex function is imsl_z_zeros_poly.

Required Arguments

int ndeg   (Input)
Degree of the polynomial.

f_complex coef[]   (Input)
Array with ndeg + 1 components containing the coefficients of the polynomial in increasing order by degree. The degree of the polynomial is

coef [n] zn + coef [n − 1] zn-1 + ¼ + coef [0]

where n = ndeg.

Return Value

A pointer to the complex array of zeros of the polynomial. To release this space, use free. If no zeros are computed, then NULL is returned.

Synopsis with Optional Arguments

#include <imsl.h>

f_complex *imsl_c_zeros_poly (int ndeg, f_complex 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_c_zeros_poly computes the n zeros of the polynomial

p(z) = anzn + an-1 zn-1 + ¼ + a1z + a0

where the coefficients ai for i = 0, 1, ¼, n are complex and n is the degree of the polynomial.

The function imsl_c_zeros_poly 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.

Examples

Example 1

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.

#include <imsl.h>

#define NDEG    3

main()
{
        f_complex       *zeros;
        f_complex coeff[NDEG + 1] = { {10.0, 0.0},
                                      {-8.0, 12.0},
                                      {-3.0, -6.0},
                                      { 1.0, 0.0} };

        zeros = imsl_c_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,         1)  (         1,         2)  (         1,         3)

Example 2

The same problem is solved with the return option.

#include <imsl.h>

#define NDEG    3

main()
{
        f_complex       zeros[3];
        f_complex coeff[NDEG + 1] = { {10.0, 0.0},
                                      {-8.0, 12.0},
                                      {-3.0, -6.0},
                                      { 1.0, 0.0} };

        imsl_c_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,         1)  (         1,         2)  (         1,         3)

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.

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