ZEROPOLY Function

Finds the zeros of a polynomial with real or complex coefficients using the companion matrix method or, optionally, the Jenkins-Traub, three-stage algorithm.

Usage

result = ZEROPOLY(coef)

Input Parameters

coef—An array containing the coefficients of the polynomial in increasing order by degree. The polynomial is of the form:

coef (n) zⁿ + coef (n – 1) z^{(n – 1)} + ¼ + coef (0)

Returned Value

result—The complex array of zeros of the polynomial.

Keywords

Companion—If present and nonzero, the companion matrix method is used. (Default: companion matrix method)

Double—If present and nonzero, double precision is used.

Jenkins_Traub—If present and nonzero, the Jenkins-Traub, three-stage algorithm is used.

Discussion

ZEROPOLY computes the n zeros of the polynomial

p (z) = a_n zⁿ + a_{n – 1}z^{(n – 1)} + ... + a₁ z + a₀

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

The default method used by ZEROPOLY is the companion matrix method. The companion matrix method is based on the fact that if C_a denotes the companion matrix associated with p(z), then
det (zI – C_a) = p(z), where I  is an  n-by-n  identity matrix. Thus,
det (z₀I – C_a) = 0  if, and only if, z₀ is a zero of p(z). This implies that computing the eigenvalues of C_a will yield the zeros of p(z). This method is thought to be more robust than the Jenkins-Traub algorithm in most cases, but the companion matrix method is not as computationally efficient. Thus, if speed is a concern, the Jenkins-Traub algorithm should be considered.

If the keyword Jenkins_Traub is set, ZEROPOLY 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.

Warning Errors

MATH_ZERO_COEFF—The first several coefficients of the polynomial are equal to zero. Several of the last roots are set to machine infinity to compensate for this problem.

MATH_FEWER_ZEROS_FOUND—Fewer than (N_ELEMENTS (coef) – 1) zeros were found. The root array contains 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.

; Set the coefficients.

coef = [-2, 4, -3, 1]

; Compute the zeros.

zeros = ZEROPOLY(coef)

; Print the complex polynomial results.

PM, zeros, Title = 'The complex zeros found are: '

; PV-WAVE prints the following:

; The complex zeros found are:

; (      1.00000,      0.00000)

; (      1.00000,     -1.00000)

; (      1.00000,      1.00000)

See Also

ROOT2POLY

For Additional Information

Jenkins, 1975.

Jenkins and Traub, 1970.