IMSL_ZEROPOLY

Syntax | Return Value | Arguments | Keywords | Discussion | Example | Errors | Version History

The IMSL_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.

Note
This routine requires an IDL Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.

Syntax

Result = IMSL_ZEROPOLY(coef [, /DOUBLE] [, COMPANION=value] [, JENKINS_TRAUB=value])

Return Value

The complex array of zeros of the polynomial.

Arguments

coef

Array containing coefficients of the polynomial in increasing order by degree. The polynomial is coef (n) zⁿ + coef (n – 1) z^{n – 1} + ... + coef (0).

Keywords

DOUBLE

If present and nonzero, double precision is used.

COMPANION

If present and nonzero, the companion matrix method is used. Default: companion matrix method

JENKINS_TRAUB

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

Discussion

The IMSL_ZEROPOLY function 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 IMSL_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) = a(z), where I  is an n x 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, then IMSL_ZEROPOLY function 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.

Example

This example finds the zeros of the third-degree polynomial:

p (z) = z³ – 3z² + 4z – 2

where z is a complex variable.

coef = [-2, 4, -3, 1] 
; Set the coefficients. 
zeros = IMSL_ZEROPOLY(coef) 
; Compute the zeros. 
PM, zeros, Title = $ 
'The complex zeros found are: ' 
; Print results. 
The complex zeros found are: 
   (      1.00000,      0.00000) 
   (      1.00000,     -1.00000) 
   (      1.00000,      1.00000) 

Errors

Warning Errors

MATH_ZERO_COEFF—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 vector contains the value for machine infinity in the locations that do not contain zeros.

Version History