C209 Zeros of a Complex Polynomial

Subroutine subprograms CPOLYZ and WPOLYZ compute the zeros of the polynominal

$P(z)=c$₀zⁿ+c₁z^n-1+...+c_n-1z+c_n

of degree n with complex coefficients $c$_k and $c$₀≠0 .

On computers other than CDC or Cray, only the double-precision version WPOLYZ is available. On CDC and Cray computers, only the single-precision version CPOLYZ is available.

Structure:

SUBROUTINE subprograms
User Entry Names: CPOLYZ, WPOLYZ
Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035)

Usage:

For $t=C$ (type COMPLEX), $t=W$ (type COMPLEX*16),

    CALL tPOLYZ(C,N,MAXIT,Z,R)

C

(type according to t) One-dimensional array of dimension (0:d), where $d\; \ge N$ , containing the coefficients $c$_k, (k = 0,1,...,n) .

N

( INTEGER) The degree n.

MAXIT

( INTEGER) The maximum number of iterations permitted.

Z

(type according to t) One-dimensional array of length $\ge N$ . On entry, $Z(1),...,Z(N)$ must contain starting approximations for the zeros $z$_i . If no starting approximations are available, the $Z(i)$ should be set to zero. On exit, $Z(i)$ contains an approximation to the zero $z$_i .

R

( REAL for $t=C$ , DOUBLE PRECISION for $t=W$ ) One-dimensional array of dimension $\ge N$ . On exit, $R(1),...,R(N)$ contain an estimated radius $r$_i

of a circle centered at $Z(i)$ within which the true zero $z$_i is expected to lie.

Notes:

Note that, because of accumulation of rounding errors, unreliable results can be obtained for large n even for well-conditioned polynomials.

Error handling:

Error C209.1: $c$₀=0 .
Error C209.2: The number of iterations exceeds MAXIT.
Error C209.3: An estimated radius $r$_i cannot be computed for a certain value of i.
In all cases, a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

References:

1. T. Pomentale, Homotopy iterative methods for polynomial equations, J. Inst. Maths. Applics. 13 (1974) 201--213.

C210

K.S. Kölbig