| Author(s): H.-H. Umstätter | Library: MATHLIB |
| Submitter: K.S. Kölbig | Submitted: 07.06.1992 |
| Language: Fortran | Revised: |
Subroutine subprogram RMULLZ and DMULLZ compute the zeros of the polynomial
of degree n with real coefficients 
and 
.
On computers other than CDC or Cray, only the double-precision version DMULLZ is available. On CDC and Cray computers, only the single-precision version RMULLZ is available.
Structure:
SUBROUTINE subprograms
User Entry Names : RMULLZ, DMULLZ
Files Referenced: Unit 6
External References: MTLMTR, ABEND
Usage:
For 
(type REAL), 
(type
DOUBLE PRECISION),
CALL tMULLZ(A,N,MAXIT,Z)
, containing the coefficients
.
, COMPLEX*16 for
) One-dimensional array of length
. On exit, 
contains an approximation
to the zero 
, listed in roughly decreasing order of 
.
Method:
The method of Muller (see Ref. 1) is used. This is based on iterated inverse quadratic interpolation followed by deflation to remove each zero as found.
Accuracy:
For well-conditioned polynomials (i.e. polynomials whose zeros
are not unduly sensitive to small errors in the coefficients), the
relative error of a computed zero of multiplicity m is of order
where d is the machine precision expressed in decimal
digits. For m>1, the m approximations to the single multiple zero
are uniformly distributed on a small circle of radius of order
around the exact zero. Therefore, if the polynomial is
well-conditioned, the true value of the multiple zero will be close to
the centre 
of this circle.
Error handling:
Error C202.1: 
.
Error C202.2: The number of iterations exceeds MAXIT.
In both cases, a message is written on Unit 6,
unless subroutine MTLSET (N002) has been called.
If the number of iterations exceeds MAXIT, those
zeros which have not been found are set to 
.
Notes:
For difficult cases which lead to too many iterations the following transformations may be applied, singly or together, to obtain a better-conditioned polynomial:
.
are clustered, or are too unsymmetrically
positioned with respect to the origin,
compute by synthetic division
(see Ref. 3) the coefficients of the polynomial whose argument is
, where 
is the arithmetic
mean of the zeros. The mean of the zeros of this new polynomial is
situated at the origin, which is where the subprogram starts searching.
Then, provided 
for most i,
will be more accurate zeros.
References:
