Integer, Intent (In)	::	n, itmax, polish
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	conv(n)
Real (Kind=nag_wp), Intent (Out)	::	berr(n), cond(n)
Complex (Kind=nag_wp), Intent (In)	::	a(0:n)
Complex (Kind=nag_wp), Intent (Out)	::	z(n)

C Header Interface

#include <nag.h>

void	c02aaf_ (const Complex a[], const Integer n, const Integer itmax, const Integer polish, Complex z[], double berr[], double cond[], Integer conv[], Integer ifail)

The routine may be called by the names c02aaf or nagf_zeros_poly_complex_fpml.

3 Description

c02aaf attempts to find all the roots of the

n

th degree complex polynomial equation

P (z) = a_{0} z^{n} + a_{1} z^{n - 1} + a_{2} z^{n - 2} + \dots + a_{n - 1} z + a_{n} = 0 .

The roots are located using a modified form of Laguerre's method, as implemented by Cameron (2018). An implicit deflation strategy is employed, which allows for high accuracy even when solving high degree polynomials. Linear (

n = 1

) and quadratic (

n = 2

) problems are obtained via the 'standard' closed formulae.

First, initial estimates of the roots are made using a method proposed by Bini (1996), which selects complex numbers along circles of suitable radii. Updates to each root approximation

z_{j}

, for

j = 1, 2, \dots, n

, are then made using the iterative formula from Petković et al. (1997):

{\hat{z}}_{j} = z_{j} - \frac{n}{G_{j} \pm \sqrt{(n - 1) (n H_{j} - G_{j}^{2})}},

where

G_{j} = \frac{P^{'} (z_{j})}{P (z_{j})} - \sum_{i = 1; ​ i \neq j}^{n} \frac{1}{(z_{j} - z_{i})} and H_{j} = - {(\frac{P^{'} (z_{j})}{P (z_{j})})}^{'} - \sum_{i = 1; ​ i \neq j}^{n} \frac{1}{{(z_{j} - z_{i})}^{2}} .

The nearest

{\hat{z}}_{j}

to the current approximation is used, by selecting the sign that maximizes the denominator of the correction term. The subtraction of the sum terms when computing

G_{j}

and

H_{j}

determines the implicit deflation strategy of the modified Laguerre method.

The relative backward error of a root approximation

z_{j}

is given by

η (z_{j}) = \frac{| P (z_{j}) |}{α (z_{j})},

where

α

is the perturbed polynomial

α (z) = \sum_{k = 0}^{n} (3.8 k + 1) | a_{n - k} | {| z |}^{k} .

A root approximation is deemed to have converged if

η (z_{j}) \leq 2 ​ ​ \times

machine precision, at which point updates of that root cease. If the stopping criterion holds, then the computed root is the exact root of a polynomial whose coefficients are no more perturbed than the floating-point computation of

P (z_{j})

The condition number of each root is also computed, as a measure of sensitivity to changes in the coefficients of the polynomial. It is given by

κ (z_{j}) = \frac{α (z_{j})}{| z_{j} | | P^{'} (z_{j}) |} .

Root approximations can be further refined with optional 'polishing' processes. A simple polishing process is provided that carries out a single iteration of Newton's method, which proves quick yet often effective. Alternatively, a compensated polishing process from Cameron and O'Neill (2019) can be applied. This iterative method combines the implicit deflation of the modified Laguerre method, with the accuracy of evaluating polynomials and their derivatives using the compensated Horner's method from Graillat et al. (2005). Compensated polishing yields approximations with a limiting accuracy as if computed in twice the working precision.

It is recommended that you read Section 9.1 for advice on selecting an appropriate polishing process.

4 References

Bini D A (1996) Numerical computation of polynomial zeros by means of Aberth's method Numerical Algorithms 13 179–200 Springer US

Cameron T R (2018) An effective implementation of a modified Laguerre method for the roots of a polynomial Numerical Algorithms Springer US https://doi.org/10.1007/s11075-018-0641-9

Cameron T R and O'Neill A (2019) On a compensated polishing technique for polynomial root solvers To be published

Graillat S, Louvet N, and Langlois P (2005) Compensated Horner scheme Technical Report Université de Perpignan Via Domitia

Petković M, Ilić S, and Tričković S (1997) A family of simultaneous zero-finding methods Computers & Mathematics with Applications (Volume 34) 10 49–59 https://doi.org/10.1016/S0898-1221(97)00206-X

Wilkinson J H (1959) The evaluation of the zeros of ill-conditioned polynomials. Part I Numerische Mathematik (Volume 1) 1 150–166 Springer-Verlag

5 Arguments

1: $a (0 : n)$ – Complex (Kind=nag_wp) array Input

On entry:

a (i)

must contain the coefficient of

z^{n - i}

, for

i = 0, 1, \dots, n

Constraint:

a (0) \neq (0.0, 0.0)

2: $n$ – Integer Input

On entry:

n

, the degree of the polynomial.

Constraint:

n \geq 1

3: $itmax$ – Integer Input

On entry: the maximum number of iterations to be performed.

Suggested value:

itmax = 30

Constraint:

itmax \geq 1

4: $polish$ – Integer Input

On entry: specifies the polishing technique used to refine root approximations.

$polish = 0$: No polishing.
$polish = 1$: Single iteration of Newton's method.
$polish = 2$: Iterative refinement using the compensated Horner's method.

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, the complex variable $a (0) = (0.0, 0.0)$ .

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 3$: On entry, $itmax = ⟨ value ⟩$ .
Constraint: $itmax \geq 1$ .

$ifail = 4$: On entry, $polish = ⟨ value ⟩$ .
Constraint: $polish = 0$ , $1$ or $2$ .

$ifail = 5$: Convergence has failed for at least one root approximation.
Check conv and consider increasing itmax.

$ifail = 6$: c02aaf encountered overflow during at least one root approximation.
Check conv and consider scaling the polynomial (see Section 9.2).

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

All roots are evaluated as accurately as possible, but because of the inherent nature of the problem complete accuracy cannot be guaranteed.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

c02aaf is not threaded in any implementation.

9 Further Comments

9.1 Selecting a Polishing Process

The choice of polishing technique ultimately depends on two factors: how well conditioned the problem is, and a preference between run time and accuracy. For a detailed analysis of the polishing techniques, see Cameron and O'Neill (2019).

Well-conditioned Problems

Simple polishing is effective in reducing the error in approximations of well-conditioned roots, doing so with a negligible increase in run time. Compensated polishing has comparable accuracy, but it is approximately ten times slower than when using simple polishing.

Simple polishing (

polish = 1

) is recommended for well-conditioned problems.

Ill-conditioned Problems

There is a dramatic difference in accuracy between the two polishing techniques for ill-conditioned polynomials. Unpolished approximations are inaccurate and simple polishing often proves ineffective. However, compensated polishing is able to reduce errors by several orders of magnitude.

Compensated polishing (

polish = 2

) is highly recommended for ill-conditioned problems.

9.2 Scaling the Polynomial

c02aaf attempts to avoid overflow conditions where possible. However, if the routine fails with

ifail = 6

, such conditions could not be avoided for the given polynomial. Use conv to identify the roots for which overflow occurred, as other approximations may still have succeeded.

Extremely large and/or small coefficients are likely to be the cause of overflow failures. In such cases, you are recommended to scale the independent variable

(z)

so that the disparity between the largest and smallest coefficient in magnitude is reduced. That is, use the routine to locate the zeros of the polynomial

s P (c z)

for some suitable values of

c

and

s

. For example, if the original polynomial was

P (z) = 2^{−100} i + 2^{100} z^{20}

, then choosing

c = 2^{−10}

and

s = 2^{100}

, for instance, would yield the scaled polynomial

i + z^{20}

, which is well-behaved relative to overflow and has zeros which are

2^{10}

times those of

P (z)

10 Example

The example program for c02aaf demonstrates two problems, given in the subroutines ex1_basic and ex2_polishing. (Note that by default, the second example is switched off because the results may be machine dependent. Edit the program in the obvious way to switch it on.)

Example 1: Basic Problem

This example finds the roots of the polynomial

a_{0} z^{5} + a_{1} z^{4} + a_{2} z^{3} + a_{3} z^{2} + a_{4} z + a_{5} = 0,

where

a_{0} = (5.0 + 6.0 i)

a_{1} = (30.0 + 20.0 i)

a_{2} = - (0.2 + 6.0 i)

a_{3} = (50.0 + 100000.0 i)

a_{4} = - (2.0 - 40.0 i)

and

a_{5} = (10.0 + 1.0 i)

Example 2: Polishing Processes

This example finds the roots of a polynomial of the form

(z - 1) (z - 2) \dots (z - n) = 0,

first proposed by Wilkinson (1959) as an example of polynomials with ill-conditioned roots, that are sensitive to small changes in the coefficients. A polishing mode is demonstrated with

n = 10

, and the maximum forward and relative backward errors of the approximations displayed.

c02aa: FL CL CPP AD PY MB

NAG FL Interfacec02aaf (poly_​complex_​fpml)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

9.1 Selecting a Polishing Process

9.2 Scaling the Polynomial

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG FL Interface
c02aaf (poly_complex_fpml)