NAG Library Routine Document

C02AFF

, is as small as possible. The method can be shown to be cubically convergent for isolated roots (real or complex) and linearly convergent for multiple roots.

The routine generates a sequence of iterates

z_{1}, z_{2}, z_{3}, \dots

, such that

|P (z_{k + 1})| < |P (z_{k})|

and ensures that

z_{k + 1} + L (z_{k + 1})

‘roughly’ lies inside a circular region of radius

|F|

about

z_{k}

known to contain a zero of

P (z)

; that is,

|L (z_{k + 1})| \leq |F|

, where

F

denotes the Fejér bound (see Marden (1966)) at the point

z_{k}

. Following Smith (1967),

F

is taken to be

\min (B, 1.445 n R)

, where

B

is an upper bound for the magnitude of the smallest zero given by

B = 1.0001 \times \min (\sqrt{n} L (z_{k}), |r_{1}|, {|a_{n} / a_{0}|}^{1 / n}),

r_{1}

is the zero

X

of smaller magnitude of the quadratic equation

\frac{P^{''} (z_{k})}{2 n (n - 1)} X^{2} + \frac{P^{'} (z_{k})}{n} X + \frac{1}{2} P (z_{k}) = 0

and the Cauchy lower bound

R

for the smallest zero is computed (using Newton's Method) as the positive root of the polynomial equation

|a_{0}| z^{n} + |a_{1}| z^{n - 1} + |a_{2}| z^{n - 2} + \dots + |a_{n - 1}| z - |a_{n}| = 0 .

Starting from the origin, successive iterates are generated according to the rule

z_{k + 1} = z_{k} + L (z_{k})

, for

k = 1, 2, 3, \dots

, and

L (z_{k})

is ‘adjusted’ so that

|P (z_{k + 1})| < |P (z_{k})|

and

|L (z_{k + 1})| \leq |F|

. The iterative procedure terminates if

P (z_{k + 1})

is smaller in absolute value than the bound on the rounding error in

P (z_{k + 1})

and the current iterate

z_{p} = z_{k + 1}

is taken to be a zero of

P (z)

. The deflated polynomial

\tilde{P} (z) = P (z) / (z - z_{p})

of degree

n - 1

is then formed, and the above procedure is repeated on the deflated polynomial until

n < 3

, whereupon the remaining roots are obtained via the ‘standard’ closed formulae for a linear (

n = 1

) or quadratic (

n = 2

) equation.

Note that C02AHF, C02AMF and C02ANF can be used to obtain the roots of a quadratic, cubic (

n = 3

) and quartic (

n = 4

) polynomial, respectively.

4 References

Marden M (1966) Geometry of polynomials Mathematical Surveys 3 American Mathematical Society, Providence, RI

Smith B T (1967) ZERPOL: a zero finding algorithm for polynomials using Laguerre's method Technical Report Department of Computer Science, University of Toronto, Canada

Thompson K W (1991) Error analysis for polynomial solvers Fortran Journal (Volume 3) 3 10–13

Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford

5 Parameters

1: A( $2$ , $N + 1$ ) – REAL (KIND=nag_wp) arrayInput: On entry: if A is declared with bounds $(2, 0 : N)$ , then $A (1, i)$ and $A (2, i)$ must contain the real and imaginary parts of $a_{i}$ (i.e., the coefficient of $z^{n - i}$ ), for $i = 0, 1, \dots, n$ .
Constraint: $A (1, 0) \neq 0.0$ or $A (2, 0) \neq 0.0$ .
2: N – INTEGERInput: On entry: $n$ , the degree of the polynomial.
Constraint: $N \geq 1$ .
3: SCAL – LOGICALInput: On entry: indicates whether or not the polynomial is to be scaled. See Section 8 for advice on when it may be preferable to set $SCAL = .FALSE.$ and for a description of the scaling strategy.

Suggested value: $SCAL = .TRUE.$ .
4: Z( $2$ ,N) – REAL (KIND=nag_wp) arrayOutput: On exit: the real and imaginary parts of the roots are stored in $Z (1, i)$ and $Z (2, i)$ respectively, for $i = 1, 2, \dots, n$ .
5: W( $4 \times (N + 1)$ ) – REAL (KIND=nag_wp) arrayWorkspace
6: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

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,	$A (1, 0) = 0.0$ and $A (2, 0) = 0.0$ ,
or	$N < 1$ .

$IFAIL = 2$: The iterative procedure has failed to converge. This error is very unlikely to occur. If it does, please contact NAG, as some basic assumption for the arithmetic has been violated. See also Section 8.

$IFAIL = 3$: Either overflow or underflow prevents the evaluation of $P (z)$ near some of its zeros. This error is very unlikely to occur. If it does, please contact NAG. See also Section 8.

7 Accuracy

All roots are evaluated as accurately as possible, but because of the inherent nature of the problem complete accuracy cannot be guaranteed. See also Section 9.

8 Further Comments

SCAL = .TRUE.

, then a scaling factor for the coefficients is chosen as a power of the base

b

of the machine so that the largest coefficient in magnitude approaches

thresh = b^{e_{\max} - p}

. You should note that no scaling is performed if the largest coefficient in magnitude exceeds

thresh

, even if

SCAL = .TRUE.

. (

b

e_{\max}

and

p

are defined in Chapter X02.)

However, with

SCAL = .TRUE.

, overflow may be encountered when the input coefficients

a_{0}, a_{1}, a_{2}, \dots, a_{n}

vary widely in magnitude, particularly on those machines for which

b^{(4 p)}

overflows. In such cases, SCAL should be set to .FALSE. and the coefficients scaled so that the largest coefficient in magnitude does not exceed

b^{(e_{\max} - 2 p)}

Even so, the scaling strategy used by C02AFF is sometimes insufficient to avoid overflow and/or underflow conditions. 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

d P (c z)

for some suitable values of

c

and

d

. For example, if the original polynomial was

P (z) = 2^{- 100} i + 2^{100} z^{20}

, then choosing

c = 2^{- 10}

and

d = 2^{100}

, for instance, would yield the scaled polynomial

i + z^{20}

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

2^{10}

times those of

P (z)

If the routine fails with

IFAIL = 2

3

, then the real and imaginary parts of any roots obtained before the failure occurred are stored in Z in the reverse order in which they were found. Let

n_{R}

denote the number of roots found before the failure occurred. Then

Z (1, n)

and

Z (2, n)

contain the real and imaginary parts of the first root found,

Z (1, n - 1)

and

Z (2, n - 1)

contain the real and imaginary parts of the second root found,

\dots, Z (1, n - n_{R} + 1)

and

Z (2, n - n_{R} + 1)

contain the real and imaginary parts of the

n_{R}

th root found. After the failure has occurred, the remaining

2 \times (n - n_{R})

elements of Z contain a large negative number (equal to

- 1 / (X02AMF () \times \sqrt{2})

9 Example

For this routine two examples are presented. There is a single example program for C02AFF, with a main program and the code to solve the two example problems given in the subroutines EX1 and EX2.

Example 1 (EX1)

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 (EX2)

This example solves the same problem as subroutine EX1, but in addition attempts to estimate the accuracy of the computed roots using a perturbation analysis. Further details can be found in Thompson (1991).

NAG Library Routine DocumentC02AFF

+− Contents

1 Purpose

2 Specification

3 Description