ZREAL

Finds the real zeros of a real function using Müller’s method.

Required Arguments

F — User-supplied FUNCTION to compute the value of the function of which a zero will be found. The form is

F(X), where

F(X), where

X – The point at which the function is evaluated. (Input)

X should not be changed by F.

X should not be changed by F.

F – The computed function value at the point X. (Output)

F must be declared EXTERNAL in the calling program.

X — A vector of length NROOT. (Output)

X contains the computed zeros.

X contains the computed zeros.

Optional Arguments

ERRABS — First stopping criterion. (Input)

A zero X(I) is accepted if ABS(F(X(I)).LT. ERRABS.

Default: ERRABS = 1.e-4 for single precision and 1.d-8 for double precision.

A zero X(I) is accepted if ABS(F(X(I)).LT. ERRABS.

Default: ERRABS = 1.e-4 for single precision and 1.d-8 for double precision.

ERRREL — Second stopping criterion is the relative error. (Input)

A zero X(I) is accepted if the relative change of two successive approximations to X(I) is less than ERRREL.

Default: ERRREL = 1.e-4 for single precision and 1.d-8 for double precision.

A zero X(I) is accepted if the relative change of two successive approximations to X(I) is less than ERRREL.

Default: ERRREL = 1.e-4 for single precision and 1.d-8 for double precision.

EPS — See ETA. (Input)

Default: EPS = 1.e-4 for single precision and 1.d-8 for double precision.

Default: EPS = 1.e-4 for single precision and 1.d-8 for double precision.

ETA — Spread criteria for multiple zeros. (Input)

If the zero X(I) has been computed and ABS(X(I) – X(J)).LT.EPS, where X(J) is a previously computed zero, then the computation is restarted with a guess equal to X(I) + ETA.

Default: ETA = .01.

If the zero X(I) has been computed and ABS(X(I) – X(J)).LT.EPS, where X(J) is a previously computed zero, then the computation is restarted with a guess equal to X(I) + ETA.

Default: ETA = .01.

NROOT — The number of zeros to be found by ZREAL. (Input)

Default: NROOT = 1.

Default: NROOT = 1.

ITMAX — The maximum allowable number of iterations per zero. (Input)

Default: ITMAX = 100.

Default: ITMAX = 100.

XGUESS — A vector of length NROOT. (Input)

XGUESS contains the initial guesses for the zeros.

Default: XGUESS = 0.0.

XGUESS contains the initial guesses for the zeros.

Default: XGUESS = 0.0.

INFO — An integer vector of length NROOT. (Output)

INFO(J) contains the number of iterations used in finding the J-th zero when convergence was achieved. If convergence was not obtained in ITMAX iterations, INFO(J) will be greater than ITMAX.

INFO(J) contains the number of iterations used in finding the J-th zero when convergence was achieved. If convergence was not obtained in ITMAX iterations, INFO(J) will be greater than ITMAX.

FORTRAN 90 Interface

Generic: CALL ZREAL (F, X [, …])

Specific: The specific interface names are S_ZREAL and D_ZREAL.

FORTRAN 77 Interface

Single: CALL ZREAL (F, ERRABS, ERRREL, EPS, ETA, NROOT, ITMAX, XGUESS, X, INFO)

Double: The double precision name is DZREAL.

Description

Routine ZREAL computes n real zeros of a real function f. Given a user-supplied function f(x) and an n-vector of initial guesses x1, x2,…, xn, the routine uses Müller’s method to locate n real zeros of f, that is, n real values of x for which f(x) = 0. The routine has two convergence criteria: the first requires that

be less than ERRABS; the second requires that the relative change of any two successive approximations to an xi be less than ERRREL. Here,

is the m-th approximation to xi. Let ERRABS be ɛ1, and ERRREL be ɛ2.The criteria may be stated mathematically as follows:

Criterion 1:

Criterion 2:

“Convergence” is the satisfaction of either criterion.

Comments

1. Informational error

Type | Code | Description |
---|---|---|

3 | 1 | Failure to converge within ITMAX iterations for at least one of the NROOT roots. |

2. Routine ZREAL always returns the last approximation for zero J in X(J). If the convergence criterion is satisfied, then INFO(J) is less than or equal to ITMAX. If the convergence criterion is not satisfied, then INFO(J) is set to ITMAX + 1.

3. The routine ZREAL assumes that there exist NROOT distinct real zeros for the function F and that they can be reached from the initial guesses supplied. The routine is designed so that convergence to any single zero cannot be obtained from two different initial guesses.

4. Scaling the X vector in the function F may be required, if any of the zeros are known to be less than one.

Example

This example finds the real zeros of the second-degree polynomial

f(x) = x2 + 2x – 6

with the initial guess (4.6, –193.3).

USE ZREAL_INT

USE WRRRN_INT

IMPLICIT NONE

! Declare variables

INTEGER NROOT

REAL EPS, ERRABS, ERRREL

PARAMETER (NROOT=2)

!

INTEGER INFO(NROOT)

REAL F, X(NROOT), XGUESS(NROOT)

EXTERNAL F

! Set values of initial guess

! XGUESS = ( 4.6 -193.3)

!

DATA XGUESS/4.6, -193.3/

!

EPS = 1.0E-5

ERRABS = 1.0E-5

ERRREL = 1.0E-5

! Find the zeros

CALL ZREAL (F, X, errabs=errabs, errrel=errrel, eps=eps, &

nroot=nroot, xguess=xguess)

!

CALL WRRRN ('The zeros are', X, 1, NROOT, 1)

!

END

!

REAL FUNCTION F (X)

REAL X

!

F = X*X + 2.0*X - 6.0

RETURN

END

Output

The zeros are

1 2

1.646 -3.646