C205 Zero of a Function of One Real Variable

Subroutine subprograms RZERO and DZERO compute, to an attempted specified accuracy, a zero of a real-valued function $f(x)$ lying in a given interval $[a,b]$ , where $f(a)*f(b)\le 0$ .

On CDC and Cray computers, the double-precision version DZERO is not available.

Structure:

SUBROUTINE subprograms
User Entry Names: RZERO, DZERO
Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035), user-supplied FUNCTION subprogram

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    CALL tZERO(A,B,X0,R,EPS,MAXF,F)

A,B

(type according to t) On entry, A and B must specify the end-points of the search interval. Unchanged on exit.

X0

(type according to t) On exit, X0 is the computed approximation to a zero $x$₀ of the function $f(x)$ .

R

(type according to t) On exit, the value of R is such that X0 $-x$₀< R, unless an error condition is detected (see Error Handling).

EPS

(type according to t) On entry, EPS must be equal to the accuracy parameter (see Accuracy). Unchanged on exit.

MAXF

( INTEGER) On entry, MAXF must be equal to the maximum permitted number of references to the function F within the iteration loop. Unchanged on exit.

F

(type according to t) Name of a user-supplied FUNCTION subprogram, declared EXTERNAL in the calling program.

$I=1$ First reference.

$I=2$ Subsequent references.

$I=3$ Final reference, before a normal ($R>0$ ) exit.

Method:

A mixed strategy is used, based on the Muller method of parabolic interpolation supplemented by bisection.

Accuracy:

The routine tries to compute a value X0 such that $|X0-\; x$₀| ≤(1 + X0) * EPS.

If this accuracy is obtained with fewer than MAXF references to the function F within the iteration loop, the subroutine exits with R positive.

Error handling:

Error C205.1: $F(A,1)*F(B,1)\; >\; 0$ . X0 is set equal to zero and R is set equal to $-2|B-A|$ .
Error C205.2: The number of calls to F exceeds MAXF. X0 is set equal to zero and R is set to $-|B-A|/2$ .
A message is written on Unit 6, unless subroutine MTLSET (N002) has been called.
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C207