*DECK SDZRO
SUBROUTINE SDZRO (AE, F, H, N, NQ, IROOT, RE, T, YH, UROUND, B, C,
8 FB, FC, Y)
C***BEGIN PROLOGUE SDZRO
C***SUBSIDIARY
C***PURPOSE SDZRO searches for a zero of a function F(N, T, Y, IROOT)
C between the given values B and C until the width of the
C interval (B, C) has collapsed to within a tolerance
C specified by the stopping criterion,
C ABS(B - C) .LE. 2.*(RW*ABS(B) + AE).
C***LIBRARY SLATEC (SDRIVE)
C***TYPE SINGLE PRECISION (SDZRO-S, DDZRO-D, CDZRO-C)
C***AUTHOR Kahaner, D. K., (NIST)
C National Institute of Standards and Technology
C Gaithersburg, MD 20899
C Sutherland, C. D., (LANL)
C Mail Stop D466
C Los Alamos National Laboratory
C Los Alamos, NM 87545
C***DESCRIPTION
C
C This is a special purpose version of ZEROIN, modified for use with
C the SDRIV package.
C
C Sandia Mathematical Program Library
C Mathematical Computing Services Division 5422
C Sandia Laboratories
C P. O. Box 5800
C Albuquerque, New Mexico 87115
C Control Data 6600 Version 4.5, 1 November 1971
C
C PARAMETERS
C F - Name of the external function, which returns a
C real result. This name must be in an
C EXTERNAL statement in the calling program.
C B - One end of the interval (B, C). The value returned for
C B usually is the better approximation to a zero of F.
C C - The other end of the interval (B, C).
C RE - Relative error used for RW in the stopping criterion.
C If the requested RE is less than machine precision,
C then RW is set to approximately machine precision.
C AE - Absolute error used in the stopping criterion. If the
C given interval (B, C) contains the origin, then a
C nonzero value should be chosen for AE.
C
C***REFERENCES L. F. Shampine and H. A. Watts, ZEROIN, a root-solving
C routine, SC-TM-70-631, Sept 1970.
C T. J. Dekker, Finding a zero by means of successive
C linear interpolation, Constructive Aspects of the
C Fundamental Theorem of Algebra, edited by B. Dejon
C and P. Henrici, 1969.
C***ROUTINES CALLED SDNTP
C***REVISION HISTORY (YYMMDD)
C 790601 DATE WRITTEN
C 900329 Initial submission to SLATEC.
C***END PROLOGUE SDZRO
INTEGER IC, IROOT, KOUNT, N, NQ
REAL A, ACBS, ACMB, AE, B, C, CMB, ER, F, FA, FB, FC,
8 H, P, Q, RE, RW, T, TOL, UROUND, Y(*), YH(N,*)
C***FIRST EXECUTABLE STATEMENT SDZRO
ER = 4.E0*UROUND
RW = MAX(RE, ER)
IC = 0
ACBS = ABS(B - C)
A = C
FA = FC
KOUNT = 0
C Perform interchange
10 IF (ABS(FC) .LT. ABS(FB)) THEN
A = B
FA = FB
B = C
FB = FC
C = A
FC = FA
END IF
CMB = 0.5E0*(C - B)
ACMB = ABS(CMB)
TOL = RW*ABS(B) + AE
C Test stopping criterion
IF (ACMB .LE. TOL) RETURN
IF (KOUNT .GT. 50) RETURN
C Calculate new iterate implicitly as
C B + P/Q, where we arrange P .GE. 0.
C The implicit form is used to prevent overflow.
P = (B - A)*FB
Q = FA - FB
IF (P .LT. 0.E0) THEN
P = -P
Q = -Q
END IF
C Update A and check for satisfactory reduction
C in the size of our bounding interval.
A = B
FA = FB
IC = IC + 1
IF (IC .GE. 4) THEN
IF (8.E0*ACMB .GE. ACBS) THEN
C Bisect
B = 0.5E0*(C + B)
GO TO 20
END IF
IC = 0
END IF
ACBS = ACMB
C Test for too small a change
IF (P .LE. ABS(Q)*TOL) THEN
C Increment by tolerance
B = B + SIGN(TOL, CMB)
C Root ought to be between
C B and (C + B)/2.
ELSE IF (P .LT. CMB*Q) THEN
C Interpolate
B = B + P/Q
ELSE
C Bisect
B = 0.5E0*(C + B)
END IF
C Have completed computation
C for new iterate B.
20 CALL SDNTP (H, 0, N, NQ, T, B, YH, Y)
FB = F(N, B, Y, IROOT)
IF (N .EQ. 0) RETURN
IF (FB .EQ. 0.E0) RETURN
KOUNT = KOUNT + 1
C
C Decide whether next step is interpolation or extrapolation
C
IF (SIGN(1.0E0, FB) .EQ. SIGN(1.0E0, FC)) THEN
C = A
FC = FA
END IF
GO TO 10
END