*DECK FZERO SUBROUTINE FZERO (F, B, C, R, RE, AE, IFLAG) C***BEGIN PROLOGUE FZERO C***PURPOSE Search for a zero of a function F(X) in a given interval C (B,C). It is designed primarily for problems where F(B) C and F(C) have opposite signs. C***LIBRARY SLATEC C***CATEGORY F1B C***TYPE SINGLE PRECISION (FZERO-S, DFZERO-D) C***KEYWORDS BISECTION, NONLINEAR EQUATIONS, ROOTS, ZEROS C***AUTHOR Shampine, L. F., (SNLA) C Watts, H. A., (SNLA) C***DESCRIPTION C C FZERO searches for a zero of a REAL function F(X) between the C given REAL values B and C until the width of the interval (B,C) C has collapsed to within a tolerance specified by the stopping C criterion, C ABS(B-C) .LE. 2.*(RW*ABS(B)+AE). C The method used is an efficient combination of bisection and the C secant rule and is due to T. J. Dekker. C C Description Of Arguments C C F :EXT - Name of the REAL external function. This name must C be in an EXTERNAL statement in the calling program. C F must be a function of one REAL argument. C C B :INOUT - One end of the REAL interval (B,C). The value C returned for B usually is the better approximation C to a zero of F. C C C :INOUT - The other end of the REAL interval (B,C) C C R :OUT - A (better) REAL guess of a zero of F which could help C in speeding up convergence. If F(B) and F(R) have C opposite signs, a root will be found in the interval C (B,R); if not, but F(R) and F(C) have opposite signs, C a root will be found in the interval (R,C); C otherwise, the interval (B,C) will be searched for a C possible root. When no better guess is known, it is C recommended that r be set to B or C, since if R is C not interior to the interval (B,C), it will be C ignored. C C RE :IN - 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 C AE :IN - Absolute error used in the stopping criterion. If C the given interval (B,C) contains the origin, then a C nonzero value should be chosen for AE. C C IFLAG :OUT - A status code. User must check IFLAG after each C call. Control returns to the user from FZERO in all C cases. C C 1 B is within the requested tolerance of a zero. C The interval (B,C) collapsed to the requested C tolerance, the function changes sign in (B,C), and C F(X) decreased in magnitude as (B,C) collapsed. C C 2 F(B) = 0. However, the interval (B,C) may not have C collapsed to the requested tolerance. C C 3 B may be near a singular point of F(X). C The interval (B,C) collapsed to the requested tol- C erance and the function changes sign in (B,C), but C F(X) increased in magnitude as (B,C) collapsed, i.e. C ABS(F(B out)) .GT. MAX(ABS(F(B in)),ABS(F(C in))) C C 4 No change in sign of F(X) was found although the C interval (B,C) collapsed to the requested tolerance. C The user must examine this case and decide whether C B is near a local minimum of F(X), or B is near a C zero of even multiplicity, or neither of these. C C 5 Too many (.GT. 500) function evaluations used. C C***REFERENCES L. F. Shampine and H. A. Watts, FZERO, a root-solving C code, Report SC-TM-70-631, Sandia Laboratories, C September 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, Wiley-Interscience, 1969. C***ROUTINES CALLED R1MACH C***REVISION HISTORY (YYMMDD) C 700901 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE FZERO REAL A,ACBS,ACMB,AE,AW,B,C,CMB,ER,FA,FB,FC,FX,FZ,P,Q,R, + RE,RW,T,TOL,Z INTEGER IC,IFLAG,KOUNT C***FIRST EXECUTABLE STATEMENT FZERO C C ER is two times the computer unit roundoff value which is defined C here by the function R1MACH. C ER = 2.0E0 * R1MACH(4) C C Initialize. C Z = R IF (R .LE. MIN(B,C) .OR. R .GE. MAX(B,C)) Z = C RW = MAX(RE,ER) AW = MAX(AE,0.E0) IC = 0 T = Z FZ = F(T) FC = FZ T = B FB = F(T) KOUNT = 2 IF (SIGN(1.0E0,FZ) .EQ. SIGN(1.0E0,FB)) GO TO 1 C = Z GO TO 2 1 IF (Z .EQ. C) GO TO 2 T = C FC = F(T) KOUNT = 3 IF (SIGN(1.0E0,FZ) .EQ. SIGN(1.0E0,FC)) GO TO 2 B = Z FB = FZ 2 A = C FA = FC ACBS = ABS(B-C) FX = MAX(ABS(FB),ABS(FC)) C 3 IF (ABS(FC) .GE. ABS(FB)) GO TO 4 C C Perform interchange. C A = B FA = FB B = C FB = FC C = A FC = FA C 4 CMB = 0.5E0*(C-B) ACMB = ABS(CMB) TOL = RW*ABS(B) + AW C C Test stopping criterion and function count. C IF (ACMB .LE. TOL) GO TO 10 IF (FB .EQ. 0.E0) GO TO 11 IF (KOUNT .GE. 500) GO TO 14 C C Calculate new iterate implicitly as B+P/Q, where we arrange C P .GE. 0. The implicit form is used to prevent overflow. C P = (B-A)*FB Q = FA - FB IF (P .GE. 0.E0) GO TO 5 P = -P Q = -Q C C Update A and check for satisfactory reduction in the size of the C bracketing interval. If not, perform bisection. C 5 A = B FA = FB IC = IC + 1 IF (IC .LT. 4) GO TO 6 IF (8.0E0*ACMB .GE. ACBS) GO TO 8 IC = 0 ACBS = ACMB C C Test for too small a change. C 6 IF (P .GT. ABS(Q)*TOL) GO TO 7 C C Increment by TOLerance. C B = B + SIGN(TOL,CMB) GO TO 9 C C Root ought to be between B and (C+B)/2. C 7 IF (P .GE. CMB*Q) GO TO 8 C C Use secant rule. C B = B + P/Q GO TO 9 C C Use bisection (C+B)/2. C 8 B = B + CMB C C Have completed computation for new iterate B. C 9 T = B FB = F(T) KOUNT = KOUNT + 1 C C Decide whether next step is interpolation or extrapolation. C IF (SIGN(1.0E0,FB) .NE. SIGN(1.0E0,FC)) GO TO 3 C = A FC = FA GO TO 3 C C Finished. Process results for proper setting of IFLAG. C 10 IF (SIGN(1.0E0,FB) .EQ. SIGN(1.0E0,FC)) GO TO 13 IF (ABS(FB) .GT. FX) GO TO 12 IFLAG = 1 RETURN 11 IFLAG = 2 RETURN 12 IFLAG = 3 RETURN 13 IFLAG = 4 RETURN 14 IFLAG = 5 RETURN END