ZEROIN

      SUBROUTINE ZEROIN(F,B,C,RE,AE,IFLAG)
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     MODIFIED TO RUN AT NBS BY D. KAHANER DIV 713
C
C     ABSTRACT
C        ZEROIN SEARCHES FOR A ZERO OF A FUNCTION F(X) BETWEEN
C        THE GIVEN VALUES B AND C UNTIL THE WIDTH OF THE INTERVAL
C        (B,C) HAS COLLAPSED TO WITHIN A TOLERANCE SPECIFIED BY
C        THE STOPPING CRITERION, ABS(B-C) .LE. 2.*(RW*ABS(B)+AE).
C
C     DESCRIPTION OF ARGUMENTS
C        F     - NAME OF THE REAL VALUED EXTERNAL FUNCTION.  THIS NAME
C                MUST BE IN AN EXTERNAL STATEMENT IN THE CALLING
C                PROGRAM.  F MUST BE A FUNCTION OF ONE REAL ARGUMENT.
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        IFLAG - RETURNS A STATUS OF THE RESULTS INDICATING WHICH
C                OF THE FOLLOWING CONDITIONS HOLD.
C                    A - ABS(B-C) .LE. 2.*(RW*ABS(B)+AE)
C                    B - F(B) * F(C) .LT. 0.
C                    C - ABS(F(B)) .LE. ABS(F(C))
C                    D - ABS(F(B   )) .LE. MAX(ABS(F(B  )),ABS(F(C  )))
C                               OUT                   IN          IN
C                    E - NUMBER OF EVALUATIONS OF F(X) .LE. 500
C               =1 INDICATES NORMAL CASE.  ALL CONDITIONS ABOVE HOLD.
C               =2 INDICATES F(B) = 0.  CONDITION A MAY NOT HOLD.
C               =3 INDICATES CONDITIONS A, B, C, AND E HOLD BUT D DOES
C                  NOT.  (B,C) PROBABLY CONTAINS A SINGULAR POINT OF F.
C               =4 INDICATES CONDITIONS A AND E HOLD BUT B DOES NOT.
C                  A LOCAL MINIMUM OF F(X) IN (B,C) MAY HAVE BEEN FOUND.
C               =5 INDICATES SEARCH WAS ABORTED WHEN CONDITION E FAILED.
C
C     REFERENCES
C       1.  L F SHAMPINE AND H A WATTS, ZEROIN, A ROOT-SOLVING CODE,
C           SC-TM-70-631, SEPT 1970.
C       2.  T J DEKKER, FINDING A ZERO BY MEANS OF SUCCESSIVE LINEAR
C           INTERPOLATION, *CONSTRUCTIVE ASPECTS OF THE FUNDAMENTAL
C           THEOREM OF ALGEBRA*, EDITED BY B DEJON AND P HENRICI, 1969.
C