*DECK DPCHCM SUBROUTINE DPCHCM (N, X, F, D, INCFD, SKIP, ISMON, IERR) C***BEGIN PROLOGUE DPCHCM C***PURPOSE Check a cubic Hermite function for monotonicity. C***LIBRARY SLATEC (PCHIP) C***CATEGORY E3 C***TYPE DOUBLE PRECISION (PCHCM-S, DPCHCM-D) C***KEYWORDS CUBIC HERMITE INTERPOLATION, MONOTONE INTERPOLATION, C PCHIP, PIECEWISE CUBIC INTERPOLATION, UTILITY ROUTINE C***AUTHOR Fritsch, F. N., (LLNL) C Computing & Mathematics Research Division C Lawrence Livermore National Laboratory C P.O. Box 808 (L-316) C Livermore, CA 94550 C FTS 532-4275, (510) 422-4275 C***DESCRIPTION C C *Usage: C C PARAMETER (INCFD = ...) C INTEGER N, ISMON(N), IERR C DOUBLE PRECISION X(N), F(INCFD,N), D(INCFD,N) C LOGICAL SKIP C C CALL DPCHCM (N, X, F, D, INCFD, SKIP, ISMON, IERR) C C *Arguments: C C N:IN is the number of data points. (Error return if N.LT.2 .) C C X:IN is a real*8 array of independent variable values. The C elements of X must be strictly increasing: C X(I-1) .LT. X(I), I = 2(1)N. C (Error return if not.) C C F:IN is a real*8 array of function values. F(1+(I-1)*INCFD) is C the value corresponding to X(I). C C D:IN is a real*8 array of derivative values. D(1+(I-1)*INCFD) is C is the value corresponding to X(I). C C INCFD:IN is the increment between successive values in F and D. C (Error return if INCFD.LT.1 .) C C SKIP:INOUT is a logical variable which should be set to C .TRUE. if the user wishes to skip checks for validity of C preceding parameters, or to .FALSE. otherwise. C This will save time in case these checks have already C been performed. C SKIP will be set to .TRUE. on normal return. C C ISMON:OUT is an integer array indicating on which intervals the C PCH function defined by N, X, F, D is monotonic. C For data interval [X(I),X(I+1)], C ISMON(I) = -3 if function is probably decreasing; C ISMON(I) = -1 if function is strictly decreasing; C ISMON(I) = 0 if function is constant; C ISMON(I) = 1 if function is strictly increasing; C ISMON(I) = 2 if function is non-monotonic; C ISMON(I) = 3 if function is probably increasing. C If ABS(ISMON)=3, this means that the D-values are near C the boundary of the monotonicity region. A small C increase produces non-monotonicity; decrease, strict C monotonicity. C The above applies to I=1(1)N-1. ISMON(N) indicates whether C the entire function is monotonic on [X(1),X(N)]. C C IERR:OUT is an error flag. C Normal return: C IERR = 0 (no errors). C "Recoverable" errors: C IERR = -1 if N.LT.2 . C IERR = -2 if INCFD.LT.1 . C IERR = -3 if the X-array is not strictly increasing. C (The ISMON-array has not been changed in any of these cases.) C NOTE: The above errors are checked in the order listed, C and following arguments have **NOT** been validated. C C *Description: C C DPCHCM: Piecewise Cubic Hermite -- Check Monotonicity. C C Checks the piecewise cubic Hermite function defined by N,X,F,D C for monotonicity. C C To provide compatibility with DPCHIM and DPCHIC, includes an C increment between successive values of the F- and D-arrays. C C *Cautions: C This provides the same capability as old DPCHMC, except that a C new output value, -3, was added February 1989. (Formerly, -3 C and +3 were lumped together in the single value 3.) Codes that C flag nonmonotonicity by "IF (ISMON.EQ.2)" need not be changed. C Codes that check via "IF (ISMON.GE.3)" should change the test to C "IF (IABS(ISMON).GE.3)". Codes that declare monotonicity via C "IF (ISMON.LE.1)" should change to "IF (IABS(ISMON).LE.1)". C C***REFERENCES F. N. Fritsch and R. E. Carlson, Monotone piecewise C cubic interpolation, SIAM Journal on Numerical Ana- C lysis 17, 2 (April 1980), pp. 238-246. C***ROUTINES CALLED DCHFCM, XERMSG C***REVISION HISTORY (YYMMDD) C 820518 DATE WRITTEN C 820804 Converted to SLATEC library version. C 831201 Reversed order of subscripts of F and D, so that the C routine will work properly when INCFD.GT.1 . (Bug!) C 870707 Corrected XERROR calls for d.p. name(s). C 890206 Corrected XERROR calls. C 890209 Added possible ISMON value of -3 and modified code so C that 1,3,-1 produces ISMON(N)=2, rather than 3. C 890306 Added caution about changed output. C 890407 Changed name from DPCHMC to DPCHCM, as requested at the C March 1989 SLATEC CML meeting, and made a few other C minor modifications necessitated by this change. C 890407 Converted to new SLATEC format. C 890407 Modified DESCRIPTION to LDOC format. C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 920429 Revised format and order of references. (WRB,FNF) C***END PROLOGUE DPCHCM C C Fortran intrinsics used: ISIGN. C Other routines used: CHFCM, XERMSG. C C ---------------------------------------------------------------------- C C Programming notes: C C An alternate organization would have separate loops for computing C ISMON(i), i=1,...,NSEG, and for the computation of ISMON(N). The C first loop can be readily parallelized, since the NSEG calls to C CHFCM are independent. The second loop can be cut short if C ISMON(N) is ever equal to 2, for it cannot be changed further. C C To produce a single precision version, simply: C a. Change DPCHCM to PCHCM wherever it occurs, C b. Change DCHFCM to CHFCM wherever it occurs, and C c. Change the double precision declarations to real. C C DECLARE ARGUMENTS. C INTEGER N, INCFD, ISMON(N), IERR DOUBLE PRECISION X(N), F(INCFD,N), D(INCFD,N) LOGICAL SKIP C C DECLARE LOCAL VARIABLES. C INTEGER I, NSEG DOUBLE PRECISION DELTA INTEGER DCHFCM C C VALIDITY-CHECK ARGUMENTS. C C***FIRST EXECUTABLE STATEMENT DPCHCM IF (SKIP) GO TO 5 C IF ( N.LT.2 ) GO TO 5001 IF ( INCFD.LT.1 ) GO TO 5002 DO 1 I = 2, N IF ( X(I).LE.X(I-1) ) GO TO 5003 1 CONTINUE SKIP = .TRUE. C C FUNCTION DEFINITION IS OK -- GO ON. C 5 CONTINUE NSEG = N - 1 DO 90 I = 1, NSEG DELTA = (F(1,I+1)-F(1,I))/(X(I+1)-X(I)) C ------------------------------- ISMON(I) = DCHFCM (D(1,I), D(1,I+1), DELTA) C ------------------------------- IF (I .EQ. 1) THEN ISMON(N) = ISMON(1) ELSE C Need to figure out cumulative monotonicity from following C "multiplication table": C C + I S M O N (I) C + -3 -1 0 1 3 2 C +------------------------+ C I -3 I -3 -3 -3 2 2 2 I C S -1 I -3 -1 -1 2 2 2 I C M 0 I -3 -1 0 1 3 2 I C O 1 I 2 2 1 1 3 2 I C N 3 I 2 2 3 3 3 2 I C (N) 2 I 2 2 2 2 2 2 I C +------------------------+ C Note that the 2 row and column are out of order so as not C to obscure the symmetry in the rest of the table. C C No change needed if equal or constant on this interval or C already declared nonmonotonic. IF ( (ISMON(I).NE.ISMON(N)) .AND. (ISMON(I).NE.0) . .AND. (ISMON(N).NE.2) ) THEN IF ( (ISMON(I).EQ.2) .OR. (ISMON(N).EQ.0) ) THEN ISMON(N) = ISMON(I) ELSE IF (ISMON(I)*ISMON(N) .LT. 0) THEN C This interval has opposite sense from curve so far. ISMON(N) = 2 ELSE C At this point, both are nonzero with same sign, and C we have already eliminated case both +-1. ISMON(N) = ISIGN (3, ISMON(N)) ENDIF ENDIF ENDIF 90 CONTINUE C C NORMAL RETURN. C IERR = 0 RETURN C C ERROR RETURNS. C 5001 CONTINUE C N.LT.2 RETURN. IERR = -1 CALL XERMSG ('SLATEC', 'DPCHCM', + 'NUMBER OF DATA POINTS LESS THAN TWO', IERR, 1) RETURN C 5002 CONTINUE C INCFD.LT.1 RETURN. IERR = -2 CALL XERMSG ('SLATEC', 'DPCHCM', 'INCREMENT LESS THAN ONE', IERR, + 1) RETURN C 5003 CONTINUE C X-ARRAY NOT STRICTLY INCREASING. IERR = -3 CALL XERMSG ('SLATEC', 'DPCHCM', + 'X-ARRAY NOT STRICTLY INCREASING', IERR, 1) RETURN C------------- LAST LINE OF DPCHCM FOLLOWS ----------------------------- END