*DECK PCHID REAL FUNCTION PCHID (N, X, F, D, INCFD, SKIP, IA, IB, IERR) C***BEGIN PROLOGUE PCHID C***PURPOSE Evaluate the definite integral of a piecewise cubic C Hermite function over an interval whose endpoints are data C points. C***LIBRARY SLATEC (PCHIP) C***CATEGORY E3, H2A1B2 C***TYPE SINGLE PRECISION (PCHID-S, DPCHID-D) C***KEYWORDS CUBIC HERMITE INTERPOLATION, NUMERICAL INTEGRATION, PCHIP, C QUADRATURE C***AUTHOR Fritsch, F. N., (LLNL) 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 PCHID: Piecewise Cubic Hermite Integrator, Data limits C C Evaluates the definite integral of the cubic Hermite function C defined by N, X, F, D over the interval [X(IA), X(IB)]. C C To provide compatibility with PCHIM and PCHIC, includes an C increment between successive values of the F- and D-arrays. C C ---------------------------------------------------------------------- C C Calling sequence: C C PARAMETER (INCFD = ...) C INTEGER N, IA, IB, IERR C REAL X(N), F(INCFD,N), D(INCFD,N) C LOGICAL SKIP C C VALUE = PCHID (N, X, F, D, INCFD, SKIP, IA, IB, IERR) C C Parameters: C C VALUE -- (output) value of the requested integral. C C N -- (input) number of data points. (Error return if N.LT.2 .) C C X -- (input) real 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 -- (input) real array of function values. F(1+(I-1)*INCFD) is C the value corresponding to X(I). C C D -- (input) real array of derivative values. D(1+(I-1)*INCFD) is C the value corresponding to X(I). C C INCFD -- (input) increment between successive values in F and D. C (Error return if INCFD.LT.1 .) C C SKIP -- (input/output) 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 (say, in PCHIM or PCHIC). C SKIP will be set to .TRUE. on return with IERR = 0 or -4. C C IA,IB -- (input) indices in X-array for the limits of integration. C both must be in the range [1,N]. (Error return if not.) C No restrictions on their relative values. C C IERR -- (output) 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 IERR = -4 if IA or IB is out of range. C (VALUE will be zero 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***REFERENCES (NONE) C***ROUTINES CALLED XERMSG C***REVISION HISTORY (YYMMDD) C 820723 DATE WRITTEN C 820804 Converted to SLATEC library version. C 870813 Minor cosmetic changes. C 890411 Added SAVE statements (Vers. 3.2). C 890531 Changed all specific intrinsics to generic. (WRB) C 890703 Corrected category record. (WRB) C 890831 Modified array declarations. (WRB) C 890831 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 930504 Corrected to set VALUE=0 when IERR.ne.0. (FNF) C***END PROLOGUE PCHID C C Programming notes: C 1. This routine uses a special formula that is valid only for C integrals whose limits coincide with data values. This is C mathematically equivalent to, but much more efficient than, C calls to CHFIE. C**End C C DECLARE ARGUMENTS. C INTEGER N, INCFD, IA, IB, IERR REAL X(*), F(INCFD,*), D(INCFD,*) LOGICAL SKIP C C DECLARE LOCAL VARIABLES. C INTEGER I, IUP, LOW REAL H, HALF, SIX, SUM, VALUE, ZERO SAVE ZERO, HALF, SIX C C INITIALIZE. C DATA ZERO /0./, HALF /0.5/, SIX /6./ C***FIRST EXECUTABLE STATEMENT PCHID VALUE = ZERO C C VALIDITY-CHECK ARGUMENTS. C 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 C C FUNCTION DEFINITION IS OK, GO ON. C 5 CONTINUE SKIP = .TRUE. IF ((IA.LT.1) .OR. (IA.GT.N)) GO TO 5004 IF ((IB.LT.1) .OR. (IB.GT.N)) GO TO 5004 IERR = 0 C C COMPUTE INTEGRAL VALUE. C IF (IA .NE. IB) THEN LOW = MIN(IA, IB) IUP = MAX(IA, IB) - 1 SUM = ZERO DO 10 I = LOW, IUP H = X(I+1) - X(I) SUM = SUM + H*( (F(1,I) + F(1,I+1)) + * (D(1,I) - D(1,I+1))*(H/SIX) ) 10 CONTINUE VALUE = HALF * SUM IF (IA .GT. IB) VALUE = -VALUE ENDIF C C NORMAL RETURN. C 5000 CONTINUE PCHID = VALUE RETURN C C ERROR RETURNS. C 5001 CONTINUE C N.LT.2 RETURN. IERR = -1 CALL XERMSG ('SLATEC', 'PCHID', + 'NUMBER OF DATA POINTS LESS THAN TWO', IERR, 1) GO TO 5000 C 5002 CONTINUE C INCFD.LT.1 RETURN. IERR = -2 CALL XERMSG ('SLATEC', 'PCHID', 'INCREMENT LESS THAN ONE', IERR, + 1) GO TO 5000 C 5003 CONTINUE C X-ARRAY NOT STRICTLY INCREASING. IERR = -3 CALL XERMSG ('SLATEC', 'PCHID', + 'X-ARRAY NOT STRICTLY INCREASING', IERR, 1) GO TO 5000 C 5004 CONTINUE C IA OR IB OUT OF RANGE RETURN. IERR = -4 CALL XERMSG ('SLATEC', 'PCHID', 'IA OR IB OUT OF RANGE', IERR, 1) GO TO 5000 C------------- LAST LINE OF PCHID FOLLOWS ------------------------------ END