*DECK PCHSP SUBROUTINE PCHSP (IC, VC, N, X, F, D, INCFD, WK, NWK, IERR) C***BEGIN PROLOGUE PCHSP C***PURPOSE Set derivatives needed to determine the Hermite represen- C tation of the cubic spline interpolant to given data, with C specified boundary conditions. C***LIBRARY SLATEC (PCHIP) C***CATEGORY E1A C***TYPE SINGLE PRECISION (PCHSP-S, DPCHSP-D) C***KEYWORDS CUBIC HERMITE INTERPOLATION, PCHIP, C PIECEWISE CUBIC INTERPOLATION, SPLINE INTERPOLATION 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 PCHSP: Piecewise Cubic Hermite Spline C C Computes the Hermite representation of the cubic spline inter- C polant to the data given in X and F satisfying the boundary C conditions specified by IC and VC. C C To facilitate two-dimensional applications, includes an increment C between successive values of the F- and D-arrays. C C The resulting piecewise cubic Hermite function may be evaluated C by PCHFE or PCHFD. C C NOTE: This is a modified version of C. de Boor's cubic spline C routine CUBSPL. C C ---------------------------------------------------------------------- C C Calling sequence: C C PARAMETER (INCFD = ...) C INTEGER IC(2), N, NWK, IERR C REAL VC(2), X(N), F(INCFD,N), D(INCFD,N), WK(NWK) C C CALL PCHSP (IC, VC, N, X, F, D, INCFD, WK, NWK, IERR) C C Parameters: C C IC -- (input) integer array of length 2 specifying desired C boundary conditions: C IC(1) = IBEG, desired condition at beginning of data. C IC(2) = IEND, desired condition at end of data. C C IBEG = 0 to set D(1) so that the third derivative is con- C tinuous at X(2). This is the "not a knot" condition C provided by de Boor's cubic spline routine CUBSPL. C < This is the default boundary condition. > C IBEG = 1 if first derivative at X(1) is given in VC(1). C IBEG = 2 if second derivative at X(1) is given in VC(1). C IBEG = 3 to use the 3-point difference formula for D(1). C (Reverts to the default b.c. if N.LT.3 .) C IBEG = 4 to use the 4-point difference formula for D(1). C (Reverts to the default b.c. if N.LT.4 .) C NOTES: C 1. An error return is taken if IBEG is out of range. C 2. For the "natural" boundary condition, use IBEG=2 and C VC(1)=0. C C IEND may take on the same values as IBEG, but applied to C derivative at X(N). In case IEND = 1 or 2, the value is C given in VC(2). C C NOTES: C 1. An error return is taken if IEND is out of range. C 2. For the "natural" boundary condition, use IEND=2 and C VC(2)=0. C C VC -- (input) real array of length 2 specifying desired boundary C values, as indicated above. C VC(1) need be set only if IC(1) = 1 or 2 . C VC(2) need be set only if IC(2) = 1 or 2 . 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 dependent variable values to be inter- C polated. F(1+(I-1)*INCFD) is value corresponding to X(I). C C D -- (output) real array of derivative values at the data points. C These values will determine the cubic spline interpolant C with the requested boundary conditions. C The value corresponding to X(I) is stored in C D(1+(I-1)*INCFD), I=1(1)N. C No other entries in D are changed. C C INCFD -- (input) increment between successive values in F and D. C This argument is provided primarily for 2-D applications. C (Error return if INCFD.LT.1 .) C C WK -- (scratch) real array of working storage. C C NWK -- (input) length of work array. C (Error return if NWK.LT.2*N .) 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 IBEG.LT.0 or IBEG.GT.4 . C IERR = -5 if IEND.LT.0 of IEND.GT.4 . C IERR = -6 if both of the above are true. C IERR = -7 if NWK is too small. C NOTE: The above errors are checked in the order listed, C and following arguments have **NOT** been validated. C (The D-array has not been changed in any of these cases.) C IERR = -8 in case of trouble solving the linear system C for the interior derivative values. C (The D-array may have been changed in this case.) C ( Do **NOT** use it! ) C C***REFERENCES Carl de Boor, A Practical Guide to Splines, Springer- C Verlag, New York, 1978, pp. 53-59. C***ROUTINES CALLED PCHDF, XERMSG C***REVISION HISTORY (YYMMDD) C 820503 DATE WRITTEN C 820804 Converted to SLATEC library version. C 870707 Minor cosmetic changes to prologue. C 890411 Added SAVE statements (Vers. 3.2). 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 920429 Revised format and order of references. (WRB,FNF) C***END PROLOGUE PCHSP C Programming notes: C C To produce a double precision version, simply: C a. Change PCHSP to DPCHSP wherever it occurs, C b. Change the real declarations to double precision, and C c. Change the constants ZERO, HALF, ... to double precision. C C DECLARE ARGUMENTS. C INTEGER IC(2), N, INCFD, NWK, IERR REAL VC(2), X(*), F(INCFD,*), D(INCFD,*), WK(2,*) C C DECLARE LOCAL VARIABLES. C INTEGER IBEG, IEND, INDEX, J, NM1 REAL G, HALF, ONE, STEMP(3), THREE, TWO, XTEMP(4), ZERO SAVE ZERO, HALF, ONE, TWO, THREE REAL PCHDF C DATA ZERO /0./, HALF /0.5/, ONE /1./, TWO /2./, THREE /3./ C C VALIDITY-CHECK ARGUMENTS. C C***FIRST EXECUTABLE STATEMENT PCHSP IF ( N.LT.2 ) GO TO 5001 IF ( INCFD.LT.1 ) GO TO 5002 DO 1 J = 2, N IF ( X(J).LE.X(J-1) ) GO TO 5003 1 CONTINUE C IBEG = IC(1) IEND = IC(2) IERR = 0 IF ( (IBEG.LT.0).OR.(IBEG.GT.4) ) IERR = IERR - 1 IF ( (IEND.LT.0).OR.(IEND.GT.4) ) IERR = IERR - 2 IF ( IERR.LT.0 ) GO TO 5004 C C FUNCTION DEFINITION IS OK -- GO ON. C IF ( NWK .LT. 2*N ) GO TO 5007 C C COMPUTE FIRST DIFFERENCES OF X SEQUENCE AND STORE IN WK(1,.). ALSO, C COMPUTE FIRST DIVIDED DIFFERENCE OF DATA AND STORE IN WK(2,.). DO 5 J=2,N WK(1,J) = X(J) - X(J-1) WK(2,J) = (F(1,J) - F(1,J-1))/WK(1,J) 5 CONTINUE C C SET TO DEFAULT BOUNDARY CONDITIONS IF N IS TOO SMALL. C IF ( IBEG.GT.N ) IBEG = 0 IF ( IEND.GT.N ) IEND = 0 C C SET UP FOR BOUNDARY CONDITIONS. C IF ( (IBEG.EQ.1).OR.(IBEG.EQ.2) ) THEN D(1,1) = VC(1) ELSE IF (IBEG .GT. 2) THEN C PICK UP FIRST IBEG POINTS, IN REVERSE ORDER. DO 10 J = 1, IBEG INDEX = IBEG-J+1 C INDEX RUNS FROM IBEG DOWN TO 1. XTEMP(J) = X(INDEX) IF (J .LT. IBEG) STEMP(J) = WK(2,INDEX) 10 CONTINUE C -------------------------------- D(1,1) = PCHDF (IBEG, XTEMP, STEMP, IERR) C -------------------------------- IF (IERR .NE. 0) GO TO 5009 IBEG = 1 ENDIF C IF ( (IEND.EQ.1).OR.(IEND.EQ.2) ) THEN D(1,N) = VC(2) ELSE IF (IEND .GT. 2) THEN C PICK UP LAST IEND POINTS. DO 15 J = 1, IEND INDEX = N-IEND+J C INDEX RUNS FROM N+1-IEND UP TO N. XTEMP(J) = X(INDEX) IF (J .LT. IEND) STEMP(J) = WK(2,INDEX+1) 15 CONTINUE C -------------------------------- D(1,N) = PCHDF (IEND, XTEMP, STEMP, IERR) C -------------------------------- IF (IERR .NE. 0) GO TO 5009 IEND = 1 ENDIF C C --------------------( BEGIN CODING FROM CUBSPL )-------------------- C C **** A TRIDIAGONAL LINEAR SYSTEM FOR THE UNKNOWN SLOPES S(J) OF C F AT X(J), J=1,...,N, IS GENERATED AND THEN SOLVED BY GAUSS ELIM- C INATION, WITH S(J) ENDING UP IN D(1,J), ALL J. C WK(1,.) AND WK(2,.) ARE USED FOR TEMPORARY STORAGE. C C CONSTRUCT FIRST EQUATION FROM FIRST BOUNDARY CONDITION, OF THE FORM C WK(2,1)*S(1) + WK(1,1)*S(2) = D(1,1) C IF (IBEG .EQ. 0) THEN IF (N .EQ. 2) THEN C NO CONDITION AT LEFT END AND N = 2. WK(2,1) = ONE WK(1,1) = ONE D(1,1) = TWO*WK(2,2) ELSE C NOT-A-KNOT CONDITION AT LEFT END AND N .GT. 2. WK(2,1) = WK(1,3) WK(1,1) = WK(1,2) + WK(1,3) D(1,1) =((WK(1,2) + TWO*WK(1,1))*WK(2,2)*WK(1,3) * + WK(1,2)**2*WK(2,3)) / WK(1,1) ENDIF ELSE IF (IBEG .EQ. 1) THEN C SLOPE PRESCRIBED AT LEFT END. WK(2,1) = ONE WK(1,1) = ZERO ELSE C SECOND DERIVATIVE PRESCRIBED AT LEFT END. WK(2,1) = TWO WK(1,1) = ONE D(1,1) = THREE*WK(2,2) - HALF*WK(1,2)*D(1,1) ENDIF C C IF THERE ARE INTERIOR KNOTS, GENERATE THE CORRESPONDING EQUATIONS AND C CARRY OUT THE FORWARD PASS OF GAUSS ELIMINATION, AFTER WHICH THE J-TH C EQUATION READS WK(2,J)*S(J) + WK(1,J)*S(J+1) = D(1,J). C NM1 = N-1 IF (NM1 .GT. 1) THEN DO 20 J=2,NM1 IF (WK(2,J-1) .EQ. ZERO) GO TO 5008 G = -WK(1,J+1)/WK(2,J-1) D(1,J) = G*D(1,J-1) * + THREE*(WK(1,J)*WK(2,J+1) + WK(1,J+1)*WK(2,J)) WK(2,J) = G*WK(1,J-1) + TWO*(WK(1,J) + WK(1,J+1)) 20 CONTINUE ENDIF C C CONSTRUCT LAST EQUATION FROM SECOND BOUNDARY CONDITION, OF THE FORM C (-G*WK(2,N-1))*S(N-1) + WK(2,N)*S(N) = D(1,N) C C IF SLOPE IS PRESCRIBED AT RIGHT END, ONE CAN GO DIRECTLY TO BACK- C SUBSTITUTION, SINCE ARRAYS HAPPEN TO BE SET UP JUST RIGHT FOR IT C AT THIS POINT. IF (IEND .EQ. 1) GO TO 30 C IF (IEND .EQ. 0) THEN IF (N.EQ.2 .AND. IBEG.EQ.0) THEN C NOT-A-KNOT AT RIGHT ENDPOINT AND AT LEFT ENDPOINT AND N = 2. D(1,2) = WK(2,2) GO TO 30 ELSE IF ((N.EQ.2) .OR. (N.EQ.3 .AND. IBEG.EQ.0)) THEN C EITHER (N=3 AND NOT-A-KNOT ALSO AT LEFT) OR (N=2 AND *NOT* C NOT-A-KNOT AT LEFT END POINT). D(1,N) = TWO*WK(2,N) WK(2,N) = ONE IF (WK(2,N-1) .EQ. ZERO) GO TO 5008 G = -ONE/WK(2,N-1) ELSE C NOT-A-KNOT AND N .GE. 3, AND EITHER N.GT.3 OR ALSO NOT-A- C KNOT AT LEFT END POINT. G = WK(1,N-1) + WK(1,N) C DO NOT NEED TO CHECK FOLLOWING DENOMINATORS (X-DIFFERENCES). D(1,N) = ((WK(1,N)+TWO*G)*WK(2,N)*WK(1,N-1) * + WK(1,N)**2*(F(1,N-1)-F(1,N-2))/WK(1,N-1))/G IF (WK(2,N-1) .EQ. ZERO) GO TO 5008 G = -G/WK(2,N-1) WK(2,N) = WK(1,N-1) ENDIF ELSE C SECOND DERIVATIVE PRESCRIBED AT RIGHT ENDPOINT. D(1,N) = THREE*WK(2,N) + HALF*WK(1,N)*D(1,N) WK(2,N) = TWO IF (WK(2,N-1) .EQ. ZERO) GO TO 5008 G = -ONE/WK(2,N-1) ENDIF C C COMPLETE FORWARD PASS OF GAUSS ELIMINATION. C WK(2,N) = G*WK(1,N-1) + WK(2,N) IF (WK(2,N) .EQ. ZERO) GO TO 5008 D(1,N) = (G*D(1,N-1) + D(1,N))/WK(2,N) C C CARRY OUT BACK SUBSTITUTION C 30 CONTINUE DO 40 J=NM1,1,-1 IF (WK(2,J) .EQ. ZERO) GO TO 5008 D(1,J) = (D(1,J) - WK(1,J)*D(1,J+1))/WK(2,J) 40 CONTINUE C --------------------( END CODING FROM CUBSPL )-------------------- C C NORMAL RETURN. C RETURN C C ERROR RETURNS. C 5001 CONTINUE C N.LT.2 RETURN. IERR = -1 CALL XERMSG ('SLATEC', 'PCHSP', + 'NUMBER OF DATA POINTS LESS THAN TWO', IERR, 1) RETURN C 5002 CONTINUE C INCFD.LT.1 RETURN. IERR = -2 CALL XERMSG ('SLATEC', 'PCHSP', 'INCREMENT LESS THAN ONE', IERR, + 1) RETURN C 5003 CONTINUE C X-ARRAY NOT STRICTLY INCREASING. IERR = -3 CALL XERMSG ('SLATEC', 'PCHSP', 'X-ARRAY NOT STRICTLY INCREASING' + , IERR, 1) RETURN C 5004 CONTINUE C IC OUT OF RANGE RETURN. IERR = IERR - 3 CALL XERMSG ('SLATEC', 'PCHSP', 'IC OUT OF RANGE', IERR, 1) RETURN C 5007 CONTINUE C NWK TOO SMALL RETURN. IERR = -7 CALL XERMSG ('SLATEC', 'PCHSP', 'WORK ARRAY TOO SMALL', IERR, 1) RETURN C 5008 CONTINUE C SINGULAR SYSTEM. C *** THEORETICALLY, THIS CAN ONLY OCCUR IF SUCCESSIVE X-VALUES *** C *** ARE EQUAL, WHICH SHOULD ALREADY HAVE BEEN CAUGHT (IERR=-3). *** IERR = -8 CALL XERMSG ('SLATEC', 'PCHSP', 'SINGULAR LINEAR SYSTEM', IERR, + 1) RETURN C 5009 CONTINUE C ERROR RETURN FROM PCHDF. C *** THIS CASE SHOULD NEVER OCCUR *** IERR = -9 CALL XERMSG ('SLATEC', 'PCHSP', 'ERROR RETURN FROM PCHDF', IERR, + 1) RETURN C------------- LAST LINE OF PCHSP FOLLOWS ------------------------------ END