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*DECK PCHSP |
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SUBROUTINE PCHSP (IC, VC, N, X, F, D, INCFD, WK, NWK, IERR) |
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C***BEGIN PROLOGUE PCHSP |
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C***PURPOSE Set derivatives needed to determine the Hermite represen- |
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C tation of the cubic spline interpolant to given data, with |
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C specified boundary conditions. |
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C***LIBRARY SLATEC (PCHIP) |
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C***CATEGORY E1A |
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C***TYPE SINGLE PRECISION (PCHSP-S, DPCHSP-D) |
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C***KEYWORDS CUBIC HERMITE INTERPOLATION, PCHIP, |
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C PIECEWISE CUBIC INTERPOLATION, SPLINE INTERPOLATION |
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C***AUTHOR Fritsch, F. N., (LLNL) |
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C Lawrence Livermore National Laboratory |
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C P.O. Box 808 (L-316) |
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C Livermore, CA 94550 |
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C FTS 532-4275, (510) 422-4275 |
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C***DESCRIPTION |
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C |
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C PCHSP: Piecewise Cubic Hermite Spline |
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C |
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C Computes the Hermite representation of the cubic spline inter- |
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C polant to the data given in X and F satisfying the boundary |
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C conditions specified by IC and VC. |
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C |
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C To facilitate two-dimensional applications, includes an increment |
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C between successive values of the F- and D-arrays. |
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C |
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C The resulting piecewise cubic Hermite function may be evaluated |
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C by PCHFE or PCHFD. |
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C |
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C NOTE: This is a modified version of C. de Boor's cubic spline |
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C routine CUBSPL. |
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C |
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C ---------------------------------------------------------------------- |
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C |
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C Calling sequence: |
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C |
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C PARAMETER (INCFD = ...) |
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C INTEGER IC(2), N, NWK, IERR |
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C REAL VC(2), X(N), F(INCFD,N), D(INCFD,N), WK(NWK) |
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C |
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C CALL PCHSP (IC, VC, N, X, F, D, INCFD, WK, NWK, IERR) |
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C |
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C Parameters: |
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C |
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C IC -- (input) integer array of length 2 specifying desired |
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C boundary conditions: |
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C IC(1) = IBEG, desired condition at beginning of data. |
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C IC(2) = IEND, desired condition at end of data. |
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C |
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C IBEG = 0 to set D(1) so that the third derivative is con- |
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C tinuous at X(2). This is the "not a knot" condition |
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C provided by de Boor's cubic spline routine CUBSPL. |
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C < This is the default boundary condition. > |
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C IBEG = 1 if first derivative at X(1) is given in VC(1). |
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C IBEG = 2 if second derivative at X(1) is given in VC(1). |
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C IBEG = 3 to use the 3-point difference formula for D(1). |
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C (Reverts to the default b.c. if N.LT.3 .) |
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C IBEG = 4 to use the 4-point difference formula for D(1). |
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C (Reverts to the default b.c. if N.LT.4 .) |
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C NOTES: |
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C 1. An error return is taken if IBEG is out of range. |
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C 2. For the "natural" boundary condition, use IBEG=2 and |
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C VC(1)=0. |
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C |
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C IEND may take on the same values as IBEG, but applied to |
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C derivative at X(N). In case IEND = 1 or 2, the value is |
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C given in VC(2). |
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C |
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C NOTES: |
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C 1. An error return is taken if IEND is out of range. |
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C 2. For the "natural" boundary condition, use IEND=2 and |
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C VC(2)=0. |
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C |
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C VC -- (input) real array of length 2 specifying desired boundary |
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C values, as indicated above. |
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C VC(1) need be set only if IC(1) = 1 or 2 . |
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C VC(2) need be set only if IC(2) = 1 or 2 . |
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C |
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C N -- (input) number of data points. (Error return if N.LT.2 .) |
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C |
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C X -- (input) real array of independent variable values. The |
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C elements of X must be strictly increasing: |
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C X(I-1) .LT. X(I), I = 2(1)N. |
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C (Error return if not.) |
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C |
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C F -- (input) real array of dependent variable values to be inter- |
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C polated. F(1+(I-1)*INCFD) is value corresponding to X(I). |
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C |
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C D -- (output) real array of derivative values at the data points. |
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C These values will determine the cubic spline interpolant |
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C with the requested boundary conditions. |
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C The value corresponding to X(I) is stored in |
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C D(1+(I-1)*INCFD), I=1(1)N. |
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C No other entries in D are changed. |
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C |
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C INCFD -- (input) increment between successive values in F and D. |
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C This argument is provided primarily for 2-D applications. |
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C (Error return if INCFD.LT.1 .) |
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C |
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C WK -- (scratch) real array of working storage. |
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C |
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C NWK -- (input) length of work array. |
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C (Error return if NWK.LT.2*N .) |
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C |
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C IERR -- (output) error flag. |
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C Normal return: |
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C IERR = 0 (no errors). |
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C "Recoverable" errors: |
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C IERR = -1 if N.LT.2 . |
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C IERR = -2 if INCFD.LT.1 . |
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C IERR = -3 if the X-array is not strictly increasing. |
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C IERR = -4 if IBEG.LT.0 or IBEG.GT.4 . |
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C IERR = -5 if IEND.LT.0 of IEND.GT.4 . |
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C IERR = -6 if both of the above are true. |
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C IERR = -7 if NWK is too small. |
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C NOTE: The above errors are checked in the order listed, |
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C and following arguments have **NOT** been validated. |
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C (The D-array has not been changed in any of these cases.) |
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C IERR = -8 in case of trouble solving the linear system |
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C for the interior derivative values. |
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C (The D-array may have been changed in this case.) |
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C ( Do **NOT** use it! ) |
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C |
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C***REFERENCES Carl de Boor, A Practical Guide to Splines, Springer- |
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C Verlag, New York, 1978, pp. 53-59. |
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C***ROUTINES CALLED PCHDF, XERMSG |
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C***REVISION HISTORY (YYMMDD) |
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C 820503 DATE WRITTEN |
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C 820804 Converted to SLATEC library version. |
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C 870707 Minor cosmetic changes to prologue. |
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C 890411 Added SAVE statements (Vers. 3.2). |
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C 890703 Corrected category record. (WRB) |
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C 890831 Modified array declarations. (WRB) |
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C 890831 REVISION DATE from Version 3.2 |
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C 891214 Prologue converted to Version 4.0 format. (BAB) |
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C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) |
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C 920429 Revised format and order of references. (WRB,FNF) |
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C***END PROLOGUE PCHSP |
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C Programming notes: |
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C |
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C To produce a double precision version, simply: |
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C a. Change PCHSP to DPCHSP wherever it occurs, |
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C b. Change the real declarations to double precision, and |
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C c. Change the constants ZERO, HALF, ... to double precision. |
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C |
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C DECLARE ARGUMENTS. |
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C |
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INTEGER IC(2), N, INCFD, NWK, IERR |
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REAL VC(2), X(*), F(INCFD,*), D(INCFD,*), WK(2,*) |
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C |
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C DECLARE LOCAL VARIABLES. |
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C |
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INTEGER IBEG, IEND, INDEX, J, NM1 |
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REAL G, HALF, ONE, STEMP(3), THREE, TWO, XTEMP(4), ZERO |
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SAVE ZERO, HALF, ONE, TWO, THREE |
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REAL PCHDF |
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C |
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DATA ZERO /0./, HALF /0.5/, ONE /1./, TWO /2./, THREE /3./ |
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C |
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C VALIDITY-CHECK ARGUMENTS. |
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C |
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C***FIRST EXECUTABLE STATEMENT PCHSP |
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IF ( N.LT.2 ) GO TO 5001 |
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IF ( INCFD.LT.1 ) GO TO 5002 |
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DO 1 J = 2, N |
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IF ( X(J).LE.X(J-1) ) GO TO 5003 |
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1 CONTINUE |
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C |
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IBEG = IC(1) |
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IEND = IC(2) |
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IERR = 0 |
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IF ( (IBEG.LT.0).OR.(IBEG.GT.4) ) IERR = IERR - 1 |
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IF ( (IEND.LT.0).OR.(IEND.GT.4) ) IERR = IERR - 2 |
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IF ( IERR.LT.0 ) GO TO 5004 |
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C |
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C FUNCTION DEFINITION IS OK -- GO ON. |
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C |
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IF ( NWK .LT. 2*N ) GO TO 5007 |
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C |
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C COMPUTE FIRST DIFFERENCES OF X SEQUENCE AND STORE IN WK(1,.). ALSO, |
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C COMPUTE FIRST DIVIDED DIFFERENCE OF DATA AND STORE IN WK(2,.). |
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DO 5 J=2,N |
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WK(1,J) = X(J) - X(J-1) |
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WK(2,J) = (F(1,J) - F(1,J-1))/WK(1,J) |
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5 CONTINUE |
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C |
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C SET TO DEFAULT BOUNDARY CONDITIONS IF N IS TOO SMALL. |
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C |
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IF ( IBEG.GT.N ) IBEG = 0 |
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IF ( IEND.GT.N ) IEND = 0 |
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C |
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C SET UP FOR BOUNDARY CONDITIONS. |
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C |
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IF ( (IBEG.EQ.1).OR.(IBEG.EQ.2) ) THEN |
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D(1,1) = VC(1) |
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ELSE IF (IBEG .GT. 2) THEN |
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C PICK UP FIRST IBEG POINTS, IN REVERSE ORDER. |
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DO 10 J = 1, IBEG |
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INDEX = IBEG-J+1 |
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C INDEX RUNS FROM IBEG DOWN TO 1. |
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XTEMP(J) = X(INDEX) |
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IF (J .LT. IBEG) STEMP(J) = WK(2,INDEX) |
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10 CONTINUE |
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C -------------------------------- |
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D(1,1) = PCHDF (IBEG, XTEMP, STEMP, IERR) |
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C -------------------------------- |
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IF (IERR .NE. 0) GO TO 5009 |
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IBEG = 1 |
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ENDIF |
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C |
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IF ( (IEND.EQ.1).OR.(IEND.EQ.2) ) THEN |
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D(1,N) = VC(2) |
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ELSE IF (IEND .GT. 2) THEN |
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C PICK UP LAST IEND POINTS. |
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DO 15 J = 1, IEND |
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INDEX = N-IEND+J |
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C INDEX RUNS FROM N+1-IEND UP TO N. |
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XTEMP(J) = X(INDEX) |
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IF (J .LT. IEND) STEMP(J) = WK(2,INDEX+1) |
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15 CONTINUE |
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C -------------------------------- |
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D(1,N) = PCHDF (IEND, XTEMP, STEMP, IERR) |
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C -------------------------------- |
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IF (IERR .NE. 0) GO TO 5009 |
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IEND = 1 |
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ENDIF |
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C |
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C --------------------( BEGIN CODING FROM CUBSPL )-------------------- |
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C |
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C **** A TRIDIAGONAL LINEAR SYSTEM FOR THE UNKNOWN SLOPES S(J) OF |
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C F AT X(J), J=1,...,N, IS GENERATED AND THEN SOLVED BY GAUSS ELIM- |
| 233 |
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C INATION, WITH S(J) ENDING UP IN D(1,J), ALL J. |
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C WK(1,.) AND WK(2,.) ARE USED FOR TEMPORARY STORAGE. |
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C |
| 236 |
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C CONSTRUCT FIRST EQUATION FROM FIRST BOUNDARY CONDITION, OF THE FORM |
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C WK(2,1)*S(1) + WK(1,1)*S(2) = D(1,1) |
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C |
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IF (IBEG .EQ. 0) THEN |
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IF (N .EQ. 2) THEN |
| 241 |
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C NO CONDITION AT LEFT END AND N = 2. |
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WK(2,1) = ONE |
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WK(1,1) = ONE |
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D(1,1) = TWO*WK(2,2) |
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ELSE |
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C NOT-A-KNOT CONDITION AT LEFT END AND N .GT. 2. |
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WK(2,1) = WK(1,3) |
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WK(1,1) = WK(1,2) + WK(1,3) |
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D(1,1) =((WK(1,2) + TWO*WK(1,1))*WK(2,2)*WK(1,3) |
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* + WK(1,2)**2*WK(2,3)) / WK(1,1) |
| 251 |
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ENDIF |
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ELSE IF (IBEG .EQ. 1) THEN |
| 253 |
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C SLOPE PRESCRIBED AT LEFT END. |
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WK(2,1) = ONE |
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WK(1,1) = ZERO |
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ELSE |
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C SECOND DERIVATIVE PRESCRIBED AT LEFT END. |
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WK(2,1) = TWO |
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WK(1,1) = ONE |
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D(1,1) = THREE*WK(2,2) - HALF*WK(1,2)*D(1,1) |
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ENDIF |
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C |
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C IF THERE ARE INTERIOR KNOTS, GENERATE THE CORRESPONDING EQUATIONS AND |
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C CARRY OUT THE FORWARD PASS OF GAUSS ELIMINATION, AFTER WHICH THE J-TH |
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C EQUATION READS WK(2,J)*S(J) + WK(1,J)*S(J+1) = D(1,J). |
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C |
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NM1 = N-1 |
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IF (NM1 .GT. 1) THEN |
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DO 20 J=2,NM1 |
| 270 |
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IF (WK(2,J-1) .EQ. ZERO) GO TO 5008 |
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G = -WK(1,J+1)/WK(2,J-1) |
| 272 |
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D(1,J) = G*D(1,J-1) |
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* + THREE*(WK(1,J)*WK(2,J+1) + WK(1,J+1)*WK(2,J)) |
| 274 |
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WK(2,J) = G*WK(1,J-1) + TWO*(WK(1,J) + WK(1,J+1)) |
| 275 |
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20 CONTINUE |
| 276 |
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ENDIF |
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C |
| 278 |
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C CONSTRUCT LAST EQUATION FROM SECOND BOUNDARY CONDITION, OF THE FORM |
| 279 |
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C (-G*WK(2,N-1))*S(N-1) + WK(2,N)*S(N) = D(1,N) |
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C |
| 281 |
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C IF SLOPE IS PRESCRIBED AT RIGHT END, ONE CAN GO DIRECTLY TO BACK- |
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C SUBSTITUTION, SINCE ARRAYS HAPPEN TO BE SET UP JUST RIGHT FOR IT |
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C AT THIS POINT. |
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IF (IEND .EQ. 1) GO TO 30 |
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C |
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IF (IEND .EQ. 0) THEN |
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IF (N.EQ.2 .AND. IBEG.EQ.0) THEN |
| 288 |
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C NOT-A-KNOT AT RIGHT ENDPOINT AND AT LEFT ENDPOINT AND N = 2. |
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D(1,2) = WK(2,2) |
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GO TO 30 |
| 291 |
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ELSE IF ((N.EQ.2) .OR. (N.EQ.3 .AND. IBEG.EQ.0)) THEN |
| 292 |
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C EITHER (N=3 AND NOT-A-KNOT ALSO AT LEFT) OR (N=2 AND *NOT* |
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C NOT-A-KNOT AT LEFT END POINT). |
| 294 |
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D(1,N) = TWO*WK(2,N) |
| 295 |
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WK(2,N) = ONE |
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IF (WK(2,N-1) .EQ. ZERO) GO TO 5008 |
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G = -ONE/WK(2,N-1) |
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ELSE |
| 299 |
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C NOT-A-KNOT AND N .GE. 3, AND EITHER N.GT.3 OR ALSO NOT-A- |
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C KNOT AT LEFT END POINT. |
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✗ |
G = WK(1,N-1) + WK(1,N) |
| 302 |
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C DO NOT NEED TO CHECK FOLLOWING DENOMINATORS (X-DIFFERENCES). |
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D(1,N) = ((WK(1,N)+TWO*G)*WK(2,N)*WK(1,N-1) |
| 304 |
|
✗ |
* + WK(1,N)**2*(F(1,N-1)-F(1,N-2))/WK(1,N-1))/G |
| 305 |
|
✗ |
IF (WK(2,N-1) .EQ. ZERO) GO TO 5008 |
| 306 |
|
✗ |
G = -G/WK(2,N-1) |
| 307 |
|
✗ |
WK(2,N) = WK(1,N-1) |
| 308 |
|
|
ENDIF |
| 309 |
|
|
ELSE |
| 310 |
|
|
C SECOND DERIVATIVE PRESCRIBED AT RIGHT ENDPOINT. |
| 311 |
|
✗ |
D(1,N) = THREE*WK(2,N) + HALF*WK(1,N)*D(1,N) |
| 312 |
|
✗ |
WK(2,N) = TWO |
| 313 |
|
✗ |
IF (WK(2,N-1) .EQ. ZERO) GO TO 5008 |
| 314 |
|
✗ |
G = -ONE/WK(2,N-1) |
| 315 |
|
|
ENDIF |
| 316 |
|
|
C |
| 317 |
|
|
C COMPLETE FORWARD PASS OF GAUSS ELIMINATION. |
| 318 |
|
|
C |
| 319 |
|
✗ |
WK(2,N) = G*WK(1,N-1) + WK(2,N) |
| 320 |
|
✗ |
IF (WK(2,N) .EQ. ZERO) GO TO 5008 |
| 321 |
|
✗ |
D(1,N) = (G*D(1,N-1) + D(1,N))/WK(2,N) |
| 322 |
|
|
C |
| 323 |
|
|
C CARRY OUT BACK SUBSTITUTION |
| 324 |
|
|
C |
| 325 |
|
|
30 CONTINUE |
| 326 |
|
✗ |
DO 40 J=NM1,1,-1 |
| 327 |
|
✗ |
IF (WK(2,J) .EQ. ZERO) GO TO 5008 |
| 328 |
|
✗ |
D(1,J) = (D(1,J) - WK(1,J)*D(1,J+1))/WK(2,J) |
| 329 |
|
✗ |
40 CONTINUE |
| 330 |
|
|
C --------------------( END CODING FROM CUBSPL )-------------------- |
| 331 |
|
|
C |
| 332 |
|
|
C NORMAL RETURN. |
| 333 |
|
|
C |
| 334 |
|
✗ |
RETURN |
| 335 |
|
|
C |
| 336 |
|
|
C ERROR RETURNS. |
| 337 |
|
|
C |
| 338 |
|
|
5001 CONTINUE |
| 339 |
|
|
C N.LT.2 RETURN. |
| 340 |
|
✗ |
IERR = -1 |
| 341 |
|
|
CALL XERMSG ('SLATEC', 'PCHSP', |
| 342 |
|
✗ |
+ 'NUMBER OF DATA POINTS LESS THAN TWO', IERR, 1) |
| 343 |
|
✗ |
RETURN |
| 344 |
|
|
C |
| 345 |
|
|
5002 CONTINUE |
| 346 |
|
|
C INCFD.LT.1 RETURN. |
| 347 |
|
✗ |
IERR = -2 |
| 348 |
|
|
CALL XERMSG ('SLATEC', 'PCHSP', 'INCREMENT LESS THAN ONE', IERR, |
| 349 |
|
✗ |
+ 1) |
| 350 |
|
✗ |
RETURN |
| 351 |
|
|
C |
| 352 |
|
|
5003 CONTINUE |
| 353 |
|
|
C X-ARRAY NOT STRICTLY INCREASING. |
| 354 |
|
✗ |
IERR = -3 |
| 355 |
|
|
CALL XERMSG ('SLATEC', 'PCHSP', 'X-ARRAY NOT STRICTLY INCREASING' |
| 356 |
|
✗ |
+ , IERR, 1) |
| 357 |
|
✗ |
RETURN |
| 358 |
|
|
C |
| 359 |
|
|
5004 CONTINUE |
| 360 |
|
|
C IC OUT OF RANGE RETURN. |
| 361 |
|
✗ |
IERR = IERR - 3 |
| 362 |
|
✗ |
CALL XERMSG ('SLATEC', 'PCHSP', 'IC OUT OF RANGE', IERR, 1) |
| 363 |
|
✗ |
RETURN |
| 364 |
|
|
C |
| 365 |
|
|
5007 CONTINUE |
| 366 |
|
|
C NWK TOO SMALL RETURN. |
| 367 |
|
✗ |
IERR = -7 |
| 368 |
|
✗ |
CALL XERMSG ('SLATEC', 'PCHSP', 'WORK ARRAY TOO SMALL', IERR, 1) |
| 369 |
|
✗ |
RETURN |
| 370 |
|
|
C |
| 371 |
|
|
5008 CONTINUE |
| 372 |
|
|
C SINGULAR SYSTEM. |
| 373 |
|
|
C *** THEORETICALLY, THIS CAN ONLY OCCUR IF SUCCESSIVE X-VALUES *** |
| 374 |
|
|
C *** ARE EQUAL, WHICH SHOULD ALREADY HAVE BEEN CAUGHT (IERR=-3). *** |
| 375 |
|
✗ |
IERR = -8 |
| 376 |
|
|
CALL XERMSG ('SLATEC', 'PCHSP', 'SINGULAR LINEAR SYSTEM', IERR, |
| 377 |
|
✗ |
+ 1) |
| 378 |
|
✗ |
RETURN |
| 379 |
|
|
C |
| 380 |
|
|
5009 CONTINUE |
| 381 |
|
|
C ERROR RETURN FROM PCHDF. |
| 382 |
|
|
C *** THIS CASE SHOULD NEVER OCCUR *** |
| 383 |
|
✗ |
IERR = -9 |
| 384 |
|
|
CALL XERMSG ('SLATEC', 'PCHSP', 'ERROR RETURN FROM PCHDF', IERR, |
| 385 |
|
✗ |
+ 1) |
| 386 |
|
✗ |
RETURN |
| 387 |
|
|
C------------- LAST LINE OF PCHSP FOLLOWS ------------------------------ |
| 388 |
|
|
END |
| 389 |
|
|
|