source: LMDZ6/trunk/libf/misc/pchsp.f90 @ 5397

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