*DECK BINT4
SUBROUTINE BINT4 (X, Y, NDATA, IBCL, IBCR, FBCL, FBCR, KNTOPT, T,
+ BCOEF, N, K, W)
C***BEGIN PROLOGUE BINT4
C***PURPOSE Compute the B-representation of a cubic spline
C which interpolates given data.
C***LIBRARY SLATEC
C***CATEGORY E1A
C***TYPE SINGLE PRECISION (BINT4-S, DBINT4-D)
C***KEYWORDS B-SPLINE, CUBIC SPLINES, DATA FITTING, INTERPOLATION
C***AUTHOR Amos, D. E., (SNLA)
C***DESCRIPTION
C
C Abstract
C BINT4 computes the B representation (T,BCOEF,N,K) of a
C cubic spline (K=4) which interpolates data (X(I)),Y(I))),
C I=1,NDATA. Parameters IBCL, IBCR, FBCL, FBCR allow the
C specification of the spline first or second derivative at
C both X(1) and X(NDATA). When this data is not specified
C by the problem, it is common practice to use a natural
C spline by setting second derivatives at X(1) and X(NDATA)
C to zero (IBCL=IBCR=2,FBCL=FBCR=0.0). The spline is defined on
C T(4) .LE. X .LE. T(N+1) with (ordered) interior knots at X(I))
C values where N=NDATA+2. The knots T(1), T(2), T(3) lie to
C the left of T(4)=X(1) and the knots T(N+2), T(N+3), T(N+4)
C lie to the right of T(N+1)=X(NDATA) in increasing order. If
C no extrapolation outside (X(1),X(NDATA)) is anticipated, the
C knots T(1)=T(2)=T(3)=T(4)=X(1) and T(N+2)=T(N+3)=T(N+4)=
C T(N+1)=X(NDATA) can be specified by KNTOPT=1. KNTOPT=2
C selects a knot placement for T(1), T(2), T(3) to make the
C first 7 knots symmetric about T(4)=X(1) and similarly for
C T(N+2), T(N+3), T(N+4) about T(N+1)=X(NDATA). KNTOPT=3
C allows the user to make his own selection, in increasing
C order, for T(1), T(2), T(3) to the left of X(1) and T(N+2),
C T(N+3), T(N+4) to the right of X(NDATA) in the work array
C W(1) through W(6). In any case, the interpolation on
C T(4) .LE. X .LE. T(N+1) by using function BVALU is unique
C for given boundary conditions.
C
C Description of Arguments
C Input
C X - X vector of abscissae of length NDATA, distinct
C and in increasing order
C Y - Y vector of ordinates of length NDATA
C NDATA - number of data points, NDATA .GE. 2
C IBCL - selection parameter for left boundary condition
C IBCL = 1 constrain the first derivative at
C X(1) to FBCL
C = 2 constrain the second derivative at
C X(1) to FBCL
C IBCR - selection parameter for right boundary condition
C IBCR = 1 constrain first derivative at
C X(NDATA) to FBCR
C IBCR = 2 constrain second derivative at
C X(NDATA) to FBCR
C FBCL - left boundary values governed by IBCL
C FBCR - right boundary values governed by IBCR
C KNTOPT - knot selection parameter
C KNTOPT = 1 sets knot multiplicity at T(4) and
C T(N+1) to 4
C = 2 sets a symmetric placement of knots
C about T(4) and T(N+1)
C = 3 sets TNP)=WNP) and T(N+1+I)=w(3+I),I=1,3
C where WNP),I=1,6 is supplied by the user
C W - work array of dimension at least 5*(NDATA+2)
C if KNTOPT=3, then W(1),W(2),W(3) are knot values to
C the left of X(1) and W(4),W(5),W(6) are knot
C values to the right of X(NDATA) in increasing
C order to be supplied by the user
C
C Output
C T - knot array of length N+4
C BCOEF - B-spline coefficient array of length N
C N - number of coefficients, N=NDATA+2
C K - order of spline, K=4
C
C Error Conditions
C Improper input is a fatal error
C Singular system of equations is a fatal error
C
C***REFERENCES D. E. Amos, Computation with splines and B-splines,
C Report SAND78-1968, Sandia Laboratories, March 1979.
C Carl de Boor, Package for calculating with B-splines,
C SIAM Journal on Numerical Analysis 14, 3 (June 1977),
C pp. 441-472.
C Carl de Boor, A Practical Guide to Splines, Applied
C Mathematics Series 27, Springer-Verlag, New York,
C 1978.
C***ROUTINES CALLED BNFAC, BNSLV, BSPVD, R1MACH, XERMSG
C***REVISION HISTORY (YYMMDD)
C 800901 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890531 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 900326 Removed duplicate information from DESCRIPTION section.
C (WRB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE BINT4
C
INTEGER I, IBCL, IBCR, IFLAG, ILB, ILEFT, IT, IUB, IW, IWP, J,
1 JW, K, KNTOPT, N, NDATA, NDM, NP, NWROW
REAL BCOEF,FBCL,FBCR,T, TOL,TXN,TX1,VNIKX,W,WDTOL,WORK,X, XL,
1 Y
REAL R1MACH
DIMENSION X(*), Y(*), T(*), BCOEF(*), W(5,*), VNIKX(4,4), WORK(15)
C***FIRST EXECUTABLE STATEMENT BINT4
WDTOL = R1MACH(4)
TOL = SQRT(WDTOL)
IF (NDATA.LT.2) GO TO 200
NDM = NDATA - 1
DO 10 I=1,NDM
IF (X(I).GE.X(I+1)) GO TO 210
10 CONTINUE
IF (IBCL.LT.1 .OR. IBCL.GT.2) GO TO 220
IF (IBCR.LT.1 .OR. IBCR.GT.2) GO TO 230
IF (KNTOPT.LT.1 .OR. KNTOPT.GT.3) GO TO 240
K = 4
N = NDATA + 2
NP = N + 1
DO 20 I=1,NDATA
T(I+3) = X(I)
20 CONTINUE
GO TO (30, 50, 90), KNTOPT
C SET UP KNOT ARRAY WITH MULTIPLICITY 4 AT X(1) AND X(NDATA)
30 CONTINUE
DO 40 I=1,3
T(4-I) = X(1)
T(NP+I) = X(NDATA)
40 CONTINUE
GO TO 110
C SET UP KNOT ARRAY WITH SYMMETRIC PLACEMENT ABOUT END POINTS
50 CONTINUE
IF (NDATA.GT.3) GO TO 70
XL = (X(NDATA)-X(1))/3.0E0
DO 60 I=1,3
T(4-I) = T(5-I) - XL
T(NP+I) = T(NP+I-1) + XL
60 CONTINUE
GO TO 110
70 CONTINUE
TX1 = X(1) + X(1)
TXN = X(NDATA) + X(NDATA)
DO 80 I=1,3
T(4-I) = TX1 - X(I+1)
T(NP+I) = TXN - X(NDATA-I)
80 CONTINUE
GO TO 110
C SET UP KNOT ARRAY LESS THAN X(1) AND GREATER THAN X(NDATA) TO BE
C SUPPLIED BY USER IN WORK LOCATIONS W(1) THROUGH W(6) WHEN KNTOPT=3
90 CONTINUE
DO 100 I=1,3
T(4-I) = W(4-I,1)
JW = MAX(1,I-1)
IW = MOD(I+2,5)+1
T(NP+I) = W(IW,JW)
IF (T(4-I).GT.T(5-I)) GO TO 250
IF (T(NP+I).LT.T(NP+I-1)) GO TO 250
100 CONTINUE
110 CONTINUE
C
DO 130 I=1,5
DO 120 J=1,N
W(I,J) = 0.0E0
120 CONTINUE
130 CONTINUE
C SET UP LEFT INTERPOLATION POINT AND LEFT BOUNDARY CONDITION FOR
C RIGHT LIMITS
IT = IBCL + 1
CALL BSPVD(T, K, IT, X(1), K, 4, VNIKX, WORK)
IW = 0
IF (ABS(VNIKX(3,1)).LT.TOL) IW = 1
DO 140 J=1,3
W(J+1,4-J) = VNIKX(4-J,IT)
W(J,4-J) = VNIKX(4-J,1)
140 CONTINUE
BCOEF(1) = Y(1)
BCOEF(2) = FBCL
C SET UP INTERPOLATION EQUATIONS FOR POINTS I=2 TO I=NDATA-1
ILEFT = 4
IF (NDM.LT.2) GO TO 170
DO 160 I=2,NDM
ILEFT = ILEFT + 1
CALL BSPVD(T, K, 1, X(I), ILEFT, 4, VNIKX, WORK)
DO 150 J=1,3
W(J+1,3+I-J) = VNIKX(4-J,1)
150 CONTINUE
BCOEF(I+1) = Y(I)
160 CONTINUE
C SET UP RIGHT INTERPOLATION POINT AND RIGHT BOUNDARY CONDITION FOR
C LEFT LIMITS(ILEFT IS ASSOCIATED WITH T(N)=X(NDATA-1))
170 CONTINUE
IT = IBCR + 1
CALL BSPVD(T, K, IT, X(NDATA), ILEFT, 4, VNIKX, WORK)
JW = 0
IF (ABS(VNIKX(2,1)).LT.TOL) JW = 1
DO 180 J=1,3
W(J+1,3+NDATA-J) = VNIKX(5-J,IT)
W(J+2,3+NDATA-J) = VNIKX(5-J,1)
180 CONTINUE
BCOEF(N-1) = FBCR
BCOEF(N) = Y(NDATA)
C SOLVE SYSTEM OF EQUATIONS
ILB = 2 - JW
IUB = 2 - IW
NWROW = 5
IWP = IW + 1
CALL BNFAC(W(IWP,1), NWROW, N, ILB, IUB, IFLAG)
IF (IFLAG.EQ.2) GO TO 190
CALL BNSLV(W(IWP,1), NWROW, N, ILB, IUB, BCOEF)
RETURN
C
C
190 CONTINUE
CALL XERMSG ('SLATEC', 'BINT4',
+ 'THE SYSTEM OF EQUATIONS IS SINGULAR', 2, 1)
RETURN
200 CONTINUE
CALL XERMSG ('SLATEC', 'BINT4', 'NDATA IS LESS THAN 2', 2, 1)
RETURN
210 CONTINUE
CALL XERMSG ('SLATEC', 'BINT4',
+ 'X VALUES ARE NOT DISTINCT OR NOT ORDERED', 2, 1)
RETURN
220 CONTINUE
CALL XERMSG ('SLATEC', 'BINT4', 'IBCL IS NOT 1 OR 2', 2, 1)
RETURN
230 CONTINUE
CALL XERMSG ('SLATEC', 'BINT4', 'IBCR IS NOT 1 OR 2', 2, 1)
RETURN
240 CONTINUE
CALL XERMSG ('SLATEC', 'BINT4', 'KNTOPT IS NOT 1, 2, OR 3', 2, 1)
RETURN
250 CONTINUE
CALL XERMSG ('SLATEC', 'BINT4',
+ 'KNOT INPUT THROUGH W ARRAY IS NOT ORDERED PROPERLY', 2, 1)
RETURN
END