*DECK DBINT4 SUBROUTINE DBINT4 (X, Y, NDATA, IBCL, IBCR, FBCL, FBCR, KNTOPT, T, + BCOEF, N, K, W) C***BEGIN PROLOGUE DBINT4 C***PURPOSE Compute the B-representation of a cubic spline C which interpolates given data. C***LIBRARY SLATEC C***CATEGORY E1A C***TYPE DOUBLE 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 **** a double precision routine **** C C DBINT4 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 C on T(4) .LE. X .LE. T(N+1) with (ordered) interior knots at C X(I) 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 DBVALU is unique C for given boundary conditions. C C Description of Arguments C C Input X,Y,FBCL,FBCR,W are double precision 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 T(I)=W(I) and T(N+1+I)=W(3+I),I=1,3 C where W(I),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 T,BCOEF are double precision 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 D1MACH, DBNFAC, DBNSLV, DBSPVD, 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 DBINT4 C INTEGER I, IBCL, IBCR, IFLAG, ILB, ILEFT, IT, IUB, IW, IWP, J, 1 JW, K, KNTOPT, N, NDATA, NDM, NP, NWROW DOUBLE PRECISION BCOEF,FBCL,FBCR,T,TOL,TXN,TX1,VNIKX,W,WDTOL, 1 WORK,X,XL,Y DOUBLE PRECISION D1MACH DIMENSION X(*), Y(*), T(*), BCOEF(*), W(5,*), VNIKX(4,4), WORK(15) C***FIRST EXECUTABLE STATEMENT DBINT4 WDTOL = D1MACH(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))/3.0D0 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) = 0.0D0 120 CONTINUE 130 CONTINUE C SET UP LEFT INTERPOLATION POINT AND LEFT BOUNDARY CONDITION FOR C RIGHT LIMITS IT = IBCL + 1 CALL DBSPVD(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 DBSPVD(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 DBSPVD(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 DBNFAC(W(IWP,1), NWROW, N, ILB, IUB, IFLAG) IF (IFLAG.EQ.2) GO TO 190 CALL DBNSLV(W(IWP,1), NWROW, N, ILB, IUB, BCOEF) RETURN C C 190 CONTINUE CALL XERMSG ('SLATEC', 'DBINT4', + 'THE SYSTEM OF EQUATIONS IS SINGULAR', 2, 1) RETURN 200 CONTINUE CALL XERMSG ('SLATEC', 'DBINT4', 'NDATA IS LESS THAN 2', 2, 1) RETURN 210 CONTINUE CALL XERMSG ('SLATEC', 'DBINT4', + 'X VALUES ARE NOT DISTINCT OR NOT ORDERED', 2, 1) RETURN 220 CONTINUE CALL XERMSG ('SLATEC', 'DBINT4', 'IBCL IS NOT 1 OR 2', 2, 1) RETURN 230 CONTINUE CALL XERMSG ('SLATEC', 'DBINT4', 'IBCR IS NOT 1 OR 2', 2, 1) RETURN 240 CONTINUE CALL XERMSG ('SLATEC', 'DBINT4', 'KNTOPT IS NOT 1, 2, OR 3', 2, + 1) RETURN 250 CONTINUE CALL XERMSG ('SLATEC', 'DBINT4', + 'KNOT INPUT THROUGH W ARRAY IS NOT ORDERED PROPERLY', 2, 1) RETURN END