Evaluates the derivative of a spline on a grid, given its B-spline representation.

Required Arguments

IDERIV — Order of the derivative to be evaluated.   (Input)
In particular, IDERIV = 0 returns the value of the spline.

XVEC —  Array of length N containing the points at which the spline is to be evaluated.   (Input)
XVEC should be strictly increasing.

KORDER — Order of the spline.   (Input)

XKNOT — Array of length NCOEF + KORDER containing the knot sequence.   (Input)
XKNOT must be nondecreasing.

BSCOEF — Array of length NCOEF containing the B-spline coefficients.   (Input)

VALUE — Array of length N containing the values of the IDERIV-th derivative of the spline at the points in XVEC.   (Output)

Optional Arguments

N — Length of vector XVEC.   (Input)
Default: N = size (XVEC,1).

NCOEF — Number of B-spline coefficients.   (Input)
Default: NCOEF = size (BSCOEF,1).

FORTRAN 90 Interface

Generic:          CALL BS1GD (IDERIV, XVEC, KORDER, XKNOT, BSCOEF, VALUE [,…])

Specific:         The specific interface names are S_BS1GD and D_BS1GD.

FORTRAN 77 Interface

Single:            CALL BS1GD (IDERIV, N, XVEC, KORDER, XKNOT, NCOEF, BSCOEF, VALUE)

Double:          The double precision name is DBS1GD.

Description

The routine BS1GD evaluates a B-spline (or its derivative) at a vector of points. That is, given a vector x of length n satisfying xi < xi + 1 for i = 1, …, n − 1, a derivative value j, and a B-spline s that is represented by a knot sequence and coefficient sequence, this routine returns the values

in the array VALUE. The functionality of this routine is the same as that of BSDER called in a loop, however BS1GD should be much more efficient. This routine converts the B-spline representation to piecewise polynomial form using the IMSL routine BSCPP, and then uses the IMSL routine PPVAL for evaluation.

Comments

1.         Workspace may be explicitly provided, if desired, by use of B21GD/DB21GD. The reference is:

CALL B21GD (IDERIV, N, XVEC, KORDER, XKNOT, NCOEF, BSCOEF, VALUE, RWK1, RWK2, IWK3, RWK4, RWK5, RWK6)

The additional arguments are as follows:

RWK1 — Real array of length KORDER * (NCOEF − KORDER + 1).

RWK2 — Real array of length NCOEF − KORDER + 2.

IWK3 — Integer array of length N.

RWK4 — Real array of length N.

RWK5 — Real array of length N.

RWK6 — Real array of length (KORDER + 3) * KORDER

2.         Informational error

Type   Code

4           5                  The points in XVEC must be strictly increasing.

Example

To illustrate the use of BS1GD, we modify the example program for BSDER. In this example, a quadratic (order 3) spline interpolant to F is computed. The values and derivatives of this spline are then compared with the exact function and derivative values. The routine BS1GD is based on the routines BSPLPP and PPVALU in de Boor (1978, page 89).

 

      USE BS1GD_INT

      USE BSINT_INT

      USE BSNAK_INT

      USE UMACH_INT

 

      IMPLICIT   NONE

      INTEGER    KORDER, NDATA, NKNOT, NFGRID

      PARAMETER  (KORDER=3, NDATA=5, NKNOT=NDATA+KORDER, NFGRID = 9)

!                                  SPECIFICATIONS FOR LOCAL VARIABLES

      INTEGER    I, NCOEF, NOUT

      REAL       ANS0(NFGRID), ANS1(NFGRID), BSCOEF(NDATA),&

                 FDATA(NDATA),&

                 X, XDATA(NDATA), XKNOT(NKNOT), XVEC(NFGRID)

!                                  SPECIFICATIONS FOR INTRINSICS

      INTRINSIC  FLOAT, SQRT

      REAL       FLOAT, SQRT

!                                  SPECIFICATIONS FOR SUBROUTINES

      REAL       DF, F

!

      F(X)  = SQRT(X)

      DF(X) = 0.5/SQRT(X)

!

      CALL UMACH (2, NOUT)

!                                  Set up interpolation points

      DO 10  I=1, NDATA

         XDATA(I) = FLOAT(I)/FLOAT(NDATA)

         FDATA(I) = F(XDATA(I))

   10 CONTINUE

      CALL BSNAK (NDATA, XDATA, KORDER, XKNOT)

!                                  Interpolate

      CALL BSINT (NDATA, XDATA, FDATA, KORDER, XKNOT, BSCOEF)

      WRITE (NOUT,99999)

!                                  Print on a finer grid

      NCOEF   = NDATA

      XVEC(1) = XDATA(1)

      DO 20  I=2, 2*NDATA - 2, 2

         XVEC(I)   = (XDATA(I/2+1)+XDATA(I/2))/2.0

         XVEC(I+1) = XDATA(I/2+1)

   20 CONTINUE

      CALL BS1GD (0, XVEC, KORDER, XKNOT, BSCOEF, ANS0)

      CALL BS1GD (1, XVEC, KORDER, XKNOT, BSCOEF, ANS1)

      DO 30  I=1, 2*NDATA - 1

         WRITE (NOUT,99998) XVEC(I), ANS0(I), F(XVEC(I)) - ANS0(I),&

                           ANS1(I), DF(XVEC(I)) - ANS1(I)

   30 CONTINUE

99998 FORMAT (' ', F6.4, 5X, F7.4, 5X, F8.4, 5X, F8.4, 5X, F8.4)

99999 FORMAT (6X, 'X', 8X, 'S(X)', 7X, 'Error', 8X, 'S''(X)', 8X,&

             'Error', /)

      END

Output

 

     X        S(X)       Error        S'(X)        Error

0.2000      0.4472       0.0000       1.0423       0.0757
0.3000      0.5456       0.0021       0.9262      -0.0133
0.4000      0.6325       0.0000       0.8101      -0.0196
0.5000      0.7077      -0.0006       0.6940       0.0131
0.6000      0.7746       0.0000       0.6446       0.0009
0.7000      0.8366       0.0001       0.5952       0.0024
0.8000      0.8944       0.0000       0.5615      -0.0025
0.9000      0.9489      -0.0002       0.5279      -0.0008
1.0000      1.0000       0.0000       0.4942       0.0058

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