This function evaluates the derivative of a cubic spline.
CSDER — Value of the IDERIV-th derivative of the polynomial at X. (Output)
IDERIV — Order of
the derivative to be evaluated. (Input)
In particular, IDERIV = 0 returns the
value of the polynomial.
X — Point at which the polynomial is to be evaluated. (Input)
BREAK — Array of
length NINTV + 1
containing the breakpoints for the piecewise cubic representation.
(Input)
BREAK must be strictly
increasing.
CSCOEF — Matrix of size 4 by NINTV + 1 containing the local coefficients of the cubic pieces. (Input)
NINTV —
Number of polynomial pieces. (Input)
Default: NINTV = size
(BREAK,1) –
1.
Generic: CSDER (IDERIV, X, BREAK, CSCOEF, CSDER [,…])
Specific: The specific interface names are S_CSDER and D_CSDER.
Single: CSDER (IDERIV, X, NINTV, BREAK, CSCOEF)
Double: The double precision function name is DCSDER.
The function CSDER evaluates the derivative of a cubic spline at a given point. It is a special case of the routine PPDER, which evaluates the derivative of a piecewise polynomial. (A cubic spline is a piecewise polynomial of order 4.) The routine PPDER is based on the routine PPVALU in de Boor (1978, page 89).
In this example, we compute a cubic spline interpolant to a function f using IMSL routine CSINT. The values of the spline and its first and second derivatives are computed using CSDER. These values can then be compared with the corresponding values of the interpolated function.
USE CSDER_INT
USE CSINT_INT
USE UMACH_INT
IMPLICIT NONE
INTEGER NDATA
PARAMETER (NDATA=10)
!
INTEGER I, NINTV, NOUT
REAL BREAK(NDATA), CDDF, CDF, CF, COS, CSCOEF(4,NDATA),&
DDF, DF, F, FDATA(NDATA), FLOAT, SIN, X,&
XDATA(NDATA)
INTRINSIC COS, FLOAT, SIN
! Define function and derivatives
F(X) = SIN(15.0*X)
DF(X) = 15.0*COS(15.0*X)
DDF(X) = -225.0*SIN(15.0*X)
! Set up a grid
DO 10 I=1, NDATA
XDATA(I) = FLOAT(I-1)/FLOAT(NDATA-1)
FDATA(I) = F(XDATA(I))
10 CONTINUE
! Compute cubic spline interpolant
CALL CSINT (XDATA, FDATA, BREAK, CSCOEF)
! Get output unit number
CALL UMACH (2, NOUT)
! Write heading
WRITE (NOUT,99999)
99999 FORMAT (9X, 'X', 8X, 'S(X)', 5X, 'Error', 6X, 'S''(X)', 5X,&
'Error', 6X, 'S''''(X)', 4X, 'Error', /)
NINTV = NDATA - 1
! Print the interpolant on a finer grid
DO 20 I=1, 2*NDATA
X = FLOAT(I-1)/FLOAT(2*NDATA-1)
CF = CSDER(0,X,BREAK,CSCOEF)
CDF = CSDER(1,X,BREAK,CSCOEF)
CDDF = CSDER(2,X,BREAK,CSCOEF)
WRITE (NOUT,'(F11.3, 3(F11.3, F11.6))') X, CF, F(X) - CF,&
CDF, DF(X) - CDF,&
CDDF, DDF(X) - CDDF
20 CONTINUE
END
X
S(X) Error
S'(X) Error
S''(X) Error
0.000
0.000 0.000000 26.285 -11.284739
-379.458 379.457794
0.053 0.902
-0.192203 8.841 1.722460
-283.411 123.664734
0.105 1.019
-0.019333 -3.548 3.425718
-187.364 -37.628586
0.158 0.617
0.081009 -10.882 0.146207
-91.317 -65.824875
0.211 -0.037
0.021155 -13.160 -1.837700
4.730 -1.062027
0.263 -0.674
-0.046945 -10.033 -0.355268
117.916 44.391640
0.316 -0.985
-0.015060 -0.719 1.086203
235.999 -11.066727
0.368 -0.682
-0.004651 11.314 -0.409097
154.861 -0.365387
0.421 0.045
-0.011915 14.708 0.284042
-25.887 18.552732
0.474 0.708
0.024292 9.508 0.702690
-143.785 -21.041260
0.526 0.978
0.020854 0.161 -0.771948
-211.402 -13.411087
0.579 0.673
0.001410 -11.394 0.322443
-163.483 11.674103
0.632 -0.064
0.015118 -14.937 -0.045511
28.856 -17.856323
0.684 -0.724
-0.019246 -8.859 -1.170871
163.866 3.435547
0.737 -0.954
-0.044143 0.301
0.554493 184.217
40.417282
0.789 -0.675
0.012143 10.307 0.928152
166.021 -16.939514
0.842 0.027
0.038176 15.015 -0.047344
12.914 -27.575521
0.895 0.764
-0.010112 11.666 -1.819128 -140.193
-29.538193
0.947 1.114
-0.116304 0.258 -1.357680
-293.301 68.905701
1.000
0.650 0.000000 -19.208
7.812407 -446.408 300.092896
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