IMSL_SPINTEG

The IMSL_SPINTEG function computes the integral of a one- or two-dimensional spline.

Note
This routine requires an IDL Advanced Math and Stats license. For more information, contact your ITT Visual Information Solutions sales or technical support representative.

Syntax

Result = IMSL_SPINTEG(a, b, spline)

Result = IMSL_SPINTEG(a, b, c, d, spline)

Return Value

If spline is a one-dimensional spline, then the returned value is the integral from a to b of spline. If spline is a two-dimensional, tensor-product spline, then the returned value is the value of the integral of spline over the rectangle [a, b] x [c, d]. If no value can be computed, NaN (Not a Number) is returned.

Arguments

If integration of a one-dimensional spline is desired, then arguments a, b, and spline are required. If integration of a two-dimensional spline is desired, then a, b, c, d, and spline are required.

a

Right endpoint of integration.

b

Left endpoint of integration.

c

Right endpoint of integration for the second variable of the tensor-product spline. This argument should only be used if spline is a two-dimensional, tensor-product spline.

d

Left endpoint of integration for the second variable of the tensor-product spline. This argument should only be used if spline is a two-dimensional, tensor-product spline.

spline

Structure that represents the spline to be integrated.

Discussion

The IMSL_SPINTEG function can be used to integrate splines of the following type:

Piecewise polynomials returned by IMSL_CSINTERP, IMSL_CSSHAPE, and IMSL_CSSMOOTH.

One-dimensional B-splines returned by IMSL_BSINTERP, IMSL_BSLSQ, and IMSL_CONLSQ.

Two-dimensional, tensor-product B-splines returned from IMSL_BSINTERP and IMSL_BSLSQ.

If s = spline is a one-dimensional piecewise polynomial or B-spline, then IMSL_SPINTEG computes:

If spline is a one-dimensional B-spline, then this function uses identity (22) of de Boor (1978, p. 115).

If s = spline is a two-dimensional, tensor-product spline, then the arguments c and d are required, and IMSL_SPINTEG computes:

This function uses the (univariate integration) identity (22) of de Boor (1978, p. 151):

where t₀≤x ≤ t_r. It assumes (for all knot sequences) that the first and last k knots are stacked; that is, t₀ = . . . = t_{k –}₁ and t_n = . . . = t_{n + k – 1}, where k is the order of the spline in the x or y direction.

Example

This example computes a cubic spline interpolant to function values. The values of the integral of this spline are then compared with the exact integral values. Since the default settings are being used, the interpolant is determined by the not-a-knot condition (de Boor 1978).

n = 21 
; Generate the data. 
x = FINDGEN(n)/(n - 1) 
f = SIN(15 * x) 
pp = IMSL_CSINTERP(x, f) 
; Compute the interpolant. 
results = FLTARR(22, 4)   
; Define an array to hold some results to be output later.   
FOR i = n/2, 3 * n/2 DO BEGIN & $ 
   x2 = i/FLOAT(2 * n - 2) & $ 
   y = IMSL_SPINTEG(0, x2, pp)  & $  
   results[i - n/2, *] = $   
     [x2, (1 - COS(15 * x2))/15, y, $   
     ABS((1 - COS(15 * x2))/15 - y)] & $  
; Loop over different limits of integration and compare the  
; results with the true answer. 
ENDFOR 
PM, results, FORMAT = '(4f12.4)', $ 
   Title  = '   X    True   Approx   Error' 
   ; Output the results. 
    
       X        True        Approx      Error 
   0.2500      0.1214      0.1215      0.0001 
   0.2750      0.1036      0.1037      0.0001 
   0.3000      0.0807      0.0808      0.0001 
   0.3250      0.0559      0.0560      0.0001 
   0.3500      0.0325      0.0327      0.0001 
   0.3750      0.0139      0.0141      0.0002 
   0.4000      0.0027      0.0028      0.0002 
   0.4250      0.0003      0.0004      0.0002 
   0.4500      0.0071      0.0073      0.0002 
   0.4750      0.0223      0.0224      0.0001 
   0.5000      0.0436      0.0437      0.0001 
   0.5250      0.0681      0.0682      0.0001 
   0.5500      0.0924      0.0925      0.0001 
   0.5750      0.1131      0.1132      0.0001 
   0.6000      0.1274      0.1275      0.0001 
   0.6250      0.1333      0.1333      0.0001 
   0.6500      0.1298      0.1299      0.0001 
   0.6750      0.1176      0.1177      0.0001 
   0.7000      0.0984      0.0985      0.0001 
   0.7250      0.0747      0.0748      0.0001 
   0.7500      0.0499      0.0500      0.0001 
   0.7750      0.0274      0.0276      0.0001

Errors

Warning Errors

MATH_SPLINE_LEFT_ENDPT—Left endpoint of x integration is not within the knot sequence. Integration occurs only from t_Order_{– 1} to b.

MATH_SPLINE_RIGHT_ENDPT—Right endpoint of x integration is not within the knot sequence. Integration occurs only from t_Order_{– 1} to a.

MATH_SPLINE_LEFT_ENDPT_1—Left endpoint of x integration is not within the knot sequence. Integration occurs only from b to t_{Spline_Space_Dim}_{– 1}.

MATH_SPLINE_RIGHT_ENDPT_1—Right endpoint of x integration is not within the knot sequence. Integration occurs only from a to t_{Spline_Space_Dim}_{– 1}.

MATH_SPLINE_LEFT_ENDPT_2—Left endpoint of y integration is not within the knot sequence. Integration occurs only from t_Order_{– 1} to d.

MATH_SPLINE_RIGHT_ENDPT_2—Right endpoint of y integration is not within the knot sequence. Integration occurs only from t_Order_{– 1} to c.

MATH_SPLINE_LEFT_ENDPT_3—Left endpoint of y integration is not within the knot sequence. Integration occurs only from d to t_{Spline_Space_Dim}_{– 1}.

MATH_SPLINE_RIGHT_ENDPT_3—Right endpoint of y integration is not within the knot sequence. Integration occurs only from c to t_{Spline_Space_Dim}_{– 1}.

Fatal Errors

MATH_KNOT_MULTIPLICITY—Multiplicity of the knots cannot exceed the order of the spline.

MATH_KNOT_NOT_INCREASING—Knots must be nondecreasing.

Version History


6.4	Introduced