Returns the value of an integral of a tensor-product spline on a rectangular domain.

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
public double Integral(
	double a,
	double b,
	double c,
	double d
)
Visual Basic (Declaration)
Public Function Integral ( _
	a As Double, _
	b As Double, _
	c As Double, _
	d As Double _
) As Double
Visual C++
public:
double Integral(
	double a, 
	double b, 
	double c, 
	double d
)

Parameters

a
Type: System..::.Double
A double specifying the lower limit for the first variable of the tensor-product spline.
b
Type: System..::.Double
A double specifying the upper limit for the first variable of the tensor-product spline.
c
Type: System..::.Double
A double specifying the lower limit for the second variable of the tensor-product spline.
d
Type: System..::.Double
A double specifying the upper limit for the second variable of the tensor-product spline.

Return Value

A double, the integral of the tensor-product spline over the rectangle [a, b] by [c, d].

Remarks

If s is the spline, then the Integral method returns

\int_a^b {\int_c^d {s\left( {x,y} \right)} } dydx
This method uses the (univariate integration) identity (22) in de Boor (1978, p. 151)
\int_{t_0 }^x {\sum\limits_{i = 0}^{n - 1} {\alpha _i } } B_{i,k} \left( \tau \right)d\tau = \sum\limits_{i = 0}^{r - 1} {\left[ {\sum\limits_{j = 0}^i {\alpha _j \frac{{t_{j + k} - t_j }}{k}} } \right]} B_{i,k + 1} \left( x \right)
where t_0 \le x \le t_r.

It assumes (for all knot sequences) that the first and last k knots are stacked, that is, t_0 = \ldots = t_{k-1} and t_n = \ldots = t_{n+k-1} , where k is the order of the spline in the x or y direction.

See Also