Click or drag to resize
Spline2D Class
Represents and evaluates tensor-product splines.
Inheritance Hierarchy

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0
Syntax
[SerializableAttribute]
public abstract class Spline2D

The Spline2D type exposes the following members.

Constructors
  NameDescription
Protected methodSpline2D
Initializes a new instance of the Spline2D class
Top
Methods
  NameDescription
Public methodDerivative(Double, Double, Int32, Int32)
Returns the value of the partial derivative of the tensor-product spline at the point (x, y).
Public methodDerivative(Double, Double, Int32, Int32)
Returns the values of the partial derivative of the tensor-product spline of an array of points.
Public methodEquals
Determines whether the specified object is equal to the current object.
(Inherited from Object.)
Protected methodFinalize
Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection.
(Inherited from Object.)
Public methodGetCoefficients
Returns the coefficients for the tensor-product spline.
Public methodGetHashCode
Serves as a hash function for a particular type.
(Inherited from Object.)
Public methodGetType
Gets the Type of the current instance.
(Inherited from Object.)
Public methodGetXKnots
Returns the knot sequences in the x-direction.
Public methodGetYKnots
Returns the knot sequences in the y-direction.
Public methodIntegral
Returns the value of an integral of a tensor-product spline on a rectangular domain.
Protected methodMemberwiseClone
Creates a shallow copy of the current Object.
(Inherited from Object.)
Public methodToString
Returns a string that represents the current object.
(Inherited from Object.)
Public methodValue(Double, Double)
Returns the value of the tensor-product spline at the point (x, y).
Public methodValue(Double, Double)
Returns the values of the tensor-product spline of an array of points.
Top
Remarks

The simplest method of obtaining multivariate interpolation and approximation functions is to take univariate methods and form a multivariate method via tensor products. In the case of two-dimensional spline interpolation, the derivation proceeds as follows: Let t_x be a knot sequence for splines of order k_x, and t_y be a knot sequence for splines of order k_y. Let N_x + k_x be the length of t_x, and N_y + k_x be the length of t_y. Then, the tensor-product spline has the following form:

\sum\limits_{m = 0}^{N_y  - 1} {\sum\limits_{n = 0}^{N_x  - 1} {c_{nm} B_{n,k_x ,t_x } \left( x \right)B_{m,k_y ,t_y } \left( y \right)} }
Given two sets of points
\left\{ {x_i } \right\}_{i = 1}^{N_x }
and
\left\{ {y_j } \right\}_{j = 1}^{N_y }
for which the corresponding univariate interpolation problem can be solved, the tensor-product interpolation problem finds the coefficients c_{nm} so that
\sum\limits_{m = 0}^{N_y  - 1} {\sum\limits_{n = 0}^{N_x  - 1} {c_{nm} B_{n,k_x ,t_x } \left( {x_i } \right)B_{m,k_y ,t_y } \left( {y_j } \right)} }  = f_{ij}
This problem can be solved efficiently by repeatedly solving univariate interpolation problems as described in de Boor (1978, p. 347).

See Also