BSpline Class

BSpline Class

Spline represents and evaluates univariate B-splines.

Inheritance Hierarchy

System.Object
  Imsl.Math.BSpline
    Imsl.Math.BsInterpolate
    Imsl.Math.BsLeastSquares

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public abstract class BSpline

<SerializableAttribute>
Public MustInherit Class BSpline

[SerializableAttribute]
public ref class BSpline abstract

[<AbstractClassAttribute>]
[<SerializableAttribute>]
type BSpline =  class end

The BSpline type exposes the following members.

Constructors

	Name	Description
	BSpline	Initializes a new instance of the BSpline class

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Methods

	Name	Description
	Derivative(Double)	Returns the value of the first derivative of the B-spline at a point.
	Derivative(Double, Int32)	Returns the value of the derivative of the B-spline at a point.
	Derivative(Double[], Int32)	Returns the value of the derivative of the B-spline at each point of an array.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Eval(Double)	Returns the value of the B-spline at a point.
	Eval(Double[])	Returns the value of the B-spline at each point of an array.
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetKnots	Returns a copy of the knot sequence.
	GetSpline	Returns a Spline representation of the B-spline.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	Integral	Returns the value of an integral of the B-spline.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Remarks

B-splines provide a particularly convenient and suitable basis for a given class of smooth ppoly functions. Such a class is specified by giving its breakpoint sequence, its order k, and the required smoothness across each of the interior breakpoints. The corresponding B-spline basis is specified by giving its knot sequence ${\bf t} \in {\bf R}^M$ . The specification rule is as follows: If the class is to have all derivatives up to and including the j-th derivative continuous across the interior breakpoint $\xi_i$ , then the number $\xi_i$ should occur k - j - 1 times in the knot sequence. Assuming that $\xi_1$ and $\xi_n$ are the endpoints of the interval of interest, choose the first k knots equal to $\xi_1$ and the last k knots equal to $\xi_n$ . This can be done because the B-splines are defined to be right continuous near $\xi_1$ and left continuous near $\xi_n$ .

When the above construction is completed, a knot sequence ${\bf t}$ of length M is generated, and there are m: = M-k B-splines of order k, for example $B_0, ..., B_{m-1}$ , spanning the ppoly functions on the interval with the indicated smoothness. That is, each ppoly function in this class has a unique representation $p = a_0B_0 + a_1B_1 + ... + a_{m-1}B_{m-1}$ as a linear combination of B-splines. A B-spline is a particularly compact ppoly function. is a nonnegative function that is nonzero only on the interval $[{\bf t}_i,{\bf t}_{i+k}]$ . More precisely, the support of the i-th B-spline is $\left[ t_i,t_{i+k}\right]$ . No ppoly function in the same class (other than the zero function) has smaller support (i.e., vanishes on more intervals) than a B-spline. This makes B-splines particularly attractive basis functions since the influence of any particular B-spline coefficient extends only over a few intervals.

Reference

Imsl.Math Namespace

Other Resources

BsInterpolate Example

BsLeastSquares Example