Cholesky Class

Cholesky Class

Cholesky factorization of a matrix of type double.

Inheritance Hierarchy

SystemObject
Imsl.MathCholesky

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

Syntax

C++

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[SerializableAttribute]
public class Cholesky

<SerializableAttribute>
Public Class Cholesky

[SerializableAttribute]
public ref class Cholesky

[<SerializableAttribute>]
type Cholesky =  class end

The Cholesky type exposes the following members.

Constructors

	Name	Description
	Cholesky	Create the Cholesky factorization of a symmetric positive definite matrix of type double.

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Methods

	Name	Description
	Downdate	Downdates the factorization by subtracting a rank-1 matrix.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	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.)
	GetR	The R matrix that results from the Cholesky factorization.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	Inverse	Returns the inverse of this matrix.
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	Solve	Solve Ax = b where A is a positive definite matrix with elements of type double.
	ToString	Returns a string that represents the current object. (Inherited from Object.)
	Update	Updates the factorization by adding a rank-1 matrix.

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Remarks

Class Cholesky is based on the LINPACK routine SCHDC; see Dongarra et al. (1979).

Before the decomposition is computed, initial elements are moved to the leading part of A and final elements to the trailing part of A. During the decomposition only rows and columns corresponding to the free elements are moved. The result of the decomposition is an lower triangular matrix R and a permutation matrix P that satisfy , where P is represented by ipvt.

The method Update is based on the LINPACK routine SCHUD; see Dongarra et al. (1979).

The Cholesky factorization of a matrix is , where R is an lower triangular matrix. Given this factorization, Downdate computes the factorization

$A - xx^T = \tilde R^T \tilde R$

Downdate determines an orthogonal matrix U as the product $G_N\ldots G_1$ of Givens rotations, such that

$U \left[ \begin{array}{l} R \\ 0 \\ \end{array} \right] = \left[ \begin{array}{l}{\tilde R} \\ x^T \\ \end{array} \right]$

By multiplying this equation by its transpose and noting that , the desired result

$R^T R - xx^T = \tilde R^T \tilde R$

is obtained.

Let a be the solution of the linear system and let

$\alpha = \sqrt {1 - \left\| a \right\|_2^2 }$

The Givens rotations, , are chosen such that

$G_1 \cdots G_N \left[ \begin{array}{l}a \\ \alpha \end{array} \right] = \left[ \begin{array}{l} 0 \\ 1 \end{array} \right]$

The , are (N + 1) * (N + 1) matrices of the form

$G_i = \left[ {\begin{array}{*{20}c}{ I_{i - 1} } & 0 & 0 & 0 \\ 0 & {c_i } & 0 & { - s_i } \\ 0 & 0 & {I_{N - i} } & 0 \\ 0 & {s_i } & 0 & {c_i } \\ \end{array}} \right]$

where

is the identity matrix of order k; and $c_i= \cos\theta _i, s_i= \sin\theta_i$ for some $\theta_i$ .

The Givens rotations are then used to form

$\tilde R,\,\,G_1 \cdots \,G_N \left[ \begin{array}{l} R \\ 0 \\ \end{array} \right] = \left[ \begin{array}{l}{\tilde R} \\ \tilde x^T \\ \end{array} \right]$

The matrix

$\tilde R$

is lower triangular and

$\tilde x = x$

because

$x = \left( {R^T 0} \right) \left[ \begin{array}{l}a \\ \alpha \\ \end{array} \right] = \left( R^T 0 \right) U^T U\left[ \begin{array}{l}a \\ \alpha \\ \end{array} \right] = \left( {\tilde R^T \tilde x} \right) \left[ \begin{array}{l}0 \\ 1 \\ \end{array} \right] = \tilde x$

Reference

Imsl.Math Namespace

Other Resources

Example