Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0
Syntax
C# |
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[SerializableAttribute] public class Cholesky |
Visual Basic (Declaration) |
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<SerializableAttribute> _ Public Class Cholesky |
Visual C++ |
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[SerializableAttribute] public ref class Cholesky |
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
Downdate determines an orthogonal matrix U as the product of Givens rotations, such that
By multiplying this equation by its transpose and noting that , the desired result
is obtained.Let a be the solution of the linear system and let
The Givens rotations, , are chosen such that
The , are (N + 1) * (N + 1) matrices of the form
where is the identity matrix of order k; and for some .The Givens rotations are then used to form
The matrix
is lower triangular and because .