Cholesky Class |
Namespace: Imsl.Math
The Cholesky type exposes the following members.
Name | Description | |
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Cholesky |
Create the Cholesky factorization of a symmetric positive definite
matrix of type double.
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Name | Description | |
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Downdate |
Downdates the factorization by subtracting a rank-1 matrix.
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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.
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GetType | Gets the Type of the current instance. (Inherited from Object.) | |
Inverse |
Returns the inverse of this matrix.
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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.
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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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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 .