Class CholeskyDecomposition

Cholesky Decomposition of a symmetric, positive definite matrix.

Inheritance

System.Object

CholeskyDecomposition

Implements

ISolverMatrixDecomposition<System.Double>

Namespace: ISynergy.Framework.Mathematics.Decompositions

Assembly: ISynergy.Framework.Mathematics.dll

Syntax

public sealed class CholeskyDecomposition : ICloneable, ISolverMatrixDecomposition<double>

Remarks

For a symmetric, positive definite matrix A, the Cholesky decomposition is a lower triangular matrix L so that A = L * L'. If the matrix is not positive definite, the constructor returns a partial decomposition and sets two internal variables that can be queried using the IsUndefined and IsPositiveDefinite properties.

Any square matrix A with non-zero pivots can be written as the product of a lower triangular matrix L and an upper triangular matrix U; this is called the LU decomposition. However, if A is symmetric and positive definite, we can choose the factors such that U is the transpose of L, and this is called the Cholesky decomposition. Both the LU and the Cholesky decomposition are used to solve systems of linear equations.

When it is applicable, the Cholesky decomposition is twice as efficient as the LU decomposition.

Constructors

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CholeskyDecomposition(Double[,], Boolean, Boolean, MatrixType)

Constructs a new Cholesky Decomposition.

Declaration

public CholeskyDecomposition(double[, ] value, bool robust = false, bool inPlace = false, MatrixType valueType = default(MatrixType))

Parameters

Type	Name	Description
System.Double[,]	value	The symmetric matrix, given in upper triangular form, to be decomposed.
System.Boolean	robust	True to perform a square-root free LDLt decomposition, false otherwise.
System.Boolean	inPlace	True to perform the decomposition in place, storing the factorization in the lower triangular part of the given matrix.
MatrixType	valueType	How to interpret the matrix given to be decomposed. Using this parameter, a lower or upper-triangular matrix can be interpreted as a symmetric matrix by assuming both lower and upper parts contain the same elements. Use this parameter in conjunction with inPlace to save memory by storing the original matrix and its decomposition at the same memory location (lower part will contain the decomposition's L matrix, upper part will contains the original matrix).

Properties

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Determinant

Gets the determinant of the decomposed matrix.

Declaration

public double Determinant { get; }

Property Value

Type	Description
System.Double

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Diagonal

Gets the one-dimensional array of diagonal elements in a LDLt decomposition.

Declaration

public double[] Diagonal { get; }

Property Value

Type	Description
System.Double[]

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DiagonalMatrix

Gets the block diagonal matrix of diagonal elements in a LDLt decomposition.

Declaration

public double[, ] DiagonalMatrix { get; }

Property Value

Type	Description
System.Double[,]

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IsPositiveDefinite

Gets whether the decomposed matrix was positive definite.

Declaration

public bool IsPositiveDefinite { get; }

Property Value

Type	Description
System.Boolean

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IsUndefined

Gets a value indicating whether the LDLt factorization has been computed successfully or if it is undefined.

Declaration

public bool IsUndefined { get; }

Property Value

Type	Description
System.Boolean	`true` if the factorization is not defined; otherwise, `false`.

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LeftTriangularFactor

Gets the left (lower) triangular factor L so that A = L * D * L'.

Declaration

public double[, ] LeftTriangularFactor { get; }

Property Value

Type	Description
System.Double[,]

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LogDeterminant

If the matrix is positive-definite, gets the log-determinant of the decomposed matrix.

Declaration

public double LogDeterminant { get; }

Property Value

Type	Description
System.Double

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Nonsingular

Gets a value indicating whether the decomposed matrix is non-singular (i.e. invertible).

Declaration

public bool Nonsingular { get; }

Property Value

Type	Description
System.Boolean

Methods

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Clone()

Creates a new object that is a copy of the current instance.

Declaration

public object Clone()

Returns

Type	Description
System.Object	A new object that is a copy of this instance.

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FromLeftTriangularMatrix(Double[,])

Creates a new Cholesky decomposition directly from an already computed left triangular matrix L.

Declaration

public static CholeskyDecomposition FromLeftTriangularMatrix(double[, ] leftTriangular)

Parameters

Type	Name	Description
System.Double[,]	leftTriangular	The left triangular matrix from a Cholesky decomposition.

Returns

Type	Description
CholeskyDecomposition

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GetInformationMatrix()

Computes (Xt * X)^1 (the inverse of the covariance matrix). This matrix can be used to determine standard errors for the coefficients when solving a linear set of equations through any of the methods.

Declaration

public double[, ] GetInformationMatrix()

Returns

Type	Description
System.Double[,]

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Inverse()

Solves a set of equation systems of type A * X = I.

Declaration

public double[, ] Inverse()

Returns

Type	Description
System.Double[,]

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InverseDiagonal(Boolean)

Computes the diagonal of the inverse of the decomposed matrix.

Declaration

public double[] InverseDiagonal(bool destroy = false)

Parameters

Type	Name	Description
System.Boolean	destroy

Returns

Type	Description
System.Double[]

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InverseDiagonal(Double[], Boolean)

Computes the diagonal of the inverse of the decomposed matrix.

Declaration

public double[] InverseDiagonal(double[] result, bool destroy = false)

Parameters

Type	Name	Description
System.Double[]	result	The array to hold the result of the computation. Should be of same length as the the diagonal of the original matrix.
System.Boolean	destroy	True to conserve memory by reusing the same space used to hold the decomposition, thus destroying it in the process. Pass false otherwise.

Returns

Type	Description
System.Double[]

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InverseTrace(Boolean)

Computes the trace of the inverse of the decomposed matrix.

Declaration

public double InverseTrace(bool destroy = false)

Parameters

Type	Name	Description
System.Boolean	destroy	True to conserve memory by reusing the same space used to hold the decomposition, thus destroying it in the process. Pass false otherwise.

Returns

Type	Description
System.Double

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Reverse()

Reverses the decomposition, reconstructing the original matrix X.

Declaration

public double[, ] Reverse()

Returns

Type	Description
System.Double[,]

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Solve(Double[])

Solves a set of equation systems of type A * x = b.

Declaration

public double[] Solve(double[] value)

Parameters

Type	Name	Description
System.Double[]	value	Right hand side column vector with as many rows as `A`.

Returns

Type	Description
System.Double[]	Vector `x` so that `L * L' * x = b`.

Exceptions

Type	Condition
System.ArgumentException	Matrix dimensions do not match.
System.NonSymmetricMatrixException	Matrix is not symmetric.
System.NonPositiveDefiniteMatrixException	Matrix is not positive-definite.

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Solve(Double[], Boolean)

Solves a set of equation systems of type A * x = b.

Declaration

public double[] Solve(double[] value, bool inPlace)

Parameters

Type	Name	Description
System.Double[]	value	Right hand side column vector with as many rows as `A`.
System.Boolean	inPlace	True to compute the solving in place, false otherwise.

Returns

Type	Description
System.Double[]	Vector `x` so that `L * L' * x = b`.

Exceptions

Type	Condition
System.ArgumentException	Matrix dimensions do not match.
System.NonSymmetricMatrixException	Matrix is not symmetric.
System.NonPositiveDefiniteMatrixException	Matrix is not positive-definite.

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Solve(Double[,])

Solves a set of equation systems of type A * X = B.

Declaration

public double[, ] Solve(double[, ] value)

Parameters

Type	Name	Description
System.Double[,]	value	Right hand side matrix with as many rows as `A` and any number of columns.

Returns

Type	Description
System.Double[,]	Matrix `X` so that `L * L' * X = B`.

Exceptions

Type	Condition
System.ArgumentException	Matrix dimensions do not match.
System.NonSymmetricMatrixException	Matrix is not symmetric.
System.NonPositiveDefiniteMatrixException	Matrix is not positive-definite.

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Solve(Double[,], Boolean)

Solves a set of equation systems of type A * X = B.

Declaration

public double[, ] Solve(double[, ] value, bool inPlace)

Parameters

Type	Name	Description
System.Double[,]	value	Right hand side matrix with as many rows as `A` and any number of columns.
System.Boolean	inPlace	True to compute the solving in place, false otherwise.

Returns

Type	Description
System.Double[,]	Matrix `X` so that `L * L' * X = B`.

Exceptions

Type	Condition
System.ArgumentException	Matrix dimensions do not match.
System.NonSymmetricMatrixException	Matrix is not symmetric.
System.NonPositiveDefiniteMatrixException	Matrix is not positive-definite.