smile.math.matrix

Class CholeskyDecomposition

java.lang.Object

smile.math.matrix.CholeskyDecomposition

public class CholeskyDecomposition
extends Object

Cholesky decomposition is a decomposition of a symmetric, positive-definite matrix into a lower triangular matrix L and the transpose of the lower triangular matrix such that A = L*L'. The lower triangular matrix is the Cholesky triangle of the original, positive-definite matrix. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.

The Cholesky decomposition is mainly used for the numerical solution of linear equations. The Cholesky decomposition is also commonly used in the Monte Carlo method for simulating systems with multiple correlated variables: The matrix of inter-variable correlations is decomposed, to give the lower-triangular L. Applying this to a vector of uncorrelated simulated shocks, u, produces a shock vector Lu with the covariance properties of the system being modeled.

Unscented Kalman filters commonly use the Cholesky decomposition to choose a set of so-called sigma points. The Kalman filter tracks the average state of a system as a vector x of length n and covariance as an n-by-n matrix P. The matrix P is always positive semi-definite, and can be decomposed into L*L'. The columns of L can be added and subtracted from the mean x to form a set of 2n vectors called sigma points. These sigma points completely capture the mean and covariance of the system state.

If the matrix is not positive definite, an exception will be thrown out from the method decompose().

Author:

Haifeng Li

Constructor Summary

Constructors

Constructor and Description
CholeskyDecomposition(double[][] A) Constructor.
CholeskyDecomposition(double[][] A, boolean overwrite) Constructor.

Method Summary

Methods

Modifier and Type	Method and Description
double	det() Returns the matrix determinant
double[][]	getL() Returns lower triangular factor.
double[][]	inverse() Returns the matrix inverse.
static CholeskyDecomposition	newInstance(double[][] L) Constructor.
void	solve(double[] b) Solve the linear system A * x = b.
void	solve(double[][] B) Solve the linear system A * X = B.
void	solve(double[][] B, double[][] X) Solve the linear system A * X = B.
void	solve(double[] b, double[] x) Solve the linear system A * x = b.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

CholeskyDecomposition

public CholeskyDecomposition(double[][] A)

Constructor. Cholesky decomposition for symmetric and positive definite matrix. A new matrix will allocated to store the decomposition.

Parameters:

A - square, symmetric matrix. Only the lower triangular part will be used in the decomposition. The user can just store this half part to save space.

Throws:

IllegalArgumentException - if the matrix is not positive definite.

CholeskyDecomposition

public CholeskyDecomposition(double[][] A,
                     boolean overwrite)

Constructor. Cholesky decomposition for symmetric and positive definite matrix. The user can specify if the decomposition takes in place, i.e. if the decomposition will be stored in the input matrix. Otherwise, a new matrix will be allocated to store the decomposition.

Parameters:

A - square symmetric matrix. Only the lower triangular part will be used in the decomposition. The user can just store this half part to save space.

overwrite - true if the decomposition will be taken in place. Otherwise, a new matrix will be allocated to store the decomposition. It is very useful in practice if the matrix is huge.

Throws:

IllegalArgumentException - if the matrix is not positive definite.

Method Detail

newInstance

public static CholeskyDecomposition newInstance(double[][] L)

Constructor. The decomposition is already available, e.g. from QR decomposition.

Parameters:

L - decomposition matrix.

getL

public double[][] getL()

Returns lower triangular factor.

det

public double det()

Returns the matrix determinant

inverse

public double[][] inverse()

Returns the matrix inverse.

solve

public void solve(double[] b)

Solve the linear system A * x = b. On output, b will be overwritten with the solution vector.

Parameters:

b - the right hand side of linear systems. On output, b will be overwritten with solution vector.

solve

public void solve(double[] b,
         double[] x)

Solve the linear system A * x = b.

Parameters:

b - the right hand side of linear systems.

x - the solution vector.

solve

public void solve(double[][] B)

Solve the linear system A * X = B. On output, B will be overwritten with the solution matrix.

Parameters:

B - the right hand side of linear systems.

solve

public void solve(double[][] B,
         double[][] X)

Solve the linear system A * X = B.

Parameters:

B - the right hand side of linear systems.

X - the solution matrix.