smile.math.matrix

Class Cholesky

java.lang.Object

smile.math.matrix.Cholesky

public class Cholesky
extends java.lang.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().

Field Summary

Fields

Modifier and Type	Field and Description
protected DenseMatrix	L Array for internal storage of decomposition.

Constructor Summary

Constructors

Constructor and Description
Cholesky(DenseMatrix L) Constructor.

Method Summary

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

Field Detail

L

protected DenseMatrix L

Array for internal storage of decomposition.

Constructor Detail

Cholesky

public Cholesky(DenseMatrix L)

Constructor.

Parameters:

L - the lower triangular part of matrix contains the Cholesky factorization.

Method Detail

getL

public DenseMatrix getL()

Returns lower triangular factor.

det

public double det()

Returns the matrix determinant

inverse

public DenseMatrix 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(DenseMatrix 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.