smile.math.matrix

Class SymmMatrix.Cholesky

java.lang.Object

smile.math.matrix.SymmMatrix.Cholesky

All Implemented Interfaces:

java.io.Serializable

Enclosing class:

SymmMatrix

public static class SymmMatrix.Cholesky
extends java.lang.Object
implements java.io.Serializable

The Cholesky decomposition of a symmetric, 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.

See Also:

Serialized Form

Field Summary

Fields

Modifier and Type	Field and Description
SymmMatrix	lu The Cholesky decomposition.

Constructor Summary

Constructors

Constructor and Description
Cholesky(SymmMatrix lu) Constructor.

Method Summary

Modifier and Type	Method and Description
double	det() Returns the matrix determinant.
Matrix	inverse() Returns the matrix inverse.
double	logdet() Returns the log of matrix determinant.
double[]	solve(double[] b) Solves the linear system A * x = b.
void	solve(Matrix B) Solves 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

lu

public final SymmMatrix lu

The Cholesky decomposition.

Constructor Detail

Cholesky

public Cholesky(SymmMatrix lu)

Constructor.

Parameters:

lu - the lower/upper triangular part of matrix contains the Cholesky factorization.

Method Detail

det

public double det()

Returns the matrix determinant.

logdet

public double logdet()

Returns the log of matrix determinant.

inverse

public Matrix inverse()

Returns the matrix inverse.

solve

public double[] solve(double[] b)

Solves the linear system A * x = b.

Parameters:

b - the right hand side of linear systems.

Returns:

the solution vector.

solve

public void solve(Matrix B)

Solves the linear system A * X = B.

Parameters:

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