Package smile.math.matrix

Class BandMatrix.Cholesky

java.lang.Object
smile.math.matrix.BandMatrix.Cholesky
All Implemented Interfaces:
Serializable
Enclosing class:
BandMatrix
public static class BandMatrix.Cholesky extends Object implements 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:

  • Field Details

    • lu

      public final BandMatrix lu
      The Cholesky decomposition.

  • Constructor Details

    • Cholesky

      public Cholesky(BandMatrix lu)
      Constructor.
      Parameters:
      lu - the lower/upper triangular part of matrix contains the Cholesky factorization.

  • Method Details

    • det

      public double det()
      Returns the matrix determinant.
      Returns:
      the matrix determinant.

    • logdet

      public double logdet()
      Returns the log of matrix determinant.
      Returns:
      the log of matrix determinant.

    • inverse

      public Matrix inverse()
      Returns the inverse of matrix.
      Returns:
      the inverse of matrix.

    • 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.