Class ComplexCholesky

java.lang.Object
org.flag4j.linalg.decompositions.Decomposition<CMatrix>
org.flag4j.linalg.decompositions.chol.Cholesky<CMatrix>
org.flag4j.linalg.decompositions.chol.ComplexCholesky
public class ComplexCholesky extends Cholesky<CMatrix>

An abstract base class for Cholesky decomposition of Hermitian (or Hermitian) positive-definite matrices.

The Cholesky decomposition factorizes a Hermitian/Hermitian, positive-definite matrix \( A \) as: \[ A = LL^{H} \] where \( L \) is a lower triangular matrix. The decomposition is primarily used for efficient numerical solutions to linear systems, computing matrix inverses, and generating samples from multivariate normal distributions.

Hermitian Verification:

This class provides an option to explicitly check whether the input matrix is Hermitian. If enforceHermitian is set to true, the implementation will verify that \( A \) satisfies \( A = A^{H} \) before performing decomposition. If set to false, the matrix is assumed to be Hermitian, no explicit check will be performed, and only the lower-diagonal entries of \( A \) are accessed.

positive-definiteness Check:

To ensure numerical stability, the algorithm verifies that all diagonal entries of \( L \) are positive. A tolerance threshold, posDefTolerance, is used to determine whether a diagonal entry is considered non-positive, indicating that the matrix is not positive-definite. This threshold can be adjusted using Cholesky.setPosDefTolerance(double).

Usage:

A typical workflow using a concrete implementation of Cholesky decomposition follows these steps:

  1. Instantiate an instance of ComplexCholesky.
  2. Call decompose(CMatrix) to compute the decomposition.
  3. Retrieve the factorized matrices using Cholesky.getL() or Cholesky.getLH().
See Also:

  • Constructor Details

    • ComplexCholesky

      public ComplexCholesky()

      Constructs a complex Cholesky decomposer.

    • ComplexCholesky

      public ComplexCholesky(boolean enforceHermitian)
      Constructs a complex Cholesky decomposer.
      Parameters:
      enforceHermitian - Flag indicating if an explicit check should be made that the matrix to be decomposed is Hermitian.
      • If true, the matrix will be explicitly verified to be Hermitian.
      • If false, no check will be made to verify the matrix is Hermitian, and it will be assumed to be.

  • Method Details

    • decompose

      public ComplexCholesky decompose(CMatrix src)
      Decompose a matrix into \( A = LL^{H \) where \( L \) is a lower triangular matrix and \( L^{H} \) is the conjugate transpose of \( L \).
      Specified by:
      decompose in class Decomposition<CMatrix>
      Parameters:
      src - The source matrix to decompose. Must be Hermitian positive-definite.
      Returns:
      A reference to this decomposer.
      Throws:
      IllegalArgumentException - If src.numRows != src.numCols or src is not Hermitian and enforceHermitian was set to true when this decomposer was instantiated.
      LinearAlgebraException - If src is not positive-definite.