Package org.apache.commons.math3.linear

Class SingularValueDecomposition

java.lang.Object
org.apache.commons.math3.linear.SingularValueDecomposition
public class SingularValueDecomposition extends Object
Calculates the compact Singular Value Decomposition of a matrix.

The Singular Value Decomposition of matrix A is a set of three matrices: U, Σ and V such that A = U × Σ × VT. Let A be a m × n matrix, then U is a m × p orthogonal matrix, Σ is a p × p diagonal matrix with positive or null elements, V is a p × n orthogonal matrix (hence VT is also orthogonal) where p=min(m,n).

This class is similar to the class with similar name from the JAMA library, with the following changes:

Since:
2.0 (changed to concrete class in 3.0)
See Also:

  • Constructor Details

    • SingularValueDecomposition

      public SingularValueDecomposition(RealMatrix matrix)
      Calculates the compact Singular Value Decomposition of the given matrix.
      Parameters:
      matrix - Matrix to decompose.

  • Method Details

    • getU

      public RealMatrix getU()
      Returns the matrix U of the decomposition.

      U is an orthogonal matrix, i.e. its transpose is also its inverse.

      Returns:
      the U matrix
      See Also:

    • getUT

      public RealMatrix getUT()
      Returns the transpose of the matrix U of the decomposition.

      U is an orthogonal matrix, i.e. its transpose is also its inverse.

      Returns:
      the U matrix (or null if decomposed matrix is singular)
      See Also:

    • getS

      public RealMatrix getS()
      Returns the diagonal matrix Σ of the decomposition.

      Σ is a diagonal matrix. The singular values are provided in non-increasing order, for compatibility with Jama.

      Returns:
      the Σ matrix

    • getSingularValues

      public double[] getSingularValues()
      Returns the diagonal elements of the matrix Σ of the decomposition.

      The singular values are provided in non-increasing order, for compatibility with Jama.

      Returns:
      the diagonal elements of the Σ matrix

    • getV

      public RealMatrix getV()
      Returns the matrix V of the decomposition.

      V is an orthogonal matrix, i.e. its transpose is also its inverse.

      Returns:
      the V matrix (or null if decomposed matrix is singular)
      See Also:

    • getVT

      public RealMatrix getVT()
      Returns the transpose of the matrix V of the decomposition.

      V is an orthogonal matrix, i.e. its transpose is also its inverse.

      Returns:
      the V matrix (or null if decomposed matrix is singular)
      See Also:

    • getCovariance

      public RealMatrix getCovariance(double minSingularValue)
      Returns the n × n covariance matrix.

      The covariance matrix is V × J × VT where J is the diagonal matrix of the inverse of the squares of the singular values.

      Parameters:
      minSingularValue - value below which singular values are ignored (a 0 or negative value implies all singular value will be used)
      Returns:
      covariance matrix
      Throws:
      IllegalArgumentException - if minSingularValue is larger than the largest singular value, meaning all singular values are ignored

    • getNorm

      public double getNorm()
      Returns the L2 norm of the matrix.

      The L2 norm is max(|A × u|2 / |u|2), where |.|2 denotes the vectorial 2-norm (i.e. the traditional euclidian norm).

      Returns:
      norm

    • getConditionNumber

      public double getConditionNumber()
      Return the condition number of the matrix.
      Returns:
      condition number of the matrix

    • getInverseConditionNumber

      public double getInverseConditionNumber()
      Computes the inverse of the condition number. In cases of rank deficiency, the condition number will become undefined.
      Returns:
      the inverse of the condition number.

    • getRank

      public int getRank()
      Return the effective numerical matrix rank.

      The effective numerical rank is the number of non-negligible singular values. The threshold used to identify non-negligible terms is max(m,n) × ulp(s1) where ulp(s1) is the least significant bit of the largest singular value.

      Returns:
      effective numerical matrix rank

    • getSolver

      public DecompositionSolver getSolver()
      Get a solver for finding the A × X = B solution in least square sense.
      Returns:
      a solver