Package org.apache.commons.math4.linear

Class SingularValueDecomposition

  • public class SingularValueDecomposition
extends java.lang.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:
    MathWorld, Wikipedia

    • Method Summary

      Modifier and Type Method Description
      double getConditionNumber()
      Return the condition number of the matrix.
      RealMatrix getCovariance​(double minSingularValue)
      Returns the n × n covariance matrix.
      double getInverseConditionNumber()
      Computes the inverse of the condition number.
      double getNorm()
      Returns the L2 norm of the matrix.
      int getRank()
      Return the effective numerical matrix rank.
      RealMatrix getS()
      Returns the diagonal matrix Σ of the decomposition.
      double[] getSingularValues()
      Returns the diagonal elements of the matrix Σ of the decomposition.
      DecompositionSolver getSolver()
      Get a solver for finding the A × X = B solution in least square sense.
      RealMatrix getU()
      Returns the matrix U of the decomposition.
      RealMatrix getUT()
      Returns the transpose of the matrix U of the decomposition.
      RealMatrix getV()
      Returns the matrix V of the decomposition.
      RealMatrix getVT()
      Returns the transpose of the matrix V of the decomposition.

      • Methods inherited from class java.lang.Object

        clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

    • Constructor Detail

      • SingularValueDecomposition

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

    • Method Detail

      • 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()

      • 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:
        getU()

      • 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()

      • 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:
        getV()

      • 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:
        java.lang.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