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:

  - the `norm2` method which has been renamed as `getNorm`,
  - the `cond` method which has been renamed as `getConditionNumber`,
  - the `rank` method which has been renamed as `getRank`,
  - a `getUT` method has been added,
  - a `getVT` method has been added,
  - a `getSolver` method has been added,
  - a `getCovariance` method has been added.
  Since:
    2.0 (changed to concrete class in 3.0)
  Version:
    $Id: SingularValueDecomposition.java 1538368 2013-11-03 13:57:37Z erans $
  See Also:
    MathWorld, Wikipedia

- - ### Constructor Summary

    Constructors
    Constructor and Description
    `SingularValueDecomposition(RealMatrix matrix)` Calculates the compact Singular Value Decomposition of the given matrix.

  - ### Method Summary

    Methods
    Modifier and Type | Method and 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:
        `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

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