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## org.apache.commons.math.linear
Class SingularValueDecompositionImpl

    java.lang.Object
      org.apache.commons.math.linear.SingularValueDecompositionImpl

All Implemented Interfaces:
  SingularValueDecomposition

    public class SingularValueDecompositionImpl

extends java.lang.Object
implements SingularValueDecomposition

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).

Since:
  2.0
Version:
  $Revision: 912413 $ $Date: 2010-02-21 16:46:12 -0500 (Sun, 21 Feb 2010) $

Nested Class Summary
`private static class` | `SingularValueDecompositionImpl.Solver` Specialized solver.

Field Summary
`private RealMatrix` | `cachedS` Cached value of S.
`private RealMatrix` | `cachedU` Cached value of U.
`private RealMatrix` | `cachedUt` Cached value of UT.
`private RealMatrix` | `cachedV` Cached value of V.
`private RealMatrix` | `cachedVt` Cached value of VT.
`private EigenDecomposition` | `eigenDecomposition` Eigen decomposition of the tridiagonal matrix.
`private int` | `m` Number of rows of the initial matrix.
`private int` | `n` Number of columns of the initial matrix.
`private double[]` | `singularValues` Singular values.

Constructor Summary
`SingularValueDecompositionImpl(RealMatrix matrix)` Calculates the compact Singular Value Decomposition of the given matrix.

Method Summary
`double` | `getConditionNumber()` Return the condition number of the matrix.
`RealMatrix` | `getCovariance(double minSingularValue)` Returns the n × n covariance matrix.
`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`

Field Detail

### m

    private int m

  Number of rows of the initial matrix.

### n

    private int n

  Number of columns of the initial matrix.

### eigenDecomposition

    private EigenDecomposition eigenDecomposition

  Eigen decomposition of the tridiagonal matrix.

### singularValues

    private double[] singularValues

  Singular values.

### cachedU

    private RealMatrix cachedU

  Cached value of U.

### cachedUt

    private RealMatrix cachedUt

  Cached value of UT.

### cachedS

    private RealMatrix cachedS

  Cached value of S.

### cachedV

    private RealMatrix cachedV

  Cached value of V.

### cachedVt

    private RealMatrix cachedVt

  Cached value of VT.

Constructor Detail

### SingularValueDecompositionImpl

    public SingularValueDecompositionImpl(RealMatrix matrix)
                                   throws InvalidMatrixException

  Calculates the compact Singular Value Decomposition of the given matrix.

  Parameters:
    `matrix` - The matrix to decompose.
  Throws:
    `InvalidMatrixException` - (wrapping a `ConvergenceException` if algorithm fails to converge

Method Detail

### getU

    public RealMatrix getU()
                    throws InvalidMatrixException

  Returns the matrix U of the decomposition.

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

  Specified by:
    `getU` in interface `SingularValueDecomposition`
  Returns:
    the U matrix
  Throws:
    `InvalidMatrixException`
  See Also:
    `SingularValueDecomposition.getUT()`

### getUT

    public RealMatrix getUT()
                     throws InvalidMatrixException

  Returns the transpose of the matrix U of the decomposition.

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

  Specified by:
    `getUT` in interface `SingularValueDecomposition`
  Returns:
    the U matrix (or null if decomposed matrix is singular)
  Throws:
    `InvalidMatrixException`
  See Also:
    `SingularValueDecomposition.getU()`

### getS

    public RealMatrix getS()
                    throws InvalidMatrixException

  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.

  Specified by:
    `getS` in interface `SingularValueDecomposition`
  Returns:
    the Σ matrix
  Throws:
    `InvalidMatrixException`

### getSingularValues

    public double[] getSingularValues()
                               throws InvalidMatrixException

  Returns the diagonal elements of the matrix Σ of the decomposition.

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

  Specified by:
    `getSingularValues` in interface `SingularValueDecomposition`
  Returns:
    the diagonal elements of the Σ matrix
  Throws:
    `InvalidMatrixException`

### getV

    public RealMatrix getV()
                    throws InvalidMatrixException

  Returns the matrix V of the decomposition.

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

  Specified by:
    `getV` in interface `SingularValueDecomposition`
  Returns:
    the V matrix (or null if decomposed matrix is singular)
  Throws:
    `InvalidMatrixException`
  See Also:
    `SingularValueDecomposition.getVT()`

### getVT

    public RealMatrix getVT()
                     throws InvalidMatrixException

  Returns the transpose of the matrix V of the decomposition.

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

  Specified by:
    `getVT` in interface `SingularValueDecomposition`
  Returns:
    the V matrix (or null if decomposed matrix is singular)
  Throws:
    `InvalidMatrixException`
  See Also:
    `SingularValueDecomposition.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.

  Specified by:
    `getCovariance` in interface `SingularValueDecomposition`
  Parameters:
    `minSingularValue` - value below which singular values are ignored (a 0 or negative value implies all singular value will be used)
  Returns:
    covariance matrix

### getNorm

    public double getNorm()
                   throws InvalidMatrixException

  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).

  Specified by:
    `getNorm` in interface `SingularValueDecomposition`
  Returns:
    norm
  Throws:
    `InvalidMatrixException`

### getConditionNumber

    public double getConditionNumber()
                              throws InvalidMatrixException

  Return the condition number of the matrix.

  Specified by:
    `getConditionNumber` in interface `SingularValueDecomposition`
  Returns:
    condition number of the matrix
  Throws:
    `InvalidMatrixException`

### getRank

    public int getRank()
                throws java.lang.IllegalStateException

  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.

  Specified by:
    `getRank` in interface `SingularValueDecomposition`
  Returns:
    effective numerical matrix rank
  Throws:
    `java.lang.IllegalStateException`

### getSolver

    public DecompositionSolver getSolver()

  Get a solver for finding the A × X = B solution in least square sense.

  Specified by:
    `getSolver` in interface `SingularValueDecomposition`
  Returns:
    a solver

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