Class VisualNumerics.math.DoubleSVD 
All Packages  Class Hierarchy  This Package  Previous  Next  Index

Class VisualNumerics.math.DoubleSVD 

java.lang.Object
   |
   +----VisualNumerics.math.DoubleSVD
public class DoubleSVD
extends Object
Singular Value Decomposition (SVD) of a rectangular matrix of type double. We denote this orthogonal factorization by A = U S VT.

Constructor Index

o DoubleSVD(double[][])
Construct the singular value decomposition of a rectangular matrix with default tolerance.
o DoubleSVD(double[][], double)
Construct the singular value decomposition of a rectangular matrix with a given tolerance.

Method Index

o info()
Return convergence information about S, U, and V.
o inverse()
Compute the generalized inverse of a real matrix.
o rank()
Return the rank of the matrix used to construct this instance.
o S()
Return the singular values of the matrix used to construct this instance in decreasing order of magnitude.
o U()
Return the left singular vectors resulting from the singular value decomposition of the matrix used to construct this instance.
o V()
Return the right singular vectors resulting from the singular value decomposition of the matrix used to construct this instance.

Constructors

o DoubleSVD 
  public DoubleSVD(double aa[][],
                   double tol)
Construct the singular value decomposition of a rectangular matrix with a given tolerance. This tolerance is used to determine rank. If the DoubleSVD constructor is called without the tol argument, the tolerance is chosen as 2.2204460492503e-14. If tol is positive, then a singular value is considered negligible if the singular value is less than or equal to tol. If tol is negative, then a singular value is considered negligible if the singular value is less than or equal to the absolute value of tol * the infinity norm of the input matrix. In the latter case, the absolute value of tol generally contains an estimate of the level of the relative error in the data.
Parameters:
aa - Input matrix of type double for which the singular value decomposition is to be computed.
tol - Input scalar of type double containing the tolerance used to determine when a singular value is negligible.
Throws: IllegalArgumentException
This exception is issued when the row lengths of input matrix a are not equal (i.e. the matrix edges are "jagged".)
Algorithm:
This constructor is based on the LINPACK routine DSVDC. Let n represent the number of rows of matrix aa and let p represent the number of columns of matrix aa. For any n x p matrix aa, there exists an n x n orthogonal matrix U and a p x p orthogonal matrix V such that UTAV = [S0], if n is greater than or equal to p, and UTAV = [S 0], if n is less than or equal to p. S is the diagonal matrix S = diag(s1,...,sm), where m=min(n,p). The scalars s1, s2, ...sm are called the singular values of matrix aa. The columns of U are called the left singular vectors of matrix aa. The columns of V are called the right singular vectors of aa.
Example: 
import VisualNumerics.math.DoubleSVD;
import VisualNumerics.math.MathException;
import java.io.*;
class testDSVD {
    public static void main(String args[]) {
      int      rank;
      double   s[], ainv[][];
      double   a[][] = {{  1.0,  0.0},
                        {  1.0,  1.0},
                        {100.0,-50.0}};
      try {
        DoubleSVD svd = new DoubleSVD(a);
        s = svd.S();
        rank = svd.rank();
        ainv = svd.inverse();
        //  ... print results ...
      } catch (IOException ioe) {
          System.err.println(ioe.toString());
      } catch (IllegalArgumentException e) {
          System.err.println(e.toString());
      } catch (MathException e) {
          System.err.println(e.toString());
      }
   }
}
o DoubleSVD 
  public DoubleSVD(double aa[][])
Construct the singular value decomposition of a rectangular matrix with default tolerance. The tolerance used is 2.2204460492503e-14. This tolerance is used to determine rank. A singular value is considered negligible if the singular value is less than or equal to this tolerance.
Parameters:
aa - Input matrix of type double for which the singular value decomposition is to be computed.
Throws: IllegalArgumentException
This exception is issued when the row lengths of input matrix a are not equal (i.e. the matrix edges are "jagged".)
Algorithm:
This constructor is based on the LINPACK routine DSVDC. Let n represent the number of rows of matrix aa and let p represent the number of columns of matrix aa. For any n x p matrix aa, there exists an n x n orthogonal matrix U and a p x p orthogonal matrix V such that UTAV = [S0], if n is greater than or equal to p, and UTAV = [S 0], if n is less than or equal to p. S is the diagonal matrix S = diag(s1,...,sm), where m=min(n,p). The scalars s1, s2, ...sm are called the singular values of matrix aa. The columns of U are called the left singular vectors of matrix aa. The columns of V
are called the right singular vectors of aa.

Methods

o rank 
  public int rank() throws MathException
Return the rank of the matrix used to construct this instance.
Returns:
An int scalar containing the rank of the matrix used to construct this instance.
Throws: MathException
This exception is issued when the rank cannot be determined because convergence was not obtained for all singular values.
Algorithm:
The DoubleSVD constructor computes the singular value decomposition of the input matrix. The estimated rank of the input matrix is the number of singular values which are larger than a tolerance t. If the tol used by the constructor is greater than zero, then t = tol. If the tol used by the constructor is less than zero, then t = (the infinity norm of input matrix aa) * |tol|.
o U 
  public double[][] U()
Return the left singular vectors resulting from the singular value decomposition of the matrix used to construct this instance.
Returns:
A matrix of type double containing the left singular vectors of the matrix used to construct this instance.
o V 
  public double[][] V()
Return the right singular vectors resulting from the singular value decomposition of the matrix used to construct this instance.
Returns:
A matrix of type double containing the right singular vectors of the matrix used to construct this instance.
o S 
  public double[] S()
Return the singular values of the matrix used to construct this instance in decreasing order of magnitude.
Returns:
A vector of type double containing the singular values of the matrix used to construct this instance in decreasing order of magnitude.
o info 
  public int info()
Return convergence information about S, U, and V.
Returns:
A scalar of type int containing information which indicates which singular values and corresponding singular vectors reached convergence. Convergence was obtained for the info, info+1, ..., min(nra,nca) singular values and their corresponding vectors. Here, nra and nca represent the number of rows and columns of the input matrix respectively.
o inverse 
  public double[][] inverse() throws MathException
Compute the generalized inverse of a real matrix.
Returns:
A matrix of type double containing the generalized inverse of the matrix used to construct this instance.
Throws: MathException
This exception is issued when the rank cannot be determined because convergence was not obtained for all singular values.
Algorithm:
This method computes the Moore-Penrose generalized inverse. The DoubleSVD constructor is used to compute the singular value decomposition of the input matrix. The resulting U and V matrices can be partitioned as U = (U1,U2) and V = (V1,V2) where
both U1 and V1 are k x k matrices. Let S1 = diag(s1,...,sm). The Moore-Penrose generalized inverse of input matrix aa is given by V1S1-1U1T. 
All Packages  Class Hierarchy  This Package  Previous  Next  Index