libvtk7-dev 7.1.1+dfsg1-2 / usr / include / vtk-7.1 / alglib / bdsvd.h

/*************************************************************************
Copyright (c) 1992-2007 The University of Tennessee.  All rights reserved.

Contributors:
    * Sergey Bochkanov (ALGLIB project). Translation from FORTRAN to
      pseudocode.

See subroutines comments for additional copyrights.

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*************************************************************************/

#ifndef _bdsvd_h
#define _bdsvd_h

#include "alglib/ap.h"

#include "alglib/rotations.h"


/*************************************************************************
Singular value decomposition of a bidiagonal matrix (extended algorithm)

The algorithm performs the singular value decomposition  of  a  bidiagonal
matrix B (upper or lower) representing it as B = Q*S*P^T, where Q and  P -
orthogonal matrices, S - diagonal matrix with non-negative elements on the
main diagonal, in descending order.

The  algorithm  finds  singular  values.  In  addition,  the algorithm can
calculate  matrices  Q  and P (more precisely, not the matrices, but their
product  with  given  matrices U and VT - U*Q and (P^T)*VT)).  Of  course,
matrices U and VT can be of any type, including identity. Furthermore, the
algorithm can calculate Q'*C (this product is calculated more  effectively
than U*Q,  because  this calculation operates with rows instead  of matrix
columns).

The feature of the algorithm is its ability to find  all  singular  values
including those which are arbitrarily close to 0  with  relative  accuracy
close to  machine precision. If the parameter IsFractionalAccuracyRequired
is set to True, all singular values will have high relative accuracy close
to machine precision. If the parameter is set to False, only  the  biggest
singular value will have relative accuracy  close  to  machine  precision.
The absolute error of other singular values is equal to the absolute error
of the biggest singular value.

Input parameters:
    D       -   main diagonal of matrix B.
                Array whose index ranges within [0..N-1].
    E       -   superdiagonal (or subdiagonal) of matrix B.
                Array whose index ranges within [0..N-2].
    N       -   size of matrix B.
    IsUpper -   True, if the matrix is upper bidiagonal.
    IsFractionalAccuracyRequired -
                accuracy to search singular values with.
    U       -   matrix to be multiplied by Q.
                Array whose indexes range within [0..NRU-1, 0..N-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..NRU-1, 0..N-1] will be multiplied by Q.
    NRU     -   number of rows in matrix U.
    C       -   matrix to be multiplied by Q'.
                Array whose indexes range within [0..N-1, 0..NCC-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..N-1, 0..NCC-1] will be multiplied by Q'.
    NCC     -   number of columns in matrix C.
    VT      -   matrix to be multiplied by P^T.
                Array whose indexes range within [0..N-1, 0..NCVT-1].
                The matrix can be bigger, in that case only the  submatrix
                [0..N-1, 0..NCVT-1] will be multiplied by P^T.
    NCVT    -   number of columns in matrix VT.

Output parameters:
    D       -   singular values of matrix B in descending order.
    U       -   if NRU>0, contains matrix U*Q.
    VT      -   if NCVT>0, contains matrix (P^T)*VT.
    C       -   if NCC>0, contains matrix Q'*C.

Result:
    True, if the algorithm has converged.
    False, if the algorithm hasn't converged (rare case).

Additional information:
    The type of convergence is controlled by the internal  parameter  TOL.
    If the parameter is greater than 0, the singular values will have
    relative accuracy TOL. If TOL<0, the singular values will have
    absolute accuracy ABS(TOL)*norm(B).
    By default, |TOL| falls within the range of 10*Epsilon and 100*Epsilon,
    where Epsilon is the machine precision. It is not  recommended  to  use
    TOL less than 10*Epsilon since this will  considerably  slow  down  the
    algorithm and may not lead to error decreasing.
History:
    * 31 March, 2007.
        changed MAXITR from 6 to 12.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1999.
*************************************************************************/
ALGLIB_EXPORT
bool rmatrixbdsvd(ap::real_1d_array& d,
     ap::real_1d_array e,
     int n,
     bool isupper,
     bool isfractionalaccuracyrequired,
     ap::real_2d_array& u,
     int nru,
     ap::real_2d_array& c,
     int ncc,
     ap::real_2d_array& vt,
     int ncvt);


/*************************************************************************
Obsolete 1-based subroutine. See RMatrixBDSVD for 0-based replacement.

History:
    * 31 March, 2007.
        changed MAXITR from 6 to 12.

  -- LAPACK routine (version 3.0) --
     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
     Courant Institute, Argonne National Lab, and Rice University
     October 31, 1999.
*************************************************************************/
ALGLIB_EXPORT
bool bidiagonalsvddecomposition(ap::real_1d_array& d,
     ap::real_1d_array e,
     int n,
     bool isupper,
     bool isfractionalaccuracyrequired,
     ap::real_2d_array& u,
     int nru,
     ap::real_2d_array& c,
     int ncc,
     ap::real_2d_array& vt,
     int ncvt);


#endif