/usr/include/vtk-5.8/alglib/bdsvd.h is in libvtk5-dev 5.8.0-5.
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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.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
met:
- Redistributions of source code must retain the above copyright
notice, this list of conditions and the following disclaimer.
- Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer listed
in this license in the documentation and/or other materials
provided with the distribution.
- Neither the name of the copyright holders nor the names of its
contributors may be used to endorse or promote products derived from
this software without specific prior written permission.
THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
"AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
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OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
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LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
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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
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