/usr/include/ortfac.h is in libalglib-dev 2.6.0-3.
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Copyright (c) 2005-2010 Sergey Bochkanov.
Additional copyrights:
1992-2007 The University of Tennessee (as indicated in subroutines
comments).
>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses
>>> END OF LICENSE >>>
*************************************************************************/
#ifndef _ortfac_h
#define _ortfac_h
#include "ap.h"
#include "ialglib.h"
#include "hblas.h"
#include "reflections.h"
#include "creflections.h"
#include "sblas.h"
#include "ablasf.h"
#include "ablas.h"
/*************************************************************************
QR decomposition of a rectangular matrix of size MxN
Input parameters:
A - matrix A whose indexes range within [0..M-1, 0..N-1].
M - number of rows in matrix A.
N - number of columns in matrix A.
Output parameters:
A - matrices Q and R in compact form (see below).
Tau - array of scalar factors which are used to form
matrix Q. Array whose index ranges within [0.. Min(M-1,N-1)].
Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
MxM, R - upper triangular (or upper trapezoid) matrix of size M x N.
The elements of matrix R are located on and above the main diagonal of
matrix A. The elements which are located in Tau array and below the main
diagonal of matrix A are used to form matrix Q as follows:
Matrix Q is represented as a product of elementary reflections
Q = H(0)*H(2)*...*H(k-1),
where k = min(m,n), and each H(i) is in the form
H(i) = 1 - tau * v * (v^T)
where tau is a scalar stored in Tau[I]; v - real vector,
so that v(0:i-1) = 0, v(i) = 1, v(i+1:m-1) stored in A(i+1:m-1,i).
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixqr(ap::real_2d_array& a, int m, int n, ap::real_1d_array& tau);
/*************************************************************************
LQ decomposition of a rectangular matrix of size MxN
Input parameters:
A - matrix A whose indexes range within [0..M-1, 0..N-1].
M - number of rows in matrix A.
N - number of columns in matrix A.
Output parameters:
A - matrices L and Q in compact form (see below)
Tau - array of scalar factors which are used to form
matrix Q. Array whose index ranges within [0..Min(M,N)-1].
Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
MxM, L - lower triangular (or lower trapezoid) matrix of size M x N.
The elements of matrix L are located on and below the main diagonal of
matrix A. The elements which are located in Tau array and above the main
diagonal of matrix A are used to form matrix Q as follows:
Matrix Q is represented as a product of elementary reflections
Q = H(k-1)*H(k-2)*...*H(1)*H(0),
where k = min(m,n), and each H(i) is of the form
H(i) = 1 - tau * v * (v^T)
where tau is a scalar stored in Tau[I]; v - real vector, so that v(0:i-1)=0,
v(i) = 1, v(i+1:n-1) stored in A(i,i+1:n-1).
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixlq(ap::real_2d_array& a, int m, int n, ap::real_1d_array& tau);
/*************************************************************************
QR decomposition of a rectangular complex matrix of size MxN
Input parameters:
A - matrix A whose indexes range within [0..M-1, 0..N-1]
M - number of rows in matrix A.
N - number of columns in matrix A.
Output parameters:
A - matrices Q and R in compact form
Tau - array of scalar factors which are used to form matrix Q. Array
whose indexes range within [0.. Min(M,N)-1]
Matrix A is represented as A = QR, where Q is an orthogonal matrix of size
MxM, R - upper triangular (or upper trapezoid) matrix of size MxN.
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
*************************************************************************/
void cmatrixqr(ap::complex_2d_array& a,
int m,
int n,
ap::complex_1d_array& tau);
/*************************************************************************
LQ decomposition of a rectangular complex matrix of size MxN
Input parameters:
A - matrix A whose indexes range within [0..M-1, 0..N-1]
M - number of rows in matrix A.
N - number of columns in matrix A.
Output parameters:
A - matrices Q and L in compact form
Tau - array of scalar factors which are used to form matrix Q. Array
whose indexes range within [0.. Min(M,N)-1]
Matrix A is represented as A = LQ, where Q is an orthogonal matrix of size
MxM, L - lower triangular (or lower trapezoid) matrix of size MxN.
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994
*************************************************************************/
void cmatrixlq(ap::complex_2d_array& a,
int m,
int n,
ap::complex_1d_array& tau);
/*************************************************************************
Partial unpacking of matrix Q from the QR decomposition of a matrix A
Input parameters:
A - matrices Q and R in compact form.
Output of RMatrixQR subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Tau - scalar factors which are used to form Q.
Output of the RMatrixQR subroutine.
QColumns - required number of columns of matrix Q. M>=QColumns>=0.
Output parameters:
Q - first QColumns columns of matrix Q.
Array whose indexes range within [0..M-1, 0..QColumns-1].
If QColumns=0, the array remains unchanged.
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixqrunpackq(const ap::real_2d_array& a,
int m,
int n,
const ap::real_1d_array& tau,
int qcolumns,
ap::real_2d_array& q);
/*************************************************************************
Unpacking of matrix R from the QR decomposition of a matrix A
Input parameters:
A - matrices Q and R in compact form.
Output of RMatrixQR subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Output parameters:
R - matrix R, array[0..M-1, 0..N-1].
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixqrunpackr(const ap::real_2d_array& a,
int m,
int n,
ap::real_2d_array& r);
/*************************************************************************
Partial unpacking of matrix Q from the LQ decomposition of a matrix A
Input parameters:
A - matrices L and Q in compact form.
Output of RMatrixLQ subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Tau - scalar factors which are used to form Q.
Output of the RMatrixLQ subroutine.
QRows - required number of rows in matrix Q. N>=QRows>=0.
Output parameters:
Q - first QRows rows of matrix Q. Array whose indexes range
within [0..QRows-1, 0..N-1]. If QRows=0, the array remains
unchanged.
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixlqunpackq(const ap::real_2d_array& a,
int m,
int n,
const ap::real_1d_array& tau,
int qrows,
ap::real_2d_array& q);
/*************************************************************************
Unpacking of matrix L from the LQ decomposition of a matrix A
Input parameters:
A - matrices Q and L in compact form.
Output of RMatrixLQ subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Output parameters:
L - matrix L, array[0..M-1, 0..N-1].
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void rmatrixlqunpackl(const ap::real_2d_array& a,
int m,
int n,
ap::real_2d_array& l);
/*************************************************************************
Partial unpacking of matrix Q from QR decomposition of a complex matrix A.
Input parameters:
A - matrices Q and R in compact form.
Output of CMatrixQR subroutine .
M - number of rows in matrix A. M>=0.
N - number of columns in matrix A. N>=0.
Tau - scalar factors which are used to form Q.
Output of CMatrixQR subroutine .
QColumns - required number of columns in matrix Q. M>=QColumns>=0.
Output parameters:
Q - first QColumns columns of matrix Q.
Array whose index ranges within [0..M-1, 0..QColumns-1].
If QColumns=0, array isn't changed.
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void cmatrixqrunpackq(const ap::complex_2d_array& a,
int m,
int n,
const ap::complex_1d_array& tau,
int qcolumns,
ap::complex_2d_array& q);
/*************************************************************************
Unpacking of matrix R from the QR decomposition of a matrix A
Input parameters:
A - matrices Q and R in compact form.
Output of CMatrixQR subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Output parameters:
R - matrix R, array[0..M-1, 0..N-1].
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void cmatrixqrunpackr(const ap::complex_2d_array& a,
int m,
int n,
ap::complex_2d_array& r);
/*************************************************************************
Partial unpacking of matrix Q from LQ decomposition of a complex matrix A.
Input parameters:
A - matrices Q and R in compact form.
Output of CMatrixLQ subroutine .
M - number of rows in matrix A. M>=0.
N - number of columns in matrix A. N>=0.
Tau - scalar factors which are used to form Q.
Output of CMatrixLQ subroutine .
QRows - required number of rows in matrix Q. N>=QColumns>=0.
Output parameters:
Q - first QRows rows of matrix Q.
Array whose index ranges within [0..QRows-1, 0..N-1].
If QRows=0, array isn't changed.
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void cmatrixlqunpackq(const ap::complex_2d_array& a,
int m,
int n,
const ap::complex_1d_array& tau,
int qrows,
ap::complex_2d_array& q);
/*************************************************************************
Unpacking of matrix L from the LQ decomposition of a matrix A
Input parameters:
A - matrices Q and L in compact form.
Output of CMatrixLQ subroutine.
M - number of rows in given matrix A. M>=0.
N - number of columns in given matrix A. N>=0.
Output parameters:
L - matrix L, array[0..M-1, 0..N-1].
-- ALGLIB routine --
17.02.2010
Bochkanov Sergey
*************************************************************************/
void cmatrixlqunpackl(const ap::complex_2d_array& a,
int m,
int n,
ap::complex_2d_array& l);
/*************************************************************************
Reduction of a rectangular matrix to bidiagonal form
The algorithm reduces the rectangular matrix A to bidiagonal form by
orthogonal transformations P and Q: A = Q*B*P.
Input parameters:
A - source matrix. array[0..M-1, 0..N-1]
M - number of rows in matrix A.
N - number of columns in matrix A.
Output parameters:
A - matrices Q, B, P in compact form (see below).
TauQ - scalar factors which are used to form matrix Q.
TauP - scalar factors which are used to form matrix P.
The main diagonal and one of the secondary diagonals of matrix A are
replaced with bidiagonal matrix B. Other elements contain elementary
reflections which form MxM matrix Q and NxN matrix P, respectively.
If M>=N, B is the upper bidiagonal MxN matrix and is stored in the
corresponding elements of matrix A. Matrix Q is represented as a
product of elementary reflections Q = H(0)*H(1)*...*H(n-1), where
H(i) = 1-tau*v*v'. Here tau is a scalar which is stored in TauQ[i], and
vector v has the following structure: v(0:i-1)=0, v(i)=1, v(i+1:m-1) is
stored in elements A(i+1:m-1,i). Matrix P is as follows: P =
G(0)*G(1)*...*G(n-2), where G(i) = 1 - tau*u*u'. Tau is stored in TauP[i],
u(0:i)=0, u(i+1)=1, u(i+2:n-1) is stored in elements A(i,i+2:n-1).
If M<N, B is the lower bidiagonal MxN matrix and is stored in the
corresponding elements of matrix A. Q = H(0)*H(1)*...*H(m-2), where
H(i) = 1 - tau*v*v', tau is stored in TauQ, v(0:i)=0, v(i+1)=1, v(i+2:m-1)
is stored in elements A(i+2:m-1,i). P = G(0)*G(1)*...*G(m-1),
G(i) = 1-tau*u*u', tau is stored in TauP, u(0:i-1)=0, u(i)=1, u(i+1:n-1)
is stored in A(i,i+1:n-1).
EXAMPLE:
m=6, n=5 (m > n): m=5, n=6 (m < n):
( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
( v1 v2 v3 v4 v5 )
Here vi and ui are vectors which form H(i) and G(i), and d and e -
are the diagonal and off-diagonal elements of matrix B.
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
September 30, 1994.
Sergey Bochkanov, ALGLIB project, translation from FORTRAN to
pseudocode, 2007-2010.
*************************************************************************/
void rmatrixbd(ap::real_2d_array& a,
int m,
int n,
ap::real_1d_array& tauq,
ap::real_1d_array& taup);
/*************************************************************************
Unpacking matrix Q which reduces a matrix to bidiagonal form.
Input parameters:
QP - matrices Q and P in compact form.
Output of ToBidiagonal subroutine.
M - number of rows in matrix A.
N - number of columns in matrix A.
TAUQ - scalar factors which are used to form Q.
Output of ToBidiagonal subroutine.
QColumns - required number of columns in matrix Q.
M>=QColumns>=0.
Output parameters:
Q - first QColumns columns of matrix Q.
Array[0..M-1, 0..QColumns-1]
If QColumns=0, the array is not modified.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackq(const ap::real_2d_array& qp,
int m,
int n,
const ap::real_1d_array& tauq,
int qcolumns,
ap::real_2d_array& q);
/*************************************************************************
Multiplication by matrix Q which reduces matrix A to bidiagonal form.
The algorithm allows pre- or post-multiply by Q or Q'.
Input parameters:
QP - matrices Q and P in compact form.
Output of ToBidiagonal subroutine.
M - number of rows in matrix A.
N - number of columns in matrix A.
TAUQ - scalar factors which are used to form Q.
Output of ToBidiagonal subroutine.
Z - multiplied matrix.
array[0..ZRows-1,0..ZColumns-1]
ZRows - number of rows in matrix Z. If FromTheRight=False,
ZRows=M, otherwise ZRows can be arbitrary.
ZColumns - number of columns in matrix Z. If FromTheRight=True,
ZColumns=M, otherwise ZColumns can be arbitrary.
FromTheRight - pre- or post-multiply.
DoTranspose - multiply by Q or Q'.
Output parameters:
Z - product of Z and Q.
Array[0..ZRows-1,0..ZColumns-1]
If ZRows=0 or ZColumns=0, the array is not modified.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdmultiplybyq(const ap::real_2d_array& qp,
int m,
int n,
const ap::real_1d_array& tauq,
ap::real_2d_array& z,
int zrows,
int zcolumns,
bool fromtheright,
bool dotranspose);
/*************************************************************************
Unpacking matrix P which reduces matrix A to bidiagonal form.
The subroutine returns transposed matrix P.
Input parameters:
QP - matrices Q and P in compact form.
Output of ToBidiagonal subroutine.
M - number of rows in matrix A.
N - number of columns in matrix A.
TAUP - scalar factors which are used to form P.
Output of ToBidiagonal subroutine.
PTRows - required number of rows of matrix P^T. N >= PTRows >= 0.
Output parameters:
PT - first PTRows columns of matrix P^T
Array[0..PTRows-1, 0..N-1]
If PTRows=0, the array is not modified.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackpt(const ap::real_2d_array& qp,
int m,
int n,
const ap::real_1d_array& taup,
int ptrows,
ap::real_2d_array& pt);
/*************************************************************************
Multiplication by matrix P which reduces matrix A to bidiagonal form.
The algorithm allows pre- or post-multiply by P or P'.
Input parameters:
QP - matrices Q and P in compact form.
Output of RMatrixBD subroutine.
M - number of rows in matrix A.
N - number of columns in matrix A.
TAUP - scalar factors which are used to form P.
Output of RMatrixBD subroutine.
Z - multiplied matrix.
Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
ZRows - number of rows in matrix Z. If FromTheRight=False,
ZRows=N, otherwise ZRows can be arbitrary.
ZColumns - number of columns in matrix Z. If FromTheRight=True,
ZColumns=N, otherwise ZColumns can be arbitrary.
FromTheRight - pre- or post-multiply.
DoTranspose - multiply by P or P'.
Output parameters:
Z - product of Z and P.
Array whose indexes range within [0..ZRows-1,0..ZColumns-1].
If ZRows=0 or ZColumns=0, the array is not modified.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdmultiplybyp(const ap::real_2d_array& qp,
int m,
int n,
const ap::real_1d_array& taup,
ap::real_2d_array& z,
int zrows,
int zcolumns,
bool fromtheright,
bool dotranspose);
/*************************************************************************
Unpacking of the main and secondary diagonals of bidiagonal decomposition
of matrix A.
Input parameters:
B - output of RMatrixBD subroutine.
M - number of rows in matrix B.
N - number of columns in matrix B.
Output parameters:
IsUpper - True, if the matrix is upper bidiagonal.
otherwise IsUpper is False.
D - the main diagonal.
Array whose index ranges within [0..Min(M,N)-1].
E - the secondary diagonal (upper or lower, depending on
the value of IsUpper).
Array index ranges within [0..Min(M,N)-1], the last
element is not used.
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixbdunpackdiagonals(const ap::real_2d_array& b,
int m,
int n,
bool& isupper,
ap::real_1d_array& d,
ap::real_1d_array& e);
/*************************************************************************
Reduction of a square matrix to upper Hessenberg form: Q'*A*Q = H,
where Q is an orthogonal matrix, H - Hessenberg matrix.
Input parameters:
A - matrix A with elements [0..N-1, 0..N-1]
N - size of matrix A.
Output parameters:
A - matrices Q and P in compact form (see below).
Tau - array of scalar factors which are used to form matrix Q.
Array whose index ranges within [0..N-2]
Matrix H is located on the main diagonal, on the lower secondary diagonal
and above the main diagonal of matrix A. The elements which are used to
form matrix Q are situated in array Tau and below the lower secondary
diagonal of matrix A as follows:
Matrix Q is represented as a product of elementary reflections
Q = H(0)*H(2)*...*H(n-2),
where each H(i) is given by
H(i) = 1 - tau * v * (v^T)
where tau is a scalar stored in Tau[I]; v - is a real vector,
so that v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) stored in A(i+2:n-1,i).
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1992
*************************************************************************/
void rmatrixhessenberg(ap::real_2d_array& a, int n, ap::real_1d_array& tau);
/*************************************************************************
Unpacking matrix Q which reduces matrix A to upper Hessenberg form
Input parameters:
A - output of RMatrixHessenberg subroutine.
N - size of matrix A.
Tau - scalar factors which are used to form Q.
Output of RMatrixHessenberg subroutine.
Output parameters:
Q - matrix Q.
Array whose indexes range within [0..N-1, 0..N-1].
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixhessenbergunpackq(const ap::real_2d_array& a,
int n,
const ap::real_1d_array& tau,
ap::real_2d_array& q);
/*************************************************************************
Unpacking matrix H (the result of matrix A reduction to upper Hessenberg form)
Input parameters:
A - output of RMatrixHessenberg subroutine.
N - size of matrix A.
Output parameters:
H - matrix H. Array whose indexes range within [0..N-1, 0..N-1].
-- ALGLIB --
2005-2010
Bochkanov Sergey
*************************************************************************/
void rmatrixhessenbergunpackh(const ap::real_2d_array& a,
int n,
ap::real_2d_array& h);
/*************************************************************************
Reduction of a symmetric matrix which is given by its higher or lower
triangular part to a tridiagonal matrix using orthogonal similarity
transformation: Q'*A*Q=T.
Input parameters:
A - matrix to be transformed
array with elements [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - storage format. If IsUpper = True, then matrix A is given
by its upper triangle, and the lower triangle is not used
and not modified by the algorithm, and vice versa
if IsUpper = False.
Output parameters:
A - matrices T and Q in compact form (see lower)
Tau - array of factors which are forming matrices H(i)
array with elements [0..N-2].
D - main diagonal of symmetric matrix T.
array with elements [0..N-1].
E - secondary diagonal of symmetric matrix T.
array with elements [0..N-2].
If IsUpper=True, the matrix Q is represented as a product of elementary
reflectors
Q = H(n-2) . . . H(2) H(0).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
A(0:i-1,i+1), and tau in TAU(i).
If IsUpper=False, the matrix Q is represented as a product of elementary
reflectors
Q = H(0) H(2) . . . H(n-2).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v1 v2 v3 ) ( d )
( d e v2 v3 ) ( e d )
( d e v3 ) ( v0 e d )
( d e ) ( v0 v1 e d )
( d ) ( v0 v1 v2 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1992
*************************************************************************/
void smatrixtd(ap::real_2d_array& a,
int n,
bool isupper,
ap::real_1d_array& tau,
ap::real_1d_array& d,
ap::real_1d_array& e);
/*************************************************************************
Unpacking matrix Q which reduces symmetric matrix to a tridiagonal
form.
Input parameters:
A - the result of a SMatrixTD subroutine
N - size of matrix A.
IsUpper - storage format (a parameter of SMatrixTD subroutine)
Tau - the result of a SMatrixTD subroutine
Output parameters:
Q - transformation matrix.
array with elements [0..N-1, 0..N-1].
-- ALGLIB --
Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void smatrixtdunpackq(const ap::real_2d_array& a,
const int& n,
const bool& isupper,
const ap::real_1d_array& tau,
ap::real_2d_array& q);
/*************************************************************************
Reduction of a Hermitian matrix which is given by its higher or lower
triangular part to a real tridiagonal matrix using unitary similarity
transformation: Q'*A*Q = T.
Input parameters:
A - matrix to be transformed
array with elements [0..N-1, 0..N-1].
N - size of matrix A.
IsUpper - storage format. If IsUpper = True, then matrix A is given
by its upper triangle, and the lower triangle is not used
and not modified by the algorithm, and vice versa
if IsUpper = False.
Output parameters:
A - matrices T and Q in compact form (see lower)
Tau - array of factors which are forming matrices H(i)
array with elements [0..N-2].
D - main diagonal of real symmetric matrix T.
array with elements [0..N-1].
E - secondary diagonal of real symmetric matrix T.
array with elements [0..N-2].
If IsUpper=True, the matrix Q is represented as a product of elementary
reflectors
Q = H(n-2) . . . H(2) H(0).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector with
v(i+1:n-1) = 0, v(i) = 1, v(0:i-1) is stored on exit in
A(0:i-1,i+1), and tau in TAU(i).
If IsUpper=False, the matrix Q is represented as a product of elementary
reflectors
Q = H(0) H(2) . . . H(n-2).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a complex scalar, and v is a complex vector with
v(0:i) = 0, v(i+1) = 1, v(i+2:n-1) is stored on exit in A(i+2:n-1,i),
and tau in TAU(i).
The contents of A on exit are illustrated by the following examples
with n = 5:
if UPLO = 'U': if UPLO = 'L':
( d e v1 v2 v3 ) ( d )
( d e v2 v3 ) ( e d )
( d e v3 ) ( v0 e d )
( d e ) ( v0 v1 e d )
( d ) ( v0 v1 v2 e d )
where d and e denote diagonal and off-diagonal elements of T, and vi
denotes an element of the vector defining H(i).
-- LAPACK routine (version 3.0) --
Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
Courant Institute, Argonne National Lab, and Rice University
October 31, 1992
*************************************************************************/
void hmatrixtd(ap::complex_2d_array& a,
int n,
bool isupper,
ap::complex_1d_array& tau,
ap::real_1d_array& d,
ap::real_1d_array& e);
/*************************************************************************
Unpacking matrix Q which reduces a Hermitian matrix to a real tridiagonal
form.
Input parameters:
A - the result of a HMatrixTD subroutine
N - size of matrix A.
IsUpper - storage format (a parameter of HMatrixTD subroutine)
Tau - the result of a HMatrixTD subroutine
Output parameters:
Q - transformation matrix.
array with elements [0..N-1, 0..N-1].
-- ALGLIB --
Copyright 2005-2010 by Bochkanov Sergey
*************************************************************************/
void hmatrixtdunpackq(const ap::complex_2d_array& a,
const int& n,
const bool& isupper,
const ap::complex_1d_array& tau,
ap::complex_2d_array& q);
#endif
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