439 lines
14 KiB
Fortran
439 lines
14 KiB
Fortran
*> \brief \b SORHR_COL
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download SORHR_COL + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sorhr_col.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sorhr_col.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sorhr_col.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE SORHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LDA, LDT, M, N, NB
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* ..
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* .. Array Arguments ..
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* REAL A( LDA, * ), D( * ), T( LDT, * )
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* ..
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> SORHR_COL takes an M-by-N real matrix Q_in with orthonormal columns
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*> as input, stored in A, and performs Householder Reconstruction (HR),
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*> i.e. reconstructs Householder vectors V(i) implicitly representing
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*> another M-by-N matrix Q_out, with the property that Q_in = Q_out*S,
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*> where S is an N-by-N diagonal matrix with diagonal entries
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*> equal to +1 or -1. The Householder vectors (columns V(i) of V) are
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*> stored in A on output, and the diagonal entries of S are stored in D.
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*> Block reflectors are also returned in T
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*> (same output format as SGEQRT).
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] M
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*> \verbatim
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*> M is INTEGER
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*> The number of rows of the matrix A. M >= 0.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of columns of the matrix A. M >= N >= 0.
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*> \endverbatim
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*>
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*> \param[in] NB
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*> \verbatim
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*> NB is INTEGER
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*> The column block size to be used in the reconstruction
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*> of Householder column vector blocks in the array A and
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*> corresponding block reflectors in the array T. NB >= 1.
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*> (Note that if NB > N, then N is used instead of NB
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*> as the column block size.)
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is REAL array, dimension (LDA,N)
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*>
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*> On entry:
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*>
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*> The array A contains an M-by-N orthonormal matrix Q_in,
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*> i.e the columns of A are orthogonal unit vectors.
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*>
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*> On exit:
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*>
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*> The elements below the diagonal of A represent the unit
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*> lower-trapezoidal matrix V of Householder column vectors
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*> V(i). The unit diagonal entries of V are not stored
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*> (same format as the output below the diagonal in A from
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*> SGEQRT). The matrix T and the matrix V stored on output
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*> in A implicitly define Q_out.
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*>
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*> The elements above the diagonal contain the factor U
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*> of the "modified" LU-decomposition:
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*> Q_in - ( S ) = V * U
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*> ( 0 )
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*> where 0 is a (M-N)-by-(M-N) zero matrix.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,M).
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*> \endverbatim
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*>
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*> \param[out] T
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*> \verbatim
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*> T is REAL array,
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*> dimension (LDT, N)
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*>
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*> Let NOCB = Number_of_output_col_blocks
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*> = CEIL(N/NB)
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*>
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*> On exit, T(1:NB, 1:N) contains NOCB upper-triangular
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*> block reflectors used to define Q_out stored in compact
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*> form as a sequence of upper-triangular NB-by-NB column
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*> blocks (same format as the output T in SGEQRT).
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*> The matrix T and the matrix V stored on output in A
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*> implicitly define Q_out. NOTE: The lower triangles
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*> below the upper-triangular blocks will be filled with
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*> zeros. See Further Details.
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*> \endverbatim
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*>
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*> \param[in] LDT
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*> \verbatim
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*> LDT is INTEGER
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*> The leading dimension of the array T.
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*> LDT >= max(1,min(NB,N)).
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*> \endverbatim
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*>
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*> \param[out] D
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*> \verbatim
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*> D is REAL array, dimension min(M,N).
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*> The elements can be only plus or minus one.
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*>
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*> D(i) is constructed as D(i) = -SIGN(Q_in_i(i,i)), where
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*> 1 <= i <= min(M,N), and Q_in_i is Q_in after performing
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*> i-1 steps of “modified” Gaussian elimination.
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*> See Further Details.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> \endverbatim
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*>
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> The computed M-by-M orthogonal factor Q_out is defined implicitly as
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*> a product of orthogonal matrices Q_out(i). Each Q_out(i) is stored in
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*> the compact WY-representation format in the corresponding blocks of
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*> matrices V (stored in A) and T.
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*>
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*> The M-by-N unit lower-trapezoidal matrix V stored in the M-by-N
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*> matrix A contains the column vectors V(i) in NB-size column
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*> blocks VB(j). For example, VB(1) contains the columns
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*> V(1), V(2), ... V(NB). NOTE: The unit entries on
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*> the diagonal of Y are not stored in A.
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*>
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*> The number of column blocks is
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*>
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*> NOCB = Number_of_output_col_blocks = CEIL(N/NB)
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*>
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*> where each block is of order NB except for the last block, which
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*> is of order LAST_NB = N - (NOCB-1)*NB.
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*>
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*> For example, if M=6, N=5 and NB=2, the matrix V is
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*>
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*>
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*> V = ( VB(1), VB(2), VB(3) ) =
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*>
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*> = ( 1 )
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*> ( v21 1 )
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*> ( v31 v32 1 )
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*> ( v41 v42 v43 1 )
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*> ( v51 v52 v53 v54 1 )
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*> ( v61 v62 v63 v54 v65 )
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*>
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*>
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*> For each of the column blocks VB(i), an upper-triangular block
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*> reflector TB(i) is computed. These blocks are stored as
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*> a sequence of upper-triangular column blocks in the NB-by-N
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*> matrix T. The size of each TB(i) block is NB-by-NB, except
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*> for the last block, whose size is LAST_NB-by-LAST_NB.
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*>
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*> For example, if M=6, N=5 and NB=2, the matrix T is
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*>
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*> T = ( TB(1), TB(2), TB(3) ) =
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*>
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*> = ( t11 t12 t13 t14 t15 )
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*> ( t22 t24 )
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*>
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*>
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*> The M-by-M factor Q_out is given as a product of NOCB
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*> orthogonal M-by-M matrices Q_out(i).
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*>
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*> Q_out = Q_out(1) * Q_out(2) * ... * Q_out(NOCB),
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*>
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*> where each matrix Q_out(i) is given by the WY-representation
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*> using corresponding blocks from the matrices V and T:
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*>
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*> Q_out(i) = I - VB(i) * TB(i) * (VB(i))**T,
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*>
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*> where I is the identity matrix. Here is the formula with matrix
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*> dimensions:
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*>
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*> Q(i){M-by-M} = I{M-by-M} -
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*> VB(i){M-by-INB} * TB(i){INB-by-INB} * (VB(i))**T {INB-by-M},
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*>
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*> where INB = NB, except for the last block NOCB
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*> for which INB=LAST_NB.
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*>
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*> =====
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*> NOTE:
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*> =====
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*>
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*> If Q_in is the result of doing a QR factorization
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*> B = Q_in * R_in, then:
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*>
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*> B = (Q_out*S) * R_in = Q_out * (S * R_in) = Q_out * R_out.
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*>
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*> So if one wants to interpret Q_out as the result
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*> of the QR factorization of B, then the corresponding R_out
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*> should be equal to R_out = S * R_in, i.e. some rows of R_in
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*> should be multiplied by -1.
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*>
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*> For the details of the algorithm, see [1].
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*>
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*> [1] "Reconstructing Householder vectors from tall-skinny QR",
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*> G. Ballard, J. Demmel, L. Grigori, M. Jacquelin, H.D. Nguyen,
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*> E. Solomonik, J. Parallel Distrib. Comput.,
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*> vol. 85, pp. 3-31, 2015.
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*> \endverbatim
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*>
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup singleOTHERcomputational
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*
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*> \par Contributors:
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* ==================
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*>
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*> \verbatim
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*>
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*> November 2019, Igor Kozachenko,
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*> Computer Science Division,
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*> University of California, Berkeley
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*>
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*> \endverbatim
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*
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* =====================================================================
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SUBROUTINE SORHR_COL( M, N, NB, A, LDA, T, LDT, D, INFO )
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IMPLICIT NONE
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDA, LDT, M, N, NB
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* ..
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* .. Array Arguments ..
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REAL A( LDA, * ), D( * ), T( LDT, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ONE, ZERO
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PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
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* ..
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* .. Local Scalars ..
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INTEGER I, IINFO, J, JB, JBTEMP1, JBTEMP2, JNB,
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$ NPLUSONE
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* ..
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* .. External Subroutines ..
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EXTERNAL SCOPY, SLAORHR_COL_GETRFNP, SSCAL, STRSM,
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$XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters
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*
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INFO = 0
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IF( M.LT.0 ) THEN
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INFO = -1
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ELSE IF( N.LT.0 .OR. N.GT.M ) THEN
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INFO = -2
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ELSE IF( NB.LT.1 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -5
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ELSE IF( LDT.LT.MAX( 1, MIN( NB, N ) ) ) THEN
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INFO = -7
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END IF
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*
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* Handle error in the input parameters.
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SORHR_COL', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( MIN( M, N ).EQ.0 ) THEN
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RETURN
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END IF
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*
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* On input, the M-by-N matrix A contains the orthogonal
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* M-by-N matrix Q_in.
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*
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* (1) Compute the unit lower-trapezoidal V (ones on the diagonal
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* are not stored) by performing the "modified" LU-decomposition.
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*
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* Q_in - ( S ) = V * U = ( V1 ) * U,
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* ( 0 ) ( V2 )
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*
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* where 0 is an (M-N)-by-N zero matrix.
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*
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* (1-1) Factor V1 and U.
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CALL SLAORHR_COL_GETRFNP( N, N, A, LDA, D, IINFO )
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*
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* (1-2) Solve for V2.
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*
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IF( M.GT.N ) THEN
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CALL STRSM( 'R', 'U', 'N', 'N', M-N, N, ONE, A, LDA,
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$ A( N+1, 1 ), LDA )
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END IF
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*
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* (2) Reconstruct the block reflector T stored in T(1:NB, 1:N)
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* as a sequence of upper-triangular blocks with NB-size column
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* blocking.
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*
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* Loop over the column blocks of size NB of the array A(1:M,1:N)
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* and the array T(1:NB,1:N), JB is the column index of a column
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* block, JNB is the column block size at each step JB.
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*
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NPLUSONE = N + 1
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DO JB = 1, N, NB
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*
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* (2-0) Determine the column block size JNB.
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*
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JNB = MIN( NPLUSONE-JB, NB )
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*
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* (2-1) Copy the upper-triangular part of the current JNB-by-JNB
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* diagonal block U(JB) (of the N-by-N matrix U) stored
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* in A(JB:JB+JNB-1,JB:JB+JNB-1) into the upper-triangular part
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* of the current JNB-by-JNB block T(1:JNB,JB:JB+JNB-1)
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* column-by-column, total JNB*(JNB+1)/2 elements.
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*
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JBTEMP1 = JB - 1
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DO J = JB, JB+JNB-1
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CALL SCOPY( J-JBTEMP1, A( JB, J ), 1, T( 1, J ), 1 )
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END DO
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*
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* (2-2) Perform on the upper-triangular part of the current
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* JNB-by-JNB diagonal block U(JB) (of the N-by-N matrix U) stored
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* in T(1:JNB,JB:JB+JNB-1) the following operation in place:
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* (-1)*U(JB)*S(JB), i.e the result will be stored in the upper-
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* triangular part of T(1:JNB,JB:JB+JNB-1). This multiplication
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* of the JNB-by-JNB diagonal block U(JB) by the JNB-by-JNB
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* diagonal block S(JB) of the N-by-N sign matrix S from the
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* right means changing the sign of each J-th column of the block
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* U(JB) according to the sign of the diagonal element of the block
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* S(JB), i.e. S(J,J) that is stored in the array element D(J).
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*
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DO J = JB, JB+JNB-1
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IF( D( J ).EQ.ONE ) THEN
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CALL SSCAL( J-JBTEMP1, -ONE, T( 1, J ), 1 )
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END IF
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END DO
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*
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* (2-3) Perform the triangular solve for the current block
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* matrix X(JB):
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*
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* X(JB) * (A(JB)**T) = B(JB), where:
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*
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* A(JB)**T is a JNB-by-JNB unit upper-triangular
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* coefficient block, and A(JB)=V1(JB), which
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* is a JNB-by-JNB unit lower-triangular block
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* stored in A(JB:JB+JNB-1,JB:JB+JNB-1).
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* The N-by-N matrix V1 is the upper part
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* of the M-by-N lower-trapezoidal matrix V
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* stored in A(1:M,1:N);
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*
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* B(JB) is a JNB-by-JNB upper-triangular right-hand
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* side block, B(JB) = (-1)*U(JB)*S(JB), and
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* B(JB) is stored in T(1:JNB,JB:JB+JNB-1);
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*
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* X(JB) is a JNB-by-JNB upper-triangular solution
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* block, X(JB) is the upper-triangular block
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* reflector T(JB), and X(JB) is stored
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* in T(1:JNB,JB:JB+JNB-1).
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*
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* In other words, we perform the triangular solve for the
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* upper-triangular block T(JB):
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*
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* T(JB) * (V1(JB)**T) = (-1)*U(JB)*S(JB).
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*
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* Even though the blocks X(JB) and B(JB) are upper-
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* triangular, the routine STRSM will access all JNB**2
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* elements of the square T(1:JNB,JB:JB+JNB-1). Therefore,
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* we need to set to zero the elements of the block
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* T(1:JNB,JB:JB+JNB-1) below the diagonal before the call
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* to STRSM.
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*
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* (2-3a) Set the elements to zero.
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*
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JBTEMP2 = JB - 2
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DO J = JB, JB+JNB-2
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DO I = J-JBTEMP2, NB
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T( I, J ) = ZERO
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END DO
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END DO
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*
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* (2-3b) Perform the triangular solve.
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*
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CALL STRSM( 'R', 'L', 'T', 'U', JNB, JNB, ONE,
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$ A( JB, JB ), LDA, T( 1, JB ), LDT )
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*
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END DO
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*
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RETURN
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*
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* End of SORHR_COL
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*
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END
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