548 lines
16 KiB
Fortran
548 lines
16 KiB
Fortran
*> \brief \b SGBBRD
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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 SGBBRD + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgbbrd.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/sgbbrd.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/sgbbrd.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 SGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
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* LDQ, PT, LDPT, C, LDC, WORK, INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER VECT
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* INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
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* ..
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* .. Array Arguments ..
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* REAL AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
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* $ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
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* ..
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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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*> SGBBRD reduces a real general m-by-n band matrix A to upper
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*> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
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*>
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*> The routine computes B, and optionally forms Q or P**T, or computes
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*> Q**T*C for a given matrix C.
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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] VECT
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*> \verbatim
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*> VECT is CHARACTER*1
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*> Specifies whether or not the matrices Q and P**T are to be
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*> formed.
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*> = 'N': do not form Q or P**T;
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*> = 'Q': form Q only;
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*> = 'P': form P**T only;
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*> = 'B': form both.
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*> \endverbatim
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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. N >= 0.
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*> \endverbatim
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*>
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*> \param[in] NCC
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*> \verbatim
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*> NCC is INTEGER
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*> The number of columns of the matrix C. NCC >= 0.
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*> \endverbatim
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*>
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*> \param[in] KL
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*> \verbatim
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*> KL is INTEGER
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*> The number of subdiagonals of the matrix A. KL >= 0.
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*> \endverbatim
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*>
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*> \param[in] KU
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*> \verbatim
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*> KU is INTEGER
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*> The number of superdiagonals of the matrix A. KU >= 0.
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*> \endverbatim
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*>
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*> \param[in,out] AB
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*> \verbatim
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*> AB is REAL array, dimension (LDAB,N)
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*> On entry, the m-by-n band matrix A, stored in rows 1 to
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*> KL+KU+1. The j-th column of A is stored in the j-th column of
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*> the array AB as follows:
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*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(m,j+kl).
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*> On exit, A is overwritten by values generated during the
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*> reduction.
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*> \endverbatim
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*>
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*> \param[in] LDAB
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*> \verbatim
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*> LDAB is INTEGER
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*> The leading dimension of the array A. LDAB >= KL+KU+1.
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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 diagonal elements of the bidiagonal matrix B.
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*> \endverbatim
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*>
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*> \param[out] E
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*> \verbatim
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*> E is REAL array, dimension (min(M,N)-1)
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*> The superdiagonal elements of the bidiagonal matrix B.
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*> \endverbatim
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*>
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*> \param[out] Q
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*> \verbatim
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*> Q is REAL array, dimension (LDQ,M)
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*> If VECT = 'Q' or 'B', the m-by-m orthogonal matrix Q.
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*> If VECT = 'N' or 'P', the array Q is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDQ
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*> \verbatim
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*> LDQ is INTEGER
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*> The leading dimension of the array Q.
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*> LDQ >= max(1,M) if VECT = 'Q' or 'B'; LDQ >= 1 otherwise.
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*> \endverbatim
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*>
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*> \param[out] PT
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*> \verbatim
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*> PT is REAL array, dimension (LDPT,N)
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*> If VECT = 'P' or 'B', the n-by-n orthogonal matrix P'.
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*> If VECT = 'N' or 'Q', the array PT is not referenced.
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*> \endverbatim
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*>
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*> \param[in] LDPT
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*> \verbatim
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*> LDPT is INTEGER
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*> The leading dimension of the array PT.
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*> LDPT >= max(1,N) if VECT = 'P' or 'B'; LDPT >= 1 otherwise.
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*> \endverbatim
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*>
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*> \param[in,out] C
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*> \verbatim
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*> C is REAL array, dimension (LDC,NCC)
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*> On entry, an m-by-ncc matrix C.
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*> On exit, C is overwritten by Q**T*C.
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*> C is not referenced if NCC = 0.
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*> \endverbatim
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*>
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*> \param[in] LDC
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*> \verbatim
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*> LDC is INTEGER
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*> The leading dimension of the array C.
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*> LDC >= max(1,M) if NCC > 0; LDC >= 1 if NCC = 0.
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is REAL array, dimension (2*max(M,N))
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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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* 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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*> \date December 2016
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*
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*> \ingroup realGBcomputational
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*
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* =====================================================================
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SUBROUTINE SGBBRD( VECT, M, N, NCC, KL, KU, AB, LDAB, D, E, Q,
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$ LDQ, PT, LDPT, C, LDC, WORK, INFO )
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*
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* -- LAPACK computational routine (version 3.7.0) --
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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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* December 2016
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*
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* .. Scalar Arguments ..
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CHARACTER VECT
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INTEGER INFO, KL, KU, LDAB, LDC, LDPT, LDQ, M, N, NCC
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* ..
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* .. Array Arguments ..
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REAL AB( LDAB, * ), C( LDC, * ), D( * ), E( * ),
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$ PT( LDPT, * ), Q( LDQ, * ), WORK( * )
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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 ZERO, ONE
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PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL WANTB, WANTC, WANTPT, WANTQ
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INTEGER I, INCA, J, J1, J2, KB, KB1, KK, KLM, KLU1,
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$ KUN, L, MINMN, ML, ML0, MN, MU, MU0, NR, NRT
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REAL RA, RB, RC, RS
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* ..
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* .. External Subroutines ..
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EXTERNAL SLARGV, SLARTG, SLARTV, SLASET, SROT, 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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* .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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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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WANTB = LSAME( VECT, 'B' )
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WANTQ = LSAME( VECT, 'Q' ) .OR. WANTB
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WANTPT = LSAME( VECT, 'P' ) .OR. WANTB
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WANTC = NCC.GT.0
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KLU1 = KL + KU + 1
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INFO = 0
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IF( .NOT.WANTQ .AND. .NOT.WANTPT .AND. .NOT.LSAME( VECT, 'N' ) )
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$ THEN
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INFO = -1
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ELSE IF( M.LT.0 ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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ELSE IF( NCC.LT.0 ) THEN
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INFO = -4
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ELSE IF( KL.LT.0 ) THEN
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INFO = -5
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ELSE IF( KU.LT.0 ) THEN
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INFO = -6
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ELSE IF( LDAB.LT.KLU1 ) THEN
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INFO = -8
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ELSE IF( LDQ.LT.1 .OR. WANTQ .AND. LDQ.LT.MAX( 1, M ) ) THEN
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INFO = -12
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ELSE IF( LDPT.LT.1 .OR. WANTPT .AND. LDPT.LT.MAX( 1, N ) ) THEN
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INFO = -14
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ELSE IF( LDC.LT.1 .OR. WANTC .AND. LDC.LT.MAX( 1, M ) ) THEN
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INFO = -16
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SGBBRD', -INFO )
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RETURN
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END IF
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*
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* Initialize Q and P**T to the unit matrix, if needed
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*
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IF( WANTQ )
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$ CALL SLASET( 'Full', M, M, ZERO, ONE, Q, LDQ )
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IF( WANTPT )
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$ CALL SLASET( 'Full', N, N, ZERO, ONE, PT, LDPT )
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*
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* Quick return if possible.
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*
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IF( M.EQ.0 .OR. N.EQ.0 )
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$ RETURN
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*
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MINMN = MIN( M, N )
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*
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IF( KL+KU.GT.1 ) THEN
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*
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* Reduce to upper bidiagonal form if KU > 0; if KU = 0, reduce
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* first to lower bidiagonal form and then transform to upper
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* bidiagonal
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*
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IF( KU.GT.0 ) THEN
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ML0 = 1
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MU0 = 2
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ELSE
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ML0 = 2
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MU0 = 1
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END IF
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*
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* Wherever possible, plane rotations are generated and applied in
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* vector operations of length NR over the index set J1:J2:KLU1.
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*
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* The sines of the plane rotations are stored in WORK(1:max(m,n))
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* and the cosines in WORK(max(m,n)+1:2*max(m,n)).
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*
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MN = MAX( M, N )
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KLM = MIN( M-1, KL )
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KUN = MIN( N-1, KU )
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KB = KLM + KUN
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KB1 = KB + 1
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INCA = KB1*LDAB
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NR = 0
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J1 = KLM + 2
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J2 = 1 - KUN
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*
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DO 90 I = 1, MINMN
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*
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* Reduce i-th column and i-th row of matrix to bidiagonal form
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*
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ML = KLM + 1
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MU = KUN + 1
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DO 80 KK = 1, KB
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J1 = J1 + KB
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J2 = J2 + KB
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*
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* generate plane rotations to annihilate nonzero elements
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* which have been created below the band
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*
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IF( NR.GT.0 )
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$ CALL SLARGV( NR, AB( KLU1, J1-KLM-1 ), INCA,
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$ WORK( J1 ), KB1, WORK( MN+J1 ), KB1 )
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*
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* apply plane rotations from the left
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*
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DO 10 L = 1, KB
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IF( J2-KLM+L-1.GT.N ) THEN
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NRT = NR - 1
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ELSE
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NRT = NR
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END IF
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IF( NRT.GT.0 )
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$ CALL SLARTV( NRT, AB( KLU1-L, J1-KLM+L-1 ), INCA,
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$ AB( KLU1-L+1, J1-KLM+L-1 ), INCA,
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$ WORK( MN+J1 ), WORK( J1 ), KB1 )
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10 CONTINUE
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*
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IF( ML.GT.ML0 ) THEN
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IF( ML.LE.M-I+1 ) THEN
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*
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* generate plane rotation to annihilate a(i+ml-1,i)
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* within the band, and apply rotation from the left
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*
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CALL SLARTG( AB( KU+ML-1, I ), AB( KU+ML, I ),
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$ WORK( MN+I+ML-1 ), WORK( I+ML-1 ),
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$ RA )
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AB( KU+ML-1, I ) = RA
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IF( I.LT.N )
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$ CALL SROT( MIN( KU+ML-2, N-I ),
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$ AB( KU+ML-2, I+1 ), LDAB-1,
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$ AB( KU+ML-1, I+1 ), LDAB-1,
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$ WORK( MN+I+ML-1 ), WORK( I+ML-1 ) )
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END IF
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NR = NR + 1
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J1 = J1 - KB1
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END IF
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*
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IF( WANTQ ) THEN
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*
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* accumulate product of plane rotations in Q
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*
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DO 20 J = J1, J2, KB1
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CALL SROT( M, Q( 1, J-1 ), 1, Q( 1, J ), 1,
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$ WORK( MN+J ), WORK( J ) )
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20 CONTINUE
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END IF
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*
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IF( WANTC ) THEN
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*
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* apply plane rotations to C
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*
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DO 30 J = J1, J2, KB1
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CALL SROT( NCC, C( J-1, 1 ), LDC, C( J, 1 ), LDC,
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$ WORK( MN+J ), WORK( J ) )
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30 CONTINUE
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END IF
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*
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IF( J2+KUN.GT.N ) THEN
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*
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* adjust J2 to keep within the bounds of the matrix
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*
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NR = NR - 1
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J2 = J2 - KB1
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END IF
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*
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DO 40 J = J1, J2, KB1
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*
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* create nonzero element a(j-1,j+ku) above the band
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* and store it in WORK(n+1:2*n)
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*
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WORK( J+KUN ) = WORK( J )*AB( 1, J+KUN )
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AB( 1, J+KUN ) = WORK( MN+J )*AB( 1, J+KUN )
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40 CONTINUE
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*
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* generate plane rotations to annihilate nonzero elements
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* which have been generated above the band
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*
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IF( NR.GT.0 )
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$ CALL SLARGV( NR, AB( 1, J1+KUN-1 ), INCA,
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$ WORK( J1+KUN ), KB1, WORK( MN+J1+KUN ),
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$ KB1 )
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*
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* apply plane rotations from the right
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*
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DO 50 L = 1, KB
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IF( J2+L-1.GT.M ) THEN
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NRT = NR - 1
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ELSE
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NRT = NR
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END IF
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IF( NRT.GT.0 )
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$ CALL SLARTV( NRT, AB( L+1, J1+KUN-1 ), INCA,
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$ AB( L, J1+KUN ), INCA,
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$ WORK( MN+J1+KUN ), WORK( J1+KUN ),
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$ KB1 )
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50 CONTINUE
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*
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IF( ML.EQ.ML0 .AND. MU.GT.MU0 ) THEN
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IF( MU.LE.N-I+1 ) THEN
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*
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* generate plane rotation to annihilate a(i,i+mu-1)
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* within the band, and apply rotation from the right
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*
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CALL SLARTG( AB( KU-MU+3, I+MU-2 ),
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$ AB( KU-MU+2, I+MU-1 ),
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$ WORK( MN+I+MU-1 ), WORK( I+MU-1 ),
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$ RA )
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AB( KU-MU+3, I+MU-2 ) = RA
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CALL SROT( MIN( KL+MU-2, M-I ),
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$ AB( KU-MU+4, I+MU-2 ), 1,
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$ AB( KU-MU+3, I+MU-1 ), 1,
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$ WORK( MN+I+MU-1 ), WORK( I+MU-1 ) )
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END IF
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NR = NR + 1
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J1 = J1 - KB1
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END IF
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*
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IF( WANTPT ) THEN
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*
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* accumulate product of plane rotations in P**T
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*
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DO 60 J = J1, J2, KB1
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CALL SROT( N, PT( J+KUN-1, 1 ), LDPT,
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$ PT( J+KUN, 1 ), LDPT, WORK( MN+J+KUN ),
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$ WORK( J+KUN ) )
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60 CONTINUE
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END IF
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*
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IF( J2+KB.GT.M ) THEN
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*
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* adjust J2 to keep within the bounds of the matrix
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*
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NR = NR - 1
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J2 = J2 - KB1
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END IF
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*
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DO 70 J = J1, J2, KB1
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*
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* create nonzero element a(j+kl+ku,j+ku-1) below the
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* band and store it in WORK(1:n)
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*
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WORK( J+KB ) = WORK( J+KUN )*AB( KLU1, J+KUN )
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AB( KLU1, J+KUN ) = WORK( MN+J+KUN )*AB( KLU1, J+KUN )
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70 CONTINUE
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*
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IF( ML.GT.ML0 ) THEN
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ML = ML - 1
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ELSE
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MU = MU - 1
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END IF
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80 CONTINUE
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90 CONTINUE
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END IF
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*
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IF( KU.EQ.0 .AND. KL.GT.0 ) THEN
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*
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* A has been reduced to lower bidiagonal form
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*
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* Transform lower bidiagonal form to upper bidiagonal by applying
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* plane rotations from the left, storing diagonal elements in D
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* and off-diagonal elements in E
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*
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DO 100 I = 1, MIN( M-1, N )
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CALL SLARTG( AB( 1, I ), AB( 2, I ), RC, RS, RA )
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D( I ) = RA
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IF( I.LT.N ) THEN
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E( I ) = RS*AB( 1, I+1 )
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AB( 1, I+1 ) = RC*AB( 1, I+1 )
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END IF
|
|
IF( WANTQ )
|
|
$ CALL SROT( M, Q( 1, I ), 1, Q( 1, I+1 ), 1, RC, RS )
|
|
IF( WANTC )
|
|
$ CALL SROT( NCC, C( I, 1 ), LDC, C( I+1, 1 ), LDC, RC,
|
|
$ RS )
|
|
100 CONTINUE
|
|
IF( M.LE.N )
|
|
$ D( M ) = AB( 1, M )
|
|
ELSE IF( KU.GT.0 ) THEN
|
|
*
|
|
* A has been reduced to upper bidiagonal form
|
|
*
|
|
IF( M.LT.N ) THEN
|
|
*
|
|
* Annihilate a(m,m+1) by applying plane rotations from the
|
|
* right, storing diagonal elements in D and off-diagonal
|
|
* elements in E
|
|
*
|
|
RB = AB( KU, M+1 )
|
|
DO 110 I = M, 1, -1
|
|
CALL SLARTG( AB( KU+1, I ), RB, RC, RS, RA )
|
|
D( I ) = RA
|
|
IF( I.GT.1 ) THEN
|
|
RB = -RS*AB( KU, I )
|
|
E( I-1 ) = RC*AB( KU, I )
|
|
END IF
|
|
IF( WANTPT )
|
|
$ CALL SROT( N, PT( I, 1 ), LDPT, PT( M+1, 1 ), LDPT,
|
|
$ RC, RS )
|
|
110 CONTINUE
|
|
ELSE
|
|
*
|
|
* Copy off-diagonal elements to E and diagonal elements to D
|
|
*
|
|
DO 120 I = 1, MINMN - 1
|
|
E( I ) = AB( KU, I+1 )
|
|
120 CONTINUE
|
|
DO 130 I = 1, MINMN
|
|
D( I ) = AB( KU+1, I )
|
|
130 CONTINUE
|
|
END IF
|
|
ELSE
|
|
*
|
|
* A is diagonal. Set elements of E to zero and copy diagonal
|
|
* elements to D.
|
|
*
|
|
DO 140 I = 1, MINMN - 1
|
|
E( I ) = ZERO
|
|
140 CONTINUE
|
|
DO 150 I = 1, MINMN
|
|
D( I ) = AB( 1, I )
|
|
150 CONTINUE
|
|
END IF
|
|
RETURN
|
|
*
|
|
* End of SGBBRD
|
|
*
|
|
END
|