*> \brief \b SLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
* LDY )
*
* .. Scalar Arguments ..
* INTEGER LDA, LDX, LDY, M, N, NB
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), D( * ), E( * ), TAUP( * ),
* $ TAUQ( * ), X( LDX, * ), Y( LDY, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SLABRD reduces the first NB rows and columns of a real general
*> m by n matrix A to upper or lower bidiagonal form by an orthogonal
*> transformation Q**T * A * P, and returns the matrices X and Y which
*> are needed to apply the transformation to the unreduced part of A.
*>
*> If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
*> bidiagonal form.
*>
*> This is an auxiliary routine called by SGEBRD
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows in the matrix A.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns in the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of leading rows and columns of A to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA,N)
*> On entry, the m by n general matrix to be reduced.
*> On exit, the first NB rows and columns of the matrix are
*> overwritten; the rest of the array is unchanged.
*> If m >= n, elements on and below the diagonal in the first NB
*> columns, with the array TAUQ, represent the orthogonal
*> matrix Q as a product of elementary reflectors; and
*> elements above the diagonal in the first NB rows, with the
*> array TAUP, represent the orthogonal matrix P as a product
*> of elementary reflectors.
*> If m < n, elements below the diagonal in the first NB
*> columns, with the array TAUQ, represent the orthogonal
*> matrix Q as a product of elementary reflectors, and
*> elements on and above the diagonal in the first NB rows,
*> with the array TAUP, represent the orthogonal matrix P as
*> a product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is REAL array, dimension (NB)
*> The diagonal elements of the first NB rows and columns of
*> the reduced matrix. D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is REAL array, dimension (NB)
*> The off-diagonal elements of the first NB rows and columns of
*> the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAUQ
*> \verbatim
*> TAUQ is REAL array, dimension (NB)
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix Q. See Further Details.
*> \endverbatim
*>
*> \param[out] TAUP
*> \verbatim
*> TAUP is REAL array, dimension (NB)
*> The scalar factors of the elementary reflectors which
*> represent the orthogonal matrix P. See Further Details.
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is REAL array, dimension (LDX,NB)
*> The m-by-nb matrix X required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,M).
*> \endverbatim
*>
*> \param[out] Y
*> \verbatim
*> Y is REAL array, dimension (LDY,NB)
*> The n-by-nb matrix Y required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDY
*> \verbatim
*> LDY is INTEGER
*> The leading dimension of the array Y. LDY >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2017
*
*> \ingroup realOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The matrices Q and P are represented as products of elementary
*> reflectors:
*>
*> Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
*>
*> Each H(i) and G(i) has the form:
*>
*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
*>
*> where tauq and taup are real scalars, and v and u are real vectors.
*>
*> If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
*> A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
*> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
*> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
*>
*> The elements of the vectors v and u together form the m-by-nb matrix
*> V and the nb-by-n matrix U**T which are needed, with X and Y, to apply
*> the transformation to the unreduced part of the matrix, using a block
*> update of the form: A := A - V*Y**T - X*U**T.
*>
*> The contents of A on exit are illustrated by the following examples
*> with nb = 2:
*>
*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
*>
*> ( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
*> ( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
*> ( v1 v2 a a a ) ( v1 1 a a a a )
*> ( v1 v2 a a a ) ( v1 v2 a a a a )
*> ( v1 v2 a a a ) ( v1 v2 a a a a )
*> ( v1 v2 a a a )
*>
*> where a denotes an element of the original matrix which is unchanged,
*> vi denotes an element of the vector defining H(i), and ui an element
*> of the vector defining G(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SLABRD( M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y,
$ LDY )
*
* -- LAPACK auxiliary routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2017
*
* .. Scalar Arguments ..
INTEGER LDA, LDX, LDY, M, N, NB
* ..
* .. Array Arguments ..
REAL A( LDA, * ), D( * ), E( * ), TAUP( * ),
$ TAUQ( * ), X( LDX, * ), Y( LDY, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
INTEGER I
* ..
* .. External Subroutines ..
EXTERNAL SGEMV, SLARFG, SSCAL
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( M.LE.0 .OR. N.LE.0 )
$ RETURN
*
IF( M.GE.N ) THEN
*
* Reduce to upper bidiagonal form
*
DO 10 I = 1, NB
*
* Update A(i:m,i)
*
CALL SGEMV( 'No transpose', M-I+1, I-1, -ONE, A( I, 1 ),
$ LDA, Y( I, 1 ), LDY, ONE, A( I, I ), 1 )
CALL SGEMV( 'No transpose', M-I+1, I-1, -ONE, X( I, 1 ),
$ LDX, A( 1, I ), 1, ONE, A( I, I ), 1 )
*
* Generate reflection Q(i) to annihilate A(i+1:m,i)
*
CALL SLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1,
$ TAUQ( I ) )
D( I ) = A( I, I )
IF( I.LT.N ) THEN
A( I, I ) = ONE
*
* Compute Y(i+1:n,i)
*
CALL SGEMV( 'Transpose', M-I+1, N-I, ONE, A( I, I+1 ),
$ LDA, A( I, I ), 1, ZERO, Y( I+1, I ), 1 )
CALL SGEMV( 'Transpose', M-I+1, I-1, ONE, A( I, 1 ), LDA,
$ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
CALL SGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
$ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL SGEMV( 'Transpose', M-I+1, I-1, ONE, X( I, 1 ), LDX,
$ A( I, I ), 1, ZERO, Y( 1, I ), 1 )
CALL SGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
$ LDA, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL SSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
*
* Update A(i,i+1:n)
*
CALL SGEMV( 'No transpose', N-I, I, -ONE, Y( I+1, 1 ),
$ LDY, A( I, 1 ), LDA, ONE, A( I, I+1 ), LDA )
CALL SGEMV( 'Transpose', I-1, N-I, -ONE, A( 1, I+1 ),
$ LDA, X( I, 1 ), LDX, ONE, A( I, I+1 ), LDA )
*
* Generate reflection P(i) to annihilate A(i,i+2:n)
*
CALL SLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ),
$ LDA, TAUP( I ) )
E( I ) = A( I, I+1 )
A( I, I+1 ) = ONE
*
* Compute X(i+1:m,i)
*
CALL SGEMV( 'No transpose', M-I, N-I, ONE, A( I+1, I+1 ),
$ LDA, A( I, I+1 ), LDA, ZERO, X( I+1, I ), 1 )
CALL SGEMV( 'Transpose', N-I, I, ONE, Y( I+1, 1 ), LDY,
$ A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
CALL SGEMV( 'No transpose', M-I, I, -ONE, A( I+1, 1 ),
$ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL SGEMV( 'No transpose', I-1, N-I, ONE, A( 1, I+1 ),
$ LDA, A( I, I+1 ), LDA, ZERO, X( 1, I ), 1 )
CALL SGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
$ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL SSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
END IF
10 CONTINUE
ELSE
*
* Reduce to lower bidiagonal form
*
DO 20 I = 1, NB
*
* Update A(i,i:n)
*
CALL SGEMV( 'No transpose', N-I+1, I-1, -ONE, Y( I, 1 ),
$ LDY, A( I, 1 ), LDA, ONE, A( I, I ), LDA )
CALL SGEMV( 'Transpose', I-1, N-I+1, -ONE, A( 1, I ), LDA,
$ X( I, 1 ), LDX, ONE, A( I, I ), LDA )
*
* Generate reflection P(i) to annihilate A(i,i+1:n)
*
CALL SLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA,
$ TAUP( I ) )
D( I ) = A( I, I )
IF( I.LT.M ) THEN
A( I, I ) = ONE
*
* Compute X(i+1:m,i)
*
CALL SGEMV( 'No transpose', M-I, N-I+1, ONE, A( I+1, I ),
$ LDA, A( I, I ), LDA, ZERO, X( I+1, I ), 1 )
CALL SGEMV( 'Transpose', N-I+1, I-1, ONE, Y( I, 1 ), LDY,
$ A( I, I ), LDA, ZERO, X( 1, I ), 1 )
CALL SGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL SGEMV( 'No transpose', I-1, N-I+1, ONE, A( 1, I ),
$ LDA, A( I, I ), LDA, ZERO, X( 1, I ), 1 )
CALL SGEMV( 'No transpose', M-I, I-1, -ONE, X( I+1, 1 ),
$ LDX, X( 1, I ), 1, ONE, X( I+1, I ), 1 )
CALL SSCAL( M-I, TAUP( I ), X( I+1, I ), 1 )
*
* Update A(i+1:m,i)
*
CALL SGEMV( 'No transpose', M-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, Y( I, 1 ), LDY, ONE, A( I+1, I ), 1 )
CALL SGEMV( 'No transpose', M-I, I, -ONE, X( I+1, 1 ),
$ LDX, A( 1, I ), 1, ONE, A( I+1, I ), 1 )
*
* Generate reflection Q(i) to annihilate A(i+2:m,i)
*
CALL SLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1,
$ TAUQ( I ) )
E( I ) = A( I+1, I )
A( I+1, I ) = ONE
*
* Compute Y(i+1:n,i)
*
CALL SGEMV( 'Transpose', M-I, N-I, ONE, A( I+1, I+1 ),
$ LDA, A( I+1, I ), 1, ZERO, Y( I+1, I ), 1 )
CALL SGEMV( 'Transpose', M-I, I-1, ONE, A( I+1, 1 ), LDA,
$ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
CALL SGEMV( 'No transpose', N-I, I-1, -ONE, Y( I+1, 1 ),
$ LDY, Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL SGEMV( 'Transpose', M-I, I, ONE, X( I+1, 1 ), LDX,
$ A( I+1, I ), 1, ZERO, Y( 1, I ), 1 )
CALL SGEMV( 'Transpose', I, N-I, -ONE, A( 1, I+1 ), LDA,
$ Y( 1, I ), 1, ONE, Y( I+1, I ), 1 )
CALL SSCAL( N-I, TAUQ( I ), Y( I+1, I ), 1 )
END IF
20 CONTINUE
END IF
RETURN
*
* End of SLABRD
*
END