*> \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 *> Download SLABRD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \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