ESMF_DLABRD Subroutine

subroutine ESMF_DLABRD(M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)

\brief \b ESMF_DLABRD \htmlonly Download ESMF_DLABRD + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DLABRD 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 ESMF_DGEBRD \endverbatim \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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date November 2011 \ingroup doubleOTHERauxiliary \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 UT 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*YT - XU*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

Arguments