ESMF_DLATRD Subroutine

subroutine ESMF_DLATRD(UPLO, N, NB, A, LDA, E, TAU, W, LDW)

\brief \b ESMF_DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation. \htmlonly Download ESMF_DLATRD + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DLATRD reduces NB rows and columns of a real symmetric matrix A to symmetric tridiagonal form by an orthogonal similarity transformation Q**T * A * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of A.

If UPLO = ‘U’, ESMF_DLATRD reduces the last NB rows and columns of a matrix, of which the upper triangle is supplied; if UPLO = ‘L’, ESMF_DLATRD reduces the first NB rows and columns of a matrix, of which the lower triangle is supplied.

This is an auxiliary routine called by ESMF_DSYTRD. \endverbatim \param[in] UPLO \verbatim UPLO is CHARACTER*1 Specifies whether the upper or lower triangular part of the symmetric matrix A is stored: = ‘U’: Upper triangular = ‘L’: Lower triangular \endverbatim

\param[in] N \verbatim N is INTEGER The order of the matrix A. \endverbatim

\param[in] NB \verbatim NB is INTEGER The number of rows and columns to be reduced. \endverbatim

\param[in,out] A \verbatim A is DOUBLE PRECISION array, dimension (LDA,N) On entry, the symmetric matrix A. If UPLO = ‘U’, the leading n-by-n upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = ‘L’, the leading n-by-n lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced. On exit: if UPLO = ‘U’, the last NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements above the diagonal with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors; if UPLO = ‘L’, the first NB columns have been reduced to tridiagonal form, with the diagonal elements overwriting the diagonal elements of A; the elements below the diagonal with the array TAU, represent the orthogonal matrix Q 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 >= (1,N). \endverbatim

\param[out] E \verbatim E is DOUBLE PRECISION array, dimension (N-1) If UPLO = ‘U’, E(n-nb:n-1) contains the superdiagonal elements of the last NB columns of the reduced matrix; if UPLO = ‘L’, E(1:nb) contains the subdiagonal elements of the first NB columns of the reduced matrix. \endverbatim

\param[out] TAU \verbatim TAU is DOUBLE PRECISION array, dimension (N-1) The scalar factors of the elementary reflectors, stored in TAU(n-nb:n-1) if UPLO = ‘U’, and in TAU(1:nb) if UPLO = ‘L’. See Further Details. \endverbatim

\param[out] W \verbatim W is DOUBLE PRECISION array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A. \endverbatim

\param[in] LDW \verbatim LDW is INTEGER The leading dimension of the array W. LDW >= max(1,N). \endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date December 2016 \ingroup doubleOTHERauxiliary \par Further Details:

\verbatim

If UPLO = ‘U’, the matrix Q is represented as a product of elementary reflectors

Q = H(n) H(n-1) . . . H(n-nb+1).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), and tau in TAU(i-1).

If UPLO = ‘L’, the matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

H(i) = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), and tau in TAU(i).

The elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the transformation to the unreduced part of the matrix, using a symmetric rank-2k update of the form: A := A - VWT - WV**T.

The contents of A on exit are illustrated by the following examples with n = 5 and nb = 2:

if UPLO = ‘U’: if UPLO = ‘L’:

( a a a v4 v5 ) ( d ) ( a a v4 v5 ) ( 1 d ) ( a 1 v5 ) ( v1 1 a ) ( d 1 ) ( v1 v2 a a ) ( d ) ( v1 v2 a a a )

where d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that is unchanged, and vi denotes an element of the vector defining H(i). \endverbatim

Arguments