version 1.3, 2010/08/06 15:28:44
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version 1.9, 2011/11/21 20:42:59
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*> \brief \b DLATRD |
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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 DLATRD + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrd.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/dlatrd.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/dlatrd.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 DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER UPLO |
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* INTEGER LDA, LDW, N, NB |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * ) |
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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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*> DLATRD reduces NB rows and columns of a real symmetric matrix A to |
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*> symmetric tridiagonal form by an orthogonal similarity |
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*> transformation Q**T * A * Q, and returns the matrices V and W which are |
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*> needed to apply the transformation to the unreduced part of A. |
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*> |
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*> If UPLO = 'U', DLATRD reduces the last NB rows and columns of a |
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*> matrix, of which the upper triangle is supplied; |
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*> if UPLO = 'L', DLATRD reduces the first NB rows and columns of a |
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*> matrix, of which the lower triangle is supplied. |
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*> |
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*> This is an auxiliary routine called by DSYTRD. |
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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] UPLO |
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*> \verbatim |
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*> UPLO is CHARACTER*1 |
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*> Specifies whether the upper or lower triangular part of the |
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*> symmetric matrix A is stored: |
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*> = 'U': Upper triangular |
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*> = 'L': Lower triangular |
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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 order of the matrix A. |
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*> \endverbatim |
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*> |
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*> \param[in] NB |
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*> \verbatim |
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*> NB is INTEGER |
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*> The number of rows and columns to be reduced. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is DOUBLE PRECISION array, dimension (LDA,N) |
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*> On entry, the symmetric matrix A. If UPLO = 'U', the leading |
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*> n-by-n upper triangular part of A contains the upper |
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*> triangular part of the matrix A, and the strictly lower |
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*> triangular part of A is not referenced. If UPLO = 'L', the |
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*> leading n-by-n lower triangular part of A contains the lower |
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*> triangular part of the matrix A, and the strictly upper |
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*> triangular part of A is not referenced. |
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*> On exit: |
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*> if UPLO = 'U', the last NB columns have been reduced to |
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*> tridiagonal form, with the diagonal elements overwriting |
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*> the diagonal elements of A; the elements above the diagonal |
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*> with the array TAU, represent the orthogonal matrix Q as a |
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*> product of elementary reflectors; |
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*> if UPLO = 'L', the first NB columns have been reduced to |
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*> tridiagonal form, with the diagonal elements overwriting |
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*> the diagonal elements of A; the elements below the diagonal |
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*> with the array TAU, represent the orthogonal matrix Q as a |
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*> product of elementary reflectors. |
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*> See Further Details. |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= (1,N). |
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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 DOUBLE PRECISION array, dimension (N-1) |
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*> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal |
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*> elements of the last NB columns of the reduced matrix; |
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*> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of |
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*> the first NB columns of the reduced matrix. |
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*> \endverbatim |
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*> |
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*> \param[out] TAU |
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*> \verbatim |
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*> TAU is DOUBLE PRECISION array, dimension (N-1) |
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*> The scalar factors of the elementary reflectors, stored in |
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*> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. |
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*> See Further Details. |
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*> \endverbatim |
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*> |
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*> \param[out] W |
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*> \verbatim |
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*> W is DOUBLE PRECISION array, dimension (LDW,NB) |
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*> The n-by-nb matrix W required to update the unreduced part |
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*> of A. |
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*> \endverbatim |
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*> |
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*> \param[in] LDW |
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*> \verbatim |
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*> LDW is INTEGER |
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*> The leading dimension of the array W. LDW >= max(1,N). |
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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 November 2011 |
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* |
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*> \ingroup doubleOTHERauxiliary |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> If UPLO = 'U', the matrix Q is represented as a product of elementary |
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*> reflectors |
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*> |
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*> Q = H(n) H(n-1) . . . H(n-nb+1). |
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*> |
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*> Each H(i) has the form |
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*> |
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*> H(i) = I - tau * v * v**T |
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*> |
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*> where tau is a real scalar, and v is a real vector with |
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*> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), |
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*> and tau in TAU(i-1). |
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*> |
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*> If UPLO = 'L', the matrix Q is represented as a product of elementary |
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*> reflectors |
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*> |
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*> Q = H(1) H(2) . . . H(nb). |
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*> |
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*> Each H(i) has the form |
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*> |
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*> H(i) = I - tau * v * v**T |
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*> |
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*> where tau is a real scalar, and v is a real vector with |
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*> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), |
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*> and tau in TAU(i). |
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*> |
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*> The elements of the vectors v together form the n-by-nb matrix V |
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*> which is needed, with W, to apply the transformation to the unreduced |
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*> part of the matrix, using a symmetric rank-2k update of the form: |
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*> A := A - V*W**T - W*V**T. |
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*> |
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*> The contents of A on exit are illustrated by the following examples |
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*> with n = 5 and nb = 2: |
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*> |
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*> if UPLO = 'U': if UPLO = 'L': |
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*> |
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*> ( a a a v4 v5 ) ( d ) |
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*> ( a a v4 v5 ) ( 1 d ) |
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*> ( a 1 v5 ) ( v1 1 a ) |
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*> ( d 1 ) ( v1 v2 a a ) |
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*> ( d ) ( v1 v2 a a a ) |
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*> |
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*> where d denotes a diagonal element of the reduced matrix, a denotes |
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*> an element of the original matrix that is unchanged, and vi denotes |
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*> an element of the vector defining H(i). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) |
SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2) -- |
* -- LAPACK auxiliary routine (version 3.4.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* November 2011 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
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DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * ) |
DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DLATRD reduces NB rows and columns of a real symmetric matrix A to |
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* symmetric tridiagonal form by an orthogonal similarity |
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* transformation Q' * A * Q, and returns the matrices V and W which are |
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* needed to apply the transformation to the unreduced part of A. |
|
* |
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* If UPLO = 'U', DLATRD reduces the last NB rows and columns of a |
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* matrix, of which the upper triangle is supplied; |
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* if UPLO = 'L', DLATRD reduces the first NB rows and columns of a |
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* matrix, of which the lower triangle is supplied. |
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* |
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* This is an auxiliary routine called by DSYTRD. |
|
* |
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* Arguments |
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* ========= |
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* |
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* UPLO (input) CHARACTER*1 |
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* Specifies whether the upper or lower triangular part of the |
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* symmetric matrix A is stored: |
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* = 'U': Upper triangular |
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* = 'L': Lower triangular |
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* |
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* N (input) INTEGER |
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* The order of the matrix A. |
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* |
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* NB (input) INTEGER |
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* The number of rows and columns to be reduced. |
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* |
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* A (input/output) DOUBLE PRECISION array, dimension (LDA,N) |
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* On entry, the symmetric matrix A. If UPLO = 'U', the leading |
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* n-by-n upper triangular part of A contains the upper |
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* triangular part of the matrix A, and the strictly lower |
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* triangular part of A is not referenced. If UPLO = 'L', the |
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* leading n-by-n lower triangular part of A contains the lower |
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* triangular part of the matrix A, and the strictly upper |
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* triangular part of A is not referenced. |
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* On exit: |
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* if UPLO = 'U', the last NB columns have been reduced to |
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* tridiagonal form, with the diagonal elements overwriting |
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* the diagonal elements of A; the elements above the diagonal |
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* with the array TAU, represent the orthogonal matrix Q as a |
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* product of elementary reflectors; |
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* if UPLO = 'L', the first NB columns have been reduced to |
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* tridiagonal form, with the diagonal elements overwriting |
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* the diagonal elements of A; the elements below the diagonal |
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* with the array TAU, represent the orthogonal matrix Q as a |
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* product of elementary reflectors. |
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* See Further Details. |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= (1,N). |
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* |
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* E (output) DOUBLE PRECISION array, dimension (N-1) |
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* If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal |
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* elements of the last NB columns of the reduced matrix; |
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* if UPLO = 'L', E(1:nb) contains the subdiagonal elements of |
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* the first NB columns of the reduced matrix. |
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* |
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* TAU (output) DOUBLE PRECISION array, dimension (N-1) |
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* The scalar factors of the elementary reflectors, stored in |
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* TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. |
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* See Further Details. |
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* |
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* W (output) DOUBLE PRECISION array, dimension (LDW,NB) |
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* The n-by-nb matrix W required to update the unreduced part |
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* of A. |
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* |
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* LDW (input) INTEGER |
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* The leading dimension of the array W. LDW >= max(1,N). |
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* |
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* Further Details |
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* =============== |
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* |
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* If UPLO = 'U', the matrix Q is represented as a product of elementary |
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* reflectors |
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* |
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* Q = H(n) H(n-1) . . . H(n-nb+1). |
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* |
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* Each H(i) has the form |
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* |
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* H(i) = I - tau * v * v' |
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* |
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* where tau is a real scalar, and v is a real vector with |
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* v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), |
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* and tau in TAU(i-1). |
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* |
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* If UPLO = 'L', the matrix Q is represented as a product of elementary |
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* reflectors |
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* |
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* Q = H(1) H(2) . . . H(nb). |
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* |
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* Each H(i) has the form |
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* |
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* H(i) = I - tau * v * v' |
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* |
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* where tau is a real scalar, and v is a real vector with |
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* v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), |
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* and tau in TAU(i). |
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* |
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* The elements of the vectors v together form the n-by-nb matrix V |
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* which is needed, with W, to apply the transformation to the unreduced |
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* part of the matrix, using a symmetric rank-2k update of the form: |
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* A := A - V*W' - W*V'. |
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* |
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* The contents of A on exit are illustrated by the following examples |
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* with n = 5 and nb = 2: |
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* |
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* if UPLO = 'U': if UPLO = 'L': |
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* |
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* ( a a a v4 v5 ) ( d ) |
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* ( a a v4 v5 ) ( 1 d ) |
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* ( a 1 v5 ) ( v1 1 a ) |
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* ( d 1 ) ( v1 v2 a a ) |
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* ( d ) ( v1 v2 a a a ) |
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* |
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* where d denotes a diagonal element of the reduced matrix, a denotes |
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* an element of the original matrix that is unchanged, and vi denotes |
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* an element of the vector defining H(i). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |