version 1.7, 2010/12/21 13:53:39
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version 1.13, 2012/12/14 14:22:41
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*> \brief \b DSYTD2 reduces a symmetric matrix to real symmetric tridiagonal form by an orthogonal similarity transformation (unblocked algorithm). |
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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 DSYTD2 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytd2.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/dsytd2.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/dsytd2.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 DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER UPLO |
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* INTEGER INFO, LDA, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) |
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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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*> DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal |
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*> form T by an orthogonal similarity transformation: Q**T * A * Q = T. |
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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. N >= 0. |
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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, if UPLO = 'U', the diagonal and first superdiagonal |
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*> of A are overwritten by the corresponding elements of the |
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*> tridiagonal matrix T, and the elements above the first |
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*> superdiagonal, with the array TAU, represent the orthogonal |
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*> matrix Q as a product of elementary reflectors; if UPLO |
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*> = 'L', the diagonal and first subdiagonal of A are over- |
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*> written by the corresponding elements of the tridiagonal |
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*> matrix T, and the elements below the first subdiagonal, with |
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*> the array TAU, represent the orthogonal matrix Q as a product |
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*> of elementary reflectors. 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 >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] D |
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*> \verbatim |
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*> D is DOUBLE PRECISION array, dimension (N) |
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*> The diagonal elements of the tridiagonal matrix T: |
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*> D(i) = A(i,i). |
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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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*> The off-diagonal elements of the tridiagonal matrix T: |
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*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. |
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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 (see Further |
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*> Details). |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -i, the i-th argument had an illegal value. |
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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 September 2012 |
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* |
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*> \ingroup doubleSYcomputational |
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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-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in |
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*> A(1:i-1,i+1), and tau in TAU(i). |
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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(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), |
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*> and tau in TAU(i). |
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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: |
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*> |
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*> if UPLO = 'U': if UPLO = 'L': |
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*> |
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*> ( d e v2 v3 v4 ) ( d ) |
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*> ( d e v3 v4 ) ( e d ) |
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*> ( d e v4 ) ( v1 e d ) |
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*> ( d e ) ( v1 v2 e d ) |
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*> ( d ) ( v1 v2 v3 e d ) |
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*> |
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*> where d and e denote diagonal and off-diagonal elements of T, and vi |
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*> denotes an element of the vector defining H(i). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) |
SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.4.2) -- |
* -- 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 |
* September 2012 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
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DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) |
DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal |
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* form T by an orthogonal similarity transformation: Q' * A * Q = T. |
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* |
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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. N >= 0. |
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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, if UPLO = 'U', the diagonal and first superdiagonal |
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* of A are overwritten by the corresponding elements of the |
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* tridiagonal matrix T, and the elements above the first |
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* superdiagonal, with the array TAU, represent the orthogonal |
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* matrix Q as a product of elementary reflectors; if UPLO |
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* = 'L', the diagonal and first subdiagonal of A are over- |
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* written by the corresponding elements of the tridiagonal |
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* matrix T, and the elements below the first subdiagonal, with |
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* the array TAU, represent the orthogonal matrix Q as a product |
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* of elementary reflectors. 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 >= max(1,N). |
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* |
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* D (output) DOUBLE PRECISION array, dimension (N) |
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* The diagonal elements of the tridiagonal matrix T: |
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* D(i) = A(i,i). |
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* |
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* E (output) DOUBLE PRECISION array, dimension (N-1) |
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* The off-diagonal elements of the tridiagonal matrix T: |
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* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. |
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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 (see Further |
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* Details). |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value. |
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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-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in |
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* A(1:i-1,i+1), and tau in TAU(i). |
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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(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), |
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* and tau in TAU(i). |
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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: |
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* |
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* if UPLO = 'U': if UPLO = 'L': |
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* |
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* ( d e v2 v3 v4 ) ( d ) |
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* ( d e v3 v4 ) ( e d ) |
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* ( d e v4 ) ( v1 e d ) |
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* ( d e ) ( v1 v2 e d ) |
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* ( d ) ( v1 v2 v3 e d ) |
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* |
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* where d and e denote diagonal and off-diagonal elements of T, and vi |
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* denotes an element of the vector defining H(i). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* |
* |
DO 10 I = N - 1, 1, -1 |
DO 10 I = N - 1, 1, -1 |
* |
* |
* Generate elementary reflector H(i) = I - tau * v * v' |
* Generate elementary reflector H(i) = I - tau * v * v**T |
* to annihilate A(1:i-1,i+1) |
* to annihilate A(1:i-1,i+1) |
* |
* |
CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI ) |
CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI ) |
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CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO, |
CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO, |
$ TAU, 1 ) |
$ TAU, 1 ) |
* |
* |
* Compute w := x - 1/2 * tau * (x'*v) * v |
* Compute w := x - 1/2 * tau * (x**T * v) * v |
* |
* |
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 ) |
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 ) |
CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 ) |
CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 ) |
* |
* |
* Apply the transformation as a rank-2 update: |
* Apply the transformation as a rank-2 update: |
* A := A - v * w' - w * v' |
* A := A - v * w**T - w * v**T |
* |
* |
CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A, |
CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A, |
$ LDA ) |
$ LDA ) |
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* |
* |
DO 20 I = 1, N - 1 |
DO 20 I = 1, N - 1 |
* |
* |
* Generate elementary reflector H(i) = I - tau * v * v' |
* Generate elementary reflector H(i) = I - tau * v * v**T |
* to annihilate A(i+2:n,i) |
* to annihilate A(i+2:n,i) |
* |
* |
CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, |
CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, |
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CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA, |
CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA, |
$ A( I+1, I ), 1, ZERO, TAU( I ), 1 ) |
$ A( I+1, I ), 1, ZERO, TAU( I ), 1 ) |
* |
* |
* Compute w := x - 1/2 * tau * (x'*v) * v |
* Compute w := x - 1/2 * tau * (x**T * v) * v |
* |
* |
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ), |
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ), |
$ 1 ) |
$ 1 ) |
CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) |
CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) |
* |
* |
* Apply the transformation as a rank-2 update: |
* Apply the transformation as a rank-2 update: |
* A := A - v * w' - w * v' |
* A := A - v * w**T - w * v**T |
* |
* |
CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, |
CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, |
$ A( I+1, I+1 ), LDA ) |
$ A( I+1, I+1 ), LDA ) |