version 1.1.1.1, 2010/01/26 15:22:46
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version 1.15, 2017/06/17 10:54:03
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*> \brief \b DSPTRD |
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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 DSPTRD + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsptrd.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/dsptrd.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/dsptrd.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 DSPTRD( UPLO, N, AP, 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, N |
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* .. |
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* .. Array Arguments .. |
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* DOUBLE PRECISION AP( * ), 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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*> DSPTRD reduces a real symmetric matrix A stored in packed form to |
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*> symmetric tridiagonal form T by an orthogonal similarity |
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*> 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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*> = 'U': Upper triangle of A is stored; |
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*> = 'L': Lower triangle of A is stored. |
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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] AP |
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*> \verbatim |
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*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) |
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*> On entry, the upper or lower triangle of the symmetric matrix |
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*> A, packed columnwise in a linear array. The j-th column of A |
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*> is stored in the array AP as follows: |
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*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; |
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*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. |
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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[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 December 2016 |
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* |
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*> \ingroup doubleOTHERcomputational |
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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 AP, |
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*> overwriting A(1:i-1,i+1), and tau is stored 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 AP, |
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*> overwriting A(i+2:n,i), and tau is stored in TAU(i). |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO ) |
SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.7.0) -- |
* -- 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 |
* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
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DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * ) |
DOUBLE PRECISION AP( * ), D( * ), E( * ), TAU( * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DSPTRD reduces a real symmetric matrix A stored in packed form to |
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* symmetric tridiagonal form T by an orthogonal similarity |
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* transformation: Q**T * A * Q = T. |
|
* |
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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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* = 'U': Upper triangle of A is stored; |
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* = 'L': Lower triangle of A is stored. |
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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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* AP (input/output) DOUBLE PRECISION array, dimension (N*(N+1)/2) |
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* On entry, the upper or lower triangle of the symmetric matrix |
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* A, packed columnwise in a linear array. The j-th column of A |
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* is stored in the array AP as follows: |
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* if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; |
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* if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. |
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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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* 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 AP, |
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* overwriting A(1:i-1,i+1), and tau is stored 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 AP, |
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* overwriting A(i+2:n,i), and tau is stored in TAU(i). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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I1 = N*( N-1 ) / 2 + 1 |
I1 = N*( N-1 ) / 2 + 1 |
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, AP( I1+I-1 ), AP( I1 ), 1, TAUI ) |
CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI ) |
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CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU, |
CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU, |
$ 1 ) |
$ 1 ) |
* |
* |
* Compute w := y - 1/2 * tau * (y'*v) * v |
* Compute w := y - 1/2 * tau * (y**T *v) * v |
* |
* |
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 ) |
ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 ) |
CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 ) |
CALL DAXPY( I, ALPHA, AP( I1 ), 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 DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP ) |
CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP ) |
* |
* |
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DO 20 I = 1, N - 1 |
DO 20 I = 1, N - 1 |
I1I1 = II + N - I + 1 |
I1I1 = II + N - I + 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, AP( II+1 ), AP( II+2 ), 1, TAUI ) |
CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI ) |
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CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1, |
CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1, |
$ ZERO, TAU( I ), 1 ) |
$ ZERO, TAU( I ), 1 ) |
* |
* |
* Compute w := y - 1/2 * tau * (y'*v) * v |
* Compute w := y - 1/2 * tau * (y**T *v) * v |
* |
* |
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ), |
ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ), |
$ 1 ) |
$ 1 ) |
CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 ) |
CALL DAXPY( N-I, ALPHA, AP( II+1 ), 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 DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, |
CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1, |
$ AP( I1I1 ) ) |
$ AP( I1I1 ) ) |