298 lines
8.5 KiB
Fortran
298 lines
8.5 KiB
Fortran
*> \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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*> \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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* =====================================================================
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SUBROUTINE DSPTRD( UPLO, N, AP, D, E, TAU, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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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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*
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* .. Parameters ..
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DOUBLE PRECISION ONE, ZERO, HALF
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PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0,
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$ HALF = 1.0D0 / 2.0D0 )
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* ..
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* .. Local Scalars ..
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LOGICAL UPPER
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INTEGER I, I1, I1I1, II
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DOUBLE PRECISION ALPHA, TAUI
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* ..
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* .. External Subroutines ..
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EXTERNAL DAXPY, DLARFG, DSPMV, DSPR2, XERBLA
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DDOT
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EXTERNAL LSAME, DDOT
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters
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*
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INFO = 0
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UPPER = LSAME( UPLO, 'U' )
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IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'DSPTRD', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.LE.0 )
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$ RETURN
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*
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IF( UPPER ) THEN
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*
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* Reduce the upper triangle of A.
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* I1 is the index in AP of A(1,I+1).
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*
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I1 = N*( N-1 ) / 2 + 1
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DO 10 I = N - 1, 1, -1
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*
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* Generate elementary reflector H(i) = I - tau * v * v**T
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* to annihilate A(1:i-1,i+1)
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*
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CALL DLARFG( I, AP( I1+I-1 ), AP( I1 ), 1, TAUI )
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E( I ) = AP( I1+I-1 )
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*
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IF( TAUI.NE.ZERO ) THEN
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*
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* Apply H(i) from both sides to A(1:i,1:i)
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*
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AP( I1+I-1 ) = ONE
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*
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* Compute y := tau * A * v storing y in TAU(1:i)
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*
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CALL DSPMV( UPLO, I, TAUI, AP, AP( I1 ), 1, ZERO, TAU,
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$ 1 )
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*
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* Compute w := y - 1/2 * tau * (y**T *v) * v
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*
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ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, AP( I1 ), 1 )
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CALL DAXPY( I, ALPHA, AP( I1 ), 1, TAU, 1 )
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*
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* Apply the transformation as a rank-2 update:
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* A := A - v * w**T - w * v**T
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*
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CALL DSPR2( UPLO, I, -ONE, AP( I1 ), 1, TAU, 1, AP )
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*
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AP( I1+I-1 ) = E( I )
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END IF
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D( I+1 ) = AP( I1+I )
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TAU( I ) = TAUI
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I1 = I1 - I
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10 CONTINUE
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D( 1 ) = AP( 1 )
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ELSE
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*
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* Reduce the lower triangle of A. II is the index in AP of
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* A(i,i) and I1I1 is the index of A(i+1,i+1).
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*
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II = 1
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DO 20 I = 1, N - 1
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I1I1 = II + N - I + 1
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*
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* Generate elementary reflector H(i) = I - tau * v * v**T
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* to annihilate A(i+2:n,i)
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*
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CALL DLARFG( N-I, AP( II+1 ), AP( II+2 ), 1, TAUI )
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E( I ) = AP( II+1 )
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*
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IF( TAUI.NE.ZERO ) THEN
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*
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* Apply H(i) from both sides to A(i+1:n,i+1:n)
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*
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AP( II+1 ) = ONE
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*
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* Compute y := tau * A * v storing y in TAU(i:n-1)
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*
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CALL DSPMV( UPLO, N-I, TAUI, AP( I1I1 ), AP( II+1 ), 1,
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$ ZERO, TAU( I ), 1 )
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*
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* Compute w := y - 1/2 * tau * (y**T *v) * v
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*
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ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, AP( II+1 ),
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$ 1 )
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CALL DAXPY( N-I, ALPHA, AP( II+1 ), 1, TAU( I ), 1 )
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*
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* Apply the transformation as a rank-2 update:
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* A := A - v * w**T - w * v**T
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*
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CALL DSPR2( UPLO, N-I, -ONE, AP( II+1 ), 1, TAU( I ), 1,
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$ AP( I1I1 ) )
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*
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AP( II+1 ) = E( I )
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END IF
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D( I ) = AP( II )
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TAU( I ) = TAUI
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II = I1I1
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20 CONTINUE
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D( N ) = AP( II )
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END IF
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*
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RETURN
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*
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* End of DSPTRD
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*
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END
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