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*> \brief \b DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an orthogonal similarity transformation.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLATRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlatrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlatrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlatrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLATRD reduces NB rows and columns of a real symmetric matrix A to
*> symmetric tridiagonal form by an orthogonal similarity
*> transformation Q**T * A * Q, and returns the matrices V and W which are
*> needed to apply the transformation to the unreduced part of A.
*>
*> If UPLO = 'U', DLATRD reduces the last NB rows and columns of a
*> matrix, of which the upper triangle is supplied;
*> if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
*> matrix, of which the lower triangle is supplied.
*>
*> This is an auxiliary routine called by DSYTRD.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the upper or lower triangular part of the
*> symmetric matrix A is stored:
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A.
*> \endverbatim
*>
*> \param[in] NB
*> \verbatim
*> NB is INTEGER
*> The number of rows and columns to be reduced.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA,N)
*> On entry, the symmetric matrix A. If UPLO = 'U', the leading
*> n-by-n upper triangular part of A contains the upper
*> triangular part of the matrix A, and the strictly lower
*> triangular part of A is not referenced. If UPLO = 'L', the
*> leading n-by-n lower triangular part of A contains the lower
*> triangular part of the matrix A, and the strictly upper
*> triangular part of A is not referenced.
*> On exit:
*> if UPLO = 'U', the last NB columns have been reduced to
*> tridiagonal form, with the diagonal elements overwriting
*> the diagonal elements of A; the elements above the diagonal
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors;
*> if UPLO = 'L', the first NB columns have been reduced to
*> tridiagonal form, with the diagonal elements overwriting
*> the diagonal elements of A; the elements below the diagonal
*> with the array TAU, represent the orthogonal matrix Q as a
*> product of elementary reflectors.
*> See Further Details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= (1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is DOUBLE PRECISION array, dimension (N-1)
*> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
*> elements of the last NB columns of the reduced matrix;
*> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
*> the first NB columns of the reduced matrix.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (N-1)
*> The scalar factors of the elementary reflectors, stored in
*> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
*> See Further Details.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*> W is DOUBLE PRECISION array, dimension (LDW,NB)
*> The n-by-nb matrix W required to update the unreduced part
*> of A.
*> \endverbatim
*>
*> \param[in] LDW
*> \verbatim
*> LDW is INTEGER
*> The leading dimension of the array W. LDW >= max(1,N).
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date September 2012
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n) H(n-1) . . . H(n-nb+1).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
*> and tau in TAU(i-1).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(nb).
*>
*> Each H(i) has the form
*>
*> H(i) = I - tau * v * v**T
*>
*> where tau is a real scalar, and v is a real vector with
*> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
*> and tau in TAU(i).
*>
*> The elements of the vectors v together form the n-by-nb matrix V
*> which is needed, with W, to apply the transformation to the unreduced
*> part of the matrix, using a symmetric rank-2k update of the form:
*> A := A - V*W**T - W*V**T.
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5 and nb = 2:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( a a a v4 v5 ) ( d )
*> ( a a v4 v5 ) ( 1 d )
*> ( a 1 v5 ) ( v1 1 a )
*> ( d 1 ) ( v1 v2 a a )
*> ( d ) ( v1 v2 a a a )
*>
*> where d denotes a diagonal element of the reduced matrix, a denotes
*> an element of the original matrix that is unchanged, and vi denotes
*> an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
*
* -- LAPACK auxiliary routine (version 3.4.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* September 2012
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER LDA, LDW, N, NB
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, HALF
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 )
* ..
* .. Local Scalars ..
INTEGER I, IW
DOUBLE PRECISION ALPHA
* ..
* .. External Subroutines ..
EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DDOT
EXTERNAL LSAME, DDOT
* ..
* .. Intrinsic Functions ..
INTRINSIC MIN
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.LE.0 )
$ RETURN
*
IF( LSAME( UPLO, 'U' ) ) THEN
*
* Reduce last NB columns of upper triangle
*
DO 10 I = N, N - NB + 1, -1
IW = I - N + NB
IF( I.LT.N ) THEN
*
* Update A(1:i,i)
*
CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ),
$ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 )
CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ),
$ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 )
END IF
IF( I.GT.1 ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(1:i-2,i)
*
CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) )
E( I-1 ) = A( I-1, I )
A( I-1, I ) = ONE
*
* Compute W(1:i-1,i)
*
CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1,
$ ZERO, W( 1, IW ), 1 )
IF( I.LT.N ) THEN
CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ),
$ LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
$ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE,
$ W( 1, IW ), 1 )
CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ),
$ LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 )
CALL DGEMV( 'No transpose', I-1, N-I, -ONE,
$ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE,
$ W( 1, IW ), 1 )
END IF
CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 )
ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1,
$ A( 1, I ), 1 )
CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 )
END IF
*
10 CONTINUE
ELSE
*
* Reduce first NB columns of lower triangle
*
DO 20 I = 1, NB
*
* Update A(i:n,i)
*
CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ),
$ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 )
CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ),
$ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 )
IF( I.LT.N ) THEN
*
* Generate elementary reflector H(i) to annihilate
* A(i+2:n,i)
*
CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
$ TAU( I ) )
E( I ) = A( I+1, I )
A( I+1, I ) = ONE
*
* Compute W(i+1:n,i)
*
CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA,
$ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW,
$ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ),
$ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA,
$ A( I+1, I ), 1, ZERO, W( 1, I ), 1 )
CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ),
$ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 )
CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 )
ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1,
$ A( I+1, I ), 1 )
CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 )
END IF
*
20 CONTINUE
END IF
*
RETURN
*
* End of DLATRD
*
END
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