*> \brief \b SSYTRD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SSYTRD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ssytrd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ssytrd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ssytrd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), D( * ), E( * ), TAU( * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SSYTRD reduces a real symmetric matrix A to real symmetric
*> tridiagonal form T by an orthogonal similarity transformation:
*> Q**T * A * Q = T.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = 'U': Upper triangle of A is stored;
*> = 'L': Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL 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 diagonal and first superdiagonal
*> of A are overwritten by the corresponding elements of the
*> tridiagonal matrix T, and the elements above the first
*> superdiagonal, with the array TAU, represent the orthogonal
*> matrix Q as a product of elementary reflectors; if UPLO
*> = 'L', the diagonal and first subdiagonal of A are over-
*> written by the corresponding elements of the tridiagonal
*> matrix T, and the elements below the first subdiagonal, 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 >= max(1,N).
*> \endverbatim
*>
*> \param[out] D
*> \verbatim
*> D is REAL array, dimension (N)
*> The diagonal elements of the tridiagonal matrix T:
*> D(i) = A(i,i).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> E is REAL array, dimension (N-1)
*> The off-diagonal elements of the tridiagonal matrix T:
*> E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is REAL array, dimension (N-1)
*> The scalar factors of the elementary reflectors (see Further
*> Details).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= 1.
*> For optimum performance LWORK >= N*NB, where NB is the
*> optimal blocksize.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup realSYcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> If UPLO = 'U', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
*> A(1:i-1,i+1), and tau in TAU(i).
*>
*> If UPLO = 'L', the matrix Q is represented as a product of elementary
*> reflectors
*>
*> Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
*> and tau in TAU(i).
*>
*> The contents of A on exit are illustrated by the following examples
*> with n = 5:
*>
*> if UPLO = 'U': if UPLO = 'L':
*>
*> ( d e v2 v3 v4 ) ( d )
*> ( d e v3 v4 ) ( e d )
*> ( d e v4 ) ( v1 e d )
*> ( d e ) ( v1 v2 e d )
*> ( d ) ( v1 v2 v3 e d )
*>
*> where d and e denote diagonal and off-diagonal elements of T, and vi
*> denotes an element of the vector defining H(i).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), D( * ), E( * ), TAU( * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ONE
PARAMETER ( ONE = 1.0E+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
$ NBMIN, NX
* ..
* .. External Subroutines ..
EXTERNAL SLATRD, SSYR2K, SSYTD2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. Executable Statements ..
*
* Test the input parameters
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -9
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Determine the block size.
*
NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 )
LWKOPT = N*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SSYTRD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 ) THEN
WORK( 1 ) = 1
RETURN
END IF
*
NX = N
IWS = 1
IF( NB.GT.1 .AND. NB.LT.N ) THEN
*
* Determine when to cross over from blocked to unblocked code
* (last block is always handled by unblocked code).
*
NX = MAX( NB, ILAENV( 3, 'SSYTRD', UPLO, N, -1, -1, -1 ) )
IF( NX.LT.N ) THEN
*
* Determine if workspace is large enough for blocked code.
*
LDWORK = N
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
*
* Not enough workspace to use optimal NB: determine the
* minimum value of NB, and reduce NB or force use of
* unblocked code by setting NX = N.
*
NB = MAX( LWORK / LDWORK, 1 )
NBMIN = ILAENV( 2, 'SSYTRD', UPLO, N, -1, -1, -1 )
IF( NB.LT.NBMIN )
$ NX = N
END IF
ELSE
NX = N
END IF
ELSE
NB = 1
END IF
*
IF( UPPER ) THEN
*
* Reduce the upper triangle of A.
* Columns 1:kk are handled by the unblocked method.
*
KK = N - ( ( N-NX+NB-1 ) / NB )*NB
DO 20 I = N - NB + 1, KK + 1, -NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL SLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
$ LDWORK )
*
* Update the unreduced submatrix A(1:i-1,1:i-1), using an
* update of the form: A := A - V*W**T - W*V**T
*
CALL SSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ),
$ LDA, WORK, LDWORK, ONE, A, LDA )
*
* Copy superdiagonal elements back into A, and diagonal
* elements into D
*
DO 10 J = I, I + NB - 1
A( J-1, J ) = E( J-1 )
D( J ) = A( J, J )
10 CONTINUE
20 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL SSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
ELSE
*
* Reduce the lower triangle of A
*
DO 40 I = 1, N - NX, NB
*
* Reduce columns i:i+nb-1 to tridiagonal form and form the
* matrix W which is needed to update the unreduced part of
* the matrix
*
CALL SLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
$ TAU( I ), WORK, LDWORK )
*
* Update the unreduced submatrix A(i+ib:n,i+ib:n), using
* an update of the form: A := A - V*W**T - W*V**T
*
CALL SSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE,
$ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
$ A( I+NB, I+NB ), LDA )
*
* Copy subdiagonal elements back into A, and diagonal
* elements into D
*
DO 30 J = I, I + NB - 1
A( J+1, J ) = E( J )
D( J ) = A( J, J )
30 CONTINUE
40 CONTINUE
*
* Use unblocked code to reduce the last or only block
*
CALL SSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
$ TAU( I ), IINFO )
END IF
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of SSYTRD
*
END