*> \brief \b DSYTRD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DSYTRD + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER UPLO * INTEGER INFO, LDA, LWORK, N * .. * .. Array Arguments .. * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), * \$ WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DSYTRD 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 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 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 DOUBLE PRECISION array, dimension (N) *> The diagonal elements of the tridiagonal matrix T: *> D(i) = A(i,i). *> \endverbatim *> *> \param[out] E *> \verbatim *> E is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N-1) *> The scalar factors of the elementary reflectors (see Further *> Details). *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION 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 doubleSYcomputational * *> \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 DSYTRD( 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 .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), \$ WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) * .. * .. Local Scalars .. LOGICAL LQUERY, UPPER INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB, \$ NBMIN, NX * .. * .. External Subroutines .. EXTERNAL DLATRD, DSYR2K, DSYTD2, 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, 'DSYTRD', UPLO, N, -1, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'DSYTRD', -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, 'DSYTRD', 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, 'DSYTRD', 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 DLATRD( 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 DSYR2K( 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 DSYTD2( 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 DLATRD( 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 DSYR2K( 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 DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ), \$ TAU( I ), IINFO ) END IF * WORK( 1 ) = LWKOPT RETURN * * End of DSYTRD * END