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dsytrd
C DSYTRD    SOURCE    BP208322  22/09/16    21:15:09     11454          *> \brief \b DSYTRD**  =========== DOCUMENTATION ===========** Online html documentation available at*            http://www.netlib.org/lapack/explore-html/**> \htmlonly*> Download DSYTRD + dependencies*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dsytrd.f">*> [TGZ]&lt;/a>*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dsytrd.f">*> [ZIP]&lt;/a>*> &lt;a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dsytrd.f">*> [TXT]&lt;/a>*> \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 ..*       REAL*8   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 REAL*8 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*8 array, dimension (N)*> The diagonal elements of the tridiagonal matrix T:*> D(i) = A(i,i).*> \endverbatim*>*> \param[out] E*> \verbatim*> E is REAL*8 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*8 array, dimension (N-1)*> The scalar factors of the elementary reflectors (see Further*> Details).*> \endverbatim*>*> \param[out] WORK*> \verbatim*> WORK is REAL*8 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*> &lt; 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 .. REAL*8 A( LDA, * ), D( * ), E( * ), TAU( * ),$                   WORK( * )*     ..**  =====================================================================**     .. Parameters ..      REAL*8   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  

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