SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO ) ! ! -- LAPACK routine (version 3.1) -- ! Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. ! November 2006 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER INFO, LDA, LWORK, N ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), & WORK( * ) ! .. ! ! Purpose ! ======= ! ! DSYTRD reduces a real symmetric matrix A to real symmetric ! tridiagonal form T by an orthogonal similarity transformation: ! Q**T * A * Q = T. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! = 'U': Upper triangle of A is stored; ! = 'L': Lower triangle of A is stored. ! ! N (input) INTEGER ! The order of the matrix A. N >= 0. ! ! A (input/output) 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. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= max(1,N). ! ! D (output) DOUBLE PRECISION array, dimension (N) ! The diagonal elements of the tridiagonal matrix T: ! D(i) = A(i,i). ! ! E (output) 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'. ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors (see Further ! Details). ! ! WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) ! On exit, if INFO = 0, WORK(1) returns the optimal LWORK. ! ! LWORK (input) 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. ! ! INFO (output) INTEGER ! = 0: successful exit ! < 0: if INFO = -i, the i-th argument had an illegal value ! ! Further Details ! =============== ! ! 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' ! ! 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' ! ! 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). ! ! ===================================================================== ! ! .. 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' - W*V' ! 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' - W*V' ! 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 SUBROUTINE DSYTRD