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SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
*
* -- LAPACK routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. 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 (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
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