ssytrd.f

          SUBROUTINE SSYTRD( 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 ..
          REAL               A( LDA, * ), D( * ), E( * ), TAU( * ),
         $                   WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  SSYTRD 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) 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.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  D       (output) REAL array, dimension (N)
    *          The diagonal elements of the tridiagonal matrix T:
    *          D(i) = A(i,i).
    *
    *  E       (output) 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'.
    *
    *  TAU     (output) REAL array, dimension (N-1)
    *          The scalar factors of the elementary reflectors (see Further
    *          Details).
    *
    *  WORK    (workspace/output) REAL 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 ..
          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' - W*V'
    *
                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' - W*V'
    *
                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