MB01YD

Symmetric rank k operation C := alpha op( A ) op( A )' + beta C, C symmetric

Purpose

  To perform the symmetric rank k operations

     C := alpha*op( A )*op( A )' + beta*C,

  where alpha and beta are scalars, C is an n-by-n symmetric matrix,
  op( A ) is an n-by-k matrix, and op( A ) is one of

     op( A ) = A   or   op( A ) = A'.

  The matrix A has l nonzero codiagonals, either upper or lower.

      SUBROUTINE MB01YD( UPLO, TRANS, N, K, L, ALPHA, BETA, A, LDA, C,
     $                   LDC, INFO )
C     .. Scalar Arguments ..
      CHARACTER          TRANS, UPLO
      INTEGER            INFO, LDA, LDC, K, L, N
      DOUBLE PRECISION   ALPHA, BETA
C     .. Array Arguments ..
      DOUBLE PRECISION   A( LDA, * ), C( LDC, * )

Mode Parameters

  UPLO    CHARACTER*1
          Specifies which triangle of the symmetric matrix C
          is given and computed, as follows:
          = 'U':  the upper triangular part is given/computed;
          = 'L':  the lower triangular part is given/computed.
          UPLO also defines the pattern of the matrix A (see below).

  TRANS   CHARACTER*1
          Specifies the form of op( A ) to be used, as follows:
          = 'N':  op( A ) = A;
          = 'T':  op( A ) = A';
          = 'C':  op( A ) = A'.

  N       (input) INTEGER
          The order of the matrix C.  N >= 0.

  K       (input) INTEGER
          The number of columns of the matrix op( A ).  K >= 0.

  L       (input) INTEGER
          If UPLO = 'U', matrix A has L nonzero subdiagonals.
          If UPLO = 'L', matrix A has L nonzero superdiagonals.
          MAX(0,NR-1) >= L >= 0, if UPLO = 'U',
          MAX(0,NC-1) >= L >= 0, if UPLO = 'L',
          where NR and NC are the numbers of rows and columns of the
          matrix A, respectively.

  ALPHA   (input) DOUBLE PRECISION
          The scalar alpha. When alpha is zero then the array A is
          not referenced.

  BETA    (input) DOUBLE PRECISION
          The scalar beta. When beta is zero then the array C need
          not be set before entry.

  A       (input) DOUBLE PRECISION array, dimension (LDA,NC), where
          NC is K when TRANS = 'N', and is N otherwise.
          If TRANS = 'N', the leading N-by-K part of this array must
          contain the matrix A, otherwise the leading K-by-N part of
          this array must contain the matrix A.
          If UPLO = 'U', only the upper triangular part and the
          first L subdiagonals are referenced, and the remaining
          subdiagonals are assumed to be zero.
          If UPLO = 'L', only the lower triangular part and the
          first L superdiagonals are referenced, and the remaining
          superdiagonals are assumed to be zero.

  LDA     INTEGER
          The leading dimension of array A.  LDA >= max(1,NR),
          where NR = N, if TRANS = 'N', and NR = K, otherwise.

  C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
          On entry with UPLO = 'U', the leading N-by-N upper
          triangular part of this array must contain the upper
          triangular part of the symmetric matrix C.
          On entry with UPLO = 'L', the leading N-by-N lower
          triangular part of this array must contain the lower
          triangular part of the symmetric matrix C.
          On exit, the leading N-by-N upper triangular part (if
          UPLO = 'U'), or lower triangular part (if UPLO = 'L'), of
          this array contains the corresponding triangular part of
          the updated matrix C.

  LDC     INTEGER
          The leading dimension of array C.  LDC >= MAX(1,N).

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value.

  The calculations are efficiently performed taking the symmetry
  and structure into account.

  The matrix A may have the following patterns, when n = 7, k = 5,
  and l = 2 are used for illustration:

  UPLO = 'U', TRANS = 'N'         UPLO = 'L', TRANS = 'N'

         [ x x x x x ]                   [ x x x 0 0 ]
         [ x x x x x ]                   [ x x x x 0 ]
         [ x x x x x ]                   [ x x x x x ]
     A = [ 0 x x x x ],              A = [ x x x x x ],
         [ 0 0 x x x ]                   [ x x x x x ]
         [ 0 0 0 x x ]                   [ x x x x x ]
         [ 0 0 0 0 x ]                   [ x x x x x ]

  UPLO = 'U', TRANS = 'T'         UPLO = 'L', TRANS = 'T'

         [ x x x x x x x ]               [ x x x 0 0 0 0 ]
         [ x x x x x x x ]               [ x x x x 0 0 0 ]
     A = [ x x x x x x x ],          A = [ x x x x x 0 0 ].
         [ 0 x x x x x x ]               [ x x x x x x 0 ]
         [ 0 0 x x x x x ]               [ x x x x x x x ]

  If N = K, the matrix A is upper or lower triangular, for L = 0,
  and upper or lower Hessenberg, for L = 1.

  This routine is a specialization of the BLAS 3 routine DSYRK.
  BLAS 1 calls are used when appropriate, instead of in-line code,
  in order to increase the efficiency. If the matrix A is full, or
  its zero triangle has small order, an optimized DSYRK code could
  be faster than MB01YD.

Program Text

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