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NAME

        ZGBMV  - perform one of the matrix-vector operations   y := alpha*A*x +
        beta*y, or y := alpha*A’*x + beta*y, or   y := alpha*conjg(  A’  )*x  +
        beta*y,
 

SYNOPSIS

        SUBROUTINE ZGBMV (  TRANS,  M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA,
                         Y, INCY )
 
            COMPLEX*16   ALPHA, BETA
 
            INTEGER      INCX, INCY, KL, KU, LDA, M, N
 
            CHARACTER*1  TRANS
 
            COMPLEX*16   A( LDA, * ), X( * ), Y( * )
 

PURPOSE

        ZGBMV  performs one of the matrix-vector operations
 
        where alpha and beta are scalars, x and y are vectors and A is an m  by
        n band matrix, with kl sub-diagonals and ku super-diagonals.
 

PARAMETERS

        TRANS  - CHARACTER*1.
               On  entry, TRANS specifies the operation to be performed as fol‐
               lows:
 
               TRANS = ’N’ or ’n’   y := alpha*A*x + beta*y.
 
               TRANS = ’T’ or ’t’   y := alpha*A’*x + beta*y.
 
               TRANS = ’C’ or ’c’   y := alpha*conjg( A’ )*x + beta*y.
 
               Unchanged on exit.
 
        M      - INTEGER.
               On entry, M specifies the number of rows of  the  matrix  A.   M
               must be at least zero.  Unchanged on exit.
 
        N      - INTEGER.
               On  entry, N specifies the number of columns of the matrix A.  N
               must be at least zero.  Unchanged on exit.
 
        KL     - INTEGER.
               On entry, KL specifies the number of sub-diagonals of the matrix
               A. KL must satisfy  0 .le. KL.  Unchanged on exit.
 
        KU     - INTEGER.
               On  entry,  KU  specifies  the  number of super-diagonals of the
               matrix A. KU must satisfy  0 .le. KU.  Unchanged on exit.
 
        ALPHA  - COMPLEX*16      .
               On entry, ALPHA specifies the scalar alpha.  Unchanged on  exit.
 
        A      - COMPLEX*16       array of DIMENSION ( LDA, n ).
               Before entry, the leading ( kl + ku + 1 ) by n part of the array
               A must contain the matrix of coefficients,  supplied  column  by
               column,  with the leading diagonal of the matrix in row ( ku + 1
               ) of the array, the first super-diagonal starting at position  2
               in  row ku, the first sub-diagonal starting at position 1 in row
               ( ku + 2 ), and so on.  Elements in the array A that do not cor‐
               respond  to elements in the band matrix (such as the top left ku
               by ku triangle) are not referenced.  The following program  seg‐
               ment  will  transfer a band matrix from conventional full matrix
               storage to band storage:
 
               DO 20, J = 1, N K = KU + 1 - J DO 10, I = MAX( 1, J - KU ), MIN(
               M,  J  +  KL  ) A( K + I, J ) = matrix( I, J ) 10    CONTINUE 20
               CONTINUE
 
               Unchanged on exit.
 
        LDA    - INTEGER.
               On entry, LDA specifies the first dimension of A as declared  in
               the calling (sub) program. LDA must be at least ( kl + ku + 1 ).
               Unchanged on exit.
 
        X      - COMPLEX*16       array of DIMENSION at least
               ( 1 + ( n - 1 )*abs( INCX ) ) when TRANS = ’N’  or  ’n’  and  at
               least  (  1  + ( m - 1 )*abs( INCX ) ) otherwise.  Before entry,
               the incremented array X must contain the vector x.  Unchanged on
               exit.
 
        INCX   - INTEGER.
               On  entry,  INCX  specifies the increment for the elements of X.
               INCX must not be zero.  Unchanged on exit.
 
        BETA   - COMPLEX*16      .
               On entry, BETA specifies the scalar beta. When BETA is  supplied
               as zero then Y need not be set on input.  Unchanged on exit.
 
        Y      - COMPLEX*16       array of DIMENSION at least
               (  1  +  (  m - 1 )*abs( INCY ) ) when TRANS = ’N’ or ’n’ and at
               least ( 1 + ( n - 1 )*abs( INCY ) )  otherwise.   Before  entry,
               the incremented array Y must contain the vector y. On exit, Y is
               overwritten by the updated vector y.
 
        INCY   - INTEGER.
               On entry, INCY specifies the increment for the  elements  of  Y.
               INCY must not be zero.  Unchanged on exit.
 
               Level 2 Blas routine.
 
               --  Written on 22-October-1986.  Jack Dongarra, Argonne National
               Lab.  Jeremy Du Croz, Nag Central Office.  Sven Hammarling,  Nag
               Central Office.  Richard Hanson, Sandia National Labs.
 

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