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_GBMV(3F)							     _GBMV(3F)

NAME
     dgbmv, sgbmv, zgbmv, cgbmv - BLAS Level Two Matrix-Vector Product

FORTRAN 77 SYNOPSIS
     subroutine dgbmv( trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy )
	   character*1	      trans
	   integer	      n, m, kl, ku, lda, incx, incy
	   double precision   alpha, beta
	   double precision   a( lda,*), x(*), y(*)

     subroutine sgbmv( trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy )
	   character*1	      trans
	   integer	      n, m, kl, ku, lda, incx, incy
	   real		      alpha, beta
	   real		      a( lda,*), x(*), y(*)

     subroutine zgbmv( trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy )
	   character*1	      trans
	   integer	      n, m, kl, ku, lda, incx, incy
	   double complex     alpha, beta
	   double complex     a( lda,*), x(*), y(*)

     subroutine cgbmv( trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy )
	   character*1	      trans
	   integer	      n, m, kl, ku, lda, incx, incy
	   complex	      alpha, beta
	   complex	      a( lda,*), x(*), y(*)

C SYNOPSIS
     void dgbmv( trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy )
	   MatrixTranspose    trans;
	   Integer	      n, m, kl, ku, lda, incx, incy;
	   double	      alpha, beta;
	   double	      (*a)[lda*n], (*x)[ n ], (*y)[ n ];

     void sgbmv( trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy )
	   MatrixTranspose    trans;
	   Integer	      n, m, kl, ku, lda, incx, incy;
	   float	      alpha, beta;
	   float	      (*a)[lda*n], (*x)[ n ], (*y)[ n ];

     void zgbmv( trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy )
	   MatrixTranspose    trans;
	   Integer	      n, m, kl, ku, lda, incx, incy;
	   Zomplex	      alpha, beta;
	   Zomplex	      (*a)[lda*n], (*x)[ n ], (*y)[ n ];

     void cgbmv( trans,m,n,kl,ku,alpha,a,lda,x,incx,beta,y,incy )
	   MatrixTranspose    trans;
	   Integer	      n, m, kl, ku, lda, incx, incy;

									Page 1

_GBMV(3F)							     _GBMV(3F)

	   Complex	      alpha, beta;
	   Complex	      (*a)[lda*n], (*x)[ n ], (*y)[ n ];

DESCRIPTION
     dgbmv, sgbmv, zgbmv and cgbmv 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,

     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   On entry, trans specifies the operation to be performed as
	     follows:

	     FORTRAN
	     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.

	     C
	     trans = NoTranspose	   y := alpha*A*x + beta*y.
	     trans = Transpose		   y := alpha*A'*x + beta*y.
	     trans = ConjugateTranspose	   y := alpha*conjg( A' )*x + beta*y.

	     Unchanged on exit.

     m	     On entry, m specifies the number of rows of rows of the matrix A.
	     m must be at least zero.
	     Unchanged on exit.

     n	     On entry, n specifies the order of the matrix A. n must be at
	     least zero.
	     Unchanged on exit.

     kl	     On entry, kl specifies the number of sub-diagonals of the matrix
	     A.	 kl must satisfy 0 .le. kl.
	     Unchanged on exit.

     ku	     On entry, ku specifies the number of super-diagonals of the
	     matrix A.	ku must satisfy	 0 .le. ku.
	     Unchanged on exit.

     alpha   On entry, alpha specifies the scalar alpha.
	     Unchanged on exit.

									Page 2

_GBMV(3F)							     _GBMV(3F)

     a	     An array containing the matrix A.

	     FORTRAN
	     Array of dimension ( lda, n ).

	     C
	     A pointer to an array of size lda*n containing the matrix A.
	     See note below about array storage convention for C.

	     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 correspond
	     to elements in the band matrix (such as the top left ku by ku
	     triangle) are not referenced.  The following program segment will
	     transfer a band matrix from conventional full matrix storage to
	     band storage:

	     FORTRAN
			    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

	     C
			 for ( j=0; j<n; j++ )
			 {
			       k = ku - j;
			       for (i = MAX(0, j-ku); i < MIN(m, j+kl); i++ )
				     a(j*lda+k+i) = Matrix(j*ldm+i);
			 }

	     Unchanged on exit.

     lda     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	     Array of size at least ( 1 + ( n - 1 )*abs( incx ) ) when trans =
	     'N' or 'n' or NoTranspose and at least ( 1 + ( m - 1 )*abs( incx
	     ) ) otherwise.  Before entry, the incremented array x must
	     contain the vector x.
	     Unchanged on exit.

     incx    On entry, incx specifies the increment for the elements of x.
	     incx must not be zero.
	     Unchanged on exit.

									Page 3

_GBMV(3F)							     _GBMV(3F)

     beta    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	     Array of size at least ( 1 + ( m - 1 )*abs( incy ) ) when trans =
	     'N' or 'n' or NoTranspose and at least ( 1 + ( n - 1 )*abs( incy
	     ) ) otherwise.

	     Before entry with beta non-zero, the incremented array y must
	     contain the vector y. On exit, y is overwritten by the updated
	     vector y.

     incy    On entry, incy specifies the increment for the elements of y.
	     incy must not be zero.
	     Unchanged on exit.

C ARRAY STORAGE CONVENTION
       The matrices  are assumed  to be stored in a  one dimensional C array
       in an analogous fashion as a Fortran array (column major). Therefore,
       the element  A(i+1,j)  of matrix A  is stored  immediately  after the
       element	A(i,j), while  A(i,j+1) is lda	elements apart from  A(i,j).
       The element A(i,j) of the matrix can be accessed directly by reference
       to  a[ (j-1)*lda + (i-1) ].

AUTHORS
	  Jack Dongarra, Argonne National Laboratory.
	  Iain Duff, AERE Harwell.
	  Jeremy Du Croz, Numerical Algorithms Group Ltd.
	  Sven Hammarling, Numerical Algorithms Group Ltd.

TUNING
	  Optimized and parallelized for SGI R3000, R4x00 and R8000 platforms.

									Page 4

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