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NAME

       STBSV - solve one of the systems of equations   A*x = b, or A’*x = b,

SYNOPSIS

       SUBROUTINE STBSV ( UPLO, TRANS, DIAG, N, K, A, LDA, X, INCX )

           INTEGER      INCX, K, LDA, N

           CHARACTER*1  DIAG, TRANS, UPLO

           REAL         A( LDA, * ), X( * )

PURPOSE

       STBSV  solves one of the systems of equations

       where  b  and  x are n element vectors and A is an n by n unit, or non-
       unit, upper or lower triangular band matrix, with ( k + 1 )  diagonals.

       No  test  for  singularity  or  near-singularity  is  included  in this
       routine. Such tests must be performed before calling this routine.

PARAMETERS

       UPLO   - CHARACTER*1.
              On entry, UPLO specifies whether the matrix is an upper or lower
              triangular matrix as follows:

              UPLO = ’U’ or ’u’   A is an upper triangular matrix.

              UPLO = ’L’ or ’l’   A is a lower triangular matrix.

              Unchanged on exit.

       TRANS  - CHARACTER*1.
              On entry, TRANS specifies the equations to be solved as follows:

              TRANS = ’N’ or ’n’   A*x = b.

              TRANS = ’T’ or ’t’   A’*x = b.

              TRANS = ’C’ or ’c’   A’*x = b.

              Unchanged on exit.

       DIAG   - CHARACTER*1.
              On entry, DIAG specifies whether or not A is unit triangular  as
              follows:

              DIAG = ’U’ or ’u’   A is assumed to be unit triangular.

              DIAG = ’N’ or ’n’   A is not assumed to be unit triangular.

              Unchanged on exit.

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

       K      - INTEGER.
              On entry with UPLO = ’U’ or  ’u’,  K  specifies  the  number  of
              super-diagonals  of  the  matrix A.  On entry with UPLO = ’L’ or
              ’l’, K specifies the number of sub-diagonals of the matrix A.  K
              must satisfy  0 .le. K.  Unchanged on exit.

       A      - REAL             array of DIMENSION ( LDA, n ).
              Before  entry with UPLO = ’U’ or ’u’, the leading ( k + 1 ) by n
              part of the array A must contain the upper triangular band  part
              of  the  matrix of coefficients, supplied column by column, with
              the leading diagonal of the matrix in row (  k  +  1  )  of  the
              array, the first super-diagonal starting at position 2 in row k,
              and so on. The top left k by k triangle of the array  A  is  not
              referenced.   The  following  program  segment  will transfer an
              upper triangular  band  matrix  from  conventional  full  matrix
              storage to band storage:

              DO 20, J = 1, N M = K + 1 - J DO 10, I = MAX( 1, J - K ), J A( M
              + I, J ) = matrix( I, J ) 10    CONTINUE 20 CONTINUE

              Before entry with UPLO = ’L’ or ’l’, the leading ( k + 1 ) by  n
              part  of the array A must contain the lower triangular band part
              of the matrix of coefficients, supplied column by  column,  with
              the  leading  diagonal  of the matrix in row 1 of the array, the
              first sub-diagonal starting at position 1 in row 2, and  so  on.
              The  bottom  right  k  by  k  triangle  of  the  array  A is not
              referenced.  The following program segment will transfer a lower
              triangular  band matrix from conventional full matrix storage to
              band storage:

              DO 20, J = 1, N M = 1 - J DO 10, I = J, MIN( N, J + K ) A(  M  +
              I, J ) = matrix( I, J ) 10    CONTINUE 20 CONTINUE

              Note  that  when  DIAG  = ’U’ or ’u’ the elements of the array A
              corresponding to the diagonal elements of  the  matrix  are  not
              referenced, but are assumed to be unity.  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  (  k  +  1  ).
              Unchanged on exit.

       X      - REAL             array of dimension at least
              (  1  +  ( n - 1 )*abs( INCX ) ).  Before entry, the incremented
              array X must contain the n element right-hand side vector b.  On
              exit, X is overwritten with the solution vector x.

       INCX   - INTEGER.
              On  entry,  INCX  specifies the increment for the elements of X.
              INCX 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.