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## NAME

```       CTBSV  -  solve one of the systems of equations   A*x = b, or A’*x = b,
or conjg( A’ )*x = b,

```

## SYNOPSIS

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

INTEGER      INCX, K, LDA, N

CHARACTER*1  DIAG, TRANS, UPLO

COMPLEX      A( LDA, * ), X( * )

```

## PURPOSE

```       CTBSV  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’   conjg( 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      - COMPLEX          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      - COMPLEX          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.
```