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dgbsv

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NAME
DGBSV - compute the solution to a real system of linear
equations A * X = B, where A is a band matrix of order N
with KL subdiagonals and KU superdiagonals, and X and B are
N-by-NRHS matrices

SYNOPSIS
SUBROUTINE DGBSV( N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB,
INFO )

INTEGER       INFO, KL, KU, LDAB, LDB, N, NRHS

INTEGER       IPIV( * )

DOUBLE        PRECISION AB( LDAB, * ), B( LDB, * )

PURPOSE
DGBSV computes the solution to a real system of linear equa-
tions A * X = B, where A is a band matrix of order N with KL
subdiagonals and KU superdiagonals, and X and B are N-by-
NRHS matrices.

The LU decomposition with partial pivoting and row inter-
changes is used to factor A as A = L * U, where L is a pro-
duct of permutation and unit lower triangular matrices with
KL subdiagonals, and U is upper triangular with KL+KU super-
diagonals.  The factored form of A is then used to solve the
system of equations A * X = B.

ARGUMENTS
N       (input) INTEGER
The number of linear equations, i.e., the order of
the matrix A.  N >= 0.

KL      (input) INTEGER
The number of subdiagonals within the band of A.  KL
>= 0.

KU      (input) INTEGER
The number of superdiagonals within the band of A.
KU >= 0.

NRHS    (input) INTEGER
The number of right hand sides, i.e., the number of
columns of the matrix B.  NRHS >= 0.

AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
On entry, the matrix A in band storage, in rows KL+1
to 2*KL+KU+1; rows 1 to KL of the array need not be
set.  The j-th column of A is stored in the j-th
column of the array AB as follows: AB(KL+KU+1+i-j,j)

= A(i,j) for max(1,j-KU)<=i<=min(N,j+KL) On exit,
details of the factorization: U is stored as an
upper triangular band matrix with KL+KU superdiago-
nals in rows 1 to KL+KU+1, and the multipliers used
during the factorization are stored in rows KL+KU+2
to 2*KL+KU+1.  See below for further details.

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >=
2*KL+KU+1.

IPIV    (output) INTEGER array, dimension (N)
The pivot indices that define the permutation matrix
P; row i of the matrix was interchanged with row
IPIV(i).

(LDB,NRHS)
B       (input/output) DOUBLE PRECISION array, dimension
On entry, the N-by-NRHS right hand side matrix B.
On exit, if INFO = 0, the N-by-NRHS solution matrix
X.

LDB     (input) INTEGER
The leading dimension of the array B.  LDB >=
max(1,N).

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
> 0:  if INFO = i, U(i,i) is exactly zero.  The fac-
torization has been completed, but the factor U is
exactly singular, and the solution has not been com-
puted.

FURTHER DETAILS
The band storage scheme is illustrated by the following
example, when M = N = 6, KL = 2, KU = 1:

On entry:                       On exit:

*    *    *    +    +    +       *    *    *   u14  u25
u36
*    *    +    +    +    +       *    *   u13  u24  u35
u46
*   a12  a23  a34  a45  a56      *   u12  u23  u34  u45
u56
a11  a22  a33  a44  a55  a66     u11  u22  u33  u44  u55
u66
a21  a32  a43  a54  a65   *      m21  m32  m43  m54  m65
*
a31  a42  a53  a64   *    *      m31  m42  m53  m64   *

*

Array elements marked * are not used by the routine; ele-
ments marked + need not be set on entry, but are required by
the routine to store elements of U because of fill-in
resulting from the row interchanges.
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