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# dpbsv

```
NAME
DPBSV - compute the solution to a real system of linear
equations  A * X = B,

SYNOPSIS
SUBROUTINE DPBSV( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO
)

CHARACTER     UPLO

INTEGER       INFO, KD, LDAB, LDB, N, NRHS

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

PURPOSE
DPBSV computes the solution to a real system of linear equa-
tions
A * X = B, where A is an N-by-N symmetric positive defin-
ite band matrix and X and B are N-by-NRHS matrices.

The Cholesky decomposition is used to factor A as
A = U**T * U,  if UPLO = 'U', or
A = L * L**T,  if UPLO = 'L',
where U is an upper triangular matrix, and L is a lower tri-
angular matrix, with the same number of superdiagonals or
subdiagonals as A.  The factored form of A is then used to
solve the system of equations A * X = B.

ARGUMENTS
UPLO    (input) CHARACTER*1
= 'U':  Upper triangle of A is stored;
= 'L':  Lower triangle of A is stored.

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

KD      (input) INTEGER
The number of superdiagonals of the matrix A if UPLO
= 'U', or the number of subdiagonals if UPLO = 'L'.
KD >= 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 upper or lower triangle of the sym-
metric band matrix A, stored in the first KD+1 rows
of the array.  The j-th column of A is stored in the
j-th column of the array AB as follows: if UPLO =

'U', AB(KD+1+i-j,j) = A(i,j) for max(1,j-KD)<=i<=j;
if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for
j<=i<=min(N,j+KD).  See below for further details.

On exit, if INFO = 0, the triangular factor U or L
from the Cholesky factorization A = U**T*U or A =
L*L**T of the band matrix A, in the same storage
format as A.

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >=
KD+1.

(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, the leading minor of order i of A
is not positive definite, so the factorization could
not be completed, and the solution has not been com-
puted.

FURTHER DETAILS
The band storage scheme is illustrated by the following
example, when N = 6, KD = 2, and UPLO = 'U':

On entry:                       On exit:

*    *   a13  a24  a35  a46      *    *   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

Similarly, if UPLO = 'L' the format of A is as follows:

On entry:                       On exit:

a11  a22  a33  a44  a55  a66     l11  l22  l33  l44  l55
l66
a21  a32  a43  a54  a65   *      l21  l32  l43  l54  l65

*
a31  a42  a53  a64   *    *      l31  l42  l53  l64   *
*

Array elements marked * are not used by the routine.
```