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
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. The LU decomposition with partial
pivoting and row interchanges is used to factor A as A = L
* U, where L is a product of permutation and unit lower
triangular matrices with KL subdiagonals, and U is upper
triangular with KL+KU superdiagonals. 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
band matrix with KL+KU superdiagonals in rows 1 to
KL+KU+1, and the multipliers used during the fac
torization 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).
B (input/output) DOUBLE PRECISION array,
dimension (LDB,NRHS)
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 ille
gal value
> 0: if INFO = i, U(i,i) is exactly zero. The
factorization has been completed, but the factor U
is exactly singular, and the solution has not been
computed.
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;
required by the routine to store elements of U because of
fill-in resulting from the row interchanges.
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