NAG Library Routine Document
F07BAF (DGBSV)
1 Purpose
F07BAF (DGBSV) computes the solution to a real system of linear equations
where
A is an
n
by
n
band matrix, with
kl subdiagonals and
ku superdiagonals, and
X and
B are
n by
r matrices.
2 Specification
| SUBROUTINE F07BAF ( | N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO) |
| INTEGER | N, KL, KU, NRHS, LDAB, IPIV(N), LDB, INFO |
| double precision | AB(LDAB,*), B(LDB,*) |
The routine may be called by its
LAPACK
name dgbsv.
3 Description
F07BAF (DGBSV) uses the LU decomposition with partial pivoting and row interchanges to factor A as A=PLU, where P is a permutation matrix, 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 AX=B.
4 References
Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999)
LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
http://www.netlib.org/lapack/lugGolub G H and Van Loan C F (1996)
Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
5 Parameters
- 1: N – INTEGERInput
-
On entry:
n, the number of linear equations, i.e., the order of the matrix A.
Constraint:
N≥0.
- 2: KL – INTEGERInput
On entry: kl, the number of subdiagonals within the band of the matrix A.
Constraint:
KL≥0.
- 3: KU – INTEGERInput
On entry: ku, the number of superdiagonals within the band of the matrix A.
Constraint:
KU≥0.
- 4: NRHS – INTEGERInput
-
On entry: r, the number of right-hand sides, i.e., the number of columns of the matrix B.
Constraint:
NRHS≥0.
- 5: AB(LDAB,*) – double precision arrayInput/Output
Note: the second dimension of the array
AB
must be at least
max1,N.
On entry: the
n by
n coefficient matrix
A.
The matrix is stored in rows
kl+1 to
2kl+ku+1; the first
kl rows need not be set, more precisely, the element
Aij must be stored in
See
Section 8 for further details.
On exit: if
INFO≥0,
AB is overwritten by details of the factorization.
The upper triangular band matrix U, with kl+ku superdiagonals, is stored in rows 1 to kl+ku+1 of the array, and the multipliers used to form the matrix L are stored in rows kl+ku+2 to 2kl+ku+1.
- 6: LDAB – INTEGERInput
-
On entry: the first dimension of the array
AB as declared in the (sub)program from which F07BAF (DGBSV) is called.
Constraint:
LDAB≥2×KL+KU+1.
- 7: IPIV(N) – INTEGER arrayOutput
On exit: if no constraints are violated, the pivot indices that define the permutation matrix P; at the ith step row i of the matrix was interchanged with row IPIVi. IPIVi=i indicates a row interchange was not required.
- 8: B(LDB,*) – double precision arrayInput/Output
Note: the second dimension of the array
B
must be at least
max1,NRHS.
On entry: the n by r right-hand side matrix B.
On exit: if INFO=0, the n by r solution matrix X.
- 9: LDB – INTEGERInput
-
On entry: the first dimension of the array
B as declared in the (sub)program from which F07BAF (DGBSV) is called.
Constraint:
LDB≥max1,N.
- 10: INFO – INTEGEROutput
On exit:
INFO=0 unless the routine detects an error (see
Section 6).
6 Error Indicators and Warnings
Errors or warnings detected by the routine:
- INFO<0
If INFO=-i, the ith argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
- INFO>0
If INFO=i, uii is exactly zero. The factorization has been completed, but the factor U is exactly singular, so the solution could not be computed.
7 Accuracy
The computed solution for a single right-hand side,
x^
, satisfies an equation of the form
where
and
ε
is the
machine precision. An approximate error bound for the computed solution is given by
where
κA
=
A-1
1
A1
, the condition number of
A
with respect to the solution of the linear equations. See Section 4.4 of
Anderson et al. (1999) for further details.
Following the use of F07BAF (DGBSV),
F07BGF (DGBCON) can be used to estimate the condition number of
A
and
F07BHF (DGBRFS) can be used to obtain approximate error bounds. Alternatives to F07BAF (DGBSV), which return condition and error estimates directly are
F04BBF and
F07BBF (DGBSVX).
8 Further Comments
The band storage scheme for the array
AB is illustrated by the following example, when
n=6
,
kl=1
, and
ku=2
. Storage of the band matrix
A
in the array
AB:
Array elements marked * need not be set and are not referenced by the routine. Array elements marked + need not be set, but are defined on exit from the routine and contain the elements
u14
,
u25
and
u36
.
The total number of floating-point operations required to solve the equations
AX=B
depends upon the pivoting required, but if
n≫kl+ku
then it is approximately bounded by
O
nkl
kl
+
ku
for the factorization and
O
n
2
kl
+
ku
r
for the solution following the factorization.
The complex analogue of this routine is
F07BNF (ZGBSV).
9 Example
This example solves the equations
where
A
is the band matrix
Details of the
LU
factorization of
A
are also output.
9.1 Program Text
Program Text (f07bafe.f)
9.2 Program Data
Program Data (f07bafe.d)
9.3 Program Results
Program Results (f07bafe.r)