NAG FL Interfacef07baf (dgbsv)

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1Purpose

f07baf computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n×n$ band matrix, with ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals, and $X$ and $B$ are $n×r$ matrices.

2Specification

Fortran Interface
 Subroutine f07baf ( n, kl, ku, nrhs, ab, ldab, ipiv, b, ldb, info)
 Integer, Intent (In) :: n, kl, ku, nrhs, ldab, ldb Integer, Intent (Out) :: ipiv(n), info Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), b(ldb,*)
#include <nag.h>
 void f07baf_ (const Integer *n, const Integer *kl, const Integer *ku, const Integer *nrhs, double ab[], const Integer *ldab, Integer ipiv[], double b[], const Integer *ldb, Integer *info)
The routine may be called by the names f07baf, nagf_lapacklin_dgbsv or its LAPACK name dgbsv.

3Description

f07baf 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 ${k}_{l}$ subdiagonals, and $U$ is upper triangular with $\left({k}_{l}+{k}_{u}\right)$ superdiagonals. The factored form of $A$ is then used to solve the system of equations $AX=B$.

4References

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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2: $\mathbf{kl}$Integer Input
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
3: $\mathbf{ku}$Integer Input
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
4: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
5: $\mathbf{ab}\left({\mathbf{ldab}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n×n$ coefficient matrix $A$.
The matrix is stored in rows ${k}_{l}+1$ to $2{k}_{l}+{k}_{u}+1$; the first ${k}_{l}$ rows need not be set, more precisely, the element ${A}_{ij}$ must be stored in
 $ab(kl+ku+1+i-j,j)=Aij for ​max(1,j-ku)≤i≤min(n,j+kl).$
See Section 9 for further details.
On exit: if ${\mathbf{info}}\ge {\mathbf{0}}$, ab is overwritten by details of the factorization.
The upper triangular band matrix $U$, with ${k}_{l}+{k}_{u}$ superdiagonals, is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows ${k}_{l}+{k}_{u}+2$ to $2{k}_{l}+{k}_{u}+1$.
6: $\mathbf{ldab}$Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07baf is called.
Constraint: ${\mathbf{ldab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
7: $\mathbf{ipiv}\left({\mathbf{n}}\right)$Integer array Output
On exit: if no constraints are violated, the pivot indices that define the permutation matrix $P$; at the $i$th step row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$. ${\mathbf{ipiv}}\left(i\right)=i$ indicates a row interchange was not required.
8: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n×r$ right-hand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n×r$ solution matrix $X$.
9: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07baf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
Element $⟨\mathit{\text{value}}⟩$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, so the solution could not be computed.

7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $(A+E) x^ = b ,$
where
 $‖E‖1 = O(ε) ‖A‖1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, 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, f07bgf can be used to estimate the condition number of $A$ and f07bhf can be used to obtain approximate error bounds. Alternatives to f07baf, which return condition and error estimates directly are f04bbf and f07bbf.

8Parallelism and Performance

f07baf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07baf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The band storage scheme for the array ab is illustrated by the following example, when $n=6$, ${k}_{l}=1$, and ${k}_{u}=2$. Storage of the band matrix $A$ in the array ab:
 $* * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 *$
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 ${u}_{14}$, ${u}_{25}$ and ${u}_{36}$.
The total number of floating-point operations required to solve the equations $AX=B$ depends upon the pivoting required, but if $n\gg {k}_{l}+{k}_{u}$ then it is approximately bounded by $\mathit{O}\left(n{k}_{l}\left({k}_{l}+{k}_{u}\right)\right)$ for the factorization and $\mathit{O}\left(n\left(2{k}_{l}+{k}_{u}\right)r\right)$ for the solution following the factorization.
The complex analogue of this routine is f07bnf.

10Example

This example solves the equations
 $Ax=b ,$
where $A$ is the band matrix
 $A = ( -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 ) and b = ( 4.42 27.13 -6.14 10.50 ) .$
Details of the $LU$ factorization of $A$ are also output.

10.1Program Text

Program Text (f07bafe.f90)

10.2Program Data

Program Data (f07bafe.d)

10.3Program Results

Program Results (f07bafe.r)