f04bbf computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n\times n$ band matrix, with ${k}_{l}$ subdiagonals and ${k}_{u}$ superdiagonals, and $X$ and $B$ are $n\times r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.
The routine may be called by the names f04bbf or nagf_linsys_real_band_solve.
3Description
The $LU$ decomposition with partial pivoting and row interchanges is used to factor $A$ as $A=PLU$, where $P$ is a permutation matrix, $L$ is the product of permutation matrices and unit lower triangular matrices with ${k}_{l}$ subdiagonals, and $U$ is upper triangular with $({k}_{l}+{k}_{u})$ 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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5Arguments
1: $\mathbf{n}$ – IntegerInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
2: $\mathbf{kl}$ – IntegerInput
On entry: the number of subdiagonals ${k}_{l}$, within the band of $A$.
Constraint:
${\mathbf{kl}}\ge 0$.
3: $\mathbf{ku}$ – IntegerInput
On entry: the number of superdiagonals ${k}_{u}$, within the band of $A$.
Constraint:
${\mathbf{ku}}\ge 0$.
4: $\mathbf{nrhs}$ – IntegerInput
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.
5: $\mathbf{ab}({\mathbf{ldab}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array ab
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: the $n\times n$ 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
On exit: if ${\mathbf{ifail}}\ge {\mathbf{0}}$, 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: $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f04bbf is called.
On exit: if ${\mathbf{ifail}}\ge {\mathbf{0}}$, 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: $\mathbf{b}({\mathbf{ldb}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nrhs}})$.
On entry: the $n\times r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the $n\times r$ solution matrix $X$.
9: $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f04bbf is called.
On exit: if no constraints are violated, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{-1}\Vert}_{1}\right)$.
11: $\mathbf{errbnd}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the forward error bound for a computed solution $\hat{x}$, such that ${\Vert \hat{x}-x\Vert}_{1}/{\Vert x\Vert}_{1}\le {\mathbf{errbnd}}$, where $\hat{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, errbnd is returned as unity.
12: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Diagonal element $\u27e8\mathit{\text{value}}\u27e9$ of the upper triangular factor is zero. The factorization has been completed, but the solution could not be computed.
${\mathbf{ifail}}={\mathbf{n}}+1$
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
${\mathbf{ifail}}=-1$
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{kl}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{kl}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, ${\mathbf{ku}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ku}}\ge 0$.
${\mathbf{ifail}}=-4$
On entry, ${\mathbf{nrhs}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
${\mathbf{ifail}}=-6$
On entry, ${\mathbf{ldab}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{kl}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{ku}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldab}}\ge 2\times {\mathbf{kl}}+{\mathbf{ku}}+1$.
${\mathbf{ifail}}=-9$
On entry, ${\mathbf{ldb}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
The integer allocatable memory required is n, and the real allocatable memory required is $3\times {\mathbf{n}}$. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
The computed solution for a single right-hand side, $\hat{x}$, satisfies an equation of the form
where $\kappa \left(A\right)={\Vert {A}^{-1}\Vert}_{1}{\Vert A\Vert}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. f04bbf uses the approximation ${\Vert E\Vert}_{1}=\epsilon {\Vert A\Vert}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999)
for further details.
8Parallelism and Performance
f04bbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04bbf 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.
9Further Comments
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:
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 $\mathit{O}\left(n{k}_{l}({k}_{l}+{k}_{u})\right)$ for the factorization and $\mathit{O}\left(n(2{k}_{l}+{k}_{u})r\right)$ for the solution following the factorization. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.