f07bbf uses the $LU$ factorization to compute the solution to a real system of linear equations
$$AX=B\text{\hspace{1em} or \hspace{1em}}{A}^{\mathrm{T}}X=B\text{,}$$
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. Error bounds on the solution and a condition estimate are also provided.
The routine may be called by the names f07bbf, nagf_lapacklin_dgbsvx or its LAPACK name dgbsvx.
3Description
f07bbf performs the following steps:
1.Equilibration
The linear system to be solved may be badly scaled. However, the system can be equilibrated as a first stage by setting ${\mathbf{fact}}=\text{'E'}$. In this case, real scaling factors are computed and these factors then determine whether the system is to be equilibrated. Equilibrated forms of the systems $AX=B$ and ${A}^{\mathrm{T}}X=B$ are
respectively, where ${D}_{R}$ and ${D}_{C}$ are diagonal matrices, with positive diagonal elements, formed from the computed scaling factors.
When equilibration is used, $A$ will be overwritten by ${D}_{R}A{D}_{C}$ and $B$ will be overwritten by ${D}_{R}B$ (or ${D}_{C}B$ when the solution of ${A}^{\mathrm{T}}X=B$ is sought).
2.Factorization
The matrix $A$, or its scaled form, is copied and factored using the $LU$ decomposition
$$A=PLU\text{,}$$
where $P$ is a permutation matrix, $L$ is a unit lower triangular matrix, and $U$ is upper triangular.
This stage can be by-passed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to f07bbf with the same matrix $A$.
3.Condition Number Estimation
The $LU$ factorization of $A$ determines whether a solution to the linear system exists. If some diagonal element of $U$ is zero, then $U$ is exactly singular, no solution exists and the routine returns with a failure. Otherwise the factorized form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision then a warning code is returned on final exit.
4.Solution
The (equilibrated) system is solved for $X$ (${D}_{C}^{-1}X$ or ${D}_{R}^{-1}X$) using the factored form of $A$ (${D}_{R}A{D}_{C}$).
5.Iterative Refinement
Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for the computed solution.
6.Construct Solution Matrix $X$
If equilibration was used, the matrix $X$ is premultiplied by ${D}_{C}$ (if ${\mathbf{trans}}=\text{'N'}$) or ${D}_{R}$ (if ${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$) so that it solves the original system before equilibration.
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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5Arguments
1: $\mathbf{fact}$ – Character(1)Input
On entry: specifies whether or not the factorized form of the matrix $A$ is supplied on entry, and if not, whether the matrix $A$ should be equilibrated before it is factorized.
${\mathbf{fact}}=\text{'F'}$
afb and ipiv contain the factorized form of $A$. If ${\mathbf{equed}}\ne \text{'N'}$, the matrix $A$ has been equilibrated with scaling factors given by r and c. ab, afb and ipiv are not modified.
${\mathbf{fact}}=\text{'N'}$
The matrix $A$ will be copied to afb and factorized.
${\mathbf{fact}}=\text{'E'}$
The matrix $A$ will be equilibrated if necessary, then copied to afb and factorized.
Constraint:
${\mathbf{fact}}=\text{'F'}$, $\text{'N'}$ or $\text{'E'}$.
2: $\mathbf{trans}$ – Character(1)Input
On entry: specifies the form of the system of equations.
${\mathbf{trans}}=\text{'N'}$
$AX=B$ (No transpose).
${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$
${A}^{\mathrm{T}}X=B$ (Transpose).
Constraint:
${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
3: $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.
4: $\mathbf{kl}$ – IntegerInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint:
${\mathbf{kl}}\ge 0$.
5: $\mathbf{ku}$ – IntegerInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint:
${\mathbf{ku}}\ge 0$.
6: $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{nrhs}}\ge 0$.
7: $\mathbf{ab}({\mathbf{ldab}},*)$ – 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 $n\times n$ coefficient matrix $A$.
The matrix is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$, more precisely, the element ${A}_{ij}$ must be stored in
9: $\mathbf{afb}({\mathbf{ldafb}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array afb
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, afb need not be set.
If ${\mathbf{fact}}=\text{'F'}$, details of the $LU$ factorization of the $n\times n$ band matrix $A$, as computed by f07bdf.
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 $2{k}_{l}+{k}_{u}+1$.
If ${\mathbf{equed}}\ne \text{'N'}$, afb is the factorized form of the equilibrated matrix $A$.
On exit: if ${\mathbf{fact}}=\text{'F'}$, afb is unchanged from entry.
Otherwise, if no constraints are violated, then if ${\mathbf{fact}}=\text{'N'}$, afb returns details of the $LU$ factorization of the band matrix $A$, and if ${\mathbf{fact}}=\text{'E'}$, afb returns details of the $LU$ factorization of the equilibrated band matrix $A$ (see the description of ab for the form of the equilibrated matrix).
10: $\mathbf{ldafb}$ – IntegerInput
On entry: the first dimension of the array afb as declared in the (sub)program from which f07bbf is called.
Note: the dimension of the array ipiv
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, ipiv need not be set.
If ${\mathbf{fact}}=\text{'F'}$, ipiv contains the pivot indices from the factorization $A=LU$, as computed by f07bdf; row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$.
On exit: if ${\mathbf{fact}}=\text{'F'}$, ipiv is unchanged from entry.
Otherwise, if no constraints are violated, ipiv contains 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.
If ${\mathbf{fact}}=\text{'N'}$, the pivot indices are those corresponding to the factorization $A=LU$ of the original matrix $A$.
If ${\mathbf{fact}}=\text{'E'}$, the pivot indices are those corresponding to the factorization of $A=LU$ of the equilibrated matrix $A$.
12: $\mathbf{equed}$ – Character(1)Input/Output
On entry: if ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, equed need not be set.
If ${\mathbf{fact}}=\text{'F'}$, equed must specify the form of the equilibration that was performed as follows:
if ${\mathbf{equed}}=\text{'N'}$, no equilibration;
if ${\mathbf{equed}}=\text{'R'}$, row equilibration, i.e., $A$ has been premultiplied by ${D}_{R}$;
if ${\mathbf{equed}}=\text{'C'}$, column equilibration, i.e., $A$ has been postmultiplied by ${D}_{C}$;
if ${\mathbf{equed}}=\text{'B'}$, both row and column equilibration, i.e., $A$ has been replaced by ${D}_{R}A{D}_{C}$.
On exit: if ${\mathbf{fact}}=\text{'F'}$, equed is unchanged from entry.
Otherwise, if no constraints are violated, equed specifies the form of equilibration that was performed as specified above.
Constraint:
if ${\mathbf{fact}}=\text{'F'}$, ${\mathbf{equed}}=\text{'N'}$, $\text{'R'}$, $\text{'C'}$ or $\text{'B'}$.
13: $\mathbf{r}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array r
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, r need not be set.
If ${\mathbf{fact}}=\text{'F'}$ and ${\mathbf{equed}}=\text{'R'}$ or $\text{'B'}$, r must contain the row scale factors for $A$, ${D}_{R}$; each element of r must be positive.
On exit: if ${\mathbf{fact}}=\text{'F'}$, r is unchanged from entry.
Otherwise, if no constraints are violated and ${\mathbf{equed}}=\text{'R'}$ or $\text{'B'}$, r contains the row scale factors for $A$, ${D}_{R}$, such that $A$ is multiplied on the left by ${D}_{R}$; each element of r is positive.
14: $\mathbf{c}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array c
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{n}})$.
On entry: if ${\mathbf{fact}}=\text{'N'}$ or $\text{'E'}$, c need not be set.
If ${\mathbf{fact}}=\text{'F'}$ and ${\mathbf{equed}}=\text{'C'}$ or $\text{'B'}$, c must contain the column scale factors for $A$, ${D}_{C}$; each element of c must be positive.
On exit: if ${\mathbf{fact}}=\text{'F'}$, c is unchanged from entry.
Otherwise, if no constraints are violated and ${\mathbf{equed}}=\text{'C'}$ or $\text{'B'}$, c contains the row scale factors for $A$, ${D}_{C}$; each element of c is positive.
15: $\mathbf{b}({\mathbf{ldb}},*)$ – 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 $n\times r$ right-hand side matrix $B$.
On exit: if ${\mathbf{equed}}=\text{'N'}$, b is not modified.
If ${\mathbf{trans}}=\text{'N'}$ and ${\mathbf{equed}}=\text{'R'}$ or $\text{'B'}$, b is overwritten by ${D}_{R}B$.
If ${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$ and ${\mathbf{equed}}=\text{'C'}$ or $\text{'B'}$, b is overwritten by ${D}_{C}B$.
16: $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07bbf is called.
17: $\mathbf{x}({\mathbf{ldx}},*)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nrhs}})$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, the $n\times r$ solution matrix $X$ to the original system of equations. Note that the arrays $A$ and $B$ are modified on exit if ${\mathbf{equed}}\ne \text{'N'}$, and the solution to the equilibrated system is ${D}_{C}^{-1}X$ if ${\mathbf{trans}}=\text{'N'}$ and ${\mathbf{equed}}=\text{'C'}$ or $\text{'B'}$, or ${D}_{R}^{-1}X$ if ${\mathbf{trans}}=\text{'T'}$ or $\text{'C'}$ and ${\mathbf{equed}}=\text{'R'}$ or $\text{'B'}$.
18: $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07bbf is called.
On exit: if no constraints are violated, an estimate of the reciprocal condition number of the matrix $A$ (after equilibration if that is performed), computed as ${\mathbf{rcond}}=1.0/\left({\Vert A\Vert}_{1}{\Vert {A}^{-1}\Vert}_{1}\right)$.
20: $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the forward error bound for each computed solution vector, such that ${\Vert {\hat{x}}_{j}-{x}_{j}\Vert}_{\infty}/{\Vert {x}_{j}\Vert}_{\infty}\le {\mathbf{ferr}}\left(j\right)$ where ${\hat{x}}_{j}$ is the $j$th column of the computed solution returned in the array x and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
21: $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the component-wise relative backward error of each computed solution vector ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).
22: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,3\times {\mathbf{n}})\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the reciprocal pivot growth factor $\mathrm{max}\left|{a}_{ij}\right|/\mathrm{max}\left|{u}_{ij}\right|$. If ${\mathbf{work}}\left(1\right)$ is much less than $1$, then the stability of the $LU$ factorization of the (equilibrated) matrix $A$ could be poor. This also means that the solution $X$, condition estimator rcond, and forward error bound ferr could be unreliable. If the factorization fails with ${\mathbf{info}}>{\mathbf{0}}\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{info}}\le \mathbf{n}$, ${\mathbf{work}}\left(1\right)$ contains the reciprocal pivot growth factor for the leading info columns of $A$.
Element $\u27e8\mathit{\text{value}}\u27e9$ of the diagonal is exactly zero.
The factorization has been completed, but the factor $U$
is exactly singular, so the solution and error bounds could not be computed.
${\mathbf{rcond}}=0.0$ is returned.
${\mathbf{info}}={\mathbf{n}}+1$
$U$ is nonsingular, but rcond is less than
machine precision, meaning that the matrix is singular to working precision.
Nevertheless, the solution and error bounds are computed because there
are a number of situations where the computed solution can be more accurate
than the value of rcond would suggest.
7Accuracy
For each right-hand side vector $b$, the computed solution $\hat{x}$ is the exact solution of a perturbed system of equations $(A+E)\hat{x}=b$, where
where
$\mathrm{cond}(A,\hat{x},b)={\Vert \left|{A}^{-1}\right|(\left|A\right|\left|\hat{x}\right|+\left|b\right|)\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}\le \mathrm{cond}\left(A\right)={\Vert \left|{A}^{-1}\right|\left|A\right|\Vert}_{\infty}\le {\kappa}_{\infty}\left(A\right)$.
If $\hat{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{berr}}\left(j\right)$ and a bound on ${\Vert x-\hat{x}\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}$ is returned in ${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.
8Parallelism and Performance
f07bbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bbf 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:
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 $\mathit{O}\left(n(2{k}_{l}+{k}_{u})r\right)$ 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. The solution is then refined, and the errors estimated, using iterative refinement; see f07bhf for information on the floating-point operations required.
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.
Estimates for the backward errors, forward errors, condition number and pivot growth are also output, together with information on the equilibration of $A$.