NAG Library Routine Document
F07BBF (DGBSVX)
1 Purpose
F07BBF (DGBSVX) uses the
$LU$ factorization to compute the solution to a real system of linear equations
where
$A$ is an
$n$ by
$n$ band matrix with
${k}_{l}$ subdiagonals and
${k}_{u}$ superdiagonals, and
$X$ and
$B$ are
$n$ by
$r$ matrices. Error bounds on the solution and a condition estimate are also provided.
2 Specification
SUBROUTINE F07BBF ( 
FACT, TRANS, N, KL, KU, NRHS, AB, LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, LDB, X, LDX, RCOND, FERR, BERR, WORK, IWORK, INFO) 
INTEGER 
N, KL, KU, NRHS, LDAB, LDAFB, IPIV(*), LDB, LDX, IWORK(N), INFO 
REAL (KIND=nag_wp) 
AB(LDAB,*), AFB(LDAFB,*), R(*), C(*), B(LDB,*), X(LDX,*), RCOND, FERR(NRHS), BERR(NRHS), WORK(max(1,3*N)) 
CHARACTER(1) 
FACT, TRANS, EQUED 

The routine may be called by its
LAPACK
name dgbsvx.
3 Description
F07BBF (DGBSVX) 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
and
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
where $P$ is a permutation matrix, $L$ is a unit lower triangular matrix, and $U$ is upper triangular.
This stage can be bypassed when a factored matrix (with scaled matrices and scaling factors) are supplied; for example, as provided by a previous call to F07BBF (DGBSVX) 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. 
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/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
5 Parameters
 1: $\mathrm{FACT}$ – 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: $\mathrm{TRANS}$ – 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: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 4: $\mathrm{KL}$ – INTEGERInput

On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint:
${\mathbf{KL}}\ge 0$.
 5: $\mathrm{KU}$ – INTEGERInput

On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint:
${\mathbf{KU}}\ge 0$.
 6: $\mathrm{NRHS}$ – INTEGERInput

On entry: $r$, the number of righthand sides, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{NRHS}}\ge 0$.
 7: $\mathrm{AB}\left({\mathbf{LDAB}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
AB
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the
$n$ by
$n$ 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
See
Section 9 for further details.
If
${\mathbf{FACT}}=\text{'F'}$ and
${\mathbf{EQUED}}\ne \text{'N'}$,
$A$ must have been equilibrated by the scaling factors in
R and/or
C.
On exit: if
${\mathbf{FACT}}=\text{'F'}$ or
$\text{'N'}$, or if
${\mathbf{FACT}}=\text{'E'}$ and
${\mathbf{EQUED}}=\text{'N'}$,
AB is not modified.
If
${\mathbf{EQUED}}\ne \text{'N'}$ then, if no constraints are violated,
$A$ is scaled as follows:
 if ${\mathbf{EQUED}}=\text{'R'}$, $A={D}_{r}A$;
 if ${\mathbf{EQUED}}=\text{'C'}$, $A=A{D}_{c}$;
 if ${\mathbf{EQUED}}=\text{'B'}$, $A={D}_{r}A{D}_{c}$.
 8: $\mathrm{LDAB}$ – INTEGERInput

On entry: the first dimension of the array
AB as declared in the (sub)program from which F07BBF (DGBSVX) is called.
Constraint:
${\mathbf{LDAB}}\ge {\mathbf{KL}}+{\mathbf{KU}}+1$.
 9: $\mathrm{AFB}\left({\mathbf{LDAFB}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
AFB
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
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
$n$ by
$n$ band matrix
$A$, as computed by
F07BDF (DGBTRF).
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: $\mathrm{LDAFB}$ – INTEGERInput

On entry: the first dimension of the array
AFB as declared in the (sub)program from which F07BBF (DGBSVX) is called.
Constraint:
${\mathbf{LDAFB}}\ge 2\times {\mathbf{KL}}+{\mathbf{KU}}+1$.
 11: $\mathrm{IPIV}\left(*\right)$ – INTEGER arrayInput/Output

Note: the dimension of the array
IPIV
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
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 (DGBTRF); 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: $\mathrm{EQUED}$ – 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: $\mathrm{R}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array
R
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
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: $\mathrm{C}\left(*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the dimension of the array
C
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: if
${\mathbf{FACT}}=\text{'N'}$ or
$\text{'E'}$,
C need not be set.
If
${\mathbf{FACT}}=\text{'F'}$ or
${\mathbf{EQUED}}=\text{'C'}$ 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: $\mathrm{B}\left({\mathbf{LDB}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output

Note: the second dimension of the array
B
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ righthand 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: $\mathrm{LDB}$ – INTEGERInput

On entry: the first dimension of the array
B as declared in the (sub)program from which F07BBF (DGBSVX) is called.
Constraint:
${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
 17: $\mathrm{X}\left({\mathbf{LDX}},*\right)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array
X
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or ${\mathbf{N}+{\mathbf{1}}}$, the $n$ by $r$ 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: $\mathrm{LDX}$ – INTEGERInput

On entry: the first dimension of the array
X as declared in the (sub)program from which F07BBF (DGBSVX) is called.
Constraint:
${\mathbf{LDX}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
 19: $\mathrm{RCOND}$ – REAL (KIND=nag_wp)Output

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: $\mathrm{FERR}\left({\mathbf{NRHS}}\right)$ – 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: $\mathrm{BERR}\left({\mathbf{NRHS}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: if ${\mathbf{INFO}}={\mathbf{0}}$ or ${\mathbf{N}+{\mathbf{1}}}$, an estimate of the componentwise 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: $\mathrm{WORK}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,3\times {\mathbf{N}}\right)\right)$ – 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
$A$.
 23: $\mathrm{IWORK}\left({\mathbf{N}}\right)$ – INTEGER arrayWorkspace

 24: $\mathrm{INFO}$ – INTEGEROutput
On exit:
${\mathbf{INFO}}=0$ unless the routine detects an error (see
Section 6).
6 Error 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.
 ${\mathbf{INFO}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{INFO}}\le {\mathbf{N}}$

Element $\u2329\mathit{\text{value}}\u232a$ 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.
7 Accuracy
For each righthand side vector
$b$, the computed solution
$\hat{x}$ is the exact solution of a perturbed system of equations
$\left(A+E\right)\hat{x}=b$, where
$c\left(n\right)$ is a modest linear function of
$n$, and
$\epsilon $ is the
machine precision. See Section 9.3 of
Higham (2002) for further details.
If
$x$ is the true solution, then the computed solution
$\hat{x}$ satisfies a forward error bound of the form
where
$\mathrm{cond}\left(A,\hat{x},b\right)={\Vert \left{A}^{1}\right\left(\leftA\right\left\hat{x}\right+\leftb\right\right)\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}\le \mathrm{cond}\left(A\right)={\Vert \left{A}^{1}\right\leftA\right\Vert}_{\infty}\le {\kappa}_{\infty}\left(A\right)$.
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.
8 Parallelism and Performance
F07BBF (DGBSVX) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F07BBF (DGBSVX) 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 implementationspecific 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:
The total number of floatingpoint 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 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 (DGBRFS) for information on the floatingpoint 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.
The complex analogue of this routine is
F07BPF (ZGBSVX).
10 Example
This example solves the equations
where
$A$ is the band matrix
Estimates for the backward errors, forward errors, condition number and pivot growth are also output, together with information on the equilibration of $A$.
10.1 Program Text
Program Text (f07bbfe.f90)
10.2 Program Data
Program Data (f07bbfe.d)
10.3 Program Results
Program Results (f07bbfe.r)