# NAG CL Interfacef07bnc (zgbsv)

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## 1Purpose

f07bnc computes the solution to a complex 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

 #include
 void f07bnc (Nag_OrderType order, Integer n, Integer kl, Integer ku, Integer nrhs, Complex ab[], Integer pdab, Integer ipiv[], Complex b[], Integer pdb, NagError *fail)
The function may be called by the names: f07bnc, nag_lapacklin_zgbsv or nag_zgbsv.

## 3Description

f07bnc 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: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{kl}$Integer Input
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
4: $\mathbf{ku}$Integer Input
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
5: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
6: $\mathbf{ab}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array ab must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdab}}×{\mathbf{n}}\right)$.
On entry: the $n×n$ coefficient matrix $A$.
This is stored as a notional two-dimensional array with row elements or column elements stored contiguously. The storage of elements ${A}_{ij}$, for row $i=1,\dots ,n$ and column $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{l}\right),\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{u}\right)$, depends on the order argument as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${A}_{ij}$ is stored as ${\mathbf{ab}}\left[\left(j-1\right)×{\mathbf{pdab}}+{\mathbf{kl}}+{\mathbf{ku}}+i-j\right]$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${A}_{ij}$ is stored as ${\mathbf{ab}}\left[\left(i-1\right)×{\mathbf{pdab}}+{\mathbf{kl}}+j-i\right]$.
See Section 9 for further details.
On exit: ab is overwritten by details of the factorization.
The elements, ${u}_{ij}$, of the upper triangular band factor $U$ with ${k}_{l}+{k}_{u}$ super-diagonals, and the multipliers, ${l}_{ij}$, used to form the lower triangular factor $L$ are stored. The elements ${u}_{ij}$, for $i=1,\dots ,n$ and $j=i,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,i+{k}_{l}+{k}_{u}\right)$, and ${l}_{ij}$, for $i=1,\dots ,n$ and $j=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{k}_{l}\right),\dots ,i$, are stored where ${A}_{ij}$ is stored on entry.
7: $\mathbf{pdab}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) of the matrix $A$ in the array ab.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
8: $\mathbf{ipiv}\left[{\mathbf{n}}\right]$Integer 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 ${\mathbf{ipiv}}\left[i-1\right]$. ${\mathbf{ipiv}}\left[i-1\right]=i$ indicates a row interchange was not required.
9: $\mathbf{b}\left[\mathit{dim}\right]$Complex Input/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n×r$ right-hand side matrix $B$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the $n×r$ solution matrix $X$.
10: $\mathbf{pdb}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{kl}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{kl}}\ge 0$.
On entry, ${\mathbf{ku}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ku}}\ge 0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pdab}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdab}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
NE_INT_3
On entry, ${\mathbf{pdab}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{kl}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ku}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SINGULAR
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 f07bnc, f07buc can be used to estimate the condition number of $A$ and f07bvc can be used to obtain approximate error bounds. Alternatives to f07bnc, which return condition and error estimates directly are f04cbc and f07bpc.

## 8Parallelism and Performance

f07bnc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07bnc 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 function. 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:
 $order=Nag_ColMajor * * * + + + * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * a0 order=Nag_RowMajor * a11 a12 a13 + a21 a22 a23 a24 + a32 a33 a34 a35 + a43 a44 a45 a46 * a54 a55 a56 * * a65 a66 * * *$
Array elements marked $*$ need not be set and are not referenced by the function. Array elements marked $+$ need not be set, but are defined on exit from the function 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 real analogue of this function is f07bac.

## 10Example

This example solves the equations
 $Ax=b ,$
where $A$ is the band matrix
 $A = ( -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00i+0.00 6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00i+0.00 -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00i+0.00 0.00i+0.00 4.48-1.09i -0.46-1.72i )$
and
 $b = ( -1.06+21.50i -22.72-53.90i 28.24-38.60i -34.56+16.73i ) .$
Details of the $LU$ factorization of $A$ are also output.

### 10.1Program Text

Program Text (f07bnce.c)

### 10.2Program Data

Program Data (f07bnce.d)

### 10.3Program Results

Program Results (f07bnce.r)