# NAG Library Function Document

## 1Purpose

nag_complex_band_lin_solve (f04cbc) computes the solution to a complex system of linear equations $AX=B$, 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. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

## 2Specification

 #include #include
 void nag_complex_band_lin_solve (Nag_OrderType order, Integer n, Integer kl, Integer ku, Integer nrhs, Complex ab[], Integer pdab, Integer ipiv[], Complex b[], Integer pdb, double *rcond, double *errbnd, NagError *fail)

## 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 $\left({k}_{l}+{k}_{u}\right)$ 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 http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## 5Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{n}$IntegerInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:    $\mathbf{kl}$IntegerInput
On entry: the number of subdiagonals ${k}_{l}$, within the band of $A$.
Constraint: ${\mathbf{kl}}\ge 0$.
4:    $\mathbf{ku}$IntegerInput
On entry: the number of superdiagonals ${k}_{u}$, within the band of $A$.
Constraint: ${\mathbf{ku}}\ge 0$.
5:    $\mathbf{nrhs}$IntegerInput
On entry: the number of right-hand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
6:    $\mathbf{ab}\left[\mathit{dim}\right]$ComplexInput/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 $n$ by $n$ 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:    $\mathbf{pdab}$IntegerInput
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:    $\mathbf{ipiv}\left[{\mathbf{n}}\right]$IntegerOutput
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, 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:    $\mathbf{b}\left[\mathit{dim}\right]$ComplexInput/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 $n$ by $r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, the $n$ by $r$ solution matrix $X$.
10:  $\mathbf{pdb}$IntegerInput
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:  $\mathbf{rcond}$double *Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
12:  $\mathbf{errbnd}$double *Output
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or NE_RCOND, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ 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.
13:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
The double allocatable memory required is n, and the Complex allocatable memory required is $2×{\mathbf{n}}$. In this case the factorization and the solution $X$ have been computed, but rcond and errbnd have not been computed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation 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, ${\mathbf{pdab}}=〈\mathit{\text{value}}〉$, ${\mathbf{kl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ku}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdab}}\ge 2×{\mathbf{kl}}+{\mathbf{ku}}+1$.
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_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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_RCOND
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
NE_SINGULAR
Diagonal element $〈\mathit{\text{value}}〉$ of the upper triangular factor is zero. The factorization has been completed, but 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
 $E1=Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. nag_complex_band_lin_solve (f04cbc) uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

nag_complex_band_lin_solve (f04cbc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_complex_band_lin_solve (f04cbc) 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 $n=5$, ${k}_{l}=2$, and ${k}_{u}=1$. Storage of the band matrix $A$ in the array ab:
 Band matrix $\mathbit{A}$ Band storage in array ab ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ $\begin{array}{lllll}{a}_{11}& {a}_{12}& & & \\ {a}_{21}& {a}_{22}& {a}_{23}& & \\ {a}_{31}& {a}_{32}& {a}_{33}& {a}_{34}& \\ & {a}_{42}& {a}_{43}& {a}_{44}& {a}_{45}\\ & & {a}_{53}& {a}_{54}& {a}_{55}\end{array}$ $\begin{array}{lllll}\text{*}& \text{*}& \text{*}& +& +\\ \text{*}& \text{*}& +& +& +\\ \text{*}& {a}_{12}& {a}_{23}& {a}_{34}& {a}_{45}\\ {a}_{11}& {a}_{22}& {a}_{33}& {a}_{44}& {a}_{55}\\ {a}_{21}& {a}_{32}& {a}_{43}& {a}_{54}& \text{*}\\ {a}_{31}& {a}_{42}& {a}_{53}& \text{*}& \text{*}\end{array}$ $\begin{array}{llllll}\text{*}& \text{*}& {a}_{11}& {a}_{12}& +& +\\ \text{*}& {a}_{21}& {a}_{22}& {a}_{23}& +& +\\ {a}_{31}& {a}_{32}& {a}_{33}& {a}_{34}& +& \text{*}\\ {a}_{42}& {a}_{43}& {a}_{44}& {a}_{45}& \text{*}& \text{*}\\ {a}_{53}& {a}_{54}& {a}_{55}& \text{*}& \text{*}& \text{*}\end{array}$
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 ${u}_{13}$, ${u}_{14}$, ${u}_{24}$, ${u}_{25}$ and ${u}_{35}$. In this example when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ the first referenced element of ab is ${\mathbf{ab}}\left[3\right]={a}_{11}$; while for ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ the first referenced element is ${\mathbf{ab}}\left[2\right]={a}_{11}$.
In general, elements ${a}_{ij}$ are stored as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${a}_{ij}$ are stored in ${\mathbf{ab}}\left[\left(j-1\right)×{\mathbf{pdab}}+{\mathbf{kl}}+{\mathbf{ku}}+i-j\right]$
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${a}_{ij}$ are stored in ${\mathbf{ab}}\left[\left(i-1\right)×{\mathbf{pdab}}+{\mathbf{kl}}+j-i\right]$
where $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,i-{\mathbf{kl}}\right)\le j\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{n}},i+{\mathbf{ku}}\right)$.
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 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.
The real analogue of nag_complex_band_lin_solve (f04cbc) is nag_real_band_lin_solve (f04bbc).

## 10Example

This example solves the equations
 $AX=B,$
where $A$ is the band matrix
 $A= -1.65+2.26i -2.05-0.85i 0.97-2.84i 0.00i+0.00 0.00+6.30i -1.48-1.75i -3.99+4.01i 0.59-0.48i 0.00i+0.00 -0.77+2.83i -1.06+1.94i 3.33-1.04i 0.00i+0.00 0.00i+0.00 4.48-1.09i -0.46-1.72i$
and
 $B= -1.06+21.50i 12.85+02.84i -22.72-53.90i -70.22+21.57i 28.24-38.60i -20.73-01.23i -34.56+16.73i 26.01+31.97i .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.

### 10.1Program Text

Program Text (f04cbce.c)

### 10.2Program Data

Program Data (f04cbce.d)

### 10.3Program Results

Program Results (f04cbce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017