# NAG Library Routine Document

## 1Purpose

f07cnf (zgtsv) computes the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices.

## 2Specification

Fortran Interface
 Subroutine f07cnf ( n, nrhs, dl, d, du, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ldb Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (Inout) :: dl(*), d(*), du(*), b(ldb,*)
#include nagmk26.h
 void f07cnf_ ( const Integer *n, const Integer *nrhs, Complex dl[], Complex d[], Complex du[], Complex b[], const Integer *ldb, Integer *info)
The routine may be called by its LAPACK name zgtsv.

## 3Description

f07cnf (zgtsv) uses Gaussian elimination with partial pivoting and row interchanges to solve the equations $AX=B$. The matrix $A$ is factorized as $A=PLU$, where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element per column, and $U$ is an upper triangular band matrix, with two superdiagonals.
Note that the equations ${A}^{\mathrm{T}}X=B$ may be solved by interchanging the order of the arguments du and dl.

## 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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
2:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
3:     $\mathbf{dl}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array dl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
On exit: if no constraints are violated, dl is overwritten by the ($n-2$) elements of the second superdiagonal of the upper triangular matrix $U$ from the $LU$ factorization of $A$, in ${\mathbf{dl}}\left(1\right),{\mathbf{dl}}\left(2\right),\dots ,{\mathbf{dl}}\left(n-2\right)$.
4:     $\mathbf{d}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array d must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: must contain the $n$ diagonal elements of the matrix $A$.
On exit: if no constraints are violated, d is overwritten by the $n$ diagonal elements of the upper triangular matrix $U$ from the $LU$ factorization of $A$.
5:     $\mathbf{du}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array du must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-1\right)$.
On entry: must contain the $\left(n-1\right)$ superdiagonal elements of the matrix $A$.
On exit: if no constraints are violated, du is overwritten by the $\left(n-1\right)$ elements of the first superdiagonal of $U$.
6:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
To solve the equations $Ax=b$, where $b$ is a single right-hand side, b may be supplied as a one-dimensional array with length ${\mathbf{ldb}}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
7:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07cnf (zgtsv) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:     $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error 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.
${\mathbf{info}}>0$
Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero, and the solution has not been computed. The factorization has not been completed unless ${\mathbf{n}}=〈\mathit{\text{value}}〉$.

## 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
 $x^-x 1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
Alternatives to f07cnf (zgtsv), which return condition and error estimates are f04ccf and f07cpf (zgtsvx).

## 8Parallelism and Performance

f07cnf (zgtsv) 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.

The total number of floating-point operations required to solve the equations $AX=B$ is proportional to $nr$.
The real analogue of this routine is f07caf (dgtsv).

## 10Example

This example solves the equations
 $Ax=b ,$
where $A$ is the tridiagonal matrix
 $A = -1.3+1.3i 2.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0-2.0i -1.3+1.3i 2.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -1.3+3.3i -1.0+1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 2.0-3.0i -0.3+4.3i 1.0-1.0i 0.0i+0.0 0.0i+0.0 0.0i+0.0 1.0+1.0i -3.3+1.3i$
and
 $b = 2.4-05.0i 3.4+18.2i -14.7+09.7i 31.9-07.7i -1.0+01.6i .$

### 10.1Program Text

Program Text (f07cnfe.f90)

### 10.2Program Data

Program Data (f07cnfe.d)

### 10.3Program Results

Program Results (f07cnfe.r)

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