# NAG Library Routine Document

## 1Purpose

f07csf (zgttrs) computes the solution to a complex system of linear equations $AX=B$ or ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$, where $A$ is an $n$ by $n$ tridiagonal matrix and $X$ and $B$ are $n$ by $r$ matrices, using the $LU$ factorization returned by f07crf (zgttrf).

## 2Specification

Fortran Interface
 Subroutine f07csf ( n, nrhs, dl, d, du, du2, ipiv, b, ldb, info)
 Integer, Intent (In) :: n, nrhs, ipiv(*), ldb Integer, Intent (Out) :: info Complex (Kind=nag_wp), Intent (In) :: dl(*), d(*), du(*), du2(*) Complex (Kind=nag_wp), Intent (Inout) :: b(ldb,*) Character (1), Intent (In) :: trans
#include nagmk26.h
 void f07csf_ (const char *trans, const Integer *n, const Integer *nrhs, const Complex dl[], const Complex d[], const Complex du[], const Complex du2[], const Integer ipiv[], Complex b[], const Integer *ldb, Integer *info, const Charlen length_trans)
The routine may be called by its LAPACK name zgttrs.

## 3Description

f07csf (zgttrs) should be preceded by a call to f07crf (zgttrf), which uses Gaussian elimination with partial pivoting and row interchanges to factorize the matrix $A$ as
 $A=PLU ,$
where $P$ is a permutation matrix, $L$ is unit lower triangular with at most one nonzero subdiagonal element in each column, and $U$ is an upper triangular band matrix, with two superdiagonals. f07csf (zgttrs) then utilizes the factorization to solve the required equations.

## 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
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1:     $\mathbf{trans}$ – Character(1)Input
On entry: specifies the equations to be solved as follows:
${\mathbf{trans}}=\text{'N'}$
Solve $AX=B$ for $X$.
${\mathbf{trans}}=\text{'T'}$
Solve ${A}^{\mathrm{T}}X=B$ for $X$.
${\mathbf{trans}}=\text{'C'}$
Solve ${A}^{\mathrm{H}}X=B$ for $X$.
Constraint: ${\mathbf{trans}}=\text{'N'}$, $\text{'T'}$ or $\text{'C'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4:     $\mathbf{dl}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
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 $\left(n-1\right)$ multipliers that define the matrix $L$ of the $LU$ factorization of $A$.
5:     $\mathbf{d}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
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 upper triangular matrix $U$ from the $LU$ factorization of $A$.
6:     $\mathbf{du}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
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 $\left(n-1\right)$ elements of the first superdiagonal of $U$.
7:     $\mathbf{du2}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array du2 must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}-2\right)$.
On entry: must contain the $\left(n-2\right)$ elements of the second superdiagonal of $U$.
8:     $\mathbf{ipiv}\left(*\right)$ – Integer arrayInput
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: must contain the $n$ pivot indices that define the permutation matrix $P$. At the $i$th step, row $i$ of the matrix was interchanged with row ${\mathbf{ipiv}}\left(i\right)$, and ${\mathbf{ipiv}}\left(i\right)$ must always be either $i$ or $\left(i+1\right)$, ${\mathbf{ipiv}}\left(i\right)=i$ indicating that a row interchange was not performed.
9:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the $n$ by $r$ matrix of right-hand sides $B$.
On exit: the $n$ by $r$ solution matrix $X$.
10:   $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07csf (zgttrs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
11:   $\mathbf{info}$ – 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.

## 7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 =OεA1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x 1 x 1 ≤ κA E1 A1 ,$
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.
Following the use of this routine f07cuf (zgtcon) can be used to estimate the condition number of $A$ and f07cvf (zgtrfs) can be used to obtain approximate error bounds.

## 8Parallelism and Performance

f07csf (zgttrs) is not threaded in any implementation.

The total number of floating-point operations required to solve the equations $AX=B$ or ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$ is proportional to $nr$.
The real analogue of this routine is f07cef (dgttrs).

## 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
 $B = 2.4-05.0i 2.7+06.9i 3.4+18.2i -6.9-05.3i -14.7+09.7i -6.0-00.6i 31.9-07.7i -3.9+09.3i -1.0+01.6i -3.0+12.2i .$

### 10.1Program Text

Program Text (f07csfe.f90)

### 10.2Program Data

Program Data (f07csfe.d)

### 10.3Program Results

Program Results (f07csfe.r)

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