# NAG Library Routine Document

## 1Purpose

f07cvf (zgtrfs) computes error bounds and refines 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) and an initial solution returned by f07csf (zgttrs). Iterative refinement is used to reduce the backward error as much as possible.

## 2Specification

Fortran Interface
 Subroutine f07cvf ( n, nrhs, dl, d, du, dlf, df, duf, du2, ipiv, b, ldb, x, ldx, ferr, berr, work, info)
 Integer, Intent (In) :: n, nrhs, ipiv(*), ldb, ldx Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Out) :: ferr(nrhs), berr(nrhs), rwork(n) Complex (Kind=nag_wp), Intent (In) :: dl(*), d(*), du(*), dlf(*), df(*), duf(*), du2(*), b(ldb,*) Complex (Kind=nag_wp), Intent (Inout) :: x(ldx,*) Complex (Kind=nag_wp), Intent (Out) :: work(2*n) Character (1), Intent (In) :: trans
#include nagmk26.h
 void f07cvf_ (const char *trans, const Integer *n, const Integer *nrhs, const Complex dl[], const Complex d[], const Complex du[], const Complex dlf[], const Complex df[], const Complex duf[], const Complex du2[], const Integer ipiv[], const Complex b[], const Integer *ldb, Complex x[], const Integer *ldx, double ferr[], double berr[], Complex work[], double rwork[], Integer *info, const Charlen length_trans)
The routine may be called by its LAPACK name zgtrfs.

## 3Description

f07cvf (zgtrfs) should normally be preceded by calls to f07crf (zgttrf) and f07csf (zgttrs). f07crf (zgttrf) 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 compute a solution, $\stackrel{^}{X}$, to the required equations. Letting $\stackrel{^}{x}$ denote a column of $\stackrel{^}{X}$, f07cvf (zgtrfs) computes a component-wise backward error, $\beta$, the smallest relative perturbation in each element of $A$ and $b$ such that $\stackrel{^}{x}$ is the exact solution of a perturbed system
 $A+E x^=b+f , with eij ≤β aij , and fj ≤β bj .$
The routine also estimates a bound for the component-wise forward error in the computed solution defined by $\mathrm{max}\left|{x}_{i}-\stackrel{^}{{x}_{i}}\right|/\mathrm{max}\left|\stackrel{^}{{x}_{i}}\right|$, where $x$ is the corresponding column of the exact solution, $X$.

## 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:     $\mathbf{trans}$ – 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 $\left(n-1\right)$ subdiagonal elements of the matrix $A$.
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 matrix $A$.
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 $\left(n-1\right)$ superdiagonal elements of the matrix $A$.
7:     $\mathbf{dlf}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array dlf 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 $A$.
8:     $\mathbf{df}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array df 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 $A$.
9:     $\mathbf{duf}\left(*\right)$ – Complex (Kind=nag_wp) arrayInput
Note: the dimension of the array duf 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 $U$.
10:   $\mathbf{du2}\left(*\right)$ – 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 $U$.
11:   $\mathbf{ipiv}\left(*\right)$ – 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.
12:   $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Complex (Kind=nag_wp) arrayInput
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 $B$.
13:   $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07cvf (zgtrfs) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
14:   $\mathbf{x}\left({\mathbf{ldx}},*\right)$ – Complex (Kind=nag_wp) arrayInput/Output
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 entry: the $n$ by $r$ initial solution matrix $X$.
On exit: the $n$ by $r$ refined solution matrix $X$.
15:   $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07cvf (zgtrfs) is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
16:   $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{\stackrel{^}{x}}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$, where ${\stackrel{^}{x}}_{j}$ 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 almost always a slight overestimate of the true error.
17:   $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
18:   $\mathbf{work}\left(2×{\mathbf{n}}\right)$ – Complex (Kind=nag_wp) arrayWorkspace
19:   $\mathbf{rwork}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
20:   $\mathbf{info}$ – 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
 $E∞=OεA∞$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x ∞ x∞ ≤ κA E∞ A∞ ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{\infty }{‖A‖}_{\infty }$, 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.
Routine f07cuf (zgtcon) can be used to estimate the condition number of $A$.

## 8Parallelism and Performance

f07cvf (zgtrfs) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07cvf (zgtrfs) 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 $AX=B$ or ${A}^{\mathrm{T}}X=B$ or ${A}^{\mathrm{H}}X=B$ is proportional to $nr$. At most five steps of iterative refinement are performed, but usually only one or two steps are required.
The real analogue of this routine is f07chf (dgtrfs).

## 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
 $B = 2.4-05.0i 2.7+06.9i 3.4+18.2i -6.9-05.3i -14.7+09.7i -6.0-00.6i 31.9-07.7i -3.9+09.3i -1.0+01.6i -3.0+12.2i .$
Estimates for the backward errors and forward errors are also output.

### 10.1Program Text

Program Text (f07cvfe.f90)

### 10.2Program Data

Program Data (f07cvfe.d)

### 10.3Program Results

Program Results (f07cvfe.r)

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