f07 Chapter Contents
f07 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zgtsv (f07cnc)

## 1  Purpose

nag_zgtsv (f07cnc) 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.

## 2  Specification

 #include #include
 void nag_zgtsv (Nag_OrderType order, Integer n, Integer nrhs, Complex dl[], Complex d[], Complex du[], Complex b[], Integer pdb, NagError *fail)

## 3  Description

nag_zgtsv (f07cnc) 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.

## 4  References

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

## 5  Arguments

1:     orderNag_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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or Nag_ColMajor.
2:     nIntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     nrhsIntegerInput
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:     dl[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, 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 constrains 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 ${\mathbf{dl}}\left[0\right],{\mathbf{dl}}\left[1\right],\dots ,{\mathbf{dl}}\left[n-3\right]$.
5:     d[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, 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 $A$.
6:     du[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, 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 $U$.
7:     b[$\mathit{dim}$]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 $r$ right-hand side matrix $B$.
On exit: if NE_NOERROR, the $n$ by $r$ solution matrix $X$.
8:     pdbIntegerInput
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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
9:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
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, ${\mathbf{pdb}}=〈\mathit{\text{value}}〉$.
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_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.
NE_SINGULAR
$U\left(〈\mathit{\text{value}}〉,〈\mathit{\text{value}}〉\right)$ is exactly zero, and the solution has not been computed. The factorization has not been completed unless ${\mathbf{n}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

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 nag_zgtsv (f07cnc), which return condition and error estimates are nag_complex_tridiag_lin_solve (f04ccc) and nag_zgtsvx (f07cpc).

The total number of floating point operations required to solve the equations $AX=B$ is proportional to $nr$.
The real analogue of this function is nag_dgtsv (f07cac).

## 9  Example

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
 $b = 2.4-05.0i 3.4+18.2i -14.7+09.7i 31.9-07.7i -1.0+01.6i .$

### 9.1  Program Text

Program Text (f07cnce.c)

### 9.2  Program Data

Program Data (f07cnce.d)

### 9.3  Program Results

Program Results (f07cnce.r)