f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_ztrexc (f08qtc)

## 1  Purpose

nag_ztrexc (f08qtc) reorders the Schur factorization of a complex general matrix.

## 2  Specification

 #include #include
 void nag_ztrexc (Nag_OrderType order, Nag_ComputeQType compq, Integer n, Complex t[], Integer pdt, Complex q[], Integer pdq, Integer ifst, Integer ilst, NagError *fail)

## 3  Description

nag_ztrexc (f08qtc) reorders the Schur factorization of a complex general matrix $A=QT{Q}^{\mathrm{H}}$, so that the diagonal element of $T$ with row index ifst is moved to row ilst.
The reordered Schur form $\stackrel{~}{T}$ is computed by a unitary similarity transformation: $\stackrel{~}{T}={Z}^{\mathrm{H}}TZ$. Optionally the updated matrix $\stackrel{~}{Q}$ of Schur vectors is computed as $\stackrel{~}{Q}=QZ$, giving $A=\stackrel{~}{Q}\stackrel{~}{T}{\stackrel{~}{Q}}^{\mathrm{H}}$.

## 4  References

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

## 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:     compqNag_ComputeQTypeInput
On entry: indicates whether the matrix $Q$ of Schur vectors is to be updated.
${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$
The matrix $Q$ of Schur vectors is updated.
${\mathbf{compq}}=\mathrm{Nag_NotQ}$
No Schur vectors are updated.
Constraint: ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$ or $\mathrm{Nag_NotQ}$.
3:     nIntegerInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     t[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array t must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdt}}×{\mathbf{n}}\right)$.
The $\left(i,j\right)$th element of the matrix $T$ is stored in
• ${\mathbf{t}}\left[\left(j-1\right)×{\mathbf{pdt}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{t}}\left[\left(i-1\right)×{\mathbf{pdt}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $n$ upper triangular matrix $T$, as returned by nag_zhseqr (f08psc).
On exit: t is overwritten by the updated matrix $\stackrel{~}{T}$.
5:     pdtIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array t.
Constraint: ${\mathbf{pdt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     q[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array q must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdq}}×{\mathbf{n}}\right)$ when ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$;
• $1$ when ${\mathbf{compq}}=\mathrm{Nag_NotQ}$.
The $\left(i,j\right)$th element of the matrix $Q$ is stored in
• ${\mathbf{q}}\left[\left(j-1\right)×{\mathbf{pdq}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{pdq}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$, q must contain the $n$ by $n$ unitary matrix $Q$ of Schur vectors.
On exit: if ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$, q contains the updated matrix of Schur vectors.
If ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, q is not referenced.
7:     pdqIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array q.
Constraints:
• if ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, ${\mathbf{pdq}}\ge 1$.
8:     ifstIntegerInput
9:     ilstIntegerInput
On entry: ifst and ilst must specify the reordering of the diagonal elements of $T$. The element with row index ifst is moved to row ilst by a sequence of exchanges between adjacent elements.
Constraint: $1\le {\mathbf{ifst}}\le {\mathbf{n}}$ and $1\le {\mathbf{ilst}}\le {\mathbf{n}}$.
10:   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_ENUM_INT_2
On entry, ${\mathbf{compq}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$, ${\mathbf{pdq}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
if ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, ${\mathbf{pdq}}\ge 1$.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdq}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdq}}>0$.
On entry, ${\mathbf{pdt}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdt}}>0$.
NE_INT_2
On entry, ${\mathbf{pdt}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
NE_INT_3
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$, ${\mathbf{ifst}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ilst}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{ifst}}\le {\mathbf{n}}$ and $1\le {\mathbf{ilst}}\le {\mathbf{n}}$.
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.

## 7  Accuracy

The computed matrix $\stackrel{~}{T}$ is exactly similar to a matrix $\left(T+E\right)$, where
 $E2 = Oε T2 ,$
and $\epsilon$ is the machine precision.
The values of the eigenvalues are never changed by the reordering.

The total number of real floating point operations is approximately $20nr$ if ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, and $40nr$ if ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$, where $r=\left|{\mathbf{ifst}}-{\mathbf{ilst}}\right|$.
The real analogue of this function is nag_dtrexc (f08qfc).

## 9  Example

This example reorders the Schur factorization of the matrix $T$ so that element ${t}_{11}$ is moved to ${t}_{44}$, where
 $T = -6.00-7.00i 0.36-0.36i -0.19+0.48i 0.88-0.25i 0.00+0.00i -5.00+2.00i -0.03-0.72i -0.23+0.13i 0.00+0.00i 0.00+0.00i 8.00-1.00i 0.94+0.53i 0.00+0.00i 0.00+0.00i 0.00+0.00i 3.00-4.00i .$

### 9.1  Program Text

Program Text (f08qtce.c)

### 9.2  Program Data

Program Data (f08qtce.d)

### 9.3  Program Results

Program Results (f08qtce.r)