f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_dtrexc (f08qfc)

## 1  Purpose

nag_dtrexc (f08qfc) reorders the Schur factorization of a real general matrix.

## 2  Specification

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

## 3  Description

nag_dtrexc (f08qfc) reorders the Schur factorization of a real general matrix $A=QT{Q}^{\mathrm{T}}$, so that the diagonal element or block of $T$ with row index ifst is moved to row ilst.
The reordered Schur form $\stackrel{~}{T}$ is computed by an orthogonal similarity transformation: $\stackrel{~}{T}={Z}^{\mathrm{T}}TZ$. Optionally the updated matrix $\stackrel{~}{Q}$ of Schur vectors is computed as $\stackrel{~}{Q}=QZ$, giving $A=\stackrel{~}{Q}\stackrel{~}{T}{\stackrel{~}{Q}}^{\mathrm{T}}$.

## 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[$\mathit{dim}$]doubleInput/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 quasi-triangular matrix $T$ in canonical Schur form, as returned by nag_dhseqr (f08pec).
On exit: t is overwritten by the updated matrix $\stackrel{~}{T}$. See also Section 8.
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[$\mathit{dim}$]doubleInput/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 $n$ orthogonal 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:     ifstInteger *Input/Output
9:     ilstInteger *Input/Output
On entry: ifst and ilst must specify the reordering of the diagonal elements or blocks of $T$. The element or block with row index ifst is moved to row ilst by a sequence of exchanges between adjacent elements or blocks.
On exit: if ifst pointed to the second row of a $2$ by $2$ block on entry, it is changed to point to the first row. ilst always points to the first row of the block in its final position (which may differ from its input value by $±1$).
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_EXCHANGE
Two adjacent diagonal elements or blocks could not be successfully exchanged. This error can only occur if the exchange involves at least one $2$ by $2$ block; it implies that the problem is very ill-conditioned, and that the eigenvalues of the two blocks are very close. On exit, $T$ may have been partially reordered, and ilst points to the first row of the current position of the block being moved; $Q$ (if requested) is updated consistently with $T$.
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.
Note that if a $2$ by $2$ diagonal block is involved in the reordering, its off-diagonal elements are in general changed; the diagonal elements and the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a $2$ by $2$ block to break into two $1$ by $1$ blocks, i.e., for a pair of complex eigenvalues to become purely real. The values of real eigenvalues however are never changed by the reordering.

The total number of floating point operations is approximately $6nr$ if ${\mathbf{compq}}=\mathrm{Nag_NotQ}$, and $12nr$ if ${\mathbf{compq}}=\mathrm{Nag_UpdateSchur}$, where $r=\left|{\mathbf{ifst}}-{\mathbf{ilst}}\right|$.
The input matrix $T$ must be in canonical Schur form, as is the output matrix $\stackrel{~}{T}$. This has the following structure.
If all the computed eigenvalues are real, $T$ is upper triangular and its diagonal elements are the eigenvalues.
If some of the computed eigenvalues form complex conjugate pairs, then $T$ has $2$ by $2$ diagonal blocks. Each diagonal block has the form
 $tii ti,i+1 ti+1,i ti+1,i+1 = α β γ α$
where $\beta \gamma <0$. The corresponding eigenvalues are $\alpha ±\sqrt{\beta \gamma }$.
The complex analogue of this function is nag_ztrexc (f08qtc).

## 9  Example

This example reorders the Schur factorization of the matrix $T$ so that the $2$ by $2$ block with row index $2$ is moved to row $1$, where
 $T = 0.80 -0.11 0.01 0.03 0.00 -0.10 0.25 0.35 0.00 -0.65 -0.10 0.20 0.00 0.00 0.00 -0.10 .$

### 9.1  Program Text

Program Text (f08qfce.c)

### 9.2  Program Data

Program Data (f08qfce.d)

### 9.3  Program Results

Program Results (f08qfce.r)