F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08QFF (DTREXC)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08QFF (DTREXC) reorders the Schur factorization of a real general matrix.

## 2  Specification

 SUBROUTINE F08QFF ( COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK, INFO)
 INTEGER N, LDT, LDQ, IFST, ILST, INFO REAL (KIND=nag_wp) T(LDT,*), Q(LDQ,*), WORK(N) CHARACTER(1) COMPQ
The routine may be called by its LAPACK name dtrexc.

## 3  Description

F08QFF (DTREXC) 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:     $\mathrm{COMPQ}$ – CHARACTER(1)Input
On entry: indicates whether the matrix $Q$ of Schur vectors is to be updated.
${\mathbf{COMPQ}}=\text{'V'}$
The matrix $Q$ of Schur vectors is updated.
${\mathbf{COMPQ}}=\text{'N'}$
No Schur vectors are updated.
Constraint: ${\mathbf{COMPQ}}=\text{'V'}$ or $\text{'N'}$.
2:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     $\mathrm{T}\left({\mathbf{LDT}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array T must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $n$ by $n$ upper quasi-triangular matrix $T$ in canonical Schur form, as returned by F08PEF (DHSEQR).
On exit: T is overwritten by the updated matrix $\stackrel{~}{T}$. See also Section 9.
4:     $\mathrm{LDT}$ – INTEGERInput
On entry: the first dimension of the array T as declared in the (sub)program from which F08QFF (DTREXC) is called.
Constraint: ${\mathbf{LDT}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
5:     $\mathrm{Q}\left({\mathbf{LDQ}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array Q must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if ${\mathbf{COMPQ}}=\text{'V'}$ and at least $1$ if ${\mathbf{COMPQ}}=\text{'N'}$.
On entry: if ${\mathbf{COMPQ}}=\text{'V'}$, Q must contain the $n$ by $n$ orthogonal matrix $Q$ of Schur vectors.
On exit: if ${\mathbf{COMPQ}}=\text{'V'}$, Q contains the updated matrix of Schur vectors.
If ${\mathbf{COMPQ}}=\text{'N'}$, Q is not referenced.
6:     $\mathrm{LDQ}$ – INTEGERInput
On entry: the first dimension of the array Q as declared in the (sub)program from which F08QFF (DTREXC) is called.
Constraints:
• if ${\mathbf{COMPQ}}=\text{'V'}$, ${\mathbf{LDQ}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• if ${\mathbf{COMPQ}}=\text{'N'}$, ${\mathbf{LDQ}}\ge 1$.
7:     $\mathrm{IFST}$ – INTEGERInput/Output
8:     $\mathrm{ILST}$ – INTEGERInput/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 $1\le {\mathbf{ILST}}\le {\mathbf{N}}$.
9:     $\mathrm{WORK}\left({\mathbf{N}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
10:   $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error 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.
${\mathbf{INFO}}=1$
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 $T$.

## 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.

## 8  Parallelism and Performance

F08QFF (DTREXC) 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 is approximately $6nr$ if ${\mathbf{COMPQ}}=\text{'N'}$, and $12nr$ if ${\mathbf{COMPQ}}=\text{'V'}$, 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 routine is F08QTF (ZTREXC).

## 10  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
 $T = 0.80 -0.11 0.01 0.03 0.00 -0.10 0.25 0.35 0.00 -0.65 -0.10 0.20 0.00 0.00 0.00 -0.10 .$

### 10.1  Program Text

Program Text (f08qffe.f90)

### 10.2  Program Data

Program Data (f08qffe.d)

### 10.3  Program Results

Program Results (f08qffe.r)