# NAG CL Interfacef08qtc (ztrexc)

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## 1Purpose

f08qtc reorders the Schur factorization of a complex general matrix.

## 2Specification

 #include
 void f08qtc (Nag_OrderType order, Nag_ComputeQType compq, Integer n, Complex t[], Integer pdt, Complex q[], Integer pdq, Integer ifst, Integer ilst, NagError *fail)
The function may be called by the names: f08qtc, nag_lapackeig_ztrexc or nag_ztrexc.

## 3Description

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 $A=\stackrel{~}{Q}\stackrel{~}{T}{\stackrel{~}{Q}}^{\mathrm{H}}$.

## 4References

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

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
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 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{compq}$Nag_ComputeQType Input
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: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{t}\left[\mathit{dim}\right]$Complex Input/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 $n×n$ upper triangular matrix $T$, as returned by f08psc.
On exit: t is overwritten by the updated matrix $\stackrel{~}{T}$.
5: $\mathbf{pdt}$Integer Input
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: $\mathbf{q}\left[\mathit{dim}\right]$Complex Input/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 $n×n$ 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: $\mathbf{pdq}$Integer Input
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: $\mathbf{ifst}$Integer Input
9: $\mathbf{ilst}$Integer Input
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: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

## 7Accuracy

The computed matrix $\stackrel{~}{T}$ is exactly similar to a matrix $\left(T+E\right)$, where
 $‖E‖2 = O(ε) ‖T‖2 ,$
and $\epsilon$ is the machine precision.
The values of the eigenvalues are never changed by the reordering.

## 8Parallelism and Performance

f08qtc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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 $r=|{\mathbf{ifst}}-{\mathbf{ilst}}|$.
The real analogue of this function is f08qfc.

## 10Example

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

### 10.1Program Text

Program Text (f08qtce.c)

### 10.2Program Data

Program Data (f08qtce.d)

### 10.3Program Results

Program Results (f08qtce.r)