# NAG C Library Function Document

## 1Purpose

nag_dtrsna (f08qlc) estimates condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix.

## 2Specification

 #include #include
 void nag_dtrsna (Nag_OrderType order, Nag_JobType job, Nag_HowManyType how_many, const Nag_Boolean select[], Integer n, const double t[], Integer pdt, const double vl[], Integer pdvl, const double vr[], Integer pdvr, double s[], double sep[], Integer mm, Integer *m, NagError *fail)

## 3Description

nag_dtrsna (f08qlc) estimates condition numbers for specified eigenvalues and/or right eigenvectors of a real upper quasi-triangular matrix $T$ in canonical Schur form. These are the same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix $A=ZT{Z}^{\mathrm{T}}$ (with orthogonal $Z$), from which $T$ may have been derived.
nag_dtrsna (f08qlc) computes the reciprocal of the condition number of an eigenvalue ${\lambda }_{i}$ as
 $si = vHu uEvE ,$
where $u$ and $v$ are the right and left eigenvectors of $T$, respectively, corresponding to ${\lambda }_{i}$. This reciprocal condition number always lies between zero (i.e., ill-conditioned) and one (i.e., well-conditioned).
An approximate error estimate for a computed eigenvalue ${\lambda }_{i}$ is then given by
 $εT si ,$
where $\epsilon$ is the machine precision.
To estimate the reciprocal of the condition number of the right eigenvector corresponding to ${\lambda }_{i}$, the function first calls nag_dtrexc (f08qfc) to reorder the eigenvalues so that ${\lambda }_{i}$ is in the leading position:
 $T =Q λi cT 0 T22 QT.$
The reciprocal condition number of the eigenvector is then estimated as ${\mathit{sep}}_{i}$, the smallest singular value of the matrix $\left({T}_{22}-{\lambda }_{i}I\right)$. This number ranges from zero (i.e., ill-conditioned) to very large (i.e., well-conditioned).
An approximate error estimate for a computed right eigenvector $u$ corresponding to ${\lambda }_{i}$ is then given by
 $εT sepi .$

## 4References

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

## 5Arguments

1:    $\mathbf{order}$Nag_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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{job}$Nag_JobTypeInput
On entry: indicates whether condition numbers are required for eigenvalues and/or eigenvectors.
${\mathbf{job}}=\mathrm{Nag_EigVals}$
Condition numbers for eigenvalues only are computed.
${\mathbf{job}}=\mathrm{Nag_EigVecs}$
Condition numbers for eigenvectors only are computed.
${\mathbf{job}}=\mathrm{Nag_DoBoth}$
Condition numbers for both eigenvalues and eigenvectors are computed.
Constraint: ${\mathbf{job}}=\mathrm{Nag_EigVals}$, $\mathrm{Nag_EigVecs}$ or $\mathrm{Nag_DoBoth}$.
3:    $\mathbf{how_many}$Nag_HowManyTypeInput
On entry: indicates how many condition numbers are to be computed.
${\mathbf{how_many}}=\mathrm{Nag_ComputeAll}$
Condition numbers for all eigenpairs are computed.
${\mathbf{how_many}}=\mathrm{Nag_ComputeSelected}$
Condition numbers for selected eigenpairs (as specified by select) are computed.
Constraint: ${\mathbf{how_many}}=\mathrm{Nag_ComputeAll}$ or $\mathrm{Nag_ComputeSelected}$.
4:    $\mathbf{select}\left[\mathit{dim}\right]$const Nag_BooleanInput
Note: the dimension, dim, of the array select must be at least
• ${\mathbf{n}}$ when ${\mathbf{how_many}}=\mathrm{Nag_ComputeSelected}$;
• otherwise select may be NULL.
On entry: specifies the eigenpairs for which condition numbers are to be computed if ${\mathbf{how_many}}=\mathrm{Nag_ComputeSelected}$. To select condition numbers for the eigenpair corresponding to the real eigenvalue ${\lambda }_{j}$, ${\mathbf{select}}\left[j-1\right]$ must be set Nag_TRUE. To select condition numbers corresponding to a complex conjugate pair of eigenvalues ${\lambda }_{j}$ and ${\lambda }_{j+1}$, ${\mathbf{select}}\left[j-1\right]$ and/or ${\mathbf{select}}\left[j\right]$ must be set to Nag_TRUE.
If ${\mathbf{how_many}}=\mathrm{Nag_ComputeAll}$, select is not referenced and may be NULL.
5:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $T$.
Constraint: ${\mathbf{n}}\ge 0$.
6:    $\mathbf{t}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array t must be at least ${\mathbf{pdt}}×{\mathbf{n}}$.
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).
7:    $\mathbf{pdt}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array t.
Constraint: ${\mathbf{pdt}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:    $\mathbf{vl}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array vl must be at least
• ${\mathbf{pdvl}}×{\mathbf{mm}}$ when ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{n}}×{\mathbf{pdvl}}$ when ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• otherwise vl may be NULL.
The $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vl}}\left[\left(j-1\right)×{\mathbf{pdvl}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vl}}\left[\left(i-1\right)×{\mathbf{pdvl}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, vl must contain the left eigenvectors of $T$ (or of any matrix $QT{Q}^{\mathrm{T}}$ with $Q$ orthogonal) corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive rows or columns of vl, as returned by nag_dhsein (f08pkc) or nag_dtrevc (f08qkc).
If ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, vl is not referenced and may be NULL.
9:    $\mathbf{pdvl}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvl}}\ge {\mathbf{n}}$;
• if ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, vl may be NULL;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvl}}\ge {\mathbf{mm}}$;
• if ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, vl may be NULL.
10:  $\mathbf{vr}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array vr must be at least
• ${\mathbf{pdvr}}×{\mathbf{mm}}$ when ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{n}}×{\mathbf{pdvr}}$ when ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• otherwise vr may be NULL.
The $\left(i,j\right)$th element of the matrix is stored in
• ${\mathbf{vr}}\left[\left(j-1\right)×{\mathbf{pdvr}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vr}}\left[\left(i-1\right)×{\mathbf{pdvr}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, vr must contain the right eigenvectors of $T$ (or of any matrix $QT{Q}^{\mathrm{T}}$ with $Q$ orthogonal) corresponding to the eigenpairs specified by how_many and select. The eigenvectors must be stored in consecutive rows or columns of vr, as returned by nag_dhsein (f08pkc) or nag_dtrevc (f08qkc).
If ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, vr is not referenced and may be NULL.
11:  $\mathbf{pdvr}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vr.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvr}}\ge {\mathbf{n}}$;
• if ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, vr may be NULL;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvr}}\ge {\mathbf{mm}}$;
• if ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, vr may be NULL.
12:  $\mathbf{s}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array s must be at least
• ${\mathbf{mm}}$ when ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$;
• otherwise s may be NULL.
On exit: the reciprocal condition numbers of the selected eigenvalues if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, stored in consecutive elements of the array. Thus ${\mathbf{s}}\left[j-1\right]$, ${\mathbf{sep}}\left[j-1\right]$ and the $j$th rows or columns of vl and vr all correspond to the same eigenpair (but not in general the $j$th eigenpair unless all eigenpairs have been selected). For a complex conjugate pair of eigenvalues, two consecutive elements of s are set to the same value.
If ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, s is not referenced and may be NULL.
13:  $\mathbf{sep}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array sep must be at least
• ${\mathbf{mm}}$ when ${\mathbf{job}}=\mathrm{Nag_EigVecs}$ or $\mathrm{Nag_DoBoth}$;
• otherwise sep may be NULL.
On exit: the estimated reciprocal condition numbers of the selected right eigenvectors if ${\mathbf{job}}=\mathrm{Nag_EigVecs}$ or $\mathrm{Nag_DoBoth}$, stored in consecutive elements of the array. For a complex eigenvector, two consecutive elements of sep are set to the same value. If the eigenvalues cannot be reordered to compute ${\mathbf{sep}}\left[j\right]$, ${\mathbf{sep}}\left[j\right]$ is set to zero; this can only occur when the true value would be very small anyway.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, sep is not referenced and may be NULL.
14:  $\mathbf{mm}$IntegerInput
On entry: the number of elements in the arrays s and sep, and the number of rows or columns (depending on the value of order) in the arrays vl and vr (if used). The precise number required, $m$, is $n$ if ${\mathbf{how_many}}=\mathrm{Nag_ComputeAll}$; if ${\mathbf{how_many}}=\mathrm{Nag_ComputeSelected}$, $m$ is obtained by counting $1$ for each selected real eigenvalue, and $2$ for each selected complex conjugate pair of eigenvalues (see select), in which case $0\le m\le n$.
Constraint: ${\mathbf{mm}}\ge {\mathbf{m}}$.
Constraint: if ${\mathbf{how_many}}=\mathrm{Nag_ComputeAll}$, ${\mathbf{mm}}\ge {\mathbf{n}}$.
15:  $\mathbf{m}$Integer *Output
On exit: $m$, the number of elements of s and/or sep actually used to store the estimated condition numbers. If ${\mathbf{how_many}}=\mathrm{Nag_ComputeAll}$, m is set to $n$.
16:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_ENUM_INT_2
On entry, ${\mathbf{how_many}}=〈\mathit{\text{value}}〉$, ${\mathbf{mm}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{how_many}}=\mathrm{Nag_ComputeAll}$, ${\mathbf{mm}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvl}}=〈\mathit{\text{value}}〉$, ${\mathbf{mm}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvl}}\ge {\mathbf{mm}}$.
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvl}}\ge {\mathbf{n}}$.
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvr}}=〈\mathit{\text{value}}〉$, ${\mathbf{mm}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvr}}\ge {\mathbf{mm}}$.
On entry, ${\mathbf{job}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvr}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvr}}\ge {\mathbf{n}}$.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pdt}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdt}}>0$.
On entry, ${\mathbf{pdvl}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdvl}}>0$.
On entry, ${\mathbf{pdvr}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdvr}}>0$.
NE_INT_2
On entry, ${\mathbf{mm}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{mm}}\ge {\mathbf{m}}$.
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_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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

The computed values ${\mathit{sep}}_{i}$ may over estimate the true value, but seldom by a factor of more than $3$.

## 8Parallelism and Performance

nag_dtrsna (f08qlc) 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.

For a description of canonical Schur form, see the document for nag_dhseqr (f08pec).
The complex analogue of this function is nag_ztrsna (f08qyc).

## 10Example

This example computes approximate error estimates for all the eigenvalues and right eigenvectors of the matrix $T$, where
 $T = 0.7995 -0.1144 0.0060 0.0336 0.0000 -0.0994 0.2478 0.3474 0.0000 -0.6483 -0.0994 0.2026 0.0000 0.0000 0.0000 -0.1007 .$

### 10.1Program Text

Program Text (f08qlce.c)

### 10.2Program Data

Program Data (f08qlce.d)

### 10.3Program Results

Program Results (f08qlce.r)