f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dtgsna (f08ylc)

## 1  Purpose

nag_dtgsna (f08ylc) estimates condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair in generalized real Schur form.

## 2  Specification

 #include #include
 void nag_dtgsna (Nag_OrderType order, Nag_JobType job, Nag_HowManyType howmny, const Nag_Boolean select[], Integer n, const double a[], Integer pda, const double b[], Integer pdb, const double vl[], Integer pdvl, const double vr[], Integer pdvr, double s[], double dif[], Integer mm, Integer *m, NagError *fail)

## 3  Description

nag_dtgsna (f08ylc) estimates condition numbers for specified eigenvalues and/or right eigenvectors of an $n$ by $n$ matrix pair $\left(S,T\right)$ in real generalized Schur form. The function actually returns estimates of the reciprocals of the condition numbers in order to avoid possible overflow.
The pair $\left(S,T\right)$ are in real generalized Schur form if $S$ is block upper triangular with $1$ by $1$ and $2$ by $2$ diagonal blocks and $T$ is upper triangular as returned, for example, by nag_dgges (f08xac) or nag_dggesx (f08xbc), or nag_dhgeqz (f08xec) with ${\mathbf{job}}=\mathrm{Nag_Schur}$. The diagonal elements, or blocks, define the generalized eigenvalues $\left({\alpha }_{\mathit{i}},{\beta }_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,n$, of the pair $\left(S,T\right)$ and the eigenvalues are given by
 $λi = αi / βi ,$
so that
 $βi S xi = αi T xi or S xi = λi T xi ,$
where ${x}_{i}$ is the corresponding (right) eigenvector.
If $S$ and $T$ are the result of a generalized Schur factorization of a matrix pair $\left(A,B\right)$
 $A = QSZT , B = QTZT$
then the eigenvalues and condition numbers of the pair $\left(S,T\right)$ are the same as those of the pair $\left(A,B\right)$.
Let $\left(\alpha ,\beta \right)\ne \left(0,0\right)$ be a simple generalized eigenvalue of $\left(A,B\right)$. Then the reciprocal of the condition number of the eigenvalue $\lambda =\alpha /\beta$ is defined as
 $sλ= yTAx 2 + yTBx 2 1/2 x2 y2 ,$
where $x$ and $y$ are the right and left eigenvectors of $\left(A,B\right)$ corresponding to $\lambda$. If both $\alpha$ and $\beta$ are zero, then $\left(A,B\right)$ is singular and $s\left(\lambda \right)=-1$ is returned.
The definition of the reciprocal of the estimated condition number of the right eigenvector $x$ and the left eigenvector $y$ corresponding to the simple eigenvalue $\lambda$ depends upon whether $\lambda$ is a real eigenvalue, or one of a complex conjugate pair.
If the eigenvalue $\lambda$ is real and $U$ and $V$ are orthogonal transformations such that
 $UT A,B V= S,T = α * 0 S22 β * 0 T22 ,$
where ${S}_{22}$ and ${T}_{22}$ are $\left(n-1\right)$ by $\left(n-1\right)$ matrices, then the reciprocal condition number is given by
 $Difx ≡ Dify = Difα,β,S22,T22 = σmin Z ,$
where ${\sigma }_{\mathrm{min}}\left(Z\right)$ denotes the smallest singular value of the $2\left(n-1\right)$ by $2\left(n-1\right)$ matrix
 $Z = α⊗I -1⊗S22 β⊗I -1⊗T22$
and $\otimes$ is the Kronecker product.
If $\lambda$ is part of a complex conjugate pair and $U$ and $V$ are orthogonal transformations such that
 $UT A,B V = S,T = S11 * 0 S22 T11 * 0 T22 ,$
where ${S}_{11}$ and ${T}_{11}$ are two by two matrices, ${S}_{22}$ and ${T}_{22}$ are $\left(n-2\right)$ by $\left(n-2\right)$ matrices, and $\left({S}_{11},{T}_{11}\right)$ corresponds to the complex conjugate eigenvalue pair $\lambda$, $\stackrel{-}{\lambda }$, then there exist unitary matrices ${U}_{1}$ and ${V}_{1}$ such that
 $U1H S11 V1 = s11 s12 0 s22 and U1H T11 V1 = t11 t12 0 t22 .$
The eigenvalues are given by $\lambda ={s}_{11}/{t}_{11}$ and $\stackrel{-}{\lambda }={s}_{22}/{t}_{22}$. Then the Frobenius norm-based, estimated reciprocal condition number is bounded by
 $Difx ≡ Dify ≤ mind1,max1, Res11 / Res22 ,d2$
where $\mathrm{Re}\left(z\right)$ denotes the real part of $z$, ${d}_{1}=\mathrm{Dif}\left(\left({s}_{11},{t}_{11}\right),\left({s}_{22},{t}_{22}\right)\right)={\sigma }_{\mathrm{min}}\left({Z}_{1}\right)$, ${Z}_{1}$ is the complex two by two matrix
 $Z1 = s11 -s22 t11 -t22 ,$
and ${d}_{2}$ is an upper bound on $\mathrm{Dif}\left(\left({S}_{11},{T}_{11}\right),\left({S}_{22},{T}_{22}\right)\right)$; i.e., an upper bound on ${\sigma }_{\mathrm{min}}\left({Z}_{2}\right)$, where ${Z}_{2}$ is the $\left(2n-2\right)$ by $\left(2n-2\right)$ matrix
 $Z2 = S11T⊗I -I⊗S22 T11T⊗I -I⊗T22 .$
See Sections 2.4.8 and 4.11 of Anderson et al. (1999) and Kågström and Poromaa (1996) for further details and information.

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Kågström B and Poromaa P (1996) LAPACK-style algorithms and software for solving the generalized Sylvester equation and estimating the separation between regular matrix pairs ACM Trans. Math. Software 22 78–103

## 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 $\mathrm{Nag_ColMajor}$.
2:     jobNag_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:     howmnyNag_HowManyTypeInput
On entry: indicates how many condition numbers are to be computed.
${\mathbf{howmny}}=\mathrm{Nag_ComputeAll}$
Condition numbers for all eigenpairs are computed.
${\mathbf{howmny}}=\mathrm{Nag_ComputeSelected}$
Condition numbers for selected eigenpairs (as specified by select) are computed.
Constraint: ${\mathbf{howmny}}=\mathrm{Nag_ComputeAll}$ or $\mathrm{Nag_ComputeSelected}$.
4:     select[$\mathit{dim}$]const Nag_BooleanInput
Note: the dimension, dim, of the array select must be at least
• ${\mathbf{n}}$ when ${\mathbf{howmny}}=\mathrm{Nag_ComputeSelected}$;
• otherwise select may be NULL.
On entry: specifies the eigenpairs for which condition numbers are to be computed if ${\mathbf{howmny}}=\mathrm{Nag_ComputeSelected}$. 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 ${\mathbf{howmny}}=\mathrm{Nag_ComputeAll}$, select is not referenced and may be NULL.
5:     nIntegerInput
On entry: $n$, the order of the matrix pair $\left(S,T\right)$.
Constraint: ${\mathbf{n}}\ge 0$.
6:     a[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array a must be at least ${\mathbf{pda}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $A$ is stored in
• ${\mathbf{a}}\left[\left(j-1\right)×{\mathbf{pda}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{pda}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the upper quasi-triangular matrix $S$.
7:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8:     b[$\mathit{dim}$]const doubleInput
Note: the dimension, dim, of the array b must be at least ${\mathbf{pdb}}×{\mathbf{n}}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the upper triangular matrix $T$.
9:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   vl[$\mathit{dim}$]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 left eigenvectors of $\left(S,T\right)$, corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vl, as returned by nag_dggev (f08wac) or nag_dtgevc (f08ykc).
If ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, vl is not referenced and may be NULL.
11:   pdvlIntegerInput
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 $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvl}}\ge {\mathbf{n}}$;
• otherwise ${\mathbf{pdvl}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvl}}\ge {\mathbf{mm}}$;
• otherwise vl may be NULL.
12:   vr[$\mathit{dim}$]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 right eigenvectors of $\left(S,T\right)$, corresponding to the eigenpairs specified by howmny and select. The eigenvectors must be stored in consecutive columns of vr, as returned by nag_dggev (f08wac) or nag_dtgevc (f08ykc).
If ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, vr is not referenced and may be NULL.
13:   pdvrIntegerInput
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 $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvr}}\ge {\mathbf{n}}$;
• otherwise ${\mathbf{pdvr}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, ${\mathbf{pdvr}}\ge {\mathbf{mm}}$;
• otherwise vr may be NULL.
14:   s[$\mathit{dim}$]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: if ${\mathbf{job}}=\mathrm{Nag_EigVals}$ or $\mathrm{Nag_DoBoth}$, the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. For a complex conjugate pair of eigenvalues two consecutive elements of s are set to the same value. Thus ${\mathbf{s}}\left[j-1\right]$, ${\mathbf{dif}}\left[j-1\right]$, and the $j$th columns of $\mathrm{VL}$ and $\mathrm{VR}$ all correspond to the same eigenpair (but not in general the $j$th eigenpair, unless all eigenpairs are selected).
If ${\mathbf{job}}=\mathrm{Nag_EigVecs}$, s is not referenced and may be NULL.
15:   dif[$\mathit{dim}$]doubleOutput
Note: the dimension, dim, of the array dif must be at least
• ${\mathbf{mm}}$ when ${\mathbf{job}}=\mathrm{Nag_EigVecs}$ or $\mathrm{Nag_DoBoth}$;
• otherwise dif may be NULL.
On exit: if ${\mathbf{job}}=\mathrm{Nag_EigVecs}$ or $\mathrm{Nag_DoBoth}$, the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements of the array. For a complex eigenvector two consecutive elements of dif are set to the same value. If the eigenvalues cannot be reordered to compute ${\mathbf{dif}}\left[j-1\right]$, ${\mathbf{dif}}\left[j-1\right]$ is set to $0$; this can only occur when the true value would be very small anyway.
If ${\mathbf{job}}=\mathrm{Nag_EigVals}$, dif is not referenced and may be NULL.
16:   mmIntegerInput
On entry: the number of elements in the arrays s and dif.
Constraints:
• if ${\mathbf{howmny}}=\mathrm{Nag_ComputeAll}$, ${\mathbf{mm}}\ge {\mathbf{n}}$;
• otherwise ${\mathbf{mm}}\ge {\mathbf{m}}$.
17:   mInteger *Output
On exit: the number of elements of the arrays s and dif used to store the specified condition numbers; for each selected real eigenvalue one element is used, and for each selected complex conjugate pair of eigenvalues, two elements are used. If ${\mathbf{howmny}}=\mathrm{Nag_ComputeAll}$, m is set to n.
18:   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, ${\mathbf{job}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvl}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{mm}}=⟨\mathit{\text{value}}⟩$.
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_ENUM_INT_3
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{mm}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{howmny}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{howmny}}=\mathrm{Nag_ComputeAll}$, ${\mathbf{mm}}\ge {\mathbf{n}}$;
otherwise ${\mathbf{mm}}\ge {\mathbf{m}}$.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}>0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
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, ${\mathbf{pda}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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.

None.

## 8  Parallelism and Performance

nag_dtgsna (f08ylc) is not threaded by NAG in any implementation.
nag_dtgsna (f08ylc) 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.

An approximate asymptotic error bound on the chordal distance between the computed eigenvalue $\stackrel{~}{\lambda }$ and the corresponding exact eigenvalue $\lambda$ is
 $χλ~,λ ≤ εA,BF / Sλ$
where $\epsilon$ is the machine precision.
An approximate asymptotic error bound for the right or left computed eigenvectors $\stackrel{~}{x}$ or $\stackrel{~}{y}$ corresponding to the right and left eigenvectors $x$ and $y$ is given by
 $θz~,z ≤ ε A,BF / Dif .$
The complex analogue of this function is nag_ztgsna (f08yyc).

## 10  Example

This example estimates condition numbers and approximate error estimates for all the eigenvalues and eigenvalues and right eigenvectors of the pair $\left(S,T\right)$ given by
 $S = 4.0 1.0 1.0 2.0 0.0 3.0 -1.0 1.0 0.0 1.0 3.0 1.0 0.0 0.0 0.0 6.0 and T= 2.0 1.0 1.0 3.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 2.0 .$
The eigenvalues and eigenvectors are computed by calling nag_dtgevc (f08ykc).

### 10.1  Program Text

Program Text (f08ylce.c)

### 10.2  Program Data

Program Data (f08ylce.d)

### 10.3  Program Results

Program Results (f08ylce.r)