# NAG Library Function Document

## 1Purpose

nag_zgeevx (f08npc) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n$ by $n$ complex nonsymmetric matrix $A$.
Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.

## 2Specification

 #include #include
 void nag_zgeevx (Nag_OrderType order, Nag_BalanceType balanc, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Nag_RCondType sense, Integer n, Complex a[], Integer pda, Complex w[], Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, Integer *ilo, Integer *ihi, double scale[], double *abnrm, double rconde[], double rcondv[], NagError *fail)

## 3Description

The right eigenvector ${v}_{j}$ of $A$ satisfies
 $A vj = λj vj$
where ${\lambda }_{j}$ is the $j$th eigenvalue of $A$. The left eigenvector ${u}_{j}$ of $A$ satisfies
 $ujH A = λj ujH$
where ${u}_{j}^{\mathrm{H}}$ denotes the conjugate transpose of ${u}_{j}$.
Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation $DA{D}^{-1}$, where $D$ is a diagonal matrix, with the aim of making its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see Section 4.8.1.2 of Anderson et al. (1999).
Following the optional balancing, the matrix $A$ is first reduced to upper Hessenberg form by means of unitary similarity transformations, and the $QR$ algorithm is then used to further reduce the matrix to upper triangular Schur form, $T$, from which the eigenvalues are computed. Optionally, the eigenvectors of $T$ are also computed and backtransformed to those of $A$.

## 4References

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
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:    $\mathbf{balanc}$Nag_BalanceTypeInput
On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.
${\mathbf{balanc}}=\mathrm{Nag_NoBalancing}$
Do not diagonally scale or permute.
${\mathbf{balanc}}=\mathrm{Nag_BalancePermute}$
Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.
${\mathbf{balanc}}=\mathrm{Nag_BalanceScale}$
Diagonally scale the matrix, i.e., replace $A$ by $DA{D}^{-1}$, where $D$ is a diagonal matrix chosen to make the rows and columns of $A$ more equal in norm. Do not permute.
${\mathbf{balanc}}=\mathrm{Nag_BalanceBoth}$
Both diagonally scale and permute $A$.
Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
Constraint: ${\mathbf{balanc}}=\mathrm{Nag_NoBalancing}$, $\mathrm{Nag_BalancePermute}$, $\mathrm{Nag_BalanceScale}$ or $\mathrm{Nag_BalanceBoth}$.
3:    $\mathbf{jobvl}$Nag_LeftVecsTypeInput
On entry: if ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, the left eigenvectors of $A$ are not computed.
If ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, the left eigenvectors of $A$ are computed.
If ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$ or $\mathrm{Nag_RCondBoth}$, jobvl must be set to ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$.
Constraint: ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$ or $\mathrm{Nag_LeftVecs}$.
4:    $\mathbf{jobvr}$Nag_RightVecsTypeInput
On entry: if ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, the right eigenvectors of $A$ are not computed.
If ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, the right eigenvectors of $A$ are computed.
If ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$ or $\mathrm{Nag_RCondBoth}$, jobvr must be set to ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$.
Constraint: ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$ or $\mathrm{Nag_RightVecs}$.
5:    $\mathbf{sense}$Nag_RCondTypeInput
On entry: determines which reciprocal condition numbers are computed.
${\mathbf{sense}}=\mathrm{Nag_NotRCond}$
None are computed.
${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$
Computed for eigenvalues only.
${\mathbf{sense}}=\mathrm{Nag_RCondEigVecs}$
Computed for right eigenvectors only.
${\mathbf{sense}}=\mathrm{Nag_RCondBoth}$
Computed for eigenvalues and right eigenvectors.
If ${\mathbf{sense}}=\mathrm{Nag_RCondEigVals}$ or $\mathrm{Nag_RCondBoth}$, both left and right eigenvectors must also be computed (${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$ and ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$).
Constraint: ${\mathbf{sense}}=\mathrm{Nag_NotRCond}$, $\mathrm{Nag_RCondEigVals}$, $\mathrm{Nag_RCondEigVecs}$ or $\mathrm{Nag_RCondBoth}$.
6:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
7:    $\mathbf{a}\left[\mathit{dim}\right]$ComplexInput/Output
Note: the dimension, dim, of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pda}}×{\mathbf{n}}\right)$.
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 $n$ by $n$ matrix $A$.
On exit: a has been overwritten. If ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$ or ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, $A$ contains the Schur form of the balanced version of the matrix $A$.
8:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:    $\mathbf{w}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array w must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: contains the computed eigenvalues.
10:  $\mathbf{vl}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array vl must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvl}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$;
• $1$ otherwise.
Where ${\mathbf{VL}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vl}}\left[\left(j-1\right)×{\mathbf{pdvl}}+i-1\right]$ 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 exit: if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, the left eigenvectors ${u}_{j}$ are stored one after another in vl, in the same order as their corresponding eigenvalues; that is ${u}_{j}={\mathbf{VL}}\left(\mathit{i},j\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, vl is not referenced.
11:  $\mathbf{pdvl}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vl.
Constraints:
• if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, ${\mathbf{pdvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvl}}\ge 1$.
12:  $\mathbf{vr}\left[\mathit{dim}\right]$ComplexOutput
Note: the dimension, dim, of the array vr must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdvr}}×{\mathbf{n}}\right)$ when ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$;
• $1$ otherwise.
Where ${\mathbf{VR}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vr}}\left[\left(j-1\right)×{\mathbf{pdvr}}+i-1\right]$ 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 exit: if ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, the right eigenvectors ${v}_{j}$ are stored one after another in vr, in the same order as their corresponding eigenvalues; that is ${v}_{j}={\mathbf{VR}}\left(\mathit{i},j\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, vr is not referenced.
13:  $\mathbf{pdvr}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array vr.
Constraints:
• if ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, ${\mathbf{pdvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{pdvr}}\ge 1$.
14:  $\mathbf{ilo}$Integer *Output
15:  $\mathbf{ihi}$Integer *Output
On exit: ilo and ihi are integer values determined when $A$ was balanced. The balanced $A$ has ${a}_{ij}=0$ if $i>j$ and $j=1,2,\dots ,{\mathbf{ilo}}-1$ or $i={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
16:  $\mathbf{scale}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array scale must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: details of the permutations and scaling factors applied when balancing $A$.
If ${p}_{j}$ is the index of the row and column interchanged with row and column $j$, and ${d}_{j}$ is the scaling factor applied to row and column $j$, then
• ${\mathbf{scale}}\left[\mathit{j}-1\right]={p}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ilo}}-1$;
• ${\mathbf{scale}}\left[\mathit{j}-1\right]={d}_{\mathit{j}}$, for $\mathit{j}={\mathbf{ilo}},\dots ,{\mathbf{ihi}}$;
• ${\mathbf{scale}}\left[\mathit{j}-1\right]={p}_{\mathit{j}}$, for $\mathit{j}={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
The order in which the interchanges are made is n to ${\mathbf{ihi}}+1$, then $1$ to ${\mathbf{ilo}}-1$.
17:  $\mathbf{abnrm}$double *Output
On exit: the $1$-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
18:  $\mathbf{rconde}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array rconde must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: ${\mathbf{rconde}}\left[j-1\right]$ is the reciprocal condition number of the $j$th eigenvalue.
19:  $\mathbf{rcondv}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, of the array rcondv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: ${\mathbf{rcondv}}\left[j-1\right]$ is the reciprocal condition number of the $j$th right eigenvector.
20:  $\mathbf{fail}$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_CONVERGENCE
The $QR$ algorithm failed to compute all the eigenvalues, and no eigenvectors or condition numbers have been computed; elements $1$ to ${\mathbf{ilo}}-1$ and $〈\mathit{\text{value}}〉$ to n of w contain eigenvalues which have converged.
NE_ENUM_INT_2
On entry, ${\mathbf{jobvl}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvl}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobvl}}=\mathrm{Nag_LeftVecs}$, ${\mathbf{pdvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvl}}\ge 1$.
On entry, ${\mathbf{jobvr}}=〈\mathit{\text{value}}〉$, ${\mathbf{pdvr}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: if ${\mathbf{jobvr}}=\mathrm{Nag_RightVecs}$, ${\mathbf{pdvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
otherwise ${\mathbf{pdvr}}\ge 1$.
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, ${\mathbf{pdvl}}=〈\mathit{\text{value}}〉$.
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: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 eigenvalues and eigenvectors are exact for a nearby matrix $\left(A+E\right)$, where
 $E2 = Oε A2 ,$
and $\epsilon$ is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

nag_zgeevx (f08npc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_zgeevx (f08npc) 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.

Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real.
The total number of floating-point operations is proportional to ${n}^{3}$.
The real analogue of this function is nag_dgeevx (f08nbc).

## 10Example

This example finds all the eigenvalues and right eigenvectors of the matrix
 $A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i ,$
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix is used. In order to compute the condition numbers of the eigenvalues, the left eigenvectors also have to be computed, but they are not printed out in this example.

### 10.1Program Text

Program Text (f08npce.c)

### 10.2Program Data

Program Data (f08npce.d)

### 10.3Program Results

Program Results (f08npce.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017