f08 Chapter Contents
f08 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_zgeev (f08nnc)

## 1  Purpose

nag_zgeev (f08nnc) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n$ by $n$ complex nonsymmetric matrix $A$.

## 2  Specification

 #include #include
 void nag_zgeev (Nag_OrderType order, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Integer n, Complex a[], Integer pda, Complex w[], Complex vl[], Integer pdvl, Complex vr[], Integer pdvr, NagError *fail)

## 3  Description

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 ${u}_{j}$.
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 $A$.

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

## 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 Nag_ColMajor.
2:     jobvlNag_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.
Constraint: ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$ or $\mathrm{Nag_LeftVecs}$.
3:     jobvrNag_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.
Constraint: ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$ or $\mathrm{Nag_RightVecs}$.
4:     nIntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:     a[$\mathit{dim}$]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.
6:     pdaIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:     w[$\mathit{dim}$]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.
8:     vl[$\mathit{dim}$]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.
9:     pdvlIntegerInput
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 ${\mathbf{pdvl}}\ge 1$.
10:   vr[$\mathit{dim}$]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.
11:   pdvrIntegerInput
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 ${\mathbf{pdvr}}\ge 1$.
12:   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_CONVERGENCE
The $QR$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $〈\mathit{\text{value}}〉$ 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.

## 7  Accuracy

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.

Each eigenvector is normalized to have Euclidean norm equal to unity and the element of largest absolute value real and positive.
The total number of floating point operations is proportional to ${n}^{3}$.
The real analogue of this function is nag_dgeev (f08nac).

## 9  Example

This example finds all the eigenvalues and right eigenvectors of the matrix
 $A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .$

### 9.1  Program Text

Program Text (f08nnce.c)

### 9.2  Program Data

Program Data (f08nnce.d)

### 9.3  Program Results

Program Results (f08nnce.r)