# NAG FL Interfacef08nnf (zgeev)

## 1Purpose

f08nnf computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n$ by $n$ complex nonsymmetric matrix $A$.

## 2Specification

Fortran Interface
 Subroutine f08nnf ( n, a, lda, w, vl, ldvl, vr, ldvr, work, info)
 Integer, Intent (In) :: n, lda, ldvl, ldvr, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: rwork(*) Complex (Kind=nag_wp), Intent (Inout) :: a(lda,*), w(*), vl(ldvl,*), vr(ldvr,*) Complex (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: jobvl, jobvr
C Header Interface
#include <nag.h>
 void f08nnf_ (const char *jobvl, const char *jobvr, const Integer *n, Complex a[], const Integer *lda, Complex w[], Complex vl[], const Integer *ldvl, Complex vr[], const Integer *ldvr, Complex work[], const Integer *lwork, double rwork[], Integer *info, const Charlen length_jobvl, const Charlen length_jobvr)
The routine may be called by the names f08nnf, nagf_lapackeig_zgeev or its LAPACK name zgeev.

## 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 ${u}_{j}$.
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 https://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: $\mathbf{jobvl}$Character(1) Input
On entry: if ${\mathbf{jobvl}}=\text{'N'}$, the left eigenvectors of $A$ are not computed.
If ${\mathbf{jobvl}}=\text{'V'}$, the left eigenvectors of $A$ are computed.
Constraint: ${\mathbf{jobvl}}=\text{'N'}$ or $\text{'V'}$.
2: $\mathbf{jobvr}$Character(1) Input
On entry: if ${\mathbf{jobvr}}=\text{'N'}$, the right eigenvectors of $A$ are not computed.
If ${\mathbf{jobvr}}=\text{'V'}$, the right eigenvectors of $A$ are computed.
Constraint: ${\mathbf{jobvr}}=\text{'N'}$ or $\text{'V'}$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{a}\left({\mathbf{lda}},*\right)$Complex (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $n$ by $n$ matrix $A$.
On exit: a has been overwritten.
5: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08nnf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6: $\mathbf{w}\left(*\right)$Complex (Kind=nag_wp) array Output
Note: the dimension 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.
7: $\mathbf{vl}\left({\mathbf{ldvl}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array vl must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvl}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobvl}}=\text{'V'}$, the left eigenvectors ${u}_{j}$ are stored one after another in the columns of vl, in the same order as their corresponding eigenvalues; that is ${u}_{j}={\mathbf{vl}}\left(:,j\right)$, the $j$th column of vl.
If ${\mathbf{jobvl}}=\text{'N'}$, vl is not referenced.
8: $\mathbf{ldvl}$Integer Input
On entry: the first dimension of the array vl as declared in the (sub)program from which f08nnf is called.
Constraints:
• if ${\mathbf{jobvl}}=\text{'V'}$, ${\mathbf{ldvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvl}}\ge 1$.
9: $\mathbf{vr}\left({\mathbf{ldvr}},*\right)$Complex (Kind=nag_wp) array Output
Note: the second dimension of the array vr must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$ if ${\mathbf{jobvr}}=\text{'V'}$, and at least $1$ otherwise.
On exit: if ${\mathbf{jobvr}}=\text{'V'}$, the right eigenvectors ${v}_{j}$ are stored one after another in the columns of vr, in the same order as their corresponding eigenvalues; that is ${v}_{j}={\mathbf{vr}}\left(:,j\right)$, the $j$th column of vr.
If ${\mathbf{jobvr}}=\text{'N'}$, vr is not referenced.
10: $\mathbf{ldvr}$Integer Input
On entry: the first dimension of the array vr as declared in the (sub)program from which f08nnf is called.
Constraints:
• if ${\mathbf{jobvr}}=\text{'V'}$, ${\mathbf{ldvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvr}}\ge 1$.
11: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Complex (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the real part of ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
12: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08nnf is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, lwork should be generally larger than the minimum, say ${\mathbf{n}}+\mathit{nb}×{\mathbf{n}}$, where $\mathit{nb}$ is the optimal block size for f08nsf.
Constraint: ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}\right)$.
13: $\mathbf{rwork}\left(*\right)$Real (Kind=nag_wp) array Workspace
Note: the dimension of the array rwork must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}\right)$.
14: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
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.

## 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

f08nnf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nnf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

## 9Further Comments

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 routine is f08naf.

## 10Example

This example finds all the eigenvalues and right eigenvectors of the matrix
 $A = -3.97-5.04i -4.11+3.70i -0.34+1.01i 1.29-0.86i 0.34-1.50i 1.52-0.43i 1.88-5.38i 3.36+0.65i 3.31-3.85i 2.50+3.45i 0.88-1.08i 0.64-1.48i -1.10+0.82i 1.81-1.59i 3.25+1.33i 1.57-3.44i .$
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

### 10.1Program Text

Program Text (f08nnfe.f90)

### 10.2Program Data

Program Data (f08nnfe.d)

### 10.3Program Results

Program Results (f08nnfe.r)