# NAG Library Routine Document

## 1Purpose

f08naf (dgeev) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n$ by $n$ real nonsymmetric matrix $A$.

## 2Specification

Fortran Interface
 Subroutine f08naf ( n, a, lda, wr, wi, vl, ldvl, vr, ldvr, work, info)
 Integer, Intent (In) :: n, lda, ldvl, ldvr, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), wr(*), wi(*), vl(ldvl,*), vr(ldvr,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: jobvl, jobvr
#include <nagmk26.h>
 void f08naf_ (const char *jobvl, const char *jobvr, const Integer *n, double a[], const Integer *lda, double wr[], double wi[], double vl[], const Integer *ldvl, double vr[], const Integer *ldvr, double work[], const Integer *lwork, Integer *info, const Charlen length_jobvl, const Charlen length_jobvr)
The routine may be called by its LAPACK name dgeev.

## 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 orthogonal similarity transformations, and the $QR$ algorithm is then used to further reduce the matrix to upper quasi-triangular Schur form, $T$, with $1$ by $1$ and $2$ by $2$ blocks on the main diagonal. The eigenvalues are computed from $T$, the $2$ by $2$ blocks corresponding to complex conjugate pairs and, optionally, the eigenvectors of $T$ are computed and backtransformed to the eigenvectors 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:     $\mathbf{jobvl}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/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:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08naf (dgeev) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
6:     $\mathbf{wr}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
7:     $\mathbf{wi}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the dimension of the arrays wr and wi must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: wr and wi contain the real and imaginary parts, respectively, of the computed eigenvalues. Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
8:     $\mathbf{vl}\left({\mathbf{ldvl}},*\right)$ – Real (Kind=nag_wp) arrayOutput
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. If the $j$th eigenvalue is real, then ${u}_{j}={\mathbf{vl}}\left(:,j\right)$, the $j$th column of vl. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then ${u}_{j}={\mathbf{vl}}\left(:,j\right)+i×{\mathbf{vl}}\left(:,j+1\right)$ and ${u}_{j+1}={\mathbf{vl}}\left(:,j\right)-i×{\mathbf{vl}}\left(:,j+1\right)$.
If ${\mathbf{jobvl}}=\text{'N'}$, vl is not referenced.
9:     $\mathbf{ldvl}$ – IntegerInput
On entry: the first dimension of the array vl as declared in the (sub)program from which f08naf (dgeev) is called.
Constraints:
• if ${\mathbf{jobvl}}=\text{'V'}$, ${\mathbf{ldvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvl}}\ge 1$.
10:   $\mathbf{vr}\left({\mathbf{ldvr}},*\right)$ – Real (Kind=nag_wp) arrayOutput
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. If the $j$th eigenvalue is real, then ${v}_{j}={\mathbf{vr}}\left(:,j\right)$, the $j$th column of vr. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then ${v}_{j}={\mathbf{vr}}\left(:,j\right)+i×{\mathbf{vr}}\left(:,j+1\right)$ and ${v}_{j+1}={\mathbf{vr}}\left(:,j\right)-i×{\mathbf{vr}}\left(:,j+1\right)$.
If ${\mathbf{jobvr}}=\text{'N'}$, vr is not referenced.
11:   $\mathbf{ldvr}$ – IntegerInput
On entry: the first dimension of the array vr as declared in the (sub)program from which f08naf (dgeev) is called.
Constraints:
• if ${\mathbf{jobvr}}=\text{'V'}$, ${\mathbf{ldvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvr}}\ge 1$.
12:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
13:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08naf (dgeev) 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 must generally be larger than the minimum, say, $4×{\mathbf{n}}+\mathit{nb}×{\mathbf{n}}$, where $\mathit{nb}$ is the optimal block size of f08nef (dgehrd).
Constraints:
• if ${\mathbf{jobvl}}=\text{'V'}$ or ${\mathbf{jobvr}}=\text{'V'}$, ${\mathbf{lwork}}\ge 4×{\mathbf{n}}$;
• otherwise ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,3×{\mathbf{n}}\right)$.
14:   $\mathbf{info}$ – IntegerOutput
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 wr and wi 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

f08naf (dgeev) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08naf (dgeev) 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.

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 complex analogue of this routine is f08nnf (zgeev).

## 10Example

This example finds all the eigenvalues and right eigenvectors of the matrix
 $A = 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 .$
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 (f08nafe.f90)

### 10.2Program Data

Program Data (f08nafe.d)

### 10.3Program Results

Program Results (f08nafe.r)