F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF08NAF (DGEEV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F08NAF (DGEEV) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n$ by $n$ real nonsymmetric matrix $A$.

## 2  Specification

 SUBROUTINE F08NAF ( JOBVL, JOBVR, N, A, LDA, WR, WI, VL, LDVL, VR, LDVR, WORK, LWORK, INFO)
 INTEGER N, LDA, LDVL, LDVR, LWORK, INFO REAL (KIND=nag_wp) A(LDA,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), WORK(max(1,LWORK)) CHARACTER(1) JOBVL, JOBVR
The routine may be called by its LAPACK name dgeev.

## 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 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 $A$.
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  Parameters

1:     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:     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:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     A(LDA,$*$) – 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:     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:     WR($*$) – REAL (KIND=nag_wp) arrayOutput
7:     WI($*$) – 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:     VL(LDVL,$*$) – 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:     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:   VR(LDVR,$*$) – 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:   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:   WORK($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{LWORK}}\right)$) – 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:   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:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the $QR$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $i+1:{\mathbf{N}}$ of WRWI contain eigenvalues which have converged.

## 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 complex analogue of this routine is F08NNF (ZGEEV).

## 9  Example

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.

### 9.1  Program Text

Program Text (f08nafe.f90)

### 9.2  Program Data

Program Data (f08nafe.d)

### 9.3  Program Results

Program Results (f08nafe.r)