f08 Chapter Contents
f08 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_dgeev (f08nac)

## 1  Purpose

nag_dgeev (f08nac) computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n$ by $n$ real nonsymmetric matrix $A$.

## 2  Specification

 #include #include
 void nag_dgeev (Nag_OrderType order, Nag_LeftVecsType jobvl, Nag_RightVecsType jobvr, Integer n, double a[], Integer pda, double wr[], double wi[], double vl[], Integer pdvl, double 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 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 $A$.

## 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:    $\mathbf{order}$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 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{jobvl}$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.
Constraint: ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$ or $\mathrm{Nag_LeftVecs}$.
3:    $\mathbf{jobvr}$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.
Constraint: ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$ or $\mathrm{Nag_RightVecs}$.
4:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
5:    $\mathbf{a}\left[\mathit{dim}\right]$doubleInput/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:    $\mathbf{pda}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array a.
Constraint: ${\mathbf{pda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
7:    $\mathbf{wr}\left[\mathit{dim}\right]$doubleOutput
8:    $\mathbf{wi}\left[\mathit{dim}\right]$doubleOutput
Note: the dimension, dim, 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.
9:    $\mathbf{vl}\left[\mathit{dim}\right]$doubleOutput
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. If the $j$th eigenvalue is real, then ${u}_{j}={\mathbf{VL}}\left(\mathit{i},j\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then ${u}_{j}={\mathbf{VL}}\left(\mathit{i},j\right)+\mathit{i}×{\mathbf{VL}}\left(\mathit{i},j+1\right)$ and ${u}_{j+1}={\mathbf{VL}}\left(\mathit{i},j\right)-\mathit{i}×{\mathbf{VL}}\left(\mathit{i},j+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{jobvl}}=\mathrm{Nag_NotLeftVecs}$, vl is not referenced.
10:  $\mathbf{pdvl}$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 ${\mathbf{pdvl}}\ge 1$.
11:  $\mathbf{vr}\left[\mathit{dim}\right]$doubleOutput
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. If the $j$th eigenvalue is real, then ${v}_{j}={\mathbf{VR}}\left(\mathit{i},j\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$. If the $j$th and $\left(j+1\right)$st eigenvalues form a complex conjugate pair, then ${v}_{j}={\mathbf{VR}}\left(\mathit{i},j\right)+\mathit{i}×{\mathbf{VR}}\left(\mathit{i},j+1\right)$ and ${v}_{j+1}={\mathbf{VR}}\left(\mathit{i},j\right)-\mathit{i}×{\mathbf{VR}}\left(\mathit{i},j+1\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{n}}$.
If ${\mathbf{jobvr}}=\mathrm{Nag_NotRightVecs}$, vr is not referenced.
12:  $\mathbf{pdvr}$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 ${\mathbf{pdvr}}\ge 1$.
13:  $\mathbf{fail}$NagError *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.
See Section 3.2.1.2 in the Essential Introduction 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 have been computed; elements $〈\mathit{\text{value}}〉$ to n of wr and wi 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 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.

## 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.

## 8  Parallelism and Performance

nag_dgeev (f08nac) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_dgeev (f08nac) 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 and positive.
The total number of floating-point operations is proportional to ${n}^{3}$.
The complex analogue of this function is nag_zgeev (f08nnc).

## 10  Example

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

### 10.1  Program Text

Program Text (f08nace.c)

### 10.2  Program Data

Program Data (f08nace.d)

### 10.3  Program Results

Program Results (f08nace.r)