# NAG FL Interfacef08nbf (dgeevx)

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## 1Purpose

f08nbf computes the eigenvalues and, optionally, the left and/or right eigenvectors for an $n×n$ real nonsymmetric matrix $A$.
Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors, reciprocal condition numbers for the eigenvalues, and reciprocal condition numbers for the right eigenvectors.

## 2Specification

Fortran Interface
 Subroutine f08nbf ( n, a, lda, wr, wi, vl, ldvl, vr, ldvr, ilo, ihi, work, info)
 Integer, Intent (In) :: n, lda, ldvl, ldvr, lwork Integer, Intent (Inout) :: iwork(*) Integer, Intent (Out) :: ilo, ihi, info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), wr(*), wi(*), vl(ldvl,*), vr(ldvr,*), scale(*), rconde(*), rcondv(*) Real (Kind=nag_wp), Intent (Out) :: abnrm, work(max(1,lwork)) Character (1), Intent (In) :: balanc, jobvl, jobvr, sense
#include <nag.h>
 void f08nbf_ (const char *balanc, const char *jobvl, const char *jobvr, const char *sense, const Integer *n, double a[], const Integer *lda, double wr[], double wi[], double vl[], const Integer *ldvl, double vr[], const Integer *ldvr, Integer *ilo, Integer *ihi, double scal[], double *abnrm, double rconde[], double rcondv[], double work[], const Integer *lwork, Integer iwork[], Integer *info, const Charlen length_balanc, const Charlen length_jobvl, const Charlen length_jobvr, const Charlen length_sense)
The routine may be called by the names f08nbf, nagf_lapackeig_dgeevx or its LAPACK name dgeevx.

## 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 ${u}_{j}$.
Balancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and applying a diagonal similarity transformation $DA{D}^{-1}$, where $D$ is a diagonal matrix, with the aim of making its rows and columns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see Section 4.8.1.2 of Anderson et al. (1999).
Following the optional balancing, 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: $\mathbf{balanc}$Character(1) Input
On entry: indicates how the input matrix should be diagonally scaled and/or permuted to improve the conditioning of its eigenvalues.
${\mathbf{balanc}}=\text{'N'}$
Do not diagonally scale or permute.
${\mathbf{balanc}}=\text{'P'}$
Perform permutations to make the matrix more nearly upper triangular. Do not diagonally scale.
${\mathbf{balanc}}=\text{'S'}$
Diagonally scale the matrix, i.e., replace $A×DA{D}^{-1}$, where $D$ is a diagonal matrix chosen to make the rows and columns of $A$ more equal in norm. Do not permute.
${\mathbf{balanc}}=\text{'B'}$
Both diagonally scale and permute $A$.
Computed reciprocal condition numbers will be for the matrix after balancing and/or permuting. Permuting does not change condition numbers (in exact arithmetic), but balancing does.
Constraint: ${\mathbf{balanc}}=\text{'N'}$, $\text{'P'}$, $\text{'S'}$ or $\text{'B'}$.
2: $\mathbf{jobvl}$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.
If ${\mathbf{sense}}=\text{'E'}$ or $\text{'B'}$, jobvl must be set to ${\mathbf{jobvl}}=\text{'V'}$.
Constraint: ${\mathbf{jobvl}}=\text{'N'}$ or $\text{'V'}$.
3: $\mathbf{jobvr}$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.
If ${\mathbf{sense}}=\text{'E'}$ or $\text{'B'}$, jobvr must be set to ${\mathbf{jobvr}}=\text{'V'}$.
Constraint: ${\mathbf{jobvr}}=\text{'N'}$ or $\text{'V'}$.
4: $\mathbf{sense}$Character(1) Input
On entry: determines which reciprocal condition numbers are computed.
${\mathbf{sense}}=\text{'N'}$
None are computed.
${\mathbf{sense}}=\text{'E'}$
Computed for eigenvalues only.
${\mathbf{sense}}=\text{'V'}$
Computed for right eigenvectors only.
${\mathbf{sense}}=\text{'B'}$
Computed for eigenvalues and right eigenvectors.
If ${\mathbf{sense}}=\text{'E'}$ or $\text{'B'}$, both left and right eigenvectors must also be computed (${\mathbf{jobvl}}=\text{'V'}$ and ${\mathbf{jobvr}}=\text{'V'}$).
Constraint: ${\mathbf{sense}}=\text{'N'}$, $\text{'E'}$, $\text{'V'}$ or $\text{'B'}$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (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 $n×n$ matrix $A$.
On exit: a has been overwritten. If ${\mathbf{jobvl}}=\text{'V'}$ or ${\mathbf{jobvr}}=\text{'V'}$, $A$ contains the real Schur form of the balanced version of the input matrix $A$.
7: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08nbf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
8: $\mathbf{wr}\left(*\right)$Real (Kind=nag_wp) array Output
9: $\mathbf{wi}\left(*\right)$Real (Kind=nag_wp) array Output
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.
10: $\mathbf{vl}\left({\mathbf{ldvl}},*\right)$Real (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. 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.
11: $\mathbf{ldvl}$Integer Input
On entry: the first dimension of the array vl as declared in the (sub)program from which f08nbf is called.
Constraints:
• if ${\mathbf{jobvl}}=\text{'V'}$, ${\mathbf{ldvl}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvl}}\ge 1$.
12: $\mathbf{vr}\left({\mathbf{ldvr}},*\right)$Real (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. 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.
13: $\mathbf{ldvr}$Integer Input
On entry: the first dimension of the array vr as declared in the (sub)program from which f08nbf is called.
Constraints:
• if ${\mathbf{jobvr}}=\text{'V'}$, ${\mathbf{ldvr}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• otherwise ${\mathbf{ldvr}}\ge 1$.
14: $\mathbf{ilo}$Integer Output
15: $\mathbf{ihi}$Integer Output
On exit: ilo and ihi are integer values determined when $A$ was balanced. The balanced $A$ has ${a}_{ij}=0$ if $i>j$ and $j=1,2,\dots ,{\mathbf{ilo}}-1$ or $i={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
16: $\mathbf{scale}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array scale must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: details of the permutations and scaling factors applied when balancing $A$.
If ${p}_{j}$ is the index of the row and column interchanged with row and column $j$, and ${d}_{j}$ is the scaling factor applied to row and column $j$, then
• ${\mathbf{scale}}\left(\mathit{j}\right)={p}_{\mathit{j}}$, for $\mathit{j}=1,2,\dots ,{\mathbf{ilo}}-1$;
• ${\mathbf{scale}}\left(\mathit{j}\right)={d}_{\mathit{j}}$, for $\mathit{j}={\mathbf{ilo}},\dots ,{\mathbf{ihi}}$;
• ${\mathbf{scale}}\left(\mathit{j}\right)={p}_{\mathit{j}}$, for $\mathit{j}={\mathbf{ihi}}+1,\dots ,{\mathbf{n}}$.
The order in which the interchanges are made is n to ${\mathbf{ihi}}+1$, then $1$ to ${\mathbf{ilo}}-1$.
17: $\mathbf{abnrm}$Real (Kind=nag_wp) Output
On exit: the $1$-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
18: $\mathbf{rconde}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array rconde must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: ${\mathbf{rconde}}\left(j\right)$ is the reciprocal condition number of the $j$th eigenvalue.
19: $\mathbf{rcondv}\left(*\right)$Real (Kind=nag_wp) array Output
Note: the dimension of the array rcondv must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On exit: ${\mathbf{rcondv}}\left(j\right)$ is the reciprocal condition number of the $j$th right eigenvector.
20: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Real (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
21: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08nbf 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, increase lwork by, say, ${\mathbf{n}}×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size for f08nef.
Constraints:
• if ${\mathbf{jobvl}}=\text{'N'}$ and ${\mathbf{jobvr}}=\text{'N'}$,
• if ${\mathbf{sense}}=\text{'N'}$, ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}\right)$;
• otherwise ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{{\mathbf{n}}}^{2}+6×{\mathbf{n}}\right)$;
• if ${\mathbf{jobvl}}=\text{'V'}$ or ${\mathbf{jobvr}}=\text{'V'}$,
• if ${\mathbf{sense}}=\text{'N'}$ or $\text{'E'}$, ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,3×{\mathbf{n}}\right)$;
• otherwise ${\mathbf{lwork}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{{\mathbf{n}}}^{2}+6×{\mathbf{n}}\right)$.
22: $\mathbf{iwork}\left(*\right)$Integer array Workspace
Note: the dimension of the array iwork must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,2×{\mathbf{n}}-1\right)$.
If ${\mathbf{sense}}=\text{'N'}$ or $\text{'E'}$, iwork is not referenced.
23: $\mathbf{info}$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 or condition numbers have been computed; elements $1$ to ${\mathbf{ilo}}-1$ and $⟨\mathit{\text{value}}⟩$ 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
 $‖E‖2 = O(ε) ‖A‖2 ,$
and $\epsilon$ is the machine precision. See Section 4.8 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f08nbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08nbf 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 f08npf.

## 10Example

This example finds all the eigenvalues and right eigenvectors of the matrix
 $A = ( 0.35 0.45 -0.14 -0.17 0.09 0.07 -0.54 0.35 -0.44 -0.33 -0.03 0.17 0.25 -0.32 -0.13 0.11 ) ,$
together with estimates of the condition number and forward error bounds for each eigenvalue and eigenvector. The option to balance the matrix is used. In order to compute the condition numbers of the eigenvalues, the left eigenvectors also have to be computed, but they are not printed out in this example.
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 (f08nbfe.f90)

### 10.2Program Data

Program Data (f08nbfe.d)

### 10.3Program Results

Program Results (f08nbfe.r)