NAG Toolbox: nag_lapack_dgeev (f08na)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{jobvl}$ – string (length ≥ 1)

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:     $\mathrm{jobvr}$ – string (length ≥ 1)

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:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array

The first dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a must be at least $\max \left(1,\mathbf{n}\right)$.

The $n$ by $n$ matrix $A$.

Optional Input Parameters

1:     $\mathrm{n}$ – int64int32nag_int scalar

Default: the first dimension of the array a and the second dimension of the array a. (An error is raised if these dimensions are not equal.)

$n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 0$.

Output Parameters

1:     $\mathrm{a}\left(\mathit{lda},:\right)$ – double array

The first dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

The second dimension of the array a will be $\max \left(1,\mathbf{n}\right)$.

2:     $\mathrm{wr}\left(:\right)$ – double array

3:     $\mathrm{wi}\left(:\right)$ – double array

The dimension of the arrays wr and wi will be $\max \left(1,\mathbf{n}\right)$

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.

4:     $\mathrm{vl}\left(\mathit{ldvl},:\right)$ – double array

The first dimension, $\mathit{ldvl}$, of the array vl will be

• if $\mathbf{jobvl}=\text{'V'}$, $\mathit{ldvl}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldvl}=1$.

The second dimension of the array vl will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvl}=\text{'V'}$ and $1$ otherwise.

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 $u_j=\mathbf{vl}\left(:,j\right)+i\times \mathbf{vl}\left(:,j+1\right)$ and $u_{j+1}=\mathbf{vl}\left(:,j\right)-i\times \mathbf{vl}\left(:,j+1\right)$.

If $\mathbf{jobvl}=\text{'N'}$, vl is not referenced.

5:     $\mathrm{vr}\left(\mathit{ldvr},:\right)$ – double array

The first dimension, $\mathit{ldvr}$, of the array vr will be

• if $\mathbf{jobvr}=\text{'V'}$, $\mathit{ldvr}=\max \left(1,\mathbf{n}\right)$;
• otherwise $\mathit{ldvr}=1$.

The second dimension of the array vr will be $\max \left(1,\mathbf{n}\right)$ if $\mathbf{jobvr}=\text{'V'}$ and $1$ otherwise.

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 $v_j=\mathbf{vr}\left(:,j\right)+i\times \mathbf{vr}\left(:,j+1\right)$ and $v_{j+1}=\mathbf{vr}\left(:,j\right)-i\times \mathbf{vr}\left(:,j+1\right)$.

If $\mathbf{jobvr}=\text{'N'}$, vr is not referenced.

6:     $\mathrm{info}$ – int64int32nag_int scalar

$\mathbf{info}=0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_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.

W  $\mathbf{info}>0$

The $QR$ algorithm failed to compute all the eigenvalues, and no eigenvectors have been computed; elements $\_$ to n of wr and wi contain eigenvalues which have converged.

Accuracy

Further Comments

Example

Open in the MATLAB editor: f08na_example

function f08na_example


  
fprintf('f08na example results\n\n');

% Matrix A
n = 4;
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];

% Eigenvalues and right eigenvectors of A
jobvl = 'No left vectors';
jobvr = 'Vectors (right)';
[~, wr, wi, ~, vr, info] = f08na( ...
                                  jobvl, jobvr, a);

fprintf('Index  Eigenvalue                Eigenvector\n');
k = 1;
conjugating = false;
for j = 1:n
  fprintf('%3d', j);
  if wi(j)==0 & ~conjugating
    fprintf('  %12.4e%15s',wr(j),' ');
    for l = 1:n
      if (l>1)
        fprintf('%32s', ' ');
      end
      fprintf('%12.4e\n',vr(l,k));
    end
    k = k + 1;
  else
    if conjugating
      pl = '-';
      mi = '+';
    else
      pl = '+';
      mi = '-';
    end
    fprintf('  %12.4e %s %10.4ei ', wr(j), pl, abs(wi(j)));
    for l = 1:n
      if (l>1)
        fprintf('%32s', ' ');
      end
      if vr(l,k+1)>0
        fprintf('%12.4e %s %10.4ei\n', vr(l,k), pl, vr(l,k+1));
      else
        fprintf('%12.4e %s %10.4ei\n', vr(l,k), mi, abs(vr(l,k+1)));
      end
    end
    if conjugating    
      k = k + 2;
    end
    conjugating = ~conjugating;
  end
  fprintf('\n');
end

f08na example results

Index  Eigenvalue                Eigenvector
  1    7.9948e-01                -6.5509e-01
                                 -5.2363e-01
                                  5.3622e-01
                                 -9.5607e-02

  2   -9.9412e-02 + 4.0079e-01i  -1.9330e-01 + 2.5463e-01i
                                  2.5186e-01 - 5.2240e-01i
                                  9.7182e-02 - 3.0838e-01i
                                  6.7595e-01 - 0.0000e+00i

  3   -9.9412e-02 - 4.0079e-01i  -1.9330e-01 - 2.5463e-01i
                                  2.5186e-01 + 5.2240e-01i
                                  9.7182e-02 + 3.0838e-01i
                                  6.7595e-01 + 0.0000e+00i

  4   -1.0066e-01                 1.2533e-01
                                  3.3202e-01
                                  5.9384e-01
                                  7.2209e-01