Additional Options

Options.Pairs
Options.Track
Options.AvoidStag
Options.LS$\underline{~}$Tol

Options.Pairs        (default 'no')

If 'yes', then jdqr searches for the complex conjugate eigenpair whenever an eigenpair has been detected. If A and B are real matrices, or the operators correspond to real matrices, then $(\bar \lambda,\bar x)$ is an eigenpair if $(\lambda,x)$ is one. Since the eigenpairs are not computed in full accuracy and since a generalized Schur decomposition is computed instead of eigenpairs, the conjugate of an approximate eigenpair may not have the required precision and jdqr may take additional iterations to obtain the conjugate pair in the desired accuracy.

Options.FixShift        (default 'no')

If Options.FixShift is scalar and Sigma(1,:) is a scalar, then jdqr takes Sigma as shift in the correction equation until the norm of the residual times Options.FixShift is less than 1. From then on, the shift is taken equal to the present approximate eigenvalue.
If Options.FixShift = 'yes' then Options.FixShift= 1.0e+3.
If Options.FixShift = 'no' is the same as Options.FixShift = 0.

Options.Track        (default '1e-4')

If the wanted eigenvalue is relatively far from the target, then the algorithm may select approximate eigenvalues that are accidentally close to the target instead of the approximate eigenvalue that is close to the wanted eigenvalue. To avoid this type of misselection, the target can be moved to an approximate eigenvalue that is close to the wanted eigenvalue. The size of the norm of the residual is used to measure the quality of the approximate eigenvalue. If norm(r)<= Options.track, then the associated approximate eigenvalue is used as target in the next iteration step.

Options.AvoidStag        (default 'no')

In some situations, the algorithm stagnates because the computed expansion vector for the search subspace belongs to the search subspace or is close to it. With 'yes', jdqr tries to remedy this type of stagnation. In the correction equation in the next iteration step, jdqr projects then on the complete search subspace rather than on the current eigenvector approximation.

Options.LS$\underline{~}$Tol        (default [1,0.7,0.7${}^2$,...])

LS$\underline{~}$Tol sets the residual reduction for the linear solver of the correction equation.
If LS$\underline{~}$Tol is a positive real then the correction equation is solved with a residual reduction of LS$\underline{~}$Tol in each Jacobi-Davidson step.
If LS$\underline{~}$Tol is a row $[a_1,a_2,\ldots,a_p]$ of positive reals then the correction equation at the $i$th Jacobi-Davidson step is solved with a residual reduction of $a_i$ provided that $i\leq p$. If $i=p+j$, then the residual reduction at the $i$th Jacobi-Davidson step is $a_p*b^j$ where $b=a_p/a_{p-1}$.
$i$ is reset to $0$ when a Schur vector is detected.
The required residual reduction is not obtained if the maximum number LS$\underline{~}$MaxIt of iteration steps of the linear solver is reached before.
The default for LS$\underline{~}$Tol is [1,0.7] if a preconditioner is used (then $a_i=0.7^{i-1}$). In the other case, the default is [0.7,0.49] (then $a_i=0.7^i$).