TMtx::EigSymGen Method

Computes all the eigenvalues, and optionally, the eigenvectors of a generalized symmetric-definite eigenproblem.

It can find solution to either of the following problems:

A X = lambda B X, A B X = lambda X, or B A X = lambda X

Here A and B are assumed to be symmetric (Hermitian) and B is also positive definite. Eigenvector are stored within V in columns.

Computes generalized eigenvalues with reduction of the symmetric-definite generalized eigenvalues/eigenvectors problem to the normal eigenvalue case.

The routine computes selected eigenvalues and optionally also eigenvectors. The problem is of type:

A x = lambda B x, A B x = lambda x, or B A x = lambda x

A and B are symmetric (Hermitian) and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying a range of values. Eigenvectors are not computed, if V is passed as nil. (NULL).

Tolerance parameter specifies the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

Tolerance + EPS / max( |a|,|b| ) ,

where EPS is the machine precision. If Tolerance is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when Tolerance is set to twice the underflow threshold, not zero. If this routine returns fails , indicating that some eigenvectors did not converge, try setting Tolerance to 2*UnderflowThreshold.

If V is assinged, VInfo contains values equal to 0 at indices for which eigenvector calculation converged. Eigenvector are stored within V in columns. The returned column count may vary between calls depending on the number of eigenvectors that converged. The eigenvectors are normalized as follows:

etAzBz, etBAz, Z**T*B*Z := I;
etABz        , Z**T*inv(B)*Z := I.

[Lapack Users Guide](Lapack Users Guide).

Computes generalized eigenvalues with reduction of the symmetric-definite generalized eigenvalues/eigenvectors problem to the normal eigenvalue case.

The routine computes selected eigenvalues and optionally also eigenvectors. The problem is of type:

A x = lambda B x, A B x = lambda x, or B A x = lambda x

A and B are symmetric (Hermitian) and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying a range of indexes of values. Eigenvectors are not computed, if V is passed as nil (NULL).

Tolerance parameter specifies the absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to

Tolerance + EPS / max( |a|,|b| ) ,

where EPS is the machine precision. If Tolerance is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.

Eigenvalues will be computed most accurately when Tolerance is set to twice the underflow threshold, not zero. If this routine returns fails , indicating that some eigenvectors did not converge, try setting Tolerance to 2*UnderflowThreshold.

The first eigenvalue index is 1 and the last index is equal to row/column count. LowerRange and UpperRange need to be specified within this interval: 1 <= LowerRange <= UpperRange <= Rows

If V is assinged, VInfo contains values equal to 0 at indices for which eigenvector calculation converged. Eigenvector are stored within V in columns. The returned column count may vary between calls depending on the number of eigenvectors that converged. The eigenvectors are normalized as follows:

etAzBz, etBAz, Z**T*B*Z := I;
etABz        , Z**T*inv(B)*Z := I.