TMtx.EigSymGen Method

Overload List

#	Signature	Description
1	void EigSymGen(TMtx B, TVec D, TMtx V, TEigGenType EigGenType)	Computes all the eigenvalues, and optionally, the eigenvectors of a generalized symmetric-definite eigenproblem.
2	void EigSymGen(TMtx B, TVec D, Double Minimum, Double Maximum, TMtx V, ref Int32[] VInfo, Double Tolerance, TEigGenType EigGenType)	Computes generalized eigenvalues with reduction of the symmetric-definite generalized eigenvalues/eigenvectors problem to the normal eigenvalue case.
3	void EigSymGen(TMtx B, TVec D, Int32 LowerRange, Int32 UpperRange, TMtx V, ref Int32[] VInfo, Double Tolerance, TEigGenType EigGenType)	Computes generalized eigenvalues with reduction of the symmetric-definite generalized eigenvalues/eigenvectors problem to the normal eigenvalue case.

Overload 1: void EigSymGen(TMtx B, TVec D, TMtx V, TEigGenType EigGenType)

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

#	Name	Type	Description
1	B	TMtx	source TMtx
2	D	TVec	source TVec
3	V	TMtx	source TMtx
4	EigGenType	TEigGenType

Result: stored in self (calling object)

Remarks:

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.

Examples

TVec D;
TMtx A;
TMtx B;
TMtx C;
MtxVec.CreateIt(out D);
MtxVec.CreateIt(out A, out B, out C);
try
    {
        D.SetIt(2  ,false,new double[] {0,2}); // vector, length 2, real
        A.SetIt(2,2,false,[1,2,
        2,4]);  // 2x2, real matrix
        // A must be symmetric
        B.SetIt(2,2,false,[1,1,
        1,2]);  // 2x2, real matrix
        // B must be symmetric and positive definite
        A.EigSymGen(B,D);       // Use A and B to find eigenvalues
    }
finally
    {
        MtxVec.FreeIt(ref A, ref B, ref C);
        MtxVec.FreeIt(ref D);
    }

See Also: TMtx.Eig

Overload 2: void EigSymGen(TMtx B, TVec D, Double Minimum, Double Maximum, TMtx V, ref Int32[] VInfo, Double Tolerance, TEigGenType EigGenType)

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

#	Name	Type	Description
1	B	TMtx	source TMtx
2	D	TVec	source TVec
3	Minimum	Double	scalar
4	Maximum	Double	scalar
5	V	TMtx	source TMtx
6	VInfo	Int32[] (ref)
7	Tolerance	Double	scalar
8	EigGenType	TEigGenType

Result: stored in self (calling object)

Remarks:

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).

Overload 3: void EigSymGen(TMtx B, TVec D, Int32 LowerRange, Int32 UpperRange, TMtx V, ref Int32[] VInfo, Double Tolerance, TEigGenType EigGenType)

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

#	Name	Type	Description
1	B	TMtx	source TMtx
2	D	TVec	source TVec
3	LowerRange	Int32
4	UpperRange	Int32
5	V	TMtx	source TMtx
6	VInfo	Int32[] (ref)
7	Tolerance	Double	scalar
8	EigGenType	TEigGenType

Result: stored in self (calling object)

Remarks:

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

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

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