psmall¶

spectral projection

Calling Sequence¶

[Q,M]=psmall(A,thres,flag)

:A real square matrix : :thres real number : :flag character string ( ‘c’ or ‘d’) : :Q,M real matrices :

Projection on eigen-subspace associated with eigenvalues with real part < thres ( flag=’c’) or with modulus < thres ( flag=’d’).

The projection is defined by Q*M, Q is full column rank, M is full row rank and M*Q=eye.

If flag=’c’, the eigenvalues of M*A*Q = eigenvalues of A with real part < thres.

If flag=’d’, the eigenvalues of M*A*Q = eigenvalues of A with magnitude < thres.

If flag=’c’ and if [Q1,M1] = full rank factorization ( fullrf) of eye()-Q*M then eigenvalues of M1*A*Q1 = eigenvalues of A with real part >= thres.

If flag=’d’ and if [Q1,M1] = full rank factorization ( fullrf) of eye()-Q*M then eigenvalues of M1*A*Q1 = eigenvalues of A with magnitude >= thres.

A=`diag`_([1,2,3]);X=`rand`_(A);A=`inv`_(X)*A*X;
[Q,M]=psmall(A,2.5,'d');
`spec`_(M*A*Q)
[Q1,M1]=`fullrf`_(`eye`_()-Q*M);
`spec`_(M1*A*Q1)

This function is based on the ordered schur form (scilab function schur).