psmall spectral projection

## Syntax

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

## Arguments

A

  real square matrix

thres

  real number

flag

  character string (`'c'` or `'d'`)

Q,M

  real matrices

## Description

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`.

## Examples

    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)

## See also

- pbig
- proj
- projspec

## Used Functions

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