Construction QT matrices
The cqt command can be used to construct QT matrices from several input data. Different syntaxes are supported.
Contents
Syntax
- T = cqt(neg, pos, E) creates the semi-infinite cqt-matrix with the symbol defined by the coefficients in neg and pos, which are expected to contain negative and positive powers of the Laurent polynomial, respectively. For consistency, it is checked that neg(1) == pos(1). An optional finite correction E can be specified.
- T = cqt(neg, pos, U, V) creates the semi-infinite cqt-matrix with the specified symbol (as above) and finite correction U * V.'.
- T = cqt(neg, pos, A, B, m, n) creates the $m \times n$ QT matrix with the specified symbol, top-left correction $A$ and bottom-right correction $B$. If one between m and n is inf then $B$ is ignored.
- T = cqt(neg, pos, U, V, W, Z, m, n) creates the $m \times n$ QT matrix with the specified symbol and finite corrections A = U * V.' and B = W * Z.'. If one between $m$ and $n$ is inf then $B$ is ignored.
- T = cqt(neg, pos) creates the semi-infinite QT matrix with the specified symbol and an empty finite correction.
- T = cqt(A) creates the semi-infinite cqt-matrix with a symbol equal to $0$ and finite correction equal to $A$.
- All the constructors can be called using the syntax: T = cqt('extended', arg1, ..., argk, v), which is equivalent to the commands T = cqt(arg1, ..., argk) and T = extend(T, v), which constructs a quasi-Toeplitz matrix plus a rank $1$ correction of the form $ev^T$, where $e$ is the vector of all ones.
- T = cqt('hankel', f) constructs the Hankel matrix with symbol given by f.
Example
A = cqt([ 1 2 3], [1 4 5 6], -ones(2)); A(1:8, 1:8)
ans =
0 3 5 6 0 0 0 0
1 0 4 5 6 0 0 0
3 2 1 4 5 6 0 0
0 3 2 1 4 5 6 0
0 0 3 2 1 4 5 6
0 0 0 3 2 1 4 5
0 0 0 0 3 2 1 4
0 0 0 0 0 3 2 1