$V = \sum_{j=0}^{\infty} A^j B A^{j'}$

V is computed by solving the corresponding discrete lyapunov equation using the doubling algorithm. See the documentation of util.solve_discrete_lyapunov for more information.

Parameters:
Aarray_like(float, ndim=2)

An n x n matrix as described above. We assume in order for convergence that the eigenvalues of $$A$$ have moduli bounded by unity

Barray_like(float, ndim=2)

An n x n matrix as described above. We assume in order for convergence that the eigenvalues of $$A$$ have moduli bounded by unity

max_itscalar(int), optional(default=50)

The maximum number of iterations

Returns:
gamma1: array_like(float, ndim=2)

Represents the value $$V$$

Computes the expected discounted quadratic sum

$q(x_0) = \mathbb{E} \Big[ \sum_{t=0}^{\infty} \beta^t x_t' H x_t \Big]$

Here $${x_t}$$ is the VAR process $$x_{t+1} = A x_t + C w_t$$ with $${x_t}$$ standard normal and $$x_0$$ the initial condition.

Parameters:
Aarray_like(float, ndim=2)

The matrix described above in description. Should be n x n

Carray_like(float, ndim=2)

The matrix described above in description. Should be n x n

Harray_like(float, ndim=2)

The matrix described above in description. Should be n x n

beta: scalar(float)

Should take a value in (0, 1)

x_0: array_like(float, ndim=1)

The initial condtion. A conformable array (of length n, or with n rows)

Returns:
q0: scalar(float)

Represents the value $$q(x_0)$$

Remarks: The formula for computing $$q(x_0)$$ is
$$q(x_0) = x_0' Q x_0 + v$$
where
• $$Q$$ is the solution to $$Q = H + \beta A' Q A$$, and

• $$v = \frac{trace(C' Q C) \beta}{(1 - \beta)}$$