"""
This module provides functions to compute quadratic sums of the form described
in the docstrings.

"""

import numpy as np
import scipy.linalg
from .matrix_eqn import solve_discrete_lyapunov

[docs]def var_quadratic_sum(A, C, H, beta, x0): r""" Computes the expected discounted quadratic sum .. math:: q(x_0) = \mathbb{E} \Big[ \sum_{t=0}^{\infty} \beta^t x_t' H x_t \Big] Here :math:{x_t} is the VAR process :math:x_{t+1} = A x_t + C w_t with :math:{x_t} standard normal and :math:x_0 the initial condition. Parameters ---------- A : array_like(float, ndim=2) The matrix described above in description. Should be n x n C : array_like(float, ndim=2) The matrix described above in description. Should be n x n H : array_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 :math:q(x_0) Remarks: The formula for computing :math:q(x_0) is :math:q(x_0) = x_0' Q x_0 + v where * :math:Q is the solution to :math:Q = H + \beta A' Q A, and * :math:v = \frac{trace(C' Q C) \beta}{(1 - \beta)} """ # == Make sure that A, C, H and x0 are array_like == # A, C, H = list(map(np.atleast_2d, (A, C, H))) x0 = np.atleast_1d(x0) # == Start computations == # Q = scipy.linalg.solve_discrete_lyapunov(np.sqrt(beta) * A.T, H) cq = np.dot(np.dot(C.T, Q), C) v = np.trace(cq) * beta / (1 - beta) q0 = np.dot(np.dot(x0.T, Q), x0) + v return q0
[docs]def m_quadratic_sum(A, B, max_it=50): r""" Computes the quadratic sum .. math:: 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 ---------- A : array_like(float, ndim=2) An n x n matrix as described above. We assume in order for convergence that the eigenvalues of :math:A have moduli bounded by unity B : array_like(float, ndim=2) An n x n matrix as described above. We assume in order for convergence that the eigenvalues of :math:A have moduli bounded by unity max_it : scalar(int), optional(default=50) The maximum number of iterations Returns ======== gamma1: array_like(float, ndim=2) Represents the value :math:V """ gamma1 = solve_discrete_lyapunov(A, B, max_it) return gamma1