matrix_eqn¶
Filename: matrix_eqn.py
This files holds several functions that are used to solve matrix equations. Currently has functionality to solve:
 Lyapunov Equations
 Ricatti Equations
 TODO: 1. See issue 47 on github repository, should add support for
 Sylvester equations 2. Fix warnings from checking conditioning of matrices

quantecon.matrix_eqn.
solve_discrete_lyapunov
(A, B, max_it=50, method='doubling')[source]¶ Computes the solution to the discrete lyapunov equation
\[AXA'  X + B = 0\]X is computed by using a doubling algorithm. In particular, we iterate to convergence on X_j with the following recursions for j = 1, 2,... starting from X_0 = B, a_0 = A:
\[ \begin{align}\begin{aligned}a_j = a_{j1} a_{j1}\\X_j = X_{j1} + a_{j1} X_{j1} a_{j1}'\end{aligned}\end{align} \]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 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 A have moduli bounded by unity
max_it : scalar(int), optional(default=50)
The maximum number of iterations
method : string, optional(default=”doubling”)
Describes the solution method to use. If it is “doubling” then uses the doubling algorithm to solve, if it is “bartelsstewart” then it uses scipy’s implementation of the BartelsStewart approach.
Returns: gamma1: array_like(float, ndim=2)
Represents the value V

quantecon.matrix_eqn.
solve_discrete_riccati
(A, B, Q, R, N=None, tolerance=1e10, max_iter=500)[source]¶ Solves the discretetime algebraic Riccati equation
X = A’XA  (N + B’XA)’(B’XB + R)^{1}(N + B’XA) + Qvia a modified structured doubling algorithm. An explanation of the algorithm can be found in the reference below.
Note that SciPy also has a discrete riccati equation solver. However it cannot handle the case where R is not invertible, or when N is nonzero. Both of these cases can be handled in the algorithm implemented below.
Parameters: A : array_like(float, ndim=2)
k x k array.
B : array_like(float, ndim=2)
k x n array
Q : array_like(float, ndim=2)
k x k, should be symmetric and nonnegative definite
R : array_like(float, ndim=2)
n x n, should be symmetric and positive definite
N : array_like(float, ndim=2)
n x k array
tolerance : scalar(float), optional(default=1e10)
The tolerance level for convergence
max_iter : scalar(int), optional(default=500)
The maximum number of iterations allowed
Returns: X : array_like(float, ndim=2)
The fixed point of the Riccati equation; a k x k array representing the approximate solution
References
Chiang, ChunYueh, HungYuan Fan, and WenWei Lin. “STRUCTURED DOUBLING ALGORITHM FOR DISCRETETIME ALGEBRAIC RICCATI EQUATIONS WITH SINGULAR CONTROL WEIGHTING MATRICES.” Taiwanese Journal of Mathematics 14, no. 3A (2010): pp935.