# compute_fp¶

Filename: compute_fp.py Authors: Thomas Sargent, John Stachurski, Daisuke Oyama

Compute an approximate fixed point of a given operator T, starting from specified initial condition v.

quantecon.compute_fp.compute_fixed_point(T, v, error_tol=0.001, max_iter=50, verbose=2, print_skip=5, method='iteration', *args, **kwargs)[source]

Computes and returns an approximate fixed point of the function T.

The default method ‘iteration’ simply iterates the function given an initial condition v and returns $$T^k v$$ when the condition $$\lVert T^k v - T^{k-1} v\rVert \leq \mathrm{error\_tol}$$ is satisfied or the number of iterations $$k$$ reaches max_iter. Provided that T is a contraction mapping or similar, $$T^k v$$ will be an approximation to the fixed point.

The method ‘imitation_game’ uses the “imitation game algorithm” developed by McLennan and Tourky [R8], which internally constructs a sequence of two-player games called imitation games and utilizes their Nash equilibria, computed by the Lemke-Howson algorithm routine. It finds an approximate fixed point of T, a point $$v^*$$ such that $$\lVert T(v) - v\rVert \leq \mathrm{error\_tol}$$, provided T is a function that satisfies the assumptions of Brouwer’s fixed point theorm, i.e., a continuous function that maps a compact and convex set to itself.

Parameters: T : callable A callable object (e.g., function) that acts on v v : object An object such that T(v) is defined; modified in place if method=’iteration’ and `v is an array error_tol : scalar(float), optional(default=1e-3) Error tolerance max_iter : scalar(int), optional(default=50) Maximum number of iterations verbose : scalar(int), optional(default=2) Level of feedback (0 for no output, 1 for warnings only, 2 for warning and residual error reports during iteration) print_skip : scalar(int), optional(default=5) How many iterations to apply between print messages (effective only when verbose=2) method : str, optional(default=’iteration’) str in {‘iteration’, ‘imitation_game’}. Method of computing an approximate fixed point args, kwargs : Other arguments and keyword arguments that are passed directly to the function T each time it is called v : object The approximate fixed point

References

 [R8] (1, 2) A. McLennan and R. Tourky, “From Imitation Games to Kakutani,” 2006.