From 13bba59d9d3cc9a508deef1e0f6176cfab73fbcd Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Mon, 15 Oct 2012 16:02:25 -0400 Subject: [PATCH] Add the fixed point method and some tests. --- fixed_point_method.m | 31 +++++++++++++++++++++++++++++++ run-tests.m | 21 +++++++++++++++++++++ 2 files changed, 52 insertions(+) create mode 100644 fixed_point_method.m diff --git a/fixed_point_method.m b/fixed_point_method.m new file mode 100644 index 0000000..c28a66e --- /dev/null +++ b/fixed_point_method.m @@ -0,0 +1,31 @@ +function [fixed_point, iterations] = fixed_point_method(g, epsilon, x0) + ## Find a fixed_point of the function `g` with initial guess x0. + ## + ## INPUTS: + ## + ## * ``g`` - The function to iterate. + ## + ## * ``epsilon`` - We stop when two successive iterations are within + ## epsilon of each other, taken under the infinity norm. halt the + ## search and return the current approximation. + ## + ## OUTPUTS: + ## + ## * ``fixed_point`` - The fixed point that we found. + ## + ## * ``iterations`` - The number of bisections that we performed + ## during the search. + ## + + iterations = 0; + prev = x0; + current = g(x0); + + while (norm(current - prev, Inf) > epsilon) + prev = current; + current = g(current); + iterations = iterations + 1; + end + + fixed_point = current; +end diff --git a/run-tests.m b/run-tests.m index 1682d18..aa285c3 100755 --- a/run-tests.m +++ b/run-tests.m @@ -51,3 +51,24 @@ expected_A = [1, 0, 0, 0, 0; ... unit_test_equals("Homework #1 problem #1 Poisson matrix is correct", ... true, ... expected_A == poisson_matrix(4, 0, 1)); + + +g = @(x) 1 + atan(x); +expected_fp = 2.1323; +tol = 1 / 10^10; +x0 = 2.4; +unit_test_equals("Homework #2 problem #5 fixed point is correct", ... + expected_fp, ... + fixed_point_method(g, tol, x0)); + + +h = 0.5; +g1 = @(u) 1 + h*exp(-u(1)^2)/(1+u(2)^2); +g2 = @(u) 0.5 + h*atan(u(1)^2 + u(2)^2); +my_g = @(u) [g1(u), g2(u)]; +tol = 1 / 10^9; +u0 = [1,1]; +expected_fp = [1.0729, 1.0821]; +unit_test_equals("Homework #3 problem #3i fixed point is correct", ... + expected_fp, ... + fixed_point_method(my_g, tol, u0)); -- 2.43.2