Data update

Task/Steffensens-method/MATLAB/steffensens-method.m

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+clear all;close all;clc;
+tol = 0.00000001; iters = 1000; stepsize = 0.1;
+test_steffensen(tol, iters, stepsize);
+
+
+% Aitken's extrapolation
+function p = aitken(f, p0)
+    p1 = f(p0);
+    p2 = f(p1);
+    p = p0 - (p1 - p0)^2 / (p2 - 2 * p1 + p0);
+end
+
+% Steffensen's method using Aitken
+function p = steffensen_aitken(f, pinit, tol, maxiter)
+    p0 = pinit;
+    p = aitken(f, p0);
+    iter = 1;
+    while abs(p - p0) > tol && iter < maxiter
+        p0 = p;
+        p = aitken(f, p0);
+        iter = iter + 1;
+    end
+    if abs(p - p0) > tol
+        p = NaN;
+    end
+end
+
+% deCasteljau function
+function result = deCasteljau(c0, c1, c2, t)
+    s = 1.0 - t;
+    result = s * (s * c0 + t * c1) + t * (s * c1 + t * c2);
+end
+
+% Helper functions
+function result = xConvexLeftParabola(t)
+    result = deCasteljau(2, -8, 2, t);
+end
+
+function result = yConvexRightParabola(t)
+    result = deCasteljau(1, 2, 3, t);
+end
+
+function result = implicit_equation(x, y)
+    result = 5 * x^2 + y - 5;
+end
+
+% Main function
+function result = f(t)
+    if isnan(t) || isinf(t)
+        result = NaN;
+    else
+        result = implicit_equation(xConvexLeftParabola(t), yConvexRightParabola(t)) + t;
+    end
+end
+
+% Test example
+function test_steffensen(tol, iters, stepsize)
+    if nargin < 1
+        tol = 0.00000001;
+    end
+    if nargin < 2
+        iters = 1000;
+    end
+    if nargin < 3
+        stepsize = 0.1;
+    end
+
+    for t0 = 0:stepsize:1.1
+        fprintf('t0 = %f : ', t0);
+        t = steffensen_aitken(@f, t0, tol, iters);
+        if isnan(t)
+            fprintf('no answer\n');
+        else
+            x = xConvexLeftParabola(t);
+            y = yConvexRightParabola(t);
+            if abs(implicit_equation(x, y)) <= tol
+                fprintf('intersection at (%f, %f)\n', x, y);
+            else
+                fprintf('spurious solution\n');
+            end
+        end
+    end
+end