Data commit

Task/Steffensens-method/Python/steffensens-method.py

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+from math import nan, isnan
+from numpy import arange
+
+
+def aitken(f, p0):
+    """ Aitken's extrapolation """
+    p1 = f(p0)
+    p2 = f(p1)
+    return p0 - (p1 - p0)**2 / (p2 - 2 * p1 + p0)
+
+def steffensen_aitken(f, pinit, tol, maxiter):
+    """ Steffensen's method using Aitken """
+    p0 = pinit
+    p = aitken(f, p0)
+    iter = 1
+    while abs(p - p0) > tol and iter < maxiter:
+        p0 = p
+        p = aitken(f, p0)
+        iter += 1
+    return nan if abs(p - p0) > tol else p
+
+def deCasteljau(c0, c1, c2, t):
+    """ deCasteljau function """
+    s = 1.0 - t
+    return s * (s * c0 + t * c1) + t * (s * c1 + t * c2)
+
+def xConvexLeftParabola(t): return deCasteljau(2, -8, 2, t)
+def yConvexRightParabola(t): return deCasteljau(1, 2, 3, t)
+def implicit_equation(x, y): return 5 * x**2 + y - 5
+
+def f(t):
+    """ Outside of NumPy arithmetic may return NoneType on overflow """
+    if type(t) == type(None):
+        return nan
+    return implicit_equation(xConvexLeftParabola(t), yConvexRightParabola(t)) + t
+
+def test_steffensen(tol=0.00000001, iters=1000, stepsize=0.1):
+    """ test the example """
+    for t0 in arange(0, 1.1, stepsize):
+        print(f't0 = {t0:0.1f} : ', end='')
+        t = steffensen_aitken(f, t0, tol, iters)
+        if isnan(t):
+            print('no answer')
+        else:
+            x = xConvexLeftParabola(t)
+            y = yConvexRightParabola(t)
+            if abs(implicit_equation(x, y)) <= tol:
+                print(f'intersection at ({x}, {y})')
+            else:
+                print('spurious solution')
+    return 0
+
+
+if __name__ == '__main__':
+
+    test_steffensen()