# coding: utf-8 # # # # # Fixed-Point Iteration # # ### Modules - Root Finding #
# By Eilif Sommer Øyre, Jonas Themsland and Jon Andreas Støvneng #
# # Last edited: March 11th 2018 # # --- # ## Introduction # # An equation $f(x) = 0$ can always be written as $g(x) = x$. Fixed-point iteration can be applied to approximate the fixed number $r = g(r)$. # # After selecting an initial guess $x_0$, the fixed-point iteration is $x_{n+1} = g(x_n)$. That is, # # \begin{equation*} # x_0 = \textrm{initial guess} \\ # x_1 = g(x_0) \\ # x_2 = g(x_1) \\ # x_3 = g(x_2) \\ # \vdots # \end{equation*} # # The algorithm is repeated until either a testing condition $e_{i+1} = \left|\:x_{i+1} - x_{i}\:\right| < \alpha$, where $\alpha$ is some tolerance limit is met, or until a fixed number of iterations $N$ is reached. Note that the method may or may not converge. Note that the choice of $g(x)$ is in general not unique. # ## Example 1 # # As a simple introductory example we use the fixed-point iteration to solve the equation # # $$\frac{1}{2}\sin(x) - x + 1 = 0.$$ # # The most natural is to use $g(x)= \sin(x)/2 + 1$ and then use the fixed-point iteration to solve $g(x)=x$. # In: import math x = 1 # initial guess N = 10 # iterations for i in range(1, N): x = math.sin(x) + 1 print("x%i:\t%.5f"%(i, x)) # ## Convergence # # For the iteration scheme to return a fixed point $r$, it needs to converge. A criterion for convergence is that the error, $e_i = \left|\:r - x_i \:\right|$, decreases for each iteration step. This means that the change from $x_i$ to $x_{i+1}$ also has to decrease for each step. The convergence criterion is explained in the following theorem : # # > **Theorem 1** Assume that $g$ is continuously differentiable, that $g(r) = r$, and that $S = \left|\:g'(r)\:\right| < 1$. Then fixed-point iteration converges linearly with rate $S$ to the fixed point $r$ for initial guesses sufficiently close to $r$. # # Note that in the example above we have $|g'(x)|=\cos(x)/2 < 1$ which means that the method will converge for all initial guesses. # # The convergence can in fact be of higher order. This is explained in the following theorem : # # > **Theorem 2** Assume that $g$ is $p$ times continuously differentiable and that the fixed-iteration converges to $r$ for some initial guess $x_0$. If $g'(r)=g''(r)=\cdots=g^{(p-1)}(r)=0$ and $g^{(p-1)}(r)=0$, then the order of convergence is $p$. # # The convergence is said to be of order $p$ when $\lim_{i\to\infty}e_{i+1}/e_{i}^p = \text{constant}$. # # **Excercise:** Prove theorem 1 using the mean value theorem and the theorem 2 using Taylor's theorem. # ## Example 2: Babylonian Method for Finding Square Roots # # As mentioned in the introduction, the choice of $g(x)$ is far from unique. We will now use fixed-point iteration to estimate the square root of a real and positive number $a$. That is, we want to solve $f(x) = x^2-a^2=0$. Two natural choices for $g(x)$ are $g_1(x)=a/x$ and $g_2(x)=x$. However, none of these converges since $|g_1'(\sqrt a)| = |g_2'(\sqrt a)| =1$. We can choose the mean as $g(x)$: # # $$g(x) = \frac{1}{2}\left(\frac{a}{x} + x\right).$$ # # In this case we have # # $$\left|\:g'(x)\:\right|_{x=\sqrt a}=\left|\:\frac{1}{2}(1-\frac{a}{x^2})\:\right|_{x=\sqrt a} = 0 < 1$$ # # The theorem above thus implies that method converges for some initial guess $x_0$. The method is in fact globally convergent due to the convexity of $f(x)$. # In: a = 0.07 # square of root x = 1 # initial guess N = 10 # iterations for i in range(1, N): x = (a/x + x)/2 print("x%i:\t%.5f"%(i, x)) # This method can also be derived from Newton's method (see the section on Newton's method below). The method was however first used by the Babylonian people long before Newton . # ## Example 3: Adding Stopping Conditions # # Underneath follows an implementation of the Fixed-Point Iteration with a testing condition $\left|\:x_{i+1} - x_i\:\right| < \alpha$. There are other stopping criteria that may be relevant such as the backward error $\left|\:f(x_a)- f(0)\right|$. We need an additional stopping criteria in case convergence fails. We therefore stop the computation if the a given number of iterations $N$ is reached. # In: def FPI(g, x, maxit=100, alpha=1e-5): """ A function using fixed-point iteration to compute an approximate solution to the fixed point problem g(r) = r. Arguments: g callable function, g(x) x float. The initial guess alpha float. Error tolerance maxit int. Maximum number of iterations Returns: array of each fixed-point approximation """ result = [x] x_next = g(x) i = 0 # Counter e = abs(x_next - x) while e > alpha : if i > maxit: print("No convergence.") break x = x_next x_next = g(x_next) result.append(x) if x - x_next > e: print("Divergence.") break e = abs(x_next - x) i += 1 return result # We want to solve the equation, # # \begin{equation*} # f(x) = x^4 + x - 1 = 0 # %\label{eq:example} # \end{equation*} # # using fixed point iteration. First, we rewrite the equation to the form, $x = g(x)$. This particular equation may be rewritten in three ways: # # $$g_1(x) = x = 1 - x^4,$$ # # $$g_2(x) = x = \sqrt{1 - x},$$ # # $$g_3(x) = x = \frac{1 + x^4}{1 + 2x^3}.$$ # # Each of these has its own convergence rate. One of the solutions is clearly somewhere between 0 and 1, so we will use $x_0=0.5$ as initial guess. # In: x_0 = 0.5 N = 100 alpha = 1e-5 def g1(x): return 1 - x**4 def g2(x): return (1 - x)**(1/4) def g3(x): return (1 + 3*x**4)/(1 + 4*x**3) def print_result(result): for i in range(len(result)): print("x%i:\t%.5f"%(i, result[i])) # In: result_g1 = FPI(g1, x_0, N, alpha) print_result(result_g1) # In: result_g2 = FPI(g2, x_0, N, alpha) print_result(result_g2) # In: result_g3 = FPI(g3, x_0, N, alpha) print_result(result_g3) # The method does not converge for $g_1$, but both $g_2$ and $g_3$ converges towards the fixed number $0.72449$, which is a solution of the equation $f(x) = 0$. The method converges much faster for $g_3$ than for $g_2$. # # This result can be verified by calculating the convergence rates $S = \left|\: g'(r) \:\right|$: # # \begin{equation*} # \left|\:g_1'(0.72449)\:\right| \approx 1.521 > 1, \\ # \left|\:g_2'(0.72449)\:\right| \approx 0.657 < 1, \\ # \left|\:g_3'(0.72449)\:\right| \approx 0.000 < 1. \\ # \end{equation*} # # Theorem 1 tells us that the method will converge for $g_2$ and $g_3$ for some inital guess. Moreover, from theorem 2 we know that the method converges linearly for $g_2$ and the order of convergence for $g_3$ is two. # # **Exercise:** Verify that the convergence for $g_2$ is linear and that the convergence of $g_3$ is of second order by using the numerical results above. # ## In comparison with the Newton-Rhapson method # # The Newton-Rhapson method is a special case of fixed-point iteration where the convergence rate is zero - the fastest possible. The method estimates the root of a differentiable function $f(x)$ by iteratively calculating the expression # # \begin{equation*} # x_{n+1} = x_n -\frac{f(x_n)}{f'(x_n)}. # \end{equation*} # # By using theorem 2, we can show that Newton's method is of second order (if $f'(r)\neq 0$). If $f''(r)=0$, then the method has a third order convergence. # # This Newton iteration may be rewritten as a fixed point iteration # # \begin{equation*} # x_{n+1} = g(x_n), \: n = 1, 2 ,3, ... # \end{equation*} # # where # # \begin{equation*} # r = g(r) = r - \frac{f(r)}{f'(r)} # %\label{eq:newton} # \end{equation*} # # Provided that the iteration converges to a fixed point $r$ of $g$. From this we obtain # # $$\frac{f(r)}{f'(r)} = 0 \: \Rightarrow \: f(r) = 0$$ # # and thus the fixed point $r$ is a root of $f$. # # The Newton's method is further discussed in the module [Root Finding - Newton-Rhapson Method](https://nbviewer.jupyter.org/urls/www.numfys.net/media/notebooks/newton_raphson_method.ipynb). # ## Final comments # # We have now used the fixed-point iteration method to solve the equation $f(x) = 0$ by rewriting it as a fixed-point problem. This is a simple method and which is easy to implement, but unlike the [Bisection Method](https://nbviewer.jupyter.org/urls/www.numfys.net/media/notebooks/bisection_method.ipynb) it only converges if the initial guess is sufficiently close to the root $r$. It is in general only locally convergent. However, the convergence rate may, or may not, be faster than that of the Bisection Method, which is 1/2. # # ## References # #  Sauer, T.: Numerical Analysis international edition, second edition, Pearson 2014 #  Gautschi, W.: Numerical Analysis, second edition, Birkhäuser 2012 (https://doi.org/10.1007/978-0-8176-8259-0)