Preface

This tutorial was made solely for the purpose of education and it was designed for students taking Applied Math 0340. It is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel. The Mathematica commands in this tutorial are all written in bold black font, while Mathematica output is in regular fonts.

Finally, you can copy and paste all commands into your Mathematica notebook, change the parameters, and run them because the tutorial is under the terms of the GNU General Public License (GPL). You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately. The tutorial accompanies the textbook Applied Differential Equations. The Primary Course by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

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Introduction to Linear Algebra

Glossary

Iterations for Nonlinear Systems

We extend the methods studied previously to the case of systems of nonlinear equations. Let us start with planar case and consider two functions
\[ f(x,y) = x^2 - 2*x - y + 1, \qquad g(x,y) = x^2 + 9*y^2 - 9 . \]
We seek a method of solution for the system of nonlinear equations
\[ f(x,y) = 0, \qquad g(x,y) = 0 . \]
By plotting implicitly these two curves, we see that the given system of equations has two roots: (0,1) and (1.88241, 0.778642).
a = Plot[y /. Solve[x^2 - 2*x - y + 1 == 0], {x, -4, 4}, AspectRatio -> 1]
b = Plot[y /. Solve[x^2 + 9*y^2 - 9 == 0], {x, -4, 4}, AspectRatio -> 1]
Show[a, b, PlotRange -> {-0.5, 1.5}]
or
ContourPlot[x^2 - 2*x - y + 1 == 0, {x, -5, 5}, {y, -3, 3}, AspectRatio -> Automatic]
NSolve[{x^2 + 9*y^2 - 9 == 0, x^2 - 2*x - y + 1 == 0}, {x, y}]

Fixed Point Iteration

We consider the problem \( {\bf x} = {\bf g} \left( {\bf x} \right) , \) where \( {\bf g}: \, \mathbb{R}^n \mapsto \mathbb{R}^n \) is the function from n-dimensional space into the same space. The fixed-point iteration is defined by selecting a vector \( {\bf x}_0 \in \mathbb{R}^n \) and defining a sequence \( {\bf x}_{k+1} = {\bf g} \left( {\bf x}_k \right) , \quad k=0,1,2,\ldots . \)

A point \( {\bf x} \in \mathbb{R}^n \) is a fixed point of \( {\bf g} \left( {\bf x} \right) \) if and only if \( {\bf x} = {\bf g} \left( {\bf x} \right) . \) A mapping \( {\bf f} \left( {\bf x} \right) \) is a contractive mapping if there is a positive number L less than 1 such that

\[ \left\| {\bf f} \left( {\bf x} \right) - {\bf f} \left( {\bf y} \right) \right\| \le L\, \| {\bf x} - {\bf y} \| , \qquad 0 < L < 1 . \]

Theorem: Let \( {\bf g} \left( {\bf x} \right) \) be a continuous function on \( \Omega = \left\{ {\bf x} = (x_1 , x_2 , \ldots x_n ) \, : \ a_i < x_i < b_i \right\} \) such that

  • \( {\bf g} \left( \Omega \right) \subset \Omega ; \)
  • \( {\bf g} \left( {\bf x} \right) \) is a contraction.
Then
  • g has a fixed point \( {\bf x} \in \Omega \) and the sequence \( {\bf x}_{k+1} = {\bf g} \left( {\bf x}_k \right) , \quad k=0,1,2,\ldots ; \quad {\bf x}_0 \in \Omega \) converges to x;
  • \( \| {\bf x}_k - {\bf x} \| \le L^k \,\| {\bf x}_0 - {\bf x} \| . \) ■

Example:

Example. We apply the fixed point iteration to find the roots of the system of nonlinear equations

\[ f(x,y) = x^2 - 2\,x - y + 1 =0, \qquad g(x,y) = x^2 + 9\,y^2 - 9 =0. \]
The first equation can be used to solve for either x or y. Depending on our choice, we can add either 18y or 2x to the second equation. So we have two formulas for iteration:
\[ \begin{cases} x &= \frac{1}{2} \left[ x^2 -y+1 \right] , \\ y &= \frac{1}{18} \left[ x^2 + 9\,y^2 -9 + 18\,y \right] , \end{cases} \qquad\mbox{or} \qquad \begin{cases} y &= \left( x -1 \right)^2 , \\ x &= \frac{1}{2} \left[ (x+1)^2 + 9\,y^2 -10 \right] . \end{cases} \]
These two system of equations are used to write the recursive formulas. Start with an initial point (0,0.5), and then compute the sequence \( \left\{ (x_k , y_k ) \right\} \) according to the formula:
\[ \begin{cases} x_{k+1} &= \frac{1}{2} \left[ x^2_k -y_k +1 \right] , \\ y_{k+1} &= \frac{1}{18} \left[ x^2_k + 9\,y^2_k -9 + 18\,y \right] . \end{cases} \]
Then we repeat calculations according to the second formula:
\[ \begin{cases} y_{k+1} &= \left( x_k -1 \right)^2 , \\ x_{k+1} &= \frac{1}{2} \left[ (x_k +1)^2 + 9\,y_k^2 -10 \right] . \end{cases} \]
x[0] = 0; y[0] = 0.5;
Do[{x[k + 1] = N[(x[k]^2 - y[k] + 1)/2], y[k + 1] = N[(x[k]^2 + 9*y[k]^2 - 9 + 18*y[k])/18]}, {k, 0, 50}]
Do[Print["i=", i, " x= ", x[i], " y= ", y[i]], {i, 0, 10}]
x[0] = 0; y[0] = 0.5;
Do[{y[k + 1] = N[(x[k] - 1)^2], x[k + 1] = N[((x[k] + 1)^2 + 9*y[k]^2 - 10)/2]}, {k, 0, 10}]
Do[Print["i=", i, " x= ", x[i], " y= ", y[i]], {i, 0, 10}]
We summarize the results of computations in the table
k  x_k  y_k    k  x_k  y_k 
0  0  0.5    0  0  0.5 
1  0.25  0.125    1  -3.375  1 
2  0.46875  -0.363715    2  2.32031   19.1406 
3  0.791721  -0.785364    3  1649.15  1.74323 
4  1.20609  -0.942142    4  1.3615*1010  2.71639*106 
5  1.6984  -0.917512    5  3.41314*1013  1.85369*1012 
6  2.40104  -0.836344    6  5.97938*1026  1.16495*1027 
7  3.80067  -0.666331         
8  8.0557  -0.141829         
9  33.0181  2.9734         
So we see that both sequences diverge from the fixed point (0,1).

If we use the starting value (1.5, 0.5), then

k  x_k  y_k    k  x_k  y_k 
0  1.5  0.5    0  1.5  0.5 
1  1.375  0.25    1  -0.75  0.25 
2  1.32031  -0.113715    2  -4.6875  3.0625 
3  1.42847  -0.510404    3  44.0039  32.3477 
4  1.77547  -0.766785    4  5716.34  1849.34 
5  2.45953  -0.797679    5  3.17342056*107  3.26652*107 
6  3.92349  -0.643461    6  5.3050884*1015  1.00706*1015 
7  8.5186  -0.0812318    7  1.86357*1031  2.8144*1031 
8  36.8239  3.45355    8     
9  676.774  84.2505    9     
So both sequences diverge. ■

We want to determine why our iterative equations were not suitable for finding the solution near both fixed points (0, 1) and (1.88241, 0.778642). To answer this question, we need to evaluate the Jacobian matrix:

\[ {\bf J} (x,y) = \begin{bmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{bmatrix} . \]

Theorem: Assume that the functions f(x,y) and g(x,y) and their first partial derivatives are continuous in a region that contain the fixed point (p,q). If the starting point is chosen sufficiently close to the fixed point and if

\[ \begin{cases} \left\vert \frac{\partial f}{\partial x}(p,q) \right\vert + \left\vert \frac{\partial f}{\partial y}(p,q) \right\vert < 1, \\ \left\vert \frac{\partial g}{\partial x}(p,q) \right\vert + \left\vert \frac{\partial g}{\partial y}(p,q) \right\vert < 1, \end{cases} \]
then the iteration
\[ x_{k+1} = f(x_k , y_k ) , \qquad y_{k+1} = g(x_k , y_k ) , \qquad k=0,1,2,\ldots ; \]
converges to the fixed point (p,q). ■

Seidel Method

An improvement analogous to the Gauss-Seiel method for linear systems, of fixe-point iteration can be made. Suppose we try to solve the system of two nonlinear equations
\[ x = f(x,y) , \qquad y= g(x,y) . \]
Staring with some initial point \( (x_0 , y_0 ) , \) we compute the sequence according to the following formulas, called Seidel method:
\[ x_{k+1} = f(x_k ,y_k ) , \qquad y_{k+1} = g(x_{k+1} ,y_k) , \qquad k=0,1,2,\ldots . \]

Wegstein Algorithm

Let \( {\bf g} \, : \, \mathbb{R}^n \mapsto \mathbb{R}^n \) be a continuous function. Suppose we are given a system of equations

\[ \begin{split} x_1 &= g_1 \left( x_1 , x_2 , \ldots , x_n \right) , \\ x_2 &= g_2 \left( x_1 , x_2 , \ldots , x_n \right) , \\ \vdots & \qquad \vdots \\ x_n &= g_n \left( x_1 , x_2 , \ldots , x_n \right) . \end{split} \]
Then Wegstein iteration equations can we written as
\[ \begin{split} x_1^{(k+1)} &= q_1 x_1^{(k)} + \left( 1- q_1 \right) g_1 \left( x_1^{(k)} , x_2^{(k)} , \ldots , x_n^{(k)} \right) , \\ x_2^{(k+1)} &= q_2 x_2^{(k)} + \left( 1- q_2 \right) g_2 \left( x_1^{(k)} , x_2^{(k)} , \ldots , x_n^{(k)} \right) , \\ \vdots & \qquad \vdots \\ x_n^{(k+1)} &= q_n x_n^{(k)} + \left( 1- q_n \right) g_n \left( x_1^{(k)} , x_2^{(k)} , \ldots , x_n^{(k)} \right) , \end{split} \]
where
\[ q_i = \dfrac{\partial g_i / \partial x_i}{\partial g_i / \partial x_i -1} \qquad\mbox{evaluated at} \quad \left( x_1^{(k)} , x_2^{(k)} , \ldots , x_n^{(k)} \right) , \qquad i=0,1,2,\ldots n; \]
Here again k denotes the number of the iteration.

In planar case, we have two equations

\[ x= f(x,y), \qquad y= g(x,y) . \]
The Wegstein iterations become
\[ \begin{cases} X_{k+1} &= q_1 X_k + \left( 1-q_1 \right) f\left( X_k , Y_k \right) , \\ Y_{k+1} &= q_2 Y_k + \left( 1-q_2 \right) g\left( X_k , Y_k \right) , \end{cases} \]
where
\[ q_1 = \frac{f_x}{f_x -1} , \qquad q_2 = \frac{g_y}{g_y -1} \]
are evaluated at the point \( \left( x_k , y_k \right) . \) Then
\[ \begin{cases} x_{k+1} &= q_1 x_k + \left( 1-q_1 \right) f\left( X_{k+1} , Y_{k+1} \right) , \\ y_{k+1} &= q_2 y_k + \left( 1-q_2 \right) g\left( X_{k+1} , Y_{k+1} \right) , \qquad k=0,1,2,\ldots . \end{cases} \]
A condition for Wegstein algorithm to be stable at a root is
\[ \left\vert \frac{f_x \,g_y}{\left( f_x -1 \right)\left( g_y -1 \right)} \right\vert < 1 . \]

Example:

Example. We apply the Wegstein algorithm to find the roots of the system of nonlinear equations

\[ x^2 - 2\,x - y + 1 =0, \qquad x^2 + 9\,y^2 - 9 =0. \]
By adding and subtracting x and y from each equation, respectively, we reduce the given problem to fixed point system of equations:
\[ x= f(x,y) = x^2 - x - y + 1 , \qquad y = g(x,y) = x^2 + 9\,y^2 - 9 +y . \]
We know that such system of equations has two real fixed points (0,1) and (1.88241, 0.778642). Their partial derivatives are \( f_x = 2x-1 \) and \( g_y = 18\,y +1 . \) At the first fixed point (0,1) we have
\[ \left\vert \frac{f_x \,g_y}{\left( f_x -1 \right)\left( g_y -1 \right)} \right\vert_{x=0, y=1} = \frac{19}{36} < 1 , \]
while at the another point (1.88241, 0.778642) its value is
\[ \left\vert \frac{f_x \,g_y}{\left( f_x -1 \right)\left( g_y -1 \right)} \right\vert_{x=1.88241, y=0.778642} = 1.67841 > 1 . \]
So we expect that the sequence of Wegstein iterations will converge to the fixed point (0,1), but it will diverge from another fixed point (1.88241, 0.778642). ■

Newton's Methods

The Newton methods have problems with the initial guess which, in general, has to be selected close to the solution. In order to avoid this problem for scalar equations we combine the bisection and Newton's method. First, we apply the bisection method to obtain a small interval that contains the root and then finish the work using Newton’s iteration.

For systems, it is known the global method called the descent method of which Newton’s iteration is a special case. Newton’s method will be applied once we get close to a root.

Let us consider the system of nonlinear algebraic equations \( {\bf f} \left( {\bf x} \right) =0 , \) where \( {\bf f} \,:\, \mathbb{R}^n \mapsto \mathbb{R}^n , \) and define the scalar multivariable function

\[ \phi ({\bf x}) = \frac{1}{2}\, \sum_{i=1}^n f_i^2 ({\bf x} ) = \frac{1}{2}\,{\bf f}^T \,{\bf f} \]
The function φ has the following properties.
  • \( \phi ({\bf x}) \ge 0 \) for all \( {\bf x} \in \mathbb{R}^n . \)
  • If x is a solution of \( {\bf f} \left( {\bf x} \right) =0 , \) then φ has a local minimum at x.
  • At an arbitrary point x0, the vector \( - \nabla \phi ({\bf x}_0 ) \) is the direction of the most rapid decrease of φ.
  • φ has infinitely many descent directions.
  • A direction u is descent direction for φ at x if and only if \( {\bf u}^T \nabla \phi ({\bf x}) < 0 . \)
  • Special descent directions:
    (a) Steepest descent method: \( {\bf u}_k = -JF^T ({\bf x}_k )\, {\bf f} ({\bf x}_k ) , \)
    (b) Newton’s method: \( {\bf u}_k = -JF^{-1} ({\bf x}_k )\, {\bf f} ({\bf x}_k ) . \) ■

For scalar problems \( f (x) =0 , \) the descent direction is given by \( - \phi' (x) = -2\,f(x) \,f' (x) \) while Newton’s method yields \( -f(x) / f' (x) \) which has the same sign as the steepest descent method.

The question that arises for all descent methods is how far should one go in a given descent direction. To answer this question, there are several techniques and conditions to guarantee convergence to a minimum of φ. For a detailed discussion consult the book by

Numerical Methods for Unconstrained Optimization and Nonlinear Equations, by J. E. Dennis, Jr. and Robert B. Schnabel. Published: 1996, ISBN: 978-0-89871-364-0
http://www.math.vt.edu/people/adjerids/homepage/teaching/F09/Math5465/chapter2.pdf

 

 

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