Introduction to Numerical Math and Matlab ... - Ohio University

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10 LAB 3. NEWTON’S METHOD AND FOR LOOPS For f(x) = x 1/3 we have x ∗ = 0 but f ′ (x ∗ ) = ∞. If you try > f = inline(’x^(1/3)’) > f1 = inline(’(1/3)*x^(-2/3)’) > x = mynewton(f,f1,0.1,10) then x explodes. For f(x) = x 2 we have x ∗ = 0 but f ′ (x ∗ ) = 0. If you try > f = inline(’x^2’) > f1 = inline(’2*x’) > x = mynewton(f,f1,1,10) then x does converge to 0, but not that rapidly. If x 0 is not close enough to x ∗ that the linear approximation (3.2) is valid, then the iteration (3.4) gives some x 1 that may or may not be any better than x 0 . If we keep iterating, then either • x n will eventually get close to x ∗ and the method will then converge (rapidly), or • the iterations will not approach x ∗ . Exercises 3.1 Use mynewton on the function f(x) = x 5 − 7, with x 0 = 2. By trial and error, what is the lowest value of n for which the program converges (stops changing). How close is the answer to the true value? Plug the program’s answer into f(x); is the value zero? 3.2 Suppose a ball with a coefficient of elasticity of .9 is dropped from a height of 2 meters onto a hard surface. Write a script program to calculate the distance traveled by the ball after n bounces. Include lots of comments. By trial and error approximate how large n must be so that total distance stops changing. Turn in the program and a brief summary of the results. 3.3 For f(x) = x 3 − 4, perform 3 iterations of Newton’s method with starting point x 0 = 2. (On paper, but use a calculator.) Calculate the true solution (x ∗ = 4 1/3 ) in Matlab and find the errors and percentage errors of x 0 , x 1 , x 2 and x 3 . Put the results in a table.
Lab 4 Controlling Error and while Loops Measuring error If we are trying to find a numerical solution of an equation f(x) = 0, then there are a few different ways we can measure the error of our approximation. The most direct way to measure the error would be as: {Error at step n} = e n = x n − x ∗ where x n is the n-th approximation and x ∗ is the true value. However, we usually do not know the value of x ∗ , or we wouldn’t be trying to approximate it. This makes it impossible to know the error directly, and so we must be more clever. For Newton’s method we have the following principle: At each step the number of significant digits roughly doubles. While this is an important statement about the error (since it means Newton’s method converges really quickly), it is somewhat hard to use in a program. Rather than measure how close x n is to x ∗ , in this and many other situations it is much more practical to measure how close the equation is to being satisfied, in other words, how close f(x n ) is to 0. We will use the quantity r n = f(x n ) − 0, called the residual, in many different situations. Most of the time we only care about the size of r n , so we look at |r n | = |f(x n )|. The if ... end statement If we have a certain tolerance for |r n | = |f(x n )|, then we can incorporate that into our Newton method program using an if ... end statement: function x = mynewton(f,f1,x0,n,tol) % Solves f(x) = 0 by doing n steps of Newton’s method starting at x0. % Inputs: f -- the function, input as an inline % f1 -- it’s derivative, input as an inline % x0 -- starting guess, a number % tol -- desired tolerance, prints a warning if |f(x)|>tol % Output: x -- the approximate solution x = x0; % set x equal to the initial guess x0 for i = 1:n % Do n times x = x - f(x)./f1(x); % Newton’s formula end r = abs(f(x)); if r > tol warning(’The desired accuracy was not attained’) end 11
Page 1 and 2: Introduction to Numerical Math and
Page 3 and 4: CONTENTS iii Lab 17. Integration: L
Page 5 and 6: Part I Matlab and Solving Equations
Page 7 and 8: 3 > plot(x,y,’*’) > plot(x,y,
Page 9 and 10: Lab 2 Matlab Programs - Input/Outpu
Page 11 and 12: 7 % mygraphs % plots the graphs of
Page 13: 9 The simplest way to accomplish th
Page 17 and 18: 13 Exercises 4.1 In Calculus we lea
Page 19 and 20: 15 method and returns not only the
Page 21 and 22: 17 function x = mysecant(f,x0,x1,n)
Page 23 and 24: Lab 7 Symbolic Computations The foc
Page 25 and 26: 21 Other useful symbolic operations
Page 27 and 28: 23 Secant methods are better for ex
Page 29 and 30: Math 344 Sample Test Problems from
Page 31 and 32: Part II Linear Algebra c○Copyrigh
Page 33 and 34: 29 > A + A You can even add a numbe
Page 35 and 36: 31 Exercises 8.1 By hand, calculate
Page 37 and 38: 33 Figure 9.1: An equilateral truss
Page 39 and 40: 35 > b = [4 15 -10]’ > x = A\b Ne
Page 41 and 42: 37 On the other hand, a matrix that
Page 43 and 44: Lab 11 Accuracy, Condition Numbers
Page 45 and 46: 41 sensitivity is measured by the c
Page 47 and 48: Lab 12 Eigenvalues and Eigenvectors
Page 49 and 50: 45 Larger Matrices For a n × n mat
Page 51 and 52: 47 Likely States of a Markov Matrix
Page 53 and 54: Review of Part II Methods and Formu
Page 55 and 56: Part III Functions and Data c○Cop
Page 57 and 58: 53 When we tried a lower degree pol
Page 59 and 60: Lab 15 Spline Interpolation A chall
Page 61 and 62: Lab 16 Least Squares Interpolation:
Page 63 and 64: 59 Note that whenever you select a
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61 ✻ ✟ ✟ ✟✟✟✟✟✟
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63 efficient, first use x = linspac
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65 ✻ ✟ ✉ ✟ ✟✟✟✟✟
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67 In Matlab there is a built-in co
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69 For another example of how meshg
Page 75 and 76:
Review of Part III Methods and Form
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Part IV Appendices c○Copyright, T
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Appendix B Glossary of Matlab Comma
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77 limit Finds two-sided limit, if
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79 > 2*A ..........................
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Bibliography [1] Ayyup and McCuen,
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