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

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18 LAB 6. SECANT METHODS formula for f ′ (x) even though one knows the formula for f(x). This is often the case when f(x) is not defined explicitly, but implicitly. There are other situations, which are very common in engineering and science, where even a formula for f(x) is not known. This happens when f(x) is the result of experiment or simulation rather than a formula. In such situations, the secant method is usually the best choice. Exercises 6.1 Use the program mysecant on f(x) = x 3 − 4. Compare the results with those of Exercises 3.3 and 5.3. Then modify the program to accept a tolerance using a while loop. Test the program and turn in the program as well as the work above. 6.2 Write a program myregfalsi based on mybisect. Use it on f(x) = x 3 − 4. Compare the results with those of Exercises 3.3 and 5.3.
Lab 7 Symbolic Computations The focus of this course is on numerical computations, i.e. calculations, usually approximations, with floating point numbers. However, Matlab can also do symbolic computations which means exact calculations using symbols as in Algebra or Calculus. You should have done some symbolic Matlab computations in your Calculus courses and in this chapter we review what you should already know. Defining functions and basic operations Before doing any symbolic computation, one must declare the variables used to be symbolic: > syms x y A function is defined by simply typing the formula: > f = cos(x) + 3*x^2 Note that coefficients must be multiplied using*. To find specific values, you must use the command subs: > subs(f,pi) This command stands for substitute, it substitutes π for x in the formula for f. This is the only way to evaluate a specific value of a symbolic function. The following command will not work: > f(pi) If we define another function: > g = exp(-y^2) then we can compose the functions: > h = compose(g,f) i.e. h(x) = g(f(x)). Since f and g are functions of different variables, their product must be a function of two variables: > k = f*g > subs(k,[x,y],[0,1]) We can do simple calculus operations, like differentiation: > f1 = diff(f) indefinite integrals (antiderivatives): > F = int(f) and definite integrals: > int(f,0,2*pi) To change a symbolic answer into a numerical answer, use the double command which stands for double precision, (not times 2): > double(ans) Note that some antiderivatives cannot be found in terms of elementary functions, for some of these 19
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 and 14: 9 The simplest way to accomplish th
Page 15 and 16: Lab 4 Controlling Error and while L
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: 17 function x = mysecant(f,x0,x1,n)
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
Page 65 and 66: 61 ✻ ✟ ✟ ✟✟✟✟✟✟
Page 67 and 68: 63 efficient, first use x = linspac
Page 69 and 70: 65 ✻ ✟ ✉ ✟ ✟✟✟✟✟
Page 71 and 72: 67 In Matlab there is a built-in co
Page 73 and 74:
69 For another example of how meshg
Page 75 and 76:
Review of Part III Methods and Form
Page 77 and 78:
Part IV Appendices c○Copyright, T
Page 79 and 80:
Appendix B Glossary of Matlab Comma
Page 81 and 82:
77 limit Finds two-sided limit, if
Page 83 and 84:
79 > 2*A ..........................
Page 85:
Bibliography [1] Ayyup and McCuen,
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