Introduction to Numerical Methods and Matlab Programming for Engineers

Introduction to Numerical Methods and Matlab Programming for Engineers
Todd Young and Martin J. Mohlenkamp
Department of Mathematics
Ohio University
Athens, OH 45701
Contents
Preface iii
I Matlab and Solving Equations 1
Lecture 1. Vectors, Functions, and Plots in Matlab 2
Lecture 2. Matlab Programs 6
Lecture 3. Newton’s Method and Loops 10
Lecture 4. Controlling Error and Conditional Statements 14
Lecture 5. The Bisection Method and Locating Roots 18
Lecture 6. Secant Methods 22
Lecture 7. Symbolic Computations 25
Review of Part I 29
II Linear Algebra 33
Lecture 8. Matrices and Matrix Operations in Matlab 34
Lecture 9. Introduction to Linear Systems 39
Lecture 10. Some Facts About Linear Systems 43
Lecture 11. Accuracy, Condition Numbers and Pivoting 46
Lecture 12. LU Decomposition 50
Lecture 13. Nonlinear Systems – Newton’s Method 53
Lecture 14. Eigenvalues and Eigenvectors 57
Lecture 15. Vibrational Modes and Frequencies 60
Lecture 16. Numerical Methods for Eigenvalues 62
vvi CONTENTS
Lecture 17. The QR Method* 66
Lecture 18. Iterative solution of linear systems* 68
Review of Part II 69
III Functions and Data 73
Lecture 19. Polynomial and Spline Interpolation 74
Lecture 20. Least Squares Fitting: Noisy Data 78
Lecture 21. Integration: Left, Right and Trapezoid Rules 82
Lecture 22. Integration: Midpoint and Simpson’s Rules 87
Lecture 23. Plotting Functions of Two Variables 91
Lecture 24. Double Integrals for Rectangles 94
Lecture 25. Double Integrals for Non-rectangles 98
Lecture 26. Gaussian Quadrature* 101
Lecture 27. Numerical Differentiation 102
Lecture 28. The Main Sources of Error 106
Review of Part III 109
IV Differential Equations 115
Lecture 29. Reduction of Higher Order Equations to Systems 116
Lecture 30. Euler Methods 120
Lecture 31. Higher Order Methods 124
Lecture 32. Multi-step Methods* 127
Lecture 33. ODE Boundary Value Problems and Finite Differences 128
Lecture 34. Finite Difference Method { Nonlinear ODE 132
Lecture 35. Parabolic PDEs – Explicit Method 135
Lecture 36. Solution Instability for the Explicit Method 140CONTENTS vii
Lecture 37. Implicit Methods 143
Lecture 38. Insulated Boundary Conditions 147
Lecture 39. Finite Difference Method for Elliptic PDEs 152
Lecture 40. Convection-Diffusion Equations* 155
Lecture 41. Finite Elements 156
Lecture 42. Determining Internal Node Values 160
Review of Part IV 163
V Appendices 167
Lecture A. Glossary of Matlab Commands
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