Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

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140 CHAPTER 10. ORDINARY DIFFERENTIAL EQUATIONS for any r ≥ 0, and t 0 ∈ (0, 2π] . Given an initial value, the parameters r and t 0 are uniquely determined. Thus the trajectory of X over time is that of an ellipse in the plane. 2 Thus the family of such ellipses, taking all r ≥ 0, form the phase curves of this ODE. We would expect the approximation of an ODE to never cross a phase curve. This is because the phase curve represents the Euler’s Method and the second order Runge-Kutta Method were used to approximate the ODE with initial value [ ] 1.5 X (0) = . 1.5 Euler’s Method was used for step size h = 0.005 for 3000 steps. The Runge-Kutta Method was employed with step size h = 0.015 for 3000 steps. The approximations are shown in Figure 10.7. Given that the actual solution of this initial value problem is an ellipse, we see that Euler’s Method performed rather poorly, spiralling out from the ellipse. This must be the case for any step size, since Euler’s Method steps in the direction tangent to the actual solution; thus every step of Euler’s Method puts the approximation on an ellipse of larger radius. Smaller stepsize minimizes this effect, but at greater computational cost. The Runge-Kutta Method performs much better, and for larger step size. At the given resolution, no spiralling is observable. Thus the Runge-Kutta Method outperforms Euler’s Method, and at lesser computational cost. 3 2 In the case r = 0, the ellipse is the “trivial” ellipse which consists of a point. 3 Because the Runge-Kutta Method of order two requires two evaluations of F , whereas Euler’s Method requires only one, per step, it is expected that they would be ‘evenly matched’ when Runge-Kutta Method uses a step size twice that of Euler’s Method.
10.3. SYSTEMS OF ODES 141 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 (a) Euler’s Method 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 -0.5 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 (b) Runge-Kutta Method Figure 10.7: The simulated phase curves of the system of Example 1<strong>0.1</strong>0 are shown for (a) Euler’s Method and (b) the Runge-Kutta Method of order 2. The Runge-Kutta Method used larger step size, but was more accurate. The actual solution to the ODE is an ellipse, thus the “spiralling” observed for Euler’s Method is spurious.
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Numerical Methods Course Notes Vers
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Contents Acknowledgments i 1 Introd
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CONTENTS v 8 Integrals and Quadratu
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Chapter 1 Introduction 1.1 Taylor
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1.1. TAYLOR’S THEOREM 3 with the
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1.3. EIGENVALUES 5 To correct this
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1.3. EIGENVALUES 7 Example Problem
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1.3. EIGENVALUES 9 for all vectors
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Chapter 2 A “Crash” Course in o
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2.1. GETTING STARTED 13 77 22 333 o
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2.2. USEFUL COMMANDS 15 octave:22>
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2.3. PROGRAMMING AND CONTROL 17 x1
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2.4. PLOTTING 19 %just plot Y plot(
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2.4. PLOTTING 21 Exercises (2.1) Wh
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Chapter 3 Solving Linear Systems A
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3.1. GAUSSIAN ELIMINATION WITH NAÏ
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3.2. PIVOTING STRATEGIES FOR GAUSSI
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3.2. PIVOTING STRATEGIES FOR GAUSSI
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3.3. LU FACTORIZATION 31 appropriat
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3.3. LU FACTORIZATION 33 We pivot o
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3.4. ITERATIVE SOLUTIONS 35 3.3.3 S
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3.4. ITERATIVE SOLUTIONS 37 3.4.2 D
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3.4. ITERATIVE SOLUTIONS 39 We star
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3.4. ITERATIVE SOLUTIONS 41 Now not
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3.4. ITERATIVE SOLUTIONS 43 Remembe
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3.4. ITERATIVE SOLUTIONS 45 is the
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Chapter 4 Finding Roots 4.1 Bisecti
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4.2. NEWTON’S METHOD 49 Algorithm
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4.2. NEWTON’S METHOD 51 4.2.2 Pro
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4.3. SECANT METHOD 53 The following
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4.3. SECANT METHOD 55 Since the roo
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4.3. SECANT METHOD 57 relation betw
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4.3. SECANT METHOD 59 (b) f(x) = x
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Chapter 5 Interpolation 5.1 Polynom
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5.1. POLYNOMIAL INTERPOLATION 63 2
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5.1. POLYNOMIAL INTERPOLATION 65 Su
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5.2. ERRORS IN POLYNOMIAL INTERPOLA
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frag replacements 5.2. ERRORS IN PO
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Chapter 6 Spline Interpolation Spli
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frag replacements 6.1. FIRST AND SE
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6.2. (NATURAL) CUBIC SPLINES 81 pro
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6.3. B SPLINES 83 The zero degree B
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6.3. B SPLINES 85 Exercises (6.1) I
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Chapter 7 Approximating Derivatives
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Page 105 and 106: 8.1. THE DEFINITE INTEGRAL 97 Examp
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Page 163 and 164: Appendix B GNU Free Documentation L
Page 165 and 166: 157 F. Include, immediately after t
Page 167 and 168: Bibliography [1] Guillaume Bal. Lec
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Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

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