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

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86 CHAPTER 6. SPLINE INTERPOLATION (6.9) Find the natural cubic spline that interpolates the data x 0 1 3 y 4 2 7 It may help to assume your answer has the form { Ax 3 + Bx 2 + Cx + 4 : 0 ≤ x < 1 S(x) = D(x − 1) 3 + E(x − 1) 2 + F (x − 1) + 2 : 1 ≤ x ≤ 3
Chapter 7 Approximating Derivatives 7.1 Finite Differences Suppose we have some blackbox function f(x) and we wish to calculate f ′ (x) at some given x. Not surprisingly, we start with Taylor’s theorem: f(x + h) = f(x) + f ′ (x)h + f ′′ (ξ)h 2 . 2 Rearranging we get f ′ f(x + h) − f(x) (x) = − f ′′ (ξ)h . h 2 Remember that ξ is between x and x + h, but its exact value is not known. If we wish to calculate f ′ (x), we cannot evaluate f ′′ (ξ), so we approximate the derivative by dropping the last term. That is, we calculate [f(x + h) − f(x)] /h as an approximation 1 to f ′ (x). In so doing, we have dropped the last term. If there is a finite bound on f ′′ (z) on the interval in question then the dropped term is bounded by a constant times h. That is, f ′ (x) = f(x + h) − f(x) h + O (h) (7.1) The error that we incur when we approximate f ′ (x) by calculating [f(x + h) − f(x)] /h is called truncation error. It has nothing to do with the kind of error that you get when you do calculations with a computer with limited precision; even if you worked in infinite precision, you would still have truncation error. The truncation error can be made small by making h small. However, as h gets smaller, precision will be lost in equation 7.1 due to subtractive cancellation. The error in calculation for small h is called roundoff error. Generally the roundoff error will increase as h decreases. Thus there is a nonzero h for which the sum of these two errors is minimized. See Example Problem 7.5 for an example of this. The truncation error for this approximation is O (h). We may want a more precise approximation. By now, you should know that any calculation starts with Taylor’s Theorem: f(x + h) = f(x) + f ′ (x)h + f ′′ (x) 2 f(x − h) = f(x) − f ′ (x)h + f ′′ (x) 2 h 2 + f ′′′ (ξ 1 ) h 3 3! h 2 − f ′′′ (ξ 2 ) h 3 3! 1 This approximation for f ′ (x) should remind you of the definition of f ′ (x) as a limit. 87
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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
Page 43 and 44: 3.4. ITERATIVE SOLUTIONS 35 3.3.3 S
Page 45 and 46: 3.4. ITERATIVE SOLUTIONS 37 3.4.2 D
Page 47 and 48: 3.4. ITERATIVE SOLUTIONS 39 We star
Page 49 and 50: 3.4. ITERATIVE SOLUTIONS 41 Now not
Page 51 and 52: 3.4. ITERATIVE SOLUTIONS 43 Remembe
Page 53 and 54: 3.4. ITERATIVE SOLUTIONS 45 is the
Page 55 and 56: Chapter 4 Finding Roots 4.1 Bisecti
Page 57 and 58: 4.2. NEWTON’S METHOD 49 Algorithm
Page 59 and 60: 4.2. NEWTON’S METHOD 51 4.2.2 Pro
Page 61 and 62: 4.3. SECANT METHOD 53 The following
Page 63 and 64: 4.3. SECANT METHOD 55 Since the roo
Page 65 and 66: 4.3. SECANT METHOD 57 relation betw
Page 67 and 68: 4.3. SECANT METHOD 59 (b) f(x) = x
Page 69 and 70: Chapter 5 Interpolation 5.1 Polynom
Page 71 and 72: 5.1. POLYNOMIAL INTERPOLATION 63 2
Page 73 and 74: 5.1. POLYNOMIAL INTERPOLATION 65 Su
Page 75 and 76: 5.2. ERRORS IN POLYNOMIAL INTERPOLA
Page 77 and 78: frag replacements 5.2. ERRORS IN PO
Page 85 and 86: Chapter 6 Spline Interpolation Spli
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Page 89 and 90: 6.2. (NATURAL) CUBIC SPLINES 81 pro
Page 91 and 92: 6.3. B SPLINES 83 The zero degree B
Page 93: 6.3. B SPLINES 85 Exercises (6.1) I
Page 97 and 98: 7.2. RICHARDSON EXTRAPOLATION 89 7.
Page 99 and 100: 7.2. RICHARDSON EXTRAPOLATION 91 7.
Page 101 and 102: 7.2. RICHARDSON EXTRAPOLATION 93 Ex
Page 103 and 104: Chapter 8 Integrals and Quadrature
Page 105 and 106: 8.1. THE DEFINITE INTEGRAL 97 Examp
Page 107 and 108: 8.2. TRAPEZOIDAL RULE 99 The compos
Page 109 and 110: frag replacements 8.2. TRAPEZOIDAL
Page 111 and 112: 8.3. ROMBERG ALGORITHM 103 then def
Page 113 and 114: 8.4. GAUSSIAN QUADRATURE 105 8.4 Ga
Page 115 and 116: 8.4. GAUSSIAN QUADRATURE 107 Exampl
Page 117 and 118: 8.4. GAUSSIAN QUADRATURE 109 Simple
Page 119 and 120: 8.4. GAUSSIAN QUADRATURE 111 Exerci
Page 121 and 122: 8.4. GAUSSIAN QUADRATURE 113 functi
Page 123 and 124: Chapter 9 Least Squares 9.1 Least S
Page 125 and 126: 9.1. LEAST SQUARES 117 The answer i
Page 127 and 128: 9.2. ORTHONORMAL BASES 119 g 0 (x 0
Page 129 and 130: frag replacements 9.2. ORTHONORMAL
Page 131 and 132: Chapter 10 Ordinary Differential Eq
Page 133 and 134: 10.1. ELEMENTARY METHODS 125 value
Page 135 and 136: frag replacements 10.1. ELEMENTARY
Page 137 and 138: 10.1. ELEMENTARY METHODS 129 It tur
Page 139 and 140: 10.2. RUNGE-KUTTA METHODS 131 Examp
Page 141 and 142: 10.2. RUNGE-KUTTA METHODS 133 We no
Page 143 and 144: 10.3. SYSTEMS OF ODES 135 Example 1
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10.3. SYSTEMS OF ODES 137 Euler’s
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10.3. SYSTEMS OF ODES 139 have to b
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10.3. SYSTEMS OF ODES 141 4.5 4 3.5
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10.3. SYSTEMS OF ODES 143 (10.9) Im
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Appendix A Old Exams A.1 First Midt
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A.3. FINAL EXAM, FALL 2003 147 •
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A.4. FIRST MIDTERM, FALL 2004 149
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A.5. SECOND MIDTERM, FALL 2004 151
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A.6. FINAL EXAM, FALL 2004 153 P5 (
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Appendix B GNU Free Documentation L
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157 F. Include, immediately after t
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Bibliography [1] Guillaume Bal. Lec
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Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

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