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

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78 CHAPTER 6. SPLINE INTERPOLATION 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 1 Figure 6.1: A linear spline. The spline is piecewise linear, and is linear between each knot. 6.1.1 First Degree Spline Accuracy As with polynomial functions, splines are used to interpolate tabulated data as well as functions. In the latter case, if the spline is being used to interpolate the function f, say, then this is equivalent to interpolating the data t t 0 t 1 . . . t n y f(t 0 ) f(t 1 ) . . . f(t n ) A function and its linear spline interpolant are shown in Figure 6.2. The spline interpolant in that figure is fairly close to the function over some of the interval in question, but it also deviates greatly from the function at other points of the interval. We are interested in finding bounds on the possible error between a function and its spline interpolant. To find the error bound, we will consider the error on a single interval of the partition, and use a little calculus. 1 Suppose p(t) is the linear polynomial interpolating f(t) at the endpoints of the subinterval [t i , t i+1 ], then for t ∈ [t i , t i+1 ] , |f(t) − p(t)| ≤ max {|f(t) − f(t i )| , |f(t) − f(t i+1 )|} . That is, |f(t) − p(t)| is no larger than the “maximum variation” of f(t) on this interval. In particular, if f ′ (t) exists and is bounded by M 1 on [t i , t i+1 ], then |f(t) − p(t)| ≤ M 1 2 (t i+1 − t i ) . Similarly, if f ′′ (t) exists and is bounded by M 2 on [t i , t i+1 ], then |f(t) − p(t)| ≤ M 2 8 (t i+1 − t i ) 2 . Over a given partition, these become, respectively 1 Although we could directly claim Theorem 5.8, it is a bit of overkill.
frag replacements 6.1. FIRST AND SECOND DEGREE SPLINES 79 6 function spline interpolant 5.5 5 4.5 4 3.5 3 2.5 2 0 0.2 0.4 0.6 0.8 1 Figure 6.2: The function f(t) = 4.1 + sin (1/(0.08t + 0.5)) is shown, along with the linear spline interpolant, for the knots 0, <strong>0.1</strong>, 0.4, 0.6, 0.75, 0.9, 1.0 For some values of t, the function and its interpolant are very close, while they vary greatly in the middle of the interval. |f(t) − p(t)| ≤ M 1 2 max 0≤i
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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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Page 39 and 40: 3.3. LU FACTORIZATION 31 appropriat
Page 41 and 42: 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: Chapter 6 Spline Interpolation Spli
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 and 94: 6.3. B SPLINES 85 Exercises (6.1) I
Page 95 and 96: Chapter 7 Approximating Derivatives
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
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Page 131 and 132: Chapter 10 Ordinary Differential Eq
Page 133 and 134: 10.1. ELEMENTARY METHODS 125 value
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10.1. ELEMENTARY METHODS 129 It tur
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10.2. RUNGE-KUTTA METHODS 131 Examp
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10.2. RUNGE-KUTTA METHODS 133 We no
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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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