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

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8 CHAPTER 1. INTRODUCTION Exercises (1.1) Suppose f ∈ O ( h k) . Show that f ∈ O (h m ) for any m with 0 < m < k. (Hint: Take h ∗ < 1.) Note this may appear counterintuitive, unless you remember that O ( h k) is a better approximation than O (h m ) when m < k. (1.2) Suppose f ∈ O ( h k) , and g ∈ O (h m ) . Show that fg ∈ O ( h k+m) . (1.3) Suppose f ∈ O ( h k) , and g ∈ O (h m ) , with m < k. Show that f + g ∈ O (h m ) . (1.4) Prove that f(h) = −3h 5 is in O ( h 5) . (1.5) Prove that f(h) = h 2 + 5h 17 is in O ( h 2) . (1.6) Prove that f(h) = h 3 is not in O ( h 4) (Hint: Proof by contradiction.) (1.7) Prove that sin(h) is in O (h). (1.8) Find a O ( h 3) approximation to sin h. (1.9) Find a O ( h 4) approximation to ln(1+h). Compare the approximate value to the actual when h = <strong>0.1</strong>. How does this approximation compare to the O ( h 3) approximate from Example 1.7 for h = <strong>0.1</strong>? (1.10) Suppose that f ∈ O ( h k) . Can you show that f ′ ∈ O ( h k−1) ? (1.11) Rewrite √ x + 1 − √ 1 to get rid of subtractive cancellation when x ≈ 0. (1.12) Rewrite √ x + 1 − √ x to get rid of subtractive cancellation when x is very large. (1.13) Use a Taylor’s expansion to rid the expression 1 − cos x of subtractive cancellation for x small. Use a O ( x 5) approximate. (1.14) Use a Taylor’s expansion to rid the expression 1 − cos 2 x of subtractive cancellation for x small. Use a O ( x 6) approximate. (1.15) Calculate cos(π/2 + 0.001) to within 8 decimal places by using the Taylor’s expansion. (1.16) Prove that if x is an eigenvector of A then αx is also an eigenvector of A, for the same eigenvalue. Here α is a nonzero real number. (1.17) Prove, by induction, that if λ is an eigenvalue of A then λ k is an eigenvalue of A k for integer k > 1. The base case was done in Example Problem 1.13. (1.18) Let B = ∑ k i=0 α iA i , where A 0 = I. Prove that if λ is an eigenvalue of A, then ∑ k i=0 α iλ i is an eigenvalue of B. Thus for polynomial p(x), p(λ) is an eigenvalue of p(A). (1.19) Suppose A is an invertible matrix with eigenvalue λ. Prove that λ −1 is an eigenvalue for A −1 . (1.20) Suppose that the eigenvalues of A are 1, 10, 100. Give the eigenvalues of B = 3A 3 − 4A 2 + I. Show that B is singular. (1.21) Show that if ‖x‖ 2 = r, then x is on a sphere centered at the origin of radius r, in R n . (1.22) If ‖x‖ 2 = 0, what does this say about vector x? (1.23) Letting x = [3 4 12] ⊤ , what is ‖x‖ 2 ? (1.24) What is the norm of ⎡ ⎤ 1 0 0 · · · 0 0 1/2 0 · · · 0 A = 0 0 1/3 · · · 0 ? ⎢ ⎣ . . . . .. ⎥ . ⎦ 0 0 0 · · · 1/n (1.25) Show that ‖A‖ 2 = 0 implies that A is the matrix of all zeros. (1.26) Show that ∥ ∥ A −1 ∥ ∥ 2 equals (1/|λ min |) , where λ min is the smallest, in absolute value, eigenvalue of A. (1.27) Suppose there is some µ > 0 such that, for a given A, ‖Av‖ 2 ≥ µ‖v‖ 2 ,
1.3. EIGENVALUES 9 for all vectors v. (a) Show that µ ≤ ‖A‖ 2 . (Should be very simple.) (b) Show that A is nonsingular. (Recall: A is singular if there is some x ≠ 0 such that Ax = 0.) (c) Show that ∥ ∥ A −1 ∥ ∥ 2 ≤ (1/µ) . (1.28) If A is singular, is it necessarily the case that ‖A‖ 2 = 0? (1.29) If ‖A‖ 2 ≥ µ > 0 does it follow that A is nonsingular?
Page 1: Numerical Methods Course Notes Vers
Page 5 and 6: Contents Acknowledgments i 1 Introd
Page 7 and 8: CONTENTS v 8 Integrals and Quadratu
Page 9 and 10: Chapter 1 Introduction 1.1 Taylor
Page 11 and 12: 1.1. TAYLOR’S THEOREM 3 with the
Page 13 and 14: 1.3. EIGENVALUES 5 To correct this
Page 15: 1.3. EIGENVALUES 7 Example Problem
Page 19 and 20: Chapter 2 A “Crash” Course in o
Page 21 and 22: 2.1. GETTING STARTED 13 77 22 333 o
Page 23 and 24: 2.2. USEFUL COMMANDS 15 octave:22>
Page 25 and 26: 2.3. PROGRAMMING AND CONTROL 17 x1
Page 27 and 28: 2.4. PLOTTING 19 %just plot Y plot(
Page 29 and 30: 2.4. PLOTTING 21 Exercises (2.1) Wh
Page 31 and 32: Chapter 3 Solving Linear Systems A
Page 33 and 34: 3.1. GAUSSIAN ELIMINATION WITH NAÏ
Page 35 and 36: 3.2. PIVOTING STRATEGIES FOR GAUSSI
Page 37 and 38: 3.2. PIVOTING STRATEGIES FOR GAUSSI
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
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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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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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7.2. RICHARDSON EXTRAPOLATION 89 7.
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7.2. RICHARDSON EXTRAPOLATION 91 7.
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7.2. RICHARDSON EXTRAPOLATION 93 Ex
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Chapter 8 Integrals and Quadrature
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8.1. THE DEFINITE INTEGRAL 97 Examp
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8.2. TRAPEZOIDAL RULE 99 The compos
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frag replacements 8.2. TRAPEZOIDAL
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8.3. ROMBERG ALGORITHM 103 then def
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8.4. GAUSSIAN QUADRATURE 105 8.4 Ga
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8.4. GAUSSIAN QUADRATURE 107 Exampl
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8.4. GAUSSIAN QUADRATURE 109 Simple
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8.4. GAUSSIAN QUADRATURE 111 Exerci
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8.4. GAUSSIAN QUADRATURE 113 functi
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Chapter 9 Least Squares 9.1 Least S
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9.1. LEAST SQUARES 117 The answer i
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9.2. ORTHONORMAL BASES 119 g 0 (x 0
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frag replacements 9.2. ORTHONORMAL
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Chapter 10 Ordinary Differential Eq
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10.1. ELEMENTARY METHODS 125 value
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frag replacements 10.1. ELEMENTARY
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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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Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

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