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

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34 CHAPTER 3. SOLVING LINEAR SYSTEMS 3.3.2 Using LU Factorizations We see that the G.E. algorithm can be used to actually calculate the LU factorization. We will look at this in more detail in another example. We now examine how we can use the LU factorization to solve the equation Ax = b, Since we have A = LU, we first solve then solve Lz = b, Ux = z. Since L is lower triangular, we can solve for z with a forward substitution. Similarly, since U is upper triangular, we can solve for x with a back substitution. We drag out the previous example (which we never got around to solving): ⎛ ⎜ ⎝ We had found the LU factorization of A as A = ⎡ ⎢ ⎣ 2 1 1 3 7 4 4 0 7 11 6 5 4 17 31 2 −1 0 7 15 1 0 0 0 2 1 0 0 3 1 1 0 1 −1 −1 1 ⎤ ⎡ ⎥ ⎢ ⎦ ⎣ ⎞ ⎟ ⎠ 2 1 1 3 0 2 −2 1 0 0 3 7 0 0 0 12 ⎤ ⎥ ⎦ So we solve We get Now we solve We get the ugly solution ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ 1 0 0 0 2 1 0 0 3 1 1 0 1 −1 −1 1 z = ⎡ ⎢ ⎣ 2 1 1 3 0 2 −2 1 0 0 3 7 0 0 0 12 z = ⎡ ⎢ ⎣ ⎤ ⎡ ⎥ ⎦ z = ⎢ ⎣ 7 −3 13 18 ⎤ ⎥ ⎦ ⎤ ⎡ ⎥ ⎦ x = ⎢ ⎣ 37 24 −17 12 5 6 3 2 ⎤ ⎥ ⎦ 7 11 31 15 7 −3 13 18 ⎤ ⎥ ⎦ ⎤ ⎥ ⎦
3.4. ITERATIVE SOLUTIONS 35 3.3.3 Some Theory We aren’t doing much proving here. The following theorem has an ugly proof in the Cheney & Kincaid [7]. Theorem 3.3. If A is an n × n matrix, and naïve Gaussian Elimination does not encounter a zero pivot, then the algorithm generates a LU factorization of A, where L is the lower triangular part of the output matrix, and U is the upper triangular part. This theorem relies on us using the fancy version of G.E., which saves the multipliers in the spots where there should be zeros. If correctly implemented, then, L is the lower triangular part but with ones put on the diagonal. This theorem is proved in Cheney & Kincaid [7]. This appears to me to be a case of something which can be better illustrated with an example or two and some informal investigation. The proof is an unillustrating index-chase–read it at your own risk. 3.3.4 Computing Inverses We consider finding the inverse of A. Since AA −1 = I, then the j th column of the inverse A −1 solves the equation Ax = e j , where e j is the column matrix of all zeros, but with a one in the j th position. Thus we can find the inverse of A by running n linear solves. Obviously we are only going to run G.E. once, to put the matrix in LU form, then run n solves using forward and backward substitutions. 3.4 Iterative Solutions Recall we are trying to solve Ax = b. We examine the computational cost of Gaussian Elimination to motivate the search for an alternative algorithm. 3.4.1 An Operation Count for Gaussian Elimination We consider the number of floating point operations (“flops”) required to solve the system Ax = b. Gaussian Elimnation first uses row operations to transform the problem into an equivalent problem of the form Ux = b ′ , where U is upper triangular. Then back substitution is used to solve for x. First we look at how many floating point operations are required to reduce ⎛ ⎞ a 11 a 12 a 13 · · · a 1n b 1 a 21 a 22 a 23 · · · a 2n b 2 a 31 a 32 a 33 · · · a 3n b 3 ⎜ ⎝ . . . . .. ⎟ . ⎠ a n1 a n2 a n3 · · · a nn b n
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 and 16: 1.3. EIGENVALUES 7 Example Problem
Page 17 and 18: 1.3. EIGENVALUES 9 for all vectors
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: 3.3. LU FACTORIZATION 33 We pivot o
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
Page 87 and 88: frag replacements 6.1. FIRST AND SE
Page 89 and 90: 6.2. (NATURAL) CUBIC SPLINES 81 pro
Page 91 and 92: 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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