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

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14 CHAPTER 2. A “CRASH” COURSE IN OCTAVE/MATLAB octave:16> P = L .* M P = 1.1669e+00 4.0557e+00 0.0000e+00 1.1119e+01 1.8499e+01 0.0000e+00 0.0000e+00 0.0000e+00 1.1120e+05 octave:17> x = M \ b x = -6.0000000 5.5000000 0.0090090 octave:18> err = M * x - b err = 0 0 0 Note the difference between L * M and L .* M; the former is matrix multiplication, the latter is element by element multiplication, i.e., (L .∗ M) i,j = L i,j M i,j . The command rand(m,n) gives an m × n matrix with each element “uniformly distributed” on [0, 1]. In line 17 we asked octave to solve the linear system by setting Mx = b, x = M\b = M −1 b. Note that you can construct matrices directly as you did vectors: octave:19> B = [1 3 4 5;2 -2 2 -2] B = 1 3 4 5 2 -2 2 -2 You can also create row vectors as a sequence, either using the form c:d or the form c:e:d, which give, respectively, c, c + 1, . . . , d, and c, c + e, . . . , d, (or something like it if e does not divide d − c) as follows: octave:20> z = 1:5 z = 1 2 3 4 5 octave:21> z = 5:(-1):1 z = 5 4 3 2 1
2.2. USEFUL COMMANDS 15 octave:22> z = 5:(-2):1 z = 5 3 1 octave:23> z = 2:3:11 z = 2 5 8 11 octave:24> z = 2:3:10 z = 2 5 8 2.2 Useful Commands Here’s a none too complete listing of useful commands in octave: • help is the most useful command. • floor(X) returns the largest integer not greater than X. If X is a vector or matrix, it computes the floor element-wise. This behavior is common in octave: many functions which we normally think of as applicable to scalars can be applied to matrices, with the result computed elementwise. • ceil(X) returns the smallest integer not less than X, computed element-wise. • sin(X), cos(X), tan(X), atan(X), sqrt(X), returns the sine, cosine, tangent, arctangent, square root of X, computed elementwise. • exp(X) returns e X , elementwise. • abs(X) returns |X| , elementwise. • norm(X) returns the norm of X; if X is a vector, this is the L 2 norm: ‖X‖ 2 = ( ∑ i X 2 i ) 1/2 , if X is a matrix, it is the matrix norm subordinate to the L 2 norm. You can compute other norms with norm(X,p) where p is a number, to get the L p norm, or with p one of Inf, -Inf, etc. • zeros(m,n) returns an m × n matrix of all zeros. • eye(m) returns the m × m identity matrix. • [m,n] = size(A) returns the number of rows, columns of A. • length(v) returns the length of vector v, or the larger dimension if v is a matrix. • diag(v) returns the diagonal matrix with vector v as diagonal. diag(M) returns as a vector, the diagonal of matrix v. Thus diag(diag(v)) is v for vector v, but diag(diag(M)) is the diagonal part of matrix M. • toeplitz(v) returns the Toeplitz matrix associated with vector v. That is ⎡ toeplitz(v) = ⎢ ⎣ v(1) v(2) v(3) · · · v(n) v(2) v(1) v(2) · · · v(n − 1) v(3) v(2) v(1) · · · v(n − 2) . . . . .. . v(n) v(n − 1) v(n − 2) · · · v(1) ⎤ ⎥ ⎦
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: 2.1. GETTING STARTED 13 77 22 333 o
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
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
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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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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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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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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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