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

More documents

Recommendations

Info

$Fractional and operational calculus with generalized fractional ...$

154 APPENDIX A. OLD EXAMS P3 (20 pnts) Let F = { f(x) = c 0 √ x + c1 l(x) + c 2 e x | c 0 , c 1 , c 2 ∈ R } , where l(x) is some black box function. We are concerned with finding the least-squares best f ∈ F to approximate the data x 0 x 1 . . . x n y 0 y 1 . . . y n (a) Set up (but do not attempt to solve) the normal equations to find the the vector c = [c 0 c 1 c 2 ] ⊤ , which determines the least squares approximant in F. (b) Now suppose that the function l(x) happens to be the Lagrange Polynomial associated with x 7 , among the values x 0 , x 1 , . . . , x n . That is, l(x j ) = δ j7 . Prove that f(x 7 ) = y 7 , where f is the least squares best approximant to the data. P4 (10 pnts) State some substantive question which you thought might appear on this exam, but did not. Answer this question (correctly).
Appendix B GNU Free Documentation License <strong>Version</strong> 1.2, November 2002 Copyright c○2000,2001,2002 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. Preamble The purpose of this License is to make a manual, textbook, or other functional and useful document ”free” in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of ”copyleft”, which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference. 1. APPLICABILITY AND DEFINITIONS This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The ”Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as ”you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. A ”Modified <strong>Version</strong>” of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language. A ”Secondary Section” is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document’s overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them. The ”Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none. The ”Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words. A ”Transparent” copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not ”Transparent” is called ”Opaque”. 155
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 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 79 and 80:
Page 81 and 82:
Page 83 and 84:
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
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
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
Page 145 and 146: 10.3. SYSTEMS OF ODES 137 Euler’s
Page 147 and 148: 10.3. SYSTEMS OF ODES 139 have to b
Page 149 and 150: 10.3. SYSTEMS OF ODES 141 4.5 4 3.5
Page 151 and 152: 10.3. SYSTEMS OF ODES 143 (10.9) Im
Page 153 and 154: Appendix A Old Exams A.1 First Midt
Page 155 and 156: A.3. FINAL EXAM, FALL 2003 147 •
Page 157 and 158: A.4. FIRST MIDTERM, FALL 2004 149
Page 159 and 160: A.5. SECOND MIDTERM, FALL 2004 151
Page 161: A.6. FINAL EXAM, FALL 2004 153 P5 (
Page 165 and 166: 157 F. Include, immediately after t
Page 167 and 168: Bibliography [1] Guillaume Bal. Lec
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?