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

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

4 CHAPTER 1. INTRODUCTION 1.2 Loss of Significance Generally speaking, a computer stores a number x as a mantissa and exponent, that is x = ±r×10 k , where r is a rational number of a given number of digits in [0.1, 1), and k is an integer in a certain range. The number of significant digits in r is usually determined by the user’s input. Operations on numbers stored in this way follow a “lowest common denominator” type of rule, i.e., precision cannot be gained but can be lost. Thus for example if you add the two quantities 0.171717 and 0.51, then the result should only have two significant digits; the precision of the first measurement is lost in the uncertainty of the second. This is as it should be. However, a loss of significance can be incurred if two nearly equal quantities are subtracted from one another. Thus if I were to direct my computer to subtract 0.177241 from 0.177589, the result would be .348 ×10 −3 , and three significant digits have been lost. This loss is called subtractive cancellation, and can often be avoided by rewriting the expression. This will be made clearer by the examples below. Errors can also occur when quantities of radically different magnitudes are summed. For example 0.1234 + 5.6789 × 10 −20 might be rounded to 0.1234 by a system that keeps only 16 significant digits. This may lead to unexpected results. The usual strategies for rewriting subtractive expressions are completing the square, factoring, or using the Taylor expansions, as the following examples illustrate. Example Problem 1.8. Consider the stability of √ x + 1 − 1 when x is near 0. Rewrite the expression to rid it of subtractive cancellation. Solution: Suppose that x = 1.2345678 × 10 −5 . Then √ x + 1 ≈ 1.000006173. If your computer (or calculator) can only keep 8 significant digits, this will be rounded to 1.0000062. When 1 is subtracted, the result is 6.2 × 10 −6 . Thus 6 significant digits have been lost from the original. To fix this, we rationalize the expression √ x + 1 − 1 = √ x + 1 − 1 √ x + 1 + 1 √ x + 1 + 1 = x + 1 − 1 √ x + 1 + 1 = x √ x + 1 + 1 . This expression has no subtractions, and so is not subject to subtractive cancelling. When x = 1.2345678 × 10 −5 , this expression evaluates approximately as 1.2345678 × 10 −5 2.0000062 on a machine with 8 digits, there is no loss of precision. ≈ 6.17281995 × 10 −6 ⊣ Note that nearly all modern computers and calculators store intermediate results of calculations in higher precision formats. This minimizes, but does not eliminate, problems like those of the previous example problem. Example Problem 1.9. Write stable code to find the roots of the equation x 2 + bx + c = 0. Solution: The usual quadratic formula is x ± = −b ± √ b 2 − 4c 2 Supposing that b ≫ c > 0, the expression in the square root might be rounded to b 2 , giving two roots x + = 0, x − = −b. The latter root is nearly correct, while the former has no correct digits.
1.3. EIGENVALUES 5 To correct this problem, multiply the numerator and denominator of x + by −b − √ b 2 − 4c to get x + = 2c −b − √ b 2 − 4c Now if b ≫ c > 0, this expression gives root x + = −c/b, which is nearly correct. This leads to the pair: x − = −b − √ b 2 − 4c 2c , x + = 2 −b − √ b 2 − 4c Note that the two roots are nearly reciprocals, and if x − is computed, x + can easily be computed with little additional work. ⊣ Example Problem 1.10. Rewrite e x − cos x to be stable when x is near 0. Solution: Look at the Taylor’s Series expansion for these functions: e x − cos x = ] ] [1 + x + x2 2! + x3 3! + x4 4! + x5 5! + . . . − [1 − x2 2! + x4 4! − x6 6! + . . . = x + x 2 + x3 3! + O ( x 5) This expression has no subtractions, and so is not subject to subtractive cancelling. Note that we propose calculating x + x 2 + x 3 /6 as an approximation of e x − cos x, which we cannot calculate exactly anyway. Since we assume x is nearly zero, the approximate should be good. If x is very close to zero, we may only have to take the first one or two terms. If x is not so close to zero, we may need to take all three terms, or even more terms of the expansion; if x is far from zero we should use some other technique. ⊣ 1.3 Eigenvalues It is assumed the reader has some familiarity with linear algebra. We review the topic of eigenvalues. Definition 1.11. A nonzero vector x is an eigenvector of a given matrix A, with corresponding eigenvalue λ if Ax = λx Subtracting the right hand side from the left and gathering terms gives (A − λI) x = 0. Since x is assumed to be nonzero, the matrix A − λI must be singular. A matrix is singular if and only if its determinant is zero. These steps are reversible, thus we claim λ is an eigenvalue if and only if det (A − λI) = 0. The left hand side can be expanded to a polynomial in λ, of degree n where A is an n×n matrix. This gives the so-called characteristic equation. Sometimes eigenvectors,-values are called characteristic vectors,-values.
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: 1.1. TAYLOR’S THEOREM 3 with the
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
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Page 39 and 40: 3.3. LU FACTORIZATION 31 appropriat
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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
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Page 51 and 52: 3.4. ITERATIVE SOLUTIONS 43 Remembe
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Page 55 and 56: Chapter 4 Finding Roots 4.1 Bisecti
Page 57 and 58: 4.2. NEWTON’S METHOD 49 Algorithm
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Page 61 and 62: 4.3. SECANT METHOD 53 The following
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4.3. SECANT METHOD 55 Since the roo
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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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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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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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