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

More documents

Recommendations

Info

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

112 CHAPTER 8. INTEGRALS AND QUADRATURE (8.10) Determine a “quadrature” rule of the form ∫ 1 −1 f(x) dx ≈ Af(0) + Bf ′ (−1) + Cf ′ (1) that is exact for polynomials of highest possible degree. What is the highest degree polynomial for which this rule is exact? (Since this rule uses derivatives of f, it does not exactly fit our definition of a quadrature rule, but it may be applicable in some situations.) (8.11) Determine a “quadrature” rule of the form ∫ 1 0 f(x) dx ≈ Af(0) + Bf ′ (0) + Cf(1) that is exact for polynomials of highest possible degree. What is the highest degree polynomial for which this rule is exact? (8.12) Consider the so-called order n Chebyshev Quadrature rule: ∫ 1 −1 f(x) dx ≈ c n n∑ f(x i ) Find the weighting c n and nodes x i for the case n = 2 and the case n = 3. For what order polynomials are these rules exact? (8.13) Find the Gaussian Quadrature rule with 2 nodes for the interval [1, 5], i.e., find a rule ∫ 5 1 i=0 f(x) dx ≈ Af(x 0 ) + Bf(x 1 ) Before you solve the problem, consider the following questions: do you expect the nodes to be the endpoints 1 and 5? do you expect the nodes to be arranged symmetrically around the midpoint of the interval? (8.14) Find the Gaussian Quadrature rule with 3 nodes for the interval [−1, 1], i.e., find a rule ∫ 1 −1 f(x) dx ≈ Af(x 0 ) + Bf(x 1 ) + Cf(x 2 ) To find the nodes x 0 , x 1 , x 2 you will have to find the zeroes of a cubic equation, which could be difficult. However, you may use the simplifying assumption that the nodes are symmetrically placed in the interval [−1, 1]. (8.15) Write code to approximate the integral of a f on [a, b] by the composite trapezoidal rule on n equal subintervals. Your m-file should have header line like: function iappx = trapezoidal(f,a,b,n) You may wish to use the code: x = a .+ (b-a) .* (0:n) ./ n; If f is defined to work on vectors element-wise, you can probably speed up your computation by using bigsum = 0.5 * ( f(x(1)) + f(x(n+1)) ) + sum( f(x(2:(n))) ); (8.16) Write code to implement the Gaussian Quadrature rule for n = 2 to integrate f on the interval [a, b]. Your m-file should have header line like: function iappx = gauss2(f,a,b) (8.17) Write code to implement composite Gaussian Quadrature based on code from the previous problem. Something like the following probably works:
8.4. GAUSSIAN QUADRATURE 113 function iappx = gaussComp(f,a,b,n) % code to approximate integral of f over n equal subintervals of [a,b] x = a .+ (b-a) .* (0:n) ./ n; iappx = 0; for i=1:n iappx += gauss2(f,x(i),x(i+1)); end Use your code to approximate the error function: erf(z) = 2 √ π ∫ z 0 e −t2 dt. Compare your results with the octave/Matlab builtin function erf. (Try help erf in octave, or see http://mathworld.wolfram.com/Erf.html) The error function is used in probability. In particular, the probability that a normal random variable is within z standard deviations from its mean is erf(z/ √ 2) Thus erf(1/ √ 2) ≈ 0.683, and erf(2/ √ 2) ≈ 0.955. These numbers should look familiar to you.
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 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: 8.4. GAUSSIAN QUADRATURE 111 Exerci
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 and 162: A.6. FINAL EXAM, FALL 2004 153 P5 (
Page 163 and 164: Appendix B GNU Free Documentation L
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 ...

Create successful ePaper yourself

Delete template?

Save as template?