MAT 2310. Computational Mathematics Wm C Bauldry Fall, 2012 Introduction to Computational Mathematics \1 + 1 = 3 for large enough values of 1." Introduction to Computational Mathematics Table of Contents I. Computer Arithmetic.......................................1 II. Control Structures.........................................27 S I. Special Topics: Computation Cost and Horner's Form....... 59 III. Numerical Differentiation.................................. 64 IV. Root Finding Algorithms................................... 77 S II. Special Topics: Modified Newton's Method................ 100 V. Numerical Integration.................................... 103 VI. Polynomial Interpolation.................................. 125 S III. Case Study: TI Calculator Numerics.......................146 VII. Projects................................................. 158 ICM i I. Computer Arithmetic Sections 1. Scientific Notation.............................................1 2. Converting to Different Bases...................................2 3. Floating Point Numbers........................................7 4. IEEE-754 Floating Point Standard..............................9 5. Maple's Floating Point Representation......................... 16 6. Error......................................................... 18 Exercises..................................................... 25 ICM ii II. Control Structures Sections 1. Control Structures............................................ 27 2. A Common Example.......................................... 33 3. Control Structures Syntax..................................... 35 1. Excel........................................................... 35 2. Maple [Sage/Xcas]............................................. 38 3. MATLAB [FreeMat/Octave/Scilab]............................. 41 4. C and Java..................................................... 44 5. TI-84...........................................................48 6.R.............................................................. 51 4. From Code to Flow Charts.................................... 54 Exercises..................................................... 55 Reference Sheet Links......................................... 58 ICM iii S I. Special Topics: Computation Cost and Horner's Form Sections 1. Introduction.................................................. 59 2. Horner's Form................................................ 60 Exercises..................................................... 63 ICM iv III. Numerical Differentiation Sections 1. Introduction.................................................. 64 2. Taylor's Theorem............................................. 65 3. Difference Methods........................................... 68 1. Forward Differences............................................. 68 2. Backward Differences............................................69 3. Centered Differences............................................ 70 Appendix I: Taylor's Theorem..................................73 Appendix II: Centered Difference Coefficients Chart............. 74 Exercises..................................................... 75 ICM v IV. Root Finding Algorithms Sections 1. The Bisection Method.........................................77 2. Newton-Raphson Method..................................... 81 3. Secant Method............................................... 85 4. Regula Falsi.................................................. 89 Appendix III: Rate of Convergence............................. 96 Exercises..................................................... 97 Links and Others..............................................99 ICM vi S II. Special Topics: Modified Newton's Method Sections 1. Introduction................................................. 100 2. Modified Newton's Method...................................101 Exercises.................................................... 102 ICM vii V. Numerical Integration Sections 1. Numerical Integration........................................ 103 2. Left Endpoint, Right Endpoint, and Midpoint Sums............104 3. Trapezoid Sums............................................. 105 4. Simpson's Rule.............................................. 107 5. Gaussian Quadrature......................................... 111 6. Gauss-Kronrod Quadrature................................... 116 7. A Menagerie of Test Integrals................................ 120 Appendix IV: Legendre & Stieltjes Polynomials for GK7;15 ...... 122 Exercises.................................................... 123 Links and Others............................................ 124 ICM viii VI. Polynomial Interpolation Sections 1. Polynomial Interpolation..................................... 125 2. Lagrange Interpolation....................................... 127 3. Interlude: Bernstein Polynomials.............................. 133 4. Newton Interpolation........................................ 136 5. Two Comparisons............................................ 140 6. Interlude: Splines............................................ 142 Exercises.................................................... 143 Links and Others............................................ 145 ICM ix S III. Case Study: TI Calculator Numerics Sections 1. Introduction................................................. 146 2. Floating Point Structure..................................... 148 3. Numeric Derivatives..........................................150 4. Numerically Finding Roots.................................... 151 5. Numeric Quadrature......................................... 152 6. Transcendental Functions.................................... 153 Appendix V:TI's Solving Algorithm........................... 154 Exercises.................................................... 155 ICM x VII. Projects The Project List One Function For All........................................ 158 • I. Control Structures • A Bit Adder in Excel......................................... 159 • The Collatz Conjecture....................................... 160 • The CORDIC Algorithm...................................... 162 • The Cost of Computing a Determinant........................ 166 II. Numerical Differentiation • Space Shuttle Acceleration . III. Root Finding • Commissioner Loeb's Demise................................. 167 • Roots of Wilkinson's \Perfidious Polynomial".................. 170 • Bernoulli's Method and Deflation............................. 173 ICM xi Projects, II The Project List IV. Numerical Integration • Fourier Power Spectrum . V. Polynomial Interpolation • Cubic B´ezier Splines in 3D....................................... ICM xii Introduction to Computational Mathematics On two occasions I have been asked, \Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?" ... I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question." | Charles Babbage Passages from the Life of a Philosopher, p. 67. ICM xiii I. Computer Arithmetic: Scientific Notation Definitions of Scientific Notation Normalized: Any numeric value can be written as p d0:d1d2d3 :::dn 10 × where 1 d0 9. ≤ ≤ Engineering: Any numeric value can be written as q n:d1d2d3 :::dm 10 × where 1 n 999 and q is a multiple of 3. ≤ ≤ Examples (NIST's`Values of Constants') • Speed of light in a vacuum: 2:99792458 108 m/s × • Newtonian constant of gravitation: 6:67384 10 11 m3=(kg s2) × − · • Avogadro's number: 6:022141 10 23 mol 1 × − − • Mass of a proton: 1:672621777 10 27 kg × − • Astronomical unit: 92:95580727 106 mi × ICM 1 { 175 Conversions Basic Base Transmogrification: Integers Binary Decimal Decimal Binary ! ! (Linear algebra version) (Algebra version) Think of the binary number as a Successively compute the bits vector of 1's and 0's. Use a dot (from right to left) product to convert to decimal. 1. bit = x mod 2 1. x2 = 101110 then set x = x=2 b c 2. Repeat until x = 0 2. x10 = 1 0 1 1 1 0 E.g., x10 = 46 h i 25 24 23 22 21 20 b0 = 0; then set x = 23 · h i b1 = 1; x = 11 5 3 2 1 b = 1; x = 5 3. x10 = 2 + 2 + 2 + 2 2 = 46 b3 = 1; x = 2 b4 = 0; x = 1 b5 = 1; x = 0 Whence x2 = 101110 ICM 2 { 175 Conversions Basic Base Transmogrification: Fractions Binary Decimal Decimal Binary ! ! (Linear algebra version) (Algebra version) Think of the binary number as a Successively compute the bits vector of 1's and 0's. Use a dot (from left to right) product to convert to decimal. 1. bit = 2x b c 1. x2 = 0:10111 then set x = frac(2x) 2. Repeat until x = 0 (or when 2. x10 = 1 0 1 1 1 reaching maximum length) h 1 2 3 4 5 i 2− 2− 2− 2− 2− E.g., x10 = 0:71875 · h i b 1 = 1; then set x = 0:43750 − 3. x = 2 1 + 2 3 + 2 4 + 2 5 b 2 = 0; x = 0:87500 10 − − − − − b 3 = 1; x = 0:75000 = 0:71875 − b 4 = 1; x = 0:50000 − b 5 = 1; x = 0:0 Stop − Whence x2 = 0:10111 ICM 3 { 175 Conversions Terminating Expansions? When does a fraction's expansion terminate? n n Base 10: A decimal fraction terminates when r = = . 10p 2p 5p · m Base 2: A binary fraction terminates when r = . 2p Examples 1 1. 10 = 0:110 = 0:000112 1 2. 3 = 0:310 = 0:012 : : 3. p2 = 1:414213562373095048810 = 1:01101010000010011112 : : 4. p = 3:141592653589793238510 = 11:0010010000111111012 ICM 4 { 175 Conversions Examples (Convert A Repeating Binary Expansion) Convert n = 0:0101101101 = 0:01012 to decimal. ··· 1. Convert the repeating block to decimal 1012 = 510 2. Rewrite n in \powers-of-two" notation 4 7 10 13 n = 5 2− + 5 2− + 5 2− + 5 2− + · · · · ··· 3. Express n as a geometric series ¥ 4 3k n = 5 2− ∑ 2− ·