Topics in Computational Mathematics Notes for Computational Mathematics (MA1611) Information Technology (AS1054) Dr G Bowtell Contents 1 Curve Sketching 1 1.1 CurveSketching ................................ 1 1.2 IncreasingandDecreasingFunction . .... 1 1.3 StationaryPoints ................................ 2 1.4 ClassificationofStationaryPoints. ...... 3 1.5 PointofInflection-DefinitionandComment . ..... 4 1.6 Asymptotes................................... 5 2 Root Finding 7 2.1 Introduction................................... 7 2.2 Existence of solution of f(x) = 0 ....................... 8 2.3 Iterative method to solve f(x) = 0 byrearrangement . 10 2.4 IterationusingExcel-Method1. ... 11 2.5 Newton’s Method to solve f(x) = 0 ...................... 12 2.6 IterationusingExcel-Method2. ... 14 2.7 SimultaneousEquations- linearand non-linear . ........ 15 2.7.1 Linearsimultaneousequations . 15 2.7.2 MatrixproductandinverseusingExcel . .. 18 2.7.3 Non-linearsimultaneousequations . ... 20 3 Financial Functions in Excel 27 3.1 Introduction................................... 27 3.2 GeometricProgression . 27 3.3 BasicCompoundInterest . 28 3.4 BasicInvestmentProblem. 29 3.5 BasicFinancialWorksheetFunctionsinExcel . ....... 31 3.6 Further Financial Worksheet Functionsin Excel . ........ 34 4 Curvefitting-InterpolationandExtrapolation 39 4.1 Introduction................................... 39 4.2 LinearSpline .................................. 42 4.3 CubicSpline-natural ............................. 45 4.4 LinearLeastSquaresFitting. ... 49 4.4.1 Linear Least Squares - Regression - Single Variable . ...... 49 4.4.2 Least Squares Criteria for Straight Line Fitting . ....... 50 i CONTENTS ii 4.4.3 Linear Least Squares - Multiple Regression - Many Variables . 52 4.4.4 Excel-LinearRegression . 54 4.4.5 GoodnessofFit ............................ 59 4.5 Non-LinearLeastSquaresFitting . ... 63 4.5.1 Polynomial trendline (y depending on a single x variable only)... 63 4.5.2 Power, Exponential and Logarithmic trendlines . ...... 64 4.5.3 GeneralFittingusingExcel . 65 5 Minitab - Descriptive Statistics 69 5.1 Introduction................................... 69 5.2 MeanandStandardDeviation. 69 5.3 SamplingProblem ............................... 70 5.4 CoefficientofVariation. .. 71 5.5 StandardErroroftheMean . 71 5.6 QuartilesandMedian ............................. 72 5.7 BoxPlots .................................... 73 5.8 Histogram.................................... 75 5.9 Minitab ..................................... 76 5.9.1 Descriptivestatistics . 77 5.9.2 BoxPlots ................................ 77 5.9.3 Histogram................................ 78 6 Powers of Matrices - Markov Chains 80 6.1 Introduction................................... 80 6.2 ConditionalProbability . ... 80 6.3 MarkovChain.................................. 82 7 Google & PageRank 86 7.1 Introduction................................... 86 7.2 PageRank.................................... 86 7.3 RandomSurfer&PageRank . .. .. .. .. .. .. .. 87 7.4 Google ..................................... 92 8 Polynomial Approximations 98 8.1 Introduction................................... 98 8.2 Taylor’s Polynomial Approximation . ..... 99 8.2.1 Linear-tangentapproximation . 100 8.2.2 Quadraticapproximation. 100 8.2.3 Generalapproximation. 101 8.2.4 Maclaurin’sExpansion-Example. 102 8.2.5 TheErrorTerm............................. 103 8.2.6 Convergence of pn(x) as n tendstoinfinity . 106 8.2.7 D’Alembert’sRatiotest . 108 8.3 PolynomialInterpolation . 109 8.3.1 DirectMethod.............................. 109 8.3.2 Lagrange’sMethod. 109 CONTENTS iii 8.3.3 DividedDifferences . 112 8.3.4 ForwardDifferences . 116 8.3.5 Newton-Gregoryformula . 119 9 Numerical Integration 121 9.1 Introduction................................... 121 9.2 Trapezoidalmethod-linearapproximation . ....... 121 9.2.1 Geometricalapproach . 122 9.2.2 Analyticalapproach:rule+errorterm . 122 9.2.3 Compositetrapezoidalrule . 124 9.2.4 Degreeofaccuracy . .. .. .. .. .. .. .. 125 9.3 Simpson’srule-quadraticapproximation . ...... 126 9.3.1 Basicformula.............................. 127 9.3.2 Errorterm................................ 128 9.3.3 CompositeSimpson’srule. 128 9.3.4 Degreeofaccuracy . .. .. .. .. .. .. .. 130 10 Simultaneous Equations 131 10.1Introduction ................................... 131 10.2 Matrixrepresentation-Rowreduction . ...... 131 10.3Partialpivotalselection. ..... 135 10.4ExistenceandUniqueness . 138 10.5Geometricalrepresentation . .... 140 PREFACE iv Preface for 2009–10 This course provides material for the Mathematical Science module Computational Math- ematics (MA1611) and part of the Actuarial Science module Information Technology (AS1054). It has been taught until last year by Dr Graham Bowtell, who wrote these copious notes. In 2009, after around 40 years at City University, Dr Bowtell retired. This year the course will essentailly follow that of previous years, although there will be some changes which will become apparent as the year progresses. The worksheets used by Dr Bowtell may or may not be used this year. However, you may find them usefull, and they are provided for you on my web page for this course. This can be found at http://www.staff.city.ac.uk/o.s.kerr/CompMaths where the material for his course can be found. Dr O.S. Kerr PREFACE v Preface This course looks simultaneously at mathematically based problems and associated pieces of software. In many instances the mathematics of a subject area is first de- veloped and then the software used to solve specific problems. By this means you are able to extend you mathematical knowledge whilst at the same time learn how to use various pieces of software. In detail the presentation of the course consists of two distinct parts. The theory is contained in the lectures which are for one session per week over twenty weeks. The use of the software and development of the mathematical ideas are dealt with in the computational mathematics computer laboratories. These laboratories are scheduled for two hours per fortnight for each Actuarial Science student and one hour per week for the Mathematics students. During the the following pieces of software are considered: Derive This is a package that carries out both algebraic and numerical procedures. For exam- ple amongst other things it is capable of solving equations, differentiating and integrating functions, manipulating matrices, calculating Taylor expansions and plotting graphs in two and three dimensions. Although the course starts with this piece of software and associated mathematical problems it will be used quite extensively over the course, es- pecially for plotting graphs. Excel Excel is used in this section of the module as well as in the VBA programming section. Here the course concentrates on the use of worksheet functions as well as looking at their mathematical background where appropriate. Minitab This is a package that is specifically designed to handle data and carry out statistical analysis. From simple diagrams, which can also be created in Excel, to the more sophis- ticated hypothesis testing and regression analysis. Google Google is a well known search engine, here we not only look at how to use it for spe- cialist searching but also how Google actually carries out it search with emphasis on how a particular page is ranked above another. This particular aspect of the working of the Google search engine involves a topic known as Linear Algebra and Markov Chains. The course will consider the underlying mathematics in a fairly elementary way with the object of giving the user a better understanding of the search process. Important Note Not all the material in these notes will be used on the course and as such the notes should be considered as a reference for the lectures and labs and not a replacement. It PREFACE vi is not likely that much or indeed any of the material from the final four chapters will be covered. Finally note that the course may also present material in a different order to which they appear in the notes. Chapter 1 Curve Sketching 1.1 Curve Sketching To sketch the curve of a given function y = f(x) the following features need to be identi- fied. Intervals on which f(x) is increasing or decreasing • Stationary points • Local maxima and minima • Points of inflection • x and y intercepts • Asymptotes • 1.2 Increasing and Decreasing Function a x1 x2 b Figure 1.1: With reference to fig 1.1 we make the following definition and deduction: 1 CHAPTER 1. CURVE SKETCHING 2 f(x) is an increasing function on the interval [a, b] if for all x and x [a, b] • 1 2 ∈ x >x f(x ) >f(x ) 2 1 ⇒ 2 1 Clearly • f(x ) f(x ) 2 − 1 > 0 x x 2 − 1 which, since as this tends to f (x ) 1 as x x , implies that f (x ) 0. As this is ′ 1 2 → 1 ′ 1 ≥ true for any x1 in the interval [a, b] we can make the following statement: f(x) increasing on [a, b] f ′(x) 0 on [a, b] ⇒ ≥ f ′(x) > 0 on [a, b] f(x) increasing on [a, b] ⇒ Similarly f(x) decreasing on [a, b] f ′(x) 0 on [a, b] ⇒ ≤ f ′(x) < 0 on [a, b] f(x) decreasing on [a, b] ⇒ 1.3 Stationary Points We now consider the special points where f ′(x) is zero. The point (x,f(x)) is a stationary point of f(x) if f (x) = 0 • ′ y y = f(x) A C f ′(x) > 0 f ′(x) < 0 f ′(x) > 0 f ′(x) > 0 B x x1 x2 x3 Figure 1.2: In fig 1.2 A, B and C are stationary points. We also note that at A and B the curve turns This leads to the following definition: 1provided f is differentiable on [a,b] CHAPTER 1. CURVE SKETCHING 3 A point about which f ′(x) changes sign is called a turning point Clearly at