VAN/SURNAME : UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA VOORNAME/FIRST NAMES : WTW26 NUMERIESE METODES WTW26 NUMERICAL METHODS EKSAMEN / EXAMINATION STUDENTENOMMER/STUDENT NUMBER : HANDTEKENING/SIGNATURE : Junie 2004 / June 2004 TYD/TIME: 90 min PUNTE/MARKS: 45 Eksterne eksaminator/ External examiner: Interne eksaminators/ Internal examiners: PUNTE Mev L Mostert Mev A Labuschagne MARKS LEES DIE VOLGENDE INSTRUK- SIES READ THE FOLLOWING IN- STRUCTIONS 1. Die vraestel bestaan uit bladsye 1 tot 11 (vrae 1 tot 9). Kontroleer of jou vraestel volledig is. 2. Doen alle krapwerk op die teenblad. Dit word nie nagesien nie.. As jy meer as die beskikbare ruimte vir n antwoord nodig het, gebruik dan ook die teenblad en dui dit asseblief duidelik aan. 4. Geen potloodwerk of enige iets wat in rooi ink gedoen is, word nagesien nie. 5. As jy korrigeerink ( Tipp-Ex ) gebruik, verbeur jy die reg om te kla oor werk wat nie nagesien is nie of wat verkeerd nagesien is. 6. Slegs nie-programmeerbare sakrekenaars word toegelaat. 1. The paper consists of pages 1 to 11 (questions 1 to 9). Check whether your paper is complete. 2. Do all scribbling on the facing page. It will not be marked.. If you need more than the available space for an answer, use the facing page and please indicate it clearly. 4. No pencil work or any work in red ink will be marked. 5. If you use correcting fluid ( Tipp-Ex ), you lose the right to question the marking or to indicate work that had not been marked. 6. Only non-programmable pocket calculators are allowed. Outeursreg voorbehou Copyright reserved 0
Vraag 1 Question 1 Beskou die vergelyking Consider the equation x = 7 24 x + 7 8 x2 + 17 12 x = g(x). Wanneer die vastepunt iterasieskema gebruik word met beginpunt x 0 = 2, word die volgende resultate verkry: When the fixed point iteration scheme is used with starting value x 0 = 2, the following results are found: k 0 1 2 x k 2 4 1 1.1 Bereken x. Watter gevolgtrekking kan u maak i.v.m. die ry iterasies? 1.2 Calculate x. What conclusion can you draw about the sequence of iterations? 2 1.2 Dit is bekend dat een van die vastepunte, sê p, in die interval [ 0.5, 0.25] lê. Aanvaar dat beide g en g stygend is op [ 0.5, 0.25]. Motiveer waarom die vastepunt iterasies sal konvergeer na p vir enige keuse van x 0 [ 0.5, 0.25]. 1.2 It is known that one of the fixed points, say p, lies in the interval [ 0.5, 0.25]. Assume that both g and g are increasing on [ 0.5, 0.25]. Motivate why the fixed point iterations will converge to p for any choice of x 0 [ 0.5, 0.25]. 1
Vraag 2 Question 2 Beskou die stelsel lineêre vergelykings Ax = b met A en b gegee deur A = 1 0 1 1 0 0 4 1 Consider the system of linear equations Ax = b with A and b given by en / and b = 1 5 2.1 Los die stelsel vergelykings op deur van Gauss-eliminasie sonder spillering gebruik te maak. Toon al u stappe. 2.1 Solve the system of linear equations using Gauss elimination without pivoting. Show all your steps. 4 2
2.2 Los die stelsel vergelykings op deur van die gegewe LU-ontbinding gebruik te maak. Toon al u stappe. A = 1 0 1 1 0 0 4 1 = 2.2 Solve the system of linear equations using the given LU decomposition. Show all your steps. 1 0 0 1 1 0 0 2 1 1 0 0 2 0 0 0 1
Vraag Question Beskou die derde-orde beginwaardeprobleem Consider the third order initial value problem tx (t) + x (t) x(t) = t 2 ; x(1) = 2; x (1) = 1; x (1) = 0 Herskryf bogenoemde beginwaardeprobleem as n stelsel van eerste orde vergelykings. Rewrite the initial value problem above as a system of first order equations. 4
Vraag 4 Question 4 Beskou die volgende beginwaardeprobleem: x (t) = y(t) y (t) = t + x(t) x(0) = 1 y(0) = 2 Consider the following initial value problem: Gebruik Heun se metode met staplengte h = 0.2 en vind n benadering vir (x(0.2), y(0.2)). Use Heun s method with step size h = 0.2 and find an approximation for (x(0.2), y(0.2)). 4 5
Vraag 5 Question 5 Beskou die beginwaardeprobleem: x = tx x(0) = 1 Consider the initial value problem: Vind n benadering vir x(0.2) met behulp van die Runge-Kutta orde 4 metode en h = 0.2. Toon al jou berekenings. Find an approximation for x(0.2) with the Runge-Kutta method of order 4 and h = 0.2. Show all your calculations. 6 6
Vraag 6 Question 6 Bereken n benadering vir die bepaalde integraal met behulp van Simpson se reël. Verdeel die interval [0, ] in ses deelintervalle, elk van dieselfde lengte. Toon al jou berekenings. 0 e (x2) dx Find an approximation for the definite integral with Simpson s rule. Divide the interval [1, ] in six subintervals, each of the same length. Show all your calculations. Vraag 7 Question 7 Hoeveel deelintervalle sal voldoende wees om die bepaalde integraal 1 x4 dx korrek te vind met die Simpsonreël tot 4 desimale syfers? Fout / Error = (b a)f (4) (c)h 4 How many subintervals will suffice to find the definite integral 1 x4 dx correct with Simpson s rule to 4 decimal digits? 180 7
Vraag 8 Question 8 Die volgende data gee die snelheid in m/s van n voorwerp op gegewe tydstippe. Dit is bekend dat die totale afstand afgelê in 4 sekondes gegee word deur 4 0 v(t)dt. The following data gives the velocity in m/s of an object at certain times. It is known that the total distance travelled in 4 seconds is given by Vind n benadering vir die afstand afgelê deur van die trapesiumreël vir integrasie gebruik te maak. Toon al jou berekenings. Find an approximation for the distance travelled using the trapezoidal rule for integration. Show all your calculations. t 0 0.25 0.5 1 2 4 v(t) 0 0.5 1.2 5 8 14 8
Vraag 9 Question 9 Dit is bekend dat die volgende stel getabuleerde datapunte een van die volgende twee verbande behoort te bevredig: It is known that the following tabulated set of data points must satisfy one of the following two relationships: y = 1 ax + b of / or y = ce dx 9.1 Toon in beide gevalle aan hoe die data gelineariseer kan word deur bogenoemde vergelykings in die vorm Y = AX + B te herskryf. 9.1 Show in both cases how the data can be linearized by rewriting the equations above in the form Y = AX + B. 2 9.2 Gebruik die inligting in die tabel en bepaal die waardes van die konstantes a, b, c en d met behulp van kleinste kwadrate krommepassing op die gelineariseerde data. 9.2 Use the information in the table and find the values of the constants a, b, c and d using least squares curve fitting on the linearized data. x y ln y 1/y x ln y x/y 1 0.8 0.22 1.250 0.22 1.250 2 0.6 0.510 1.667 1.022. 0.5 0.69 2.000 2.079 6.000 4 0.4 0.916 2.500.665 10.000 10 2. 2.42 7.417 6.989 20.58 Berekening van a en b: Calculation of a and b: 9
Berekening van c en d: Calculation of c and d: 6 9. Bereken die fout vir beide die passings. 9. Find the error for both fittings. Fout / Error = 1 n [f(x k ) y k ] 2 n k=1 2 9.4 Watter een van die twee passings is die beter passing? Motiveer u antwoord. 9.4 Which one of the two fittings is the better one? Motivate your answer. 1 10