November 2005 TYD/TIME: 90 min PUNTE / MARKS: 35 VAN/SURNAME: VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER: HANDTEKENING/SIGNATURE:

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1 UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS WTW NUMERIESE METODES / NUMERICAL METHODS EKSAMEN / EXAMINATION VAN/SURNAME: VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER: HANDTEKENING/SIGNATURE: November 2005 TYD/TIME: 90 min PUNTE / MARKS: 35 PUNTE / MARKS LEES DIE VOLGENDE INSTRUKSIES 1. Die vraestel bestaan uit bladsye 1 tot 9 (vrae 1 tot 9). Kontroleer of jou vraestel volledig is. 2. Doen alle krapwerk op die teenblad. Dit word nie nagesien nie. 3. 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. READ THE FOLLOWING INSTRUCTIONS 1. The paper consists of pages 1 to 9 (questions 1 to 9). Check whether your paper is complete. 2. Do all scribbling on the facing page. It will not be marked. 3. 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. Outeursreg voorbehou Copyright reserved

2 LEES DIE VOLGENDE INSTRUKSIES NOUKEURIG VOORDAT U DIE VRAE BEANTWOORD: READ THE FOLLOWING INSTRUCTIONS THOROUGHLY BEFORE YOU ANSWER THE QUESTIONS: 1. Toon al jou werk en motiveer elke bewering. Toon in die besonder aan hoe stellings toegepas word. 2. Rond alle resultate, insluitende tussenstappe af na vier beduidende / tellende syfers. 3. Indien iterasiewaardes bereken word, gebruik die afgeronde waarde vir die volgende iterasie. 1. Show all your work and justify each statement. In particular show how theorems are applied. 2. Round your results, including intermediate steps to four significant digits. 3. When iteration values are calculated, use the rounded value for the following iteration. 4. Toon al jou berekenings duidelik. 4. Show all your calculations clearly. 5. Motiveer jou bewerings, hetsy teoreties, met sketse of enige ander toepaslike metode. 6. Slegs die volgende nie-programmeerbare sakrekenaars word toegelaat: CASIO FX 82 en SHARP EL Motivate your statements, theoretically, with graphs or any other applicable method. 6. Only the following non-programmable pocket calculators are allowed: CASIO FX 82 and SHARP EL 531.

3 Vraag 1 Question 1 Die vergelyking The equation f(x) = 100x 2 x = 0 het n wortel in die interval [0, 1]. Neem x 0 = 0 en x 1 = 1 en gebruik die secant metode om die wortel te benader. Gebruik die stopkriterium has a root in the interval [0, 1]. Take x 0 = 0 and x 1 = 1 and use the secant method to approximate the root. Use the stop criterion f(x k ) < 10 3, waar x k n benadering vir die wortel is. where x k is an approximation for the root. 4 1

4 Vraag 2 Question 2 Beskou die beginwaardeprobleem Consider the initial value problem y = ty(4 y); y(0) = 1. Benader y(0.4) met Taylor se metode van orde 2 en twee tydstappe. Approximate y(0.4) with Taylor s method of order 2 and two time steps. 4 2

5 Vraag 3 Question 3 Skryf die gegewe beginwaardeprobleem as n eerste-orde beginwaardeprobleem vir stelsels en benader u(0.2) met Euler se algoritme. Gebruik 2 tydstappe. Write the given initial value problem as a first order initial value problem for systems and approximate u(0.2) with Euler s algorithm. Use 2 time steps. u + u 0.1u 3 = 0; u(0) = 1, u (0) = 0 3 3

6 Vraag 4 Question 4 Beskou die beginwaardeprobleem Consider the initial value problem u = v; u(0) = 0 v = t + u; v(0) = 0 Gebruik die Runge-Kutta metode van orde 4 met h = 0.2 en vind n benadering vir (u(0.2), v(0.2)). Use the Runge-Kutta method of order 4 with h = 0.2 and find an approximation for (u(0.2), v(0.2)). 4 4

7 Vraag 5 Question 5 Bepaal die LU-faktorisering van die matriks A. A = Find the LU-decomposition of the matrix A. 3 5

8 Vraag 6 Question 6 Beskou die stelsel lineêre vergelykings Consider the system of linear equations 2x y = 0 x + 2y z = 4 y + 2z = 0 As w 0 = (x 0, y 0, z 0 ) = (0, 0, 0), bereken die Gauss-Seidel iterasies w 1 en w 2 asook A w 2 b, waar If w 0 = (x 0, y 0, z 0 ) = (0, 0, 0), calculate the Gauss-Seidel iterations w 1 and w 2 as well as A w 2 b, where w k = (x k, y k, z k ). 5 6

9 Vraag 7 Question 7 Die stelsel nie-lineêre vergelykings The system of nonlinear equations 3x 2 x 3 y = 0 x + y 2 = 1 het n oplossing naby (0, 0). Neem has a solution near (0, 0). Let w (0) = (x 0, y 0 ) = (0, 0) en gebruik Newton se metode om w (1) en w (2) te bepaal. and use Newton s method to compute w (1) and w (2). 5 7

10 Vraag 8 Question 8 Gestel die funksie g is differensieerbaar, g(c) = c, g is kontinu in c en g (c) < 1. Veronderstel die ry {x n } gegenereer met vastepunt iterasie konvergeer na c en e n = c x n. Bewys dat e n+1 e n n Suppose the function g is differentiable, g(c) = c, g is continuous at c and g (c) < 1. Suppose the sequence {x n } generated by fixed point iteration converges to c and e n = c x n. Prove that g (c). 3 8

11 Vraag 9 Question 9 Die stelsel vergelykings The system of equations x 1 = x 1 x2 1 + x x 2 = x 1 x2 1 + x het n oplossing in die vierkant waar 4 x 1 5 en 0 x 2 1. Gebruik die teorie om te bewys dat vastepunt iterasies n ry sal genereer wat na hierdie oplossing konvergeer mits die beginpunt na genoeg is aan die oplossing. has a solution in the square where 4 x 1 5 and 0 x 2 1. Use the theory to prove that fixed point iteration will generate a sequence that converges to this solution if the starting point is close enough to it. 4 9

JUNE 2005 TYD/TIME: 90 min PUNTE / MARKS: 50 VAN/SURNAME: VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER:

JUNE 2005 TYD/TIME: 90 min PUNTE / MARKS: 50 VAN/SURNAME: VOORNAME/FIRST NAMES: STUDENTENOMMER/STUDENT NUMBER: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS WTW 63 - NUMERIESE METHODE / NUMERICAL METHODS EKSAMEN / EXAMINATION