This preview shows pages 1–5. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Department of Mathematics Solutions for the exam paper for course 157 sat in June 2002. Solution to Question 1 Taylor series for f ( a ) = 0 0 = f ( a ) = f ( x + ( a x )) = f ( x ) + ( a x ) f ( x ) + 1 2 ( x a ) 2 f ( x ) + ··· and if ( x a ) is small 0 = f ( a ) ≈ f ( x ) + ( a x ) f ( x ) therefore a ≈ x f ( x ) f ( x ) This is made the basis of the Newton Raphson method by choosing an approximation x i to a and getting a new approximation x i +1 by assuming equality in the above expression. That is x i +1 = x i f ( x i ) f ( x i ) i x i f = x 3 2 f = 3 * x * x x i +1 1 . 00000 1 . 00000 3 . 00000 1 . 33333 1 1 . 33333 . 37037 5 . 33333 1 . 26389 2 1 . 26389 . 01896 4 . 79225 1 . 25993 3 1 . 25993 . 00006 4 . 76229 1 . 25992 4 1 . 25993 1 . 25993 Thus the cube root of 2 to four decimal places is 1.2599. Rounding error . A rounding error occurs when a number is rounded from a long string of digits to a shorter string of digits. Truncation error . A truncation error occurs when an infinite process is replaced by a finite process to obtain an estimate of the answer. Cancellation . Cancellation occurs when there is a loss of significant figures in computing A B because the first n digits of A and B agree where n is large compared to the number of accurate digits in A and/or B . In this process the values of S i get smaller and smaller. The process con verges to π obtaining six decimal places of accuracy at the 11 th iteration. However, the value of the q 4 ( S i ) 2 gets closer and closer to 2 and there is cancellation in the term 2 q 4 ( S i ) 2 . This is becoming evident by the 17 th iteration and the erratic behaviour is typical of the random errors introduced by cancellation. By iteration 27, S i has become so small that 4 ( S i ) 2 is computed exactly as 4, its square root is computed exactly as 2 and the next S i has the value 0. Solution to Question 2 The Gaussian elimination process gives...
View
Full
Document
This note was uploaded on 09/26/2011 for the course MACM 201 taught by Professor Marnimishna during the Spring '09 term at Simon Fraser.
 Spring '09
 MarniMishna

Click to edit the document details