E3Solns - Department of Mathematics Solutions for the exam...

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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...
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E3Solns - Department of Mathematics Solutions for the exam...

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