Exam3 - Department of Mathematics The exam paper for course...

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Department of Mathematics The exam paper for course 157 in June 2002. Rubric Answer three questions.
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Question 1 Using Taylor Series, or otherwise, derive the Newton Raphson method. Use the Newton Raphson method to find the cube root of 2 by finding the root of x 3 - 2 using 1 as the initial estimate. Define the terms rounding error , truncation error and cancellation . Consider the iteration S i +1 = r 2 - q 4 - ( S i ) 2 , S 0 = 2 . It can be proved that the values S i satisfy 2 i +1 S i -→ π as i -→ ∞ . Using the Matlab command S=sqrt(2), I=0 followed by repeated use of the Matlab command I=I+1,S=sqrt(2-sqrt(4-S*S)),P=S*2ˆ(I+1) produced the table of results I S P 1 0 . 765367 e 0 3 . 061467 2 0 . 390180 e 0 3 . 121445 3 0 . 196034 e 0 3 . 136548 . . . . . . approaching π 10 0 . 153398 e - 2 3 . 141592 11 0 . 766990 e - 3 3 . 141593 12 0 . 383495 e - 3 3 . 141593 . . . . . . same value 16 0 . 239685 e - 5 3 . 141593 17 0 . 119842 e - 5 3 . 141594 18 0 . 599212 e - 6 3 . 141597 .
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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.

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Exam3 - Department of Mathematics The exam paper for course...

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