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# hwk4 - 2(a Use Lagrange formula to ﬁnd the unique...

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Homework # 4 Due Thursday, 02/07 1. In this problem, we will see that sometimes it is possible to avoid loosing significant digits (the main idea is to replace substraction by some other operations). (a) Show that the two roots of x 2 - 26 x + 1 are 13 + 168 and 13 - 168. (not too hard!) (b) Suppose your computer work with 5 digits decimal machine numbers. How are 13 and 168 stored in your computer (give s , e and f ) (c) What happen when you ask your computer to do 13 - 168? What is the result? How is it stored? (d) What is the closest 5 digits decimal machine number from the actual value of 13 - 168? How is it stored? How many signifant digits were lost in the above computation? (e) Prove that 13 - 168 = 1 13 + 168 (f) What happen when you ask your computer to do 1 / (13 + 168)? What is the result? How is it stored? How many significant digits are lost.
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Unformatted text preview: 2. (a) Use Lagrange formula to ﬁnd the unique polynomial of degree ≤ 2 which passes through the three points (0,1), (-1,2) and (1,3). (b) Do some algebra to write the polynomial as follow: p ( x ) = a 2 x 2 + a 1 x + a (c) What is the Vandermonde matrix X associated with this interpolation problem? (d) Check that check that the vector [ a , a 1 , a 2 ] T is a solution of X a a 1 a 2 = 1 2 3 3. We have seen in class that, if n points are given, then there is a unique polynomial of degree ≤ n-1 going through these points. Take two points, let say (0,0) and (1,1). (a) What is the only polynomial of degree ≤ 1 going through these two points? (b) Find two diﬀerent polynomials of degree 2 going through these two points. 1...
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