SelfAssessment2 - g x we know that the error of the n-th...

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Self-Assessment 2 – Math 104A, Winter 2009 Answer the following questions without looking in the book. If you do not feel comfortable answering them, read the corresponding sections in the book, and then solve the problems, again without looking in the book. 1. What does it mean that a polynomial p ( x ) interpolates a function f ( x ) at nodes x 0 , x 1 , . . . , x n ? 2. What is the difference between the Newton Interpolation Polynomial and the Lagrange Interpolation Polynomial? 3. Given ( x 0 , f ( x 0 )) , . . . , ( x n , f ( x n )), how many polynomials can you find that interpolate f at those values? How many polynomials of degree n ? 4. What is the error made when a function f is approximated by the Lagrange interpolation polynomial? 5. What do the Newton divided Differences represent? 6. What is Neville’s algorithm? 7. From the definition of fixed-point iteration with iteratin function
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Unformatted text preview: g ( x ), we know that the error of the n-th iterate satisfies e n = x n-p = g ( x n-1 )-g ( p ) . We saw in class that if g ( p ) = 0 and g 00 ( x ) is continuous at x = p , the iteration converges quadratically. State conditions under which one can expect an iteration to converge cubically. 8. Prove: If f ( x ) is a polynomial of degree ≤ n , then the polynomial of degree ≤ n which interpolaties f ( x ) at x , x 1 , . . . , x n is f ( x ) itself. 9. A table of values of cos x is required so that linear interpolation will yield six decimal-place accuracy for any value of x in [0 , π ]. Assuming that the tabular values are to be equally spaced, what is the minimum number of entries needed in the table? 1...
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This note was uploaded on 12/26/2011 for the course MATH 104a taught by Professor Staff during the Fall '08 term at UCSB.

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