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hw16_solutions

hw16_solutions - Homework#6 Section 3.9 4 for all |)| for...

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Homework #6 Section 3.9 4. ( ) ( ) for all ( ) | ( )| for all We can apply Theorem 3.5 to get that a unique fixed point exists, and the iteration ( ) converges, with the error satisfying | | | | Section 3.10 10. ( ) substituting the value of function, its first and second derivative into equation 3.45 one gets ( ) Section 4.1 First we compute the values of the function at the nodes: Now we construct the quadratic polynomial: ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) First we compute the values of the function at the nodes: Now we construct the polynomial: ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )
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3. Now we construct the polynomial: ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) 9. Define ( ) ( ) ∑ ( ) ( ) ( ) Since everything on the right side is a polynomial of degree , so is But, again, ( ) for the nodes , therefore ( ) for all which provides the desired result.
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10. Let ( ) and note that this is a polynomial of degree 0. By the previous problem, we have that ( ) ∑ ( ) ( ) ( ) ∑ ( ) ( ) and we are done. Section 4.2
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