hw16_solutions

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

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Unformatted text preview: 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 Now we construct the polynomial: at the nodes: () ) )( )( ) ) ( ( )( )( ) ) ( ( )( )( ) ) 3. polynomial: () Now we construct the ( ( )( )( ) ) ( ( )( )( ) ) ( ( )( )( ) ) 9. Define () () ∑ () ()( ) Since everything on the right side is a polynomial of degree again, ( ) for the nodes , therefore ( ) provides the desired result. , so is But, for all which 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 The divided difference coefficients are 1, -0.5, 1/8, -1/24. So the polynomial is () () ( ) ( )( √ ) )( )( ) First we compute the values of the function at the nodes: () ( Divided differences are ( ) ( )( ) ( ) . () First we compute the values of the function at the nodes: () The correct code should return the following graphs of the error: Section 4.7 () () values of function at the nodes: Linear interpolation: () { Quadratic interpolation: () { . We evaluate the Since the each polynomial piece matches the nodal data it must be correct, because of the uniqueness of polynomial interpolation. Problems not from the book. () ( ), () ( ) () ( ) The MATLAB code for Newton’s method can be found in previous HW. () The correct code returns the following graphs for evenly spaced nodes (4,8,16). () The correct code returns the following graphs for evenly spaced nodes (4,8,16). Note that the oscillations actually increase with the numbers of nodes. Why is this happening? This is so-called Runge’s phenomenon. You might want to check literature or Wikipedia for more information about the subject. ...
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This note was uploaded on 01/15/2012 for the course MAE 107 taught by Professor Rottman during the Spring '08 term at UCSD.

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