applied numerical analysis and modeling
UCLA Summer 2018
cee103 – UCLA – Taciroglu, Margulis
Homework #3 Due at midnight on Friday, July 20, 2018
Problem 1. Consider the following data points
Find the following interpolating polynomials and use MATLAB to graph the interpolants and the data points. (a) The piecewise linear Lagrange interpolating polynomial l(x). (b) The piecewise quadratic Lagrange interpolating polynomial q(x). (c) Newtons divided difference interpolating 4() pxof degree 4 . (d) The natural cubic spline s(x).
Problem 2. Let 1 ( ) (1 ) f x x − =+ and let 0 0 x = , 1 1 x = , and 2 2 x = . Determine the divided difference
01 [ , ] f x x and 0 1 2 [ , , ] f x x x . Based on these divided differences, find the quadratic interpolating polynomial 2() pxthat interpolates () fxat the given 0 x , 1 x , and 2 x . Moreover, simplify 2() pxas much as possible.
Problem 3. Find the cubic spline that satisfies the conditions and use MATLAB to plot it. (0) 0s = , (1) 0 s = , (2) 2 s = , (0) 0 s = and (0) 2 s =
Problem 4. The following table lists the displacements i u , as obtained by a numerical method, at given points i x (for all {0,1,2,…n} i ) of an elastic bar with axial rigidity EA subjected to a uniaxial distributed load P(x) and clamped at 0 x = .
Write a MATLAB code to (a) Determine the piecewise linear Lagrange interpolating polynomial (i.e. the displacement function) u(x). (b) Determine the (normal) strain function ( ) ( ) x u x = in terms of the first derivative of the piecewise linear Lagrange interpolating polynomial () ux , as obtained in Part a) (i.e. by differentiating the Lagrange basis functions). (c) Smooth the (discontinuous) strains () x according to the following algorithm: 1. At each i x , with {0,1,2,…,n 1} i−, determine the averaged strain i as the mean of the (constant) strain () x obtained in [) i-1 i x ,x and 1[) ii x ,x+ . 2. At each i x , with {0, } in , determine the averaged strain i as () ix .
Homework UCLA cee103 applied numerical analysis & computing Summer 2018
cee103 – UCLA – Taciroglu, Margulis
3. Use the same Lagrange basis functions as determined in Part a) and the averaged strains () ix (for all {0,1,2,…,n} i ) to find a piecewise linear interpolating polynomial that gives the smoothed strains () x . (d) Graph the displacement function u(x), the strains () x , the smoothed strains () x , and the errors ( ) ( ) ( ) exact E x x x =− and ( ) ( ) ( ) exact E x x x =− with 321 ( ) (4 3 12 9) 10exact x x x x = − − + . Moreover, give a brief statement about the errors
