October 16 2001 CS412 Spring 97 Prof Ron Exam 1 My Name is My i d is My seat during the exam time is Row Seat Grading table not to filled by the student Question 1 A B C D Tot Question 2 A B C D Tot Question 3 A B C D Tot Question 4 A B C D Tot 1 1 25 5 7 8 5 a Apply one interation of the fixed point algorithm to the equation x sin x 5 starting with x0 6 b Without interating further do you expect the fixed point iterations to converge Explain c Suggest an alternative method for solving the same equation which in your opinion may be better Reason your choice and implement the first iteration of your method with x 0 6 d Write the output of the matlab code C 7 0 1 0 1 0 C 2 2 2 25 10 5 5 5 You are given four data values 0 1 1 0 2 1 and 4 27 and are asked to interpolate the data by a cubic polynomial i e a polynomial in 3 a Construct a divided difference table suitable for that purpose b Using that table find the polynomial interpolant How would you check that you have the correct result c Your friend Jim tells you that there exists also a quadratic polynomial that interpolates these data Is he right or wrong Explain d Your friend Tina tells you that there exists a polynomial of exact degree 6 that interpolates these data Is she right or wrong Explain 3 3 30 8 10 7 5 a The error bound in cubic Hermite interpolation at k points is five times smaller than the error bound for spline interpolation at those points Why would one then use spline interpolation b You are asked to approximate the function f x e x x3 on the interval 0 1 by cubic Hermite interpolation on equidistant partition How many subintervals should you use in order to have an error 10 6 4 c In b where do you expect the error to be larger near x 1 or near x 0 Explain d Explain why polynomial interpolation is considered global while cubic Hermite interpolation is considered local 5 4 30 10 5 7 8 a You have a calculator that performs only the four basic arithematic operations You wish to find x 281 3 and know that x 3 Suggest an efficient method for doing that and iterate once with your method b Here is a 5 line code that implements your method from a Complete the second and fourth line xold 3 xnew 4 xold xnew 1E 6 xold xnew xnew end c Write the output of the following matlab code x 2 0 2 1 y 2 1 x y d Write a matlab code preferably without a loop that evaluates the polynomial x 2 x 3 x 4 x 51 at x 2 4 6