CSC_349A_EXAM_QUESTIONS1

# CSC_349A_EXAM_QUESTIONS1 - COMPUTER SCIENCE 349A SAMPLE...

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COMPUTER SCIENCE 349A SAMPLE EXAM QUESTIONS WITH SOLUTIONS PARTS 3, 5, 6 , 7 PART 3. 3.1 Suppose that a computer program, using the Gaussian elimination algorithm, is to be written to accurately solve a system of linear equations b Ax = , where A is an arbitrary n n × nonsingular matrix. Give two reasons why it is necessary to incorporate a pivoting strategy (such as partial pivoting) into the algorithm. 3.2 Let ⎡− = = 3 5 . 0 1 , 2 2 0 1 0 1 0 1 2 b A and suppose that b A x 1 = . Use Naïve Gaussian Elimination (Gaussian elimination without pivoting) to compute x . Do not compute 1 A . Show all of your work. 3.3 Consider the following system of linear equations b Ax = : 0 4 2 2 3 2 4 2 2 3 2 1 3 2 1 3 2 = + + = + = + x x x x x x x x Specify the augmented matrix for this linear system, and use Gaussian elimination with partial pivoting to compute the solution vector x . Show all of your work. 3.4 Suppose that the following MATLAB statement has been executed. A = [-1 2 3 4 ; 5 6 7 8 ; 9 8 7 6 ; 5 4 3 1 ] ; Specify 1 or 2 MATLAB statements that could be used to efficiently compute the second column vector of 1 A . Do not compute the entire matrix 1 A .

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3.5 Let 2 n , let A denote an n n × nonsingular, upper triangular matrix: Ο = nn n n n a a a a a a a a a a A M O L L L 3 33 2 23 22 1 13 12 11 , and let T n y y y y y ) , , , , ( 3 2 1 K = denote a column vector with n entries. The most efficient way to compute y A x 1 = is to use the back-substitution algorithm. Assuming that n , A and y are specified, write a MATLAB function M-file function x = solve ( n , A , y ) that will compute y A x 1 = using the back-substitution algorithm. Note: do not use the MATLAB operator \ or the MATLAB function inv . PART 5. 5.1 Consider the following data: 3 2 1 1 0 1 ) ( i i x f x Suppose that a function ) ( x g of the form x x e c e c c x g 2 1 0 ) ( + + = is to be determined so that ) ( x g interpolates the above data at the specified points i x . Write down a system of linear equations (in matrix/vector form b Ac = ) whose solution will give the values of the unknowns 2 1 0 and , c c c that solve this interpolation problem. Note: Leave your answer in terms of e and powers of e . Do not solve the resultant linear system. 5.2 (a) Give the Lagrange form of the quadratic ( 2 = n ) interpolating polynomial ) ( x P that interpolates x e x f = ) ( a t 4 . 0 and 2 . 0 , 0 = = = x x x .
Note: Do not numerically evaluate ) ( x f ; instead, give your answer in terms of values such as 2 . 0 e . Also, do not simplify the expression for ) ( x P . (b) Using the error term for polynomial interpolation, determine a good upper bound for 1 . 0 ) 1 . 0 ( e P . Note: Do not determine an upper bound for x e x P ) ( for all ] 4 . 0 , 0 [ x , only for 1 . 0 = x .

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CSC_349A_EXAM_QUESTIONS1 - COMPUTER SCIENCE 349A SAMPLE...

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