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Unformatted text preview: Discussion 2 EE 103 October 5, 2010 EE 103  Discussion 2 Review • You can multiply two matrices A and B provided their dimensions are compatible, which means the number of columns of A equals the number of rows of B . Suppose A and B are compatible, e.g., A has size m × p and B has size p × n . Then the product matrix C = AB is the m × n matrix with elements c ij = p X k =1 a ak b kj = a i 1 b 1 j + ··· + a ip b pj . • The cost of a matrix algorithm is often expressed by giving the total number of flops required to carry it out. One flop is defined as one ad dition, subtraction, multiplication, or division of two floatingpoint numbers. To evaluate the complexity of an algorithm, we count the total number of flops, express it as a function of the dimensions of the matrices and vectors involved, and simplify by only keeping the leading terms. 1. (2.3) Shift Matrices. a. Give a simple description in words of the function f ( x ) = Ax , where A is the 5 × 5 matrix A = 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 . b. Same question for f ( x ) = A k x , for k = 2 , 3 , 4 , 5 . ( A k is the k th power of A : A 2 = AA , A 3 = A 2 A = AAA , etc.) c. Same question for f ( x ) = A T x . 2. Exercise from 225. Evaluate y = ABx two ways ( A and B are n × n , x is a vector). 1 Discussion 2 EE 103 a. y = ( AB ) x . Coded in Matlab as: C = A * B; y=C * x; b. y = A ( Bx ) . Coded in Matlab as: z = B * x; y=A * z; Both methods give the same answer, but which method is faster? 3. (4.1) Polynomial interpolation. In this problem we construct polynomials p ( t ) = x 1 + x 2 t + ··· + x n 1 t n 2 + x n t n 1 of degree 5, 10, and 15 (i.e., for n = 6 , 11 , 16 ), that interpolate the function f ( t ) = 1 / (1 + 25 t 2 ) over the interval [ 1 , 1] . For each value of n , we com pute the interpolating polynomial as follows. We first generate n + 1 pairs ( t i ,y i ) , using the Matlab commands t = linspace(1,1,n)’; y = 1./(1+25 * t.ˆ2); This produces two vectors: a vector t with n elements t i , equally spaced in the interval [ 1 , 1] , and a vector y with elements y i = f ( t i ) . We then solve a set of linear equations 1 t 1 ··· t n 2 1 t n 1 1 1 t 2 ··· t n 2 2 t n 1 2 . . . . . . . . . . . . 1 t n 1 ··· t n 2 n 1 t n 1 n 1 1 t n ··· t n 2 n t n 1 n x 1 x 2 . . . x n 1 x n = y 1 y 2 . . . y n 1 y n to find the coefficients x i . Calculate the three polynomials (for n = 6 ,n = 11 ,n = 16 ). Plot the three polynomials and the function f on the interval [ 1 , 1] . What do your con clude about the effect of increasing the degree of the interpolating polyno mial?...
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This note was uploaded on 02/03/2011 for the course EE 103 taught by Professor Vandenberghe,lieven during the Spring '08 term at UCLA.
 Spring '08
 VANDENBERGHE,LIEVEN

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