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103_1_TomsHandouts

103_1_TomsHandouts - Discussion 2 October 5 2010 EE 103 EE...

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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 floating-point 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 2-25. Evaluate y = ABx two ways ( A and B are n × n , x is a vector). 1
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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? 2
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Discussion 3 EE 103 October 5, 2010 EE 103 - Discussion 3 Review A matrix A is positive definite if it is symmetric and x T Ax > 0 for all nonzero x . Similarly, A is positive semidefinite if it is symmetric and x T Ax 0 for all nonzero x .
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