a2W10 - AMATH 341 / CM 271 / CS 371 Assignment 2 Due :...

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Assignment 2 Due : Wednesday January 27, 2010 1. The matrix factor L that emerges from Gaussian elimination with partial pivoting is almost always surprisingly well conditioned. The reason for this is not fully understood. (a) Explain why all the entries of L below the diagonal have absolute value at most 1, provided partial pivoting is used. (b) Generate random lower triangular matrices with 1’s on the diagonal and uniformly dis- tributed random numbers in the interval [ 1 , 1] below the diagonal. (Forming a matrix of this kind can be done using the rand , tril , and eye functions in Matlab. Since rand gen- erates random numbers uniformly distributed in [0 , 1], you should use 2*rand(n,n)-1 to obtain the requested distribution.) Make a plot of condition number versus matrix size n . Use a logarithmic scale for the y-axis (condition number). The functions cond and semilogy will be helpful. Use values of n up to n = 200 and explain the apparent flattening (within fluctuations) of the plot. (c) Generate random lower triangular matrices of the same structure using the following approach: form a random matrix A (using rand ) and then extract its L factor from Gaussian elimination by invoking either the built-in function lu or the function lutx from Moler’s book. Make the same plot for this distribution as in part (b). (d) Hand in: listings of your m-files and the two requested plots. In addition, describe qualitatively the difference between the two plots, which should be immediately apparent. 2. Let L be an n × n lower triangular matrix with all nonzero entries on the diagonal. (a) Explain why the following inequality holds: for all i = 1 , . . . , n , L ≥ | L ( i, i ) | . (b) Give a formula for the ( i, i ) entry of L 1 in terms of the entries of L . Explain your formula by considering the equation Lx = e i , where x is the i th column of L 1 and e i is the i th column of I . (c) Combine (a) and (b) to derive the following inequality: κ ( L ) max i =1 ,...,n | L ( i, i ) | min i =1 ,...,n | L ( i, i ) | . 3. It can be shown that the definitions of the 1-, 2-, or -norms of a vector x and of a matrix A given in lecture have the property that A = max Ax x : x ̸ = 0 , which alternatively can be used to define the matrix norm in terms of the corresponding vector norm. The norm is then called an “induced” or “operator” norm. (The Frobenius norm is not an induced norm.) (a) Let m = min {∥ Ax / x : x ̸ = 0 } . Assuming m ̸ = 0, show that m = 1 / A 1 . Use the operator norm definition so that your result applies to all operator norms. [Hint: apply a carefully chosen change of variables to x .] (b) Establish the following identity, which holds for all invertible matrices A and all operator norms: A ∥∥ A
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This note was uploaded on 06/21/2010 for the course CS 5212 taught by Professor Papoulia,katerina during the Spring '10 term at Waterloo.

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a2W10 - AMATH 341 / CM 271 / CS 371 Assignment 2 Due :...

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