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UMD AMSC 460 - Quiz #5

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AMSC/CMSC 460 Quiz 5 , Fall 2007Show all work. You may leave arithmetic expressions in any form thata calculator could evaluate. By putting your name on this paper, you agreeto abide by the university’s code of academic integrity in completing thequiz. Use no books, calculators, cellphones, communication with others,scratchpaper, etc.Name1. In the last in-class exercise, you experimented with taking the discreteFourier transform (or discrete cosine transform) of the sunspot data andthen zeroing out some small components.1a. (5) What do the components that are large in magnitude tell you aboutsunspots?1b. (5) Give two reasons for zeroing out small components of the transformof a series of data like that for sunspots.1−6 −4 −2 0 2 4 6x 10−4−6−4−20246x 10−4Residual r = (A)x − br1r2−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−0.1−0.08−0.06−0.04−0.0200.020.040.060.080.1Change in solution = x − xe(x−xe)1(x−xe)22. (10) Consider a linear system Ax = b where A is 2 × 2. Suppose weperturb the problem, by changing it to (A + E)xe= b, where kEk1/kAk1=10−4≡ δ. We repeat this experiment 10,000 times with different values of E(all with kEk1/kAk1≈ δ) to get the figures above. Assuming thatkxk1= 1,use the following fact to estimate the condition number κ of A:kx − xekkxk≤κ(A)1 −


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