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--- gibson:teaching:fall-2012:iam961:hw3 [2012/10/24 10:28]
gibson
+++ gibson:teaching:fall-2012:iam961:hw3 [2012/10/24 11:08] (current)
gibson
@@ Line 83: / Line 83: @@
-Now see how well the computed QR decomp does what it should!
+Now see how well the computed QR decomp does what it should
+by computing **five error measures**!
 <code>
@@ Line 129: / Line 130: @@
 ''qr_cgs'', ''qr_mgs'', and ''qr_house'' codes for the QR decomposition.**
-Based on your results, how well do each of the three QR algorithms work? Make
+Use script files to automate the above calculations. Put the generation of
-some general observations based on the plots and the computed errors for the three cases.
-Make a table of the 5 error measures for the 3 algorithms. Keep in mind that errors of
-order 10^-16 are very good, 10^-8 are ok, and 10^0 (one) are very bad, and that
-the order of magnitude (exponent of 10) is all that matters.
-Hint: Use script files to automate the above calculations. Put the generation of
 ''A,x,b'' in one script file (or maybe function) and the others in another script
 so that you can rerun the whole series of calculations for different QR algorithms
 just by changing ''qr_cgs'' to ''qr_mgs'' or ''qr_house''.
+Run the script a few times for each algorithm. Note that the computed errors might
+vary over one or two orders of magnitude based on the particular random ''A,x,b''.
+Again, running the script a few times for each algorithm, record **the nearest usual
+power of ten for each error**, and make a table of the 5 error measures for the 3
+algorithms.
-**3:** For the ambitious: Note that if you rerun the above commands repeatedly for
+Based on your results, how well do each of the three QR algorithms work? Make
-different randomly constructed ''A,x,b'', you might get errors ranging over a couple
+some general observations based on the plots and the error table. Keep in mind that
-different orders of magnitude. It's actually best to run the tests repeatedly and
+errors of order 10^-16 are superb, 10^-8 are ok, and 10^0 (one) are very bad, and that
-take an average of the errors. Put the entire sequence of above commands (minus the plots) in a
+the order of magnitude (exponent of 10) is all that matters.
-''for'' loop over, say, 100 trials, and reduce the dimension of the matrices to say
-''m=30'' so they run faster. Use a geometric mean rather than the arithmetic mean, i.e.
+**3:** For the ambitious: To be more precise about the averaging over random ''A,x,b'',
+run the tests repeatedly and take an average of the errors. Put the entire sequence
+of above commands (minus the plots) in a ''for'' loop over, say, 100 trials, and reduce
+the dimension of the matrices to say ''m=30'' so they run faster. Then compute the //geometric mean//
+errors as follows
 <code>
@@ Line 161: / Line 167: @@
 Put the various calculated geometric-mean errors in a table and base your discussion
 of the behavior of the three algorithms on the geometic means rather than the
-particular-case answers of **3**.
+estimated order-of-magnitude errors of **3**.
 **4:** For those seeking world domination: calculate the geometric-mean errors for
-each algorithm as a function of condition number, and produce plots for each error
+each algorithm as a function of condition number, and produce log-log plots for each error
 type with three lines, one each for CGS, MGS, and Householder, with condition number
 ranging from 1 to 10^16.
+Turn in your codes, the error table, one inner-product matrix plot for each algorithm,
+and your discussion of the behavior of the three algorithms.

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