unh2010:iam931:hw3
====== Differences ====== This shows you the differences between two versions of the page.
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unh2010:iam931:hw3 [2010/11/01 05:40] gibson |
unh2010:iam931:hw3 [2010/11/01 10:06] (current) gibson |
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| - | {{:unh2010:iam931:hw3:lebesgueconst.png?400}} {{:unh2010:iam931:hw3:polyinterp.png?400}} | + | {{:unh2010:iam931:hw3:lebesgueconst.png?400}} |
| (b) The inf-norm of ''A'' is plotted as a function of ''n'' on the left. %%(c)%% The inf-norm condition | (b) The inf-norm of ''A'' is plotted as a function of ''n'' on the left. %%(c)%% The inf-norm condition | ||
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| Note that the structure of the driver program and eleftwards/rightwards functions ends up | Note that the structure of the driver program and eleftwards/rightwards functions ends up | ||
| calculating the same partial sums many times over, but I'm optimizing for my time, not CPU time. | calculating the same partial sums many times over, but I'm optimizing for my time, not CPU time. | ||
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| ====== ex 16.2 ====== | ====== ex 16.2 ====== | ||
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| {{:unh2010:iam931:svderrs.png?400}} {{:unh2010:iam931:qrerrs.png?400}} | {{:unh2010:iam931:svderrs.png?400}} {{:unh2010:iam931:qrerrs.png?400}} | ||
| - | These plots show (left) the normwise errors of a numerically computed SVD of randomly constructed matrices A with known SVD factors, as a function of the condition number of A, (right) ditto for QR decomposition. For the SVD, USV' = A, the errors in the computed unitary factors U and V appear to be bounded above linearly with condition number, suggesting that the accuracy of the SVD computation follows the error scaling law for backward stable algorithms (Trefethen eqn 15.1) --even though on dimensionality grounds the computation of f: A -> U,S,V cannot backward stable! Also, the singular values are computed to nearly machine precision, even when the condition number is O(10^18). | + | (b) These plots show (left) the normwise errors of a numerically computed SVD of randomly constructed matrices A with known SVD factors (with the signs of the columns of U and V fixed), as a function of the condition number of A, (right) ditto for QR decomposition. For the SVD, USV' = A, the errors in the computed unitary factors U and V appear to be bounded above linearly with condition number, suggesting that the accuracy of the SVD computation follows the error scaling law for backward stable algorithms (Trefethen eqn 15.1) --even though on dimensionality grounds the computation of f: A -> U,S,V cannot backward stable! Also, the singular values are computed to nearly machine precision, even when the condition number is O(10^18). |
| The error scalings of the computed Q,R factors of the QR decomposition are similar to those of the SVD, though they are an order of magnitude or so worse. Also, whereas the errors in the SVD factors are sometimes spread well below the apparent upper bound, the errors in Q and R are almost always within an order of magnitude of linear-in-condition number | The error scalings of the computed Q,R factors of the QR decomposition are similar to those of the SVD, though they are an order of magnitude or so worse. Also, whereas the errors in the SVD factors are sometimes spread well below the apparent upper bound, the errors in Q and R are almost always within an order of magnitude of linear-in-condition number | ||
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