tag:blogger.com,1999:blog-1352688022620133175.post5797328146168972506..comments2015-02-26T15:16:49.050+00:00Comments on The NAG Blog: How do I know I'm getting the right answer?Katie O'Harehttp://www.blogger.com/profile/09366741271809330805noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-1352688022620133175.post-12974503226672639012014-02-27T14:48:05.516+00:002014-02-27T14:48:05.516+00:00Thank you, Edvin.Thank you, Edvin.michelenoreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-50708652801477208812014-02-27T13:34:12.930+00:002014-02-27T13:34:12.930+00:00Hi Michele,
For things like matrix multiplication,...Hi Michele,<br />For things like matrix multiplication, the model described in the Accuracy and Stability book enables us to bound the forward error analytically, as you've described, by looking at how rounding errors propagate through the computation.<br /><br />In some cases, the same ideas can be applied to matrix functions. For example the 'Schur' algorithm for computing the square root of a matrix can have its error bounded in this way. Unfortunately, for more complicated algorithms, such as those used to compute matrix exponentials and logarithms, full rounding error analyses giving useful error bounds are not yet available. This is why numerical testing is so important.<br /><br />So to answer your final question, people in the field certainly do this sort of analysis, but it's tricky for really sophisticated algorithms.Edvinhttp://www.blogger.com/profile/00958470341412529745noreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-51141352187774488532014-02-27T11:34:30.586+00:002014-02-27T11:34:30.586+00:00Thank you.
Your say that you can check the algori...Thank you.<br /><br />Your say that you can check the algorithm's result on your test matrix A by using numerics in higher precision. But the claim is that you have developed an algorithm with better precision, i.e. on a _generic_ input matrix.<br /><br />Is there an analytical way of testing the algorithm precision, (even to a precision closed to machine precision) using the machine arithmetic model, such as the method described in Higham's Accuracy and Stability book (chapter 2)?<br /><br />Do people in your field carry out this sort of analysis? I ask because I have no experience in numerical analysis.michelenoreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-78941474799831724992014-02-26T12:07:51.627+00:002014-02-26T12:07:51.627+00:00To get an accurate value for the forward error in ...To get an accurate value for the forward error in general (as<br />opposed to for specific matrices with known exponential) you will<br />need to use higher precision computation, or somehow simulate it.<br />If the original computation was done in IEEE single precision,<br />then using double for the calculation of the forward error would<br />usually be fine. If the original was done in double precision,<br />you'd either need IEEE extended precision (which is hard to get<br />hold of reliably, since most computing systems don't give direct<br />access to it), or some sort of quad precision hardware or emulation.<br />Mick Ponthttp://www.blogger.com/profile/15149585002054415615noreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-76085337519665558942014-02-25T11:11:34.072+00:002014-02-25T11:11:34.072+00:00The forward error of 2.5e-16 was computed making u...The forward error of 2.5e-16 was computed making use of the known result. For this high precision, can you still evaluate the algorithm forward and backward error using the rules for IEEE arithmetics? Or are you forced to compute the errors of specific matrices? Also you talk about size, which I take it means order of magnitude. Again, when you reach 1e-16, can one still assume that the coefficient (for the order of magnitude) is of order 1?Unknownhttp://www.blogger.com/profile/10890884325091935105noreply@blogger.comtag:blogger.com,1999:blog-1352688022620133175.post-20305436792169770092013-09-25T09:01:36.451+01:002013-09-25T09:01:36.451+01:00Maybe you need more testing.Maybe you need more testing.stonehttp://bestrecumbentexercisebikes.us/noreply@blogger.com