mm-3069 === Subject: Re: Kahan summation and matrix multiply > this has a considerable overhead and hence is not used in blas. > IMO, this point of view is way too common among numerical analysts. > A 4x penalty might be acceptable to a lot of people. I have numerical > analysis texts from the 1960s and 1970s that sometimes give a > method and then say it is impractical because of a 4x speed penalty. > Computers gain 4x in speed every 5 years or so. So if a method was > 4x too slow 5 years ago, then it should be accept today, for all those > problems that were being run 5 years ago. It makes you wonder whether the advantages of eg. pivoting in LU decomposition might be achieved by Kahan-type additions. Or what constitutes too expensive, when choosing which methods to implement. It seems you might want either the fastest that is possible, or the most numerically robust that is possible - and everything in between is where you go when you want to have an argument over which does what and how this works and why and when... ;) === Subject: Re: Kahan summation and matrix multiply === >Subject: Re: Kahan summation and matrix multiply >> this has a considerable overhead and hence is not used in blas. >> IMO, this point of view is way too common among numerical analysts. >> A 4x penalty might be acceptable to a lot of people. I have numerical >> analysis texts from the 1960s and 1970s that sometimes give a >> method and then say it is impractical because of a 4x speed penalty. >> Computers gain 4x in speed every 5 years or so. So if a method was >> 4x too slow 5 years ago, then it should be accept today, for all those >> problems that were being run 5 years ago. >It makes you wonder whether the advantages of eg. pivoting in LU >decomposition might be achieved by Kahan-type additions. Or what >constitutes too expensive, when choosing which methods to implement. >It seems you might want either the fastest that is possible, or the >most numerically robust that is possible - and everything in between >is where you go when you want to have an argument over which does what >and how this works and why and when... ;) Just switch to double precision. As Kahan has said, its is the triumph of electrical engineering over mathematical analysis. Or just do what used to be known as fl_2 arithemetic where you use a double precision accumulator for a single precision inner product. === Subject: Reducible Matrices by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA7KDIS05275; permutes rows and columns independently to achieve the finest upper triangular form of the matrix; but what I need is a symmetric permutation (i.e., permutation vector of rows = permutation vector of columns); only this amounts to a relabeling of the graph nodes. I don't see how to achieve this with dmperm.m. Carlos === Subject: Pythagorean Triplets examination by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id hA876IR16258; That tested each coordinate on a form if they where part of a Pythagorean Triplet. At first i saw a lot of lines forming from the dots. My conclution: lots of p.t.'s are of the form (an)^2+(bn)^2=(cn)^2; where a, b and c are another p.t. After i removed all the dots of that form i got an intresting pathern. It reminds me of the sun-flower-pathern. Look for your selves: http://mizardx.gq.nu/unique.gif I've zoomed out 5:1 and made the dots 5 pixels big. === Subject: Re: GMRES/CGS/QMR for Singular Matrices >> There is a transpose-free QMR (TFQMR) available in qmrpack on Netlib. > Which is not as the name suggests an implementation of QMR without using > transposes, but rather CGS with residual smoothing (Walker and Zhou, > Weiss) applied to it. > V. Well, that is something I didn't realize. The QMRPACK documentation simply states In [4], Freund introduced the transpose-free QMR method (TFQMR), which alleviates the need for the transpose...., which made it appear to be a true QMR. Doing a bit of searching I see you mentioned this in the Templates book under Survey of recent methods, and that it has the same breakdown as CGS. I should have caught that before. I apologize for posting misleading information. === Subject: Re: Newton-Raphson to calc inverse function? > Hi All, > I have a function of the form: > t(p,q,c) = {Xp + Yq + Zc if t < Tmax > {Tmax otherwise > and a relation > q = t^{-1}(p,c) > where t^{-1}(p,c) is the inverse of t(p,q,c) in q > Apparently t^{-1}(p,c) can be calculated numerically > using the Newton-Raphson method but I can't see > how it is applied. Any hints or references appreciated. > rgds > rob I suggest the same method by which you can apply Newton-Raphson to calculate the inverse of a number A, using only multiplications and additions (it was widely used when machines lacked a hardware divide instruction). Start with f(x) = A - 1/x = 0 this leads to x_{n+1} = ( 2 - A * x_n) * x_n . The correct starting value is essential to keep the iteration from diverging: if A > 1 choose x_0 < 1 and vice-versa. You might also want to look at Regula Falsi for this problem. -- Julian V. Noble Professor Emeritus of Physics jvn@lessspamformother.virginia.edu ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- Bertolt Brecht, Galileo.