mm-425 === 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 LUdecomposition might be achieved by Kahan-type additions. Or whatconstitutes too expensive, when choosing which methods to implement.It seems you might want either the fastest that is possible, or themost numerically robust that is possible - and everything in betweenis where you go when you want to have an argument over which does whatand how this works and why and when... ;) === 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 triumphof electrical engineering over mathematical analysis. Or just do whatused to be known as fl_2 arithemetic where you use a double precisionaccumulator 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;Thank you for your reply. It seems to me that the DM procedurepermutes rows and columns independently to achieve the finest uppertriangular form of the matrix; but what I need is a symmetricpermutation (i.e., permutation vector of rows = permutation vector ofcolumns); only this amounts to a relabeling of the graph nodes. Idon't see how to achieve this with dmperm.m.=== === 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 aPythagorean Triplet. At first i saw a lot of lines forming from thedots. 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.gifI'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 documentationsimply states In [4],Freund introduced the transpose-free QMR method (TFQMR), which alleviatesthe 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 bookunder Survey of recent methods, and that it has the same breakdown asCGS. 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> robI suggest the same method by which you can apply Newton-Raphson tocalculate the inverse of a number A, using only multiplications andadditions (it was widely used when machines lacked a hardware divideinstruction).Start with f(x) = A - 1/x = 0this leads to x_{n+1} = ( 2 - A * x_n) * x_n .The correct starting value is essential to keep the iteration fromdiverging: if A > 1 choose x_0 < 1 and vice-versa.You might also want to look at Regula Falsi for this problem. ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/