mm-483 Subject: Re: When should I stop iterative method?> if the norm of matrices is induced by the chosen norm of vectors, then> ||B|| is a possible value of q.> If it is hard to compute ||B||, then any q about which we can prove> (or someone already proved) that ||B|| <= q < 1, is safe.so, its almost all about correctly choosed criteria, norms, tolerance,safety factors and similar... I choose and then I hope that it will workcorrectly for my problem... damn, i hate it when i must to choose because imso unsure and unconfident in myself... ; )concluding from last replies i already choosed/created my criteria (i checkresiduum), and im thanking you on your replies, but im curious about onething:in equation x_(k+1) = B * x_k + c spectral radius rho (=max|eigenvalues(B)| ) must be less than 1, if i want that this iterationmethod convergate (spelling?), so i could simply use:q=rho, because you said i must be sure that q<1, and in this case is alwaysrho=q<1, because if rho=q>1 this method wouldnt convergate and then it isntneeded to calculate q because method (choosed matrix B) isnt good...my (I hope, last) question:I know that method will (surely) convergate ifrho=max|eigenvalues(B)|<1,so could I use q=rho because in this case im sure that q=rho<1...?Maybe it wouldnt be the best criteria (q=rho), but could i use it? Is itlogical? To improve my result (if q<1, but is near to 1) i could use q=rho/2or something like that...? === Subject: Re: Best fit of rectangles in a bigger rectangle already having prefilled rectangles. I think i can make it interesting if i can let u know where iamtrying to apply it , a webpage with banner ads filled and text messageand some blanks spaces, what my goal is to fill up the rest of thefree space withthe ads that i could possibly help without altering theoriginal advertisemsnts. so anything u could point me will be reallyappreciated.also i will try to post in the other group> i need a alogrithm for computing the the locations of the rectangles> that can be best fitted in another rectangle. The other rectangle is> already been filled with rectangles with permanent non-changeable> locations and size.> The Algorithm should be able to fit other rectangles given their size> and to fit as close as to the other rectangles if possible without> colliding with other each other.> any programs or logic would be appreciates. please let me know if u> need more details.> raja> This sounds like it relates to getting the maximum yield of pieces out> of a larger piece of material (except that in that case usually all the> pieces would be relocatable, I think). I know manufacturers are faced> with this problem, but I don't know what the stat-of-the-art for> solving it is. Perhaps a post to sci.op-research would be in order. === Subject: Re: Best fit of rectangles in a bigger rectangle already having prefilled rectangles>.. I think i can make it interesting if i can let u know where iam>trying to apply it , a webpage with banner ads filled and text message>and some blanks spaces, what my goal is to fill up the rest of the>free space withthe ads that i could possibly help without altering the>original advertisemsnts. so anything u could point me will be really>appreciated.Google rectangle packing algorithm === Subject: Re: Use of Walsh Functions in Circuit Analysis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3H21nu08101;Paul Bodenstabprb@fc.hp.comPlease see this papers and you can find the real use of WalshFunctions for circuit analysis.1) K.Balachandran and K.Murugesan, Analysis of Electronic Circuitsusing the Single Term Walsh Series Approach, International Journal ofElectronics, Vol.69, No.3, 1990, pp 327-3322) K.Balachandran and K.Murugesan, Analysis of Transistor CircuitsUsing the Single Term Walsh Series Technique, International Journal of Electronics, Vol.71, No.3, 1991, pp 397-401.3) K.Murugesan, D.Paul Dhayabaran and E.C.Henry Amirtharaj, A Studyof second-order state-space systems of time-invariant and time varyingtransistor circuits using the STWS technique, International Journal of Electronics, Vol.89, No.4, 2002, pp 305-315.Dr.K.Murugesan === Subject: Re: SVD performance, Matlab vs. Lapack by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3H21pg08168;> I'm making a lot (many thousands) of singular value decompositionsof> medium-sized matrices ( e.g. 100 x 50 ) >What do you need these for?I want to experiment with regularizing these matrices beforeinversion. The physical problem relates to acoustics. Anyway, thesubject here is performance, how to efficiently decompose complexmatrices. === Subject: Re: SIMPLEC, SIMPLER or PISO? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3H21nw08097; HY/I FOUND YOUR MAIL ON THE NET , AND I 've understood that y've theSIMPLER CODE,please, would you like to be so kind as to send me thecode because i'm preparing a project and i need this code, please helpus and i 'll be so happy to your help.thinks yours : === Subject: How to apply Levenberg-MarquardtHi all:I am keen in applying LM in a feedback loop to a 2D video algorithm(prediction loop).Any ideas on how to apply it?Partho ChoudhuryRedmond WA === Subject: Re: How to apply Levenberg-Marquardtinstead try to use my method-you don't need a good initial point like LMyou will need to convert from basic into your favourite code> http://mathforum.org/epigone/sci.math.num-analysis/prunflekhay /3693ED6F.7BB5391E@home.comPaul> Hi all:> I am keen in applying LM in a feedback loop to a 2D video algorithm> (prediction loop).> Any ideas on how to apply it?> Partho Choudhury> Redmond WA === Subject: Simple question about if in Mathematica 5.0In mathematica help is written:If[condition, t, f] gives t if condition evaluates to True, and f if itevaluates to False. If[condition, t, f, u] gives u if condition evaluates to neither True norFalse.For[i = 1, i <= 5, i++,if[i < 3, Print[True=, i], Print[False=, i], Print[Undefined=, i]]];and i think it should write:True=1, True=2, False=3, False=4, False=5,but Mathematica prints:True=1, False=1, Undefined=1,True=2, False=2, Undefined=2,True=3, False=3, Undefined=3,True=4, False=4, Undefined=4,True=5, False=5, Undefined=5,WHY??? === Subject: Re: Simple question about if in Mathematica 5.0I think you mistyped your program statement. All Mathematica commands beginin upper case. Try changing if[i < 3, ... to If[i < 3, ...> In mathematica help is written:> If[condition, t, f] gives t if condition evaluates to True, and f if it> evaluates to False.> If[condition, t, f, u] gives u if condition evaluates to neither True nor> False.> For[i = 1, i <= 5, i++,> if[i < 3, Print[True=, i], Print[False=, i], Print[Undefined=, i]]> ];> and i think it should write:> True=1, True=2, False=3, False=4, False=5,> but Mathematica prints:> True=1, False=1, Undefined=1,> True=2, False=2, Undefined=2,> True=3, False=3, Undefined=3,> True=4, False=4, Undefined=4,> True=5, False=5, Undefined=5,> WHY??? === Subject: Re: Simple question about if in Mathematica 5.0> I think you mistyped your program statement. All Mathematica commandsbegin> in upper case. Try changing if[i < 3, ... to If[i < 3, ...in other cases Mathematica warns user about similar names of functions,variables... but this time i didnt get any warning... : ( ... anyway, now icorrected it and it WORKS! : ) === Subject: Re: Yikes! I've Broken Incommensurability!> In sci.math.num-analysis, ken> :> [...]> In any event, if A=1 and B=1, how does C=sqrt(2)> become rational by moving the point on the semicircle> around, or mutating the radius?> [...]It's in the Proof I posted.Given a circle with radius sqr(2), becauseA^2 + B^2 = C^2, always,[A[ca] + B[ca]] = sqr(2)in which A[ca] and B[ca] are EquivalentIntegers =in the Number system that'sestablished in the Proof=, measurable inthe same Units of central angle, eachone unit, but positional.It's what I'm 'excited ' about - a whole newNumber system, that's extremely-rich, andin which all numbers are Integers is estab-lished in the proof.What establishes this Integer-ness is thecentral angle [ca] that [for the purposes of thisdiscussion] is set, and used, unchanged, through-out a calculation. All of the new Number sys-tem's units are in terms of the central anglethat's employed.It's a bit like Complex Numbers, A[ca] andB[ca] are conjugated pairs, but it's all Inte-gers[ca].It's [another] whole new way of doing Maths.Very flexible - because each central angleestablishes a different set of Integers.And, obviously, above, they map Exactly totraditional Integers.Basically, it's why, as one scales simple sine-and cosine-based plots smaller and smaller, nodiscontinuities will ever show up, even thoughthere're standard difficult numbers all over theplace, in-there.It's part of the bridge, between simple Geom-etry and higher-level Maths, that I discussedwhile discussing my Proof of Fermat's LastTheorem.What I envision is that, through bi-directionaltransformations, these new Integers can beMapped into the 'sticky-parts' of all manner ofProblems, Exactly, thereby, enabling folks toproceed beyond the 'sticky-parts'.All I'm doing, at this 'time', is saying, Hey,here's some new Integers. [I've been usingthem for as long as I can remember, withoutrealizing that they were, in fact, Integers thatcan, infact, transform to traditional Numbers.In my Proof of Fermat's Last Theorem, forinstance, I Proved that, Given:X^2 + Y^2 = Z^2for all n > 2, either X[ca] or Y[ca] is al-ways > 1 and [ca] is always > 0.] === Subject: Re: Yikes! I've Broken Incommensurability!In sci.math.num-analysis, ken:> In sci.math.num-analysis, ken> :> [...]> In any event, if A=1 and B=1, how does C=sqrt(2)> become rational by moving the point on the semicircle> around, or mutating the radius?> [...]> It's in the Proof I posted.> Given a circle with radius sqr(2), becausesqr(2) = 4. Did you mean sqrt(2)? Admittedlyit depends on the language package; the only placeI've personally seen sqr() is in a variant of Pascal.> A^2 + B^2 = C^2, always,Given the constraints of the circle, yes.> [A[ca] + B[ca]] = sqr(2)> in which A[ca] and B[ca] are EquivalentIntegers =in the Number system that's> established in the Proof=, measurable in> the same Units of central angle, eachone unit, but positional.> It's what I'm 'excited ' about - a whole new> Number system, that's extremely-rich, and> in which all numbers are Integers is estab-> lished in the proof.> What establishes this Integer-ness is the> central angle [ca] that [for the purposes of this> discussion] is set, and used, unchanged, through-> out a calculation. All of the new Number sys-> tem's units are in terms of the central angle> that's employed.> It's a bit like Complex Numbers, A[ca] and> B[ca] are conjugated pairs, but it's all Inte-> gers[ca].> It's [another] whole new way of doing Maths.> Very flexible - because each central angle> establishes a different set of Integers.> And, obviously, above, they map Exactly to> traditional Integers.If they map exactly then you might have a problem.In the realm of standard integers:Assume A/B = sqrt(2), and A and B are integers with no common factors(i.e., no D > 1 such that A/D and B/D are integers).Then A^2/B^2 = 2and therefore A is even.Write A = 2*Cthen 4*C^2/B^2 = 2or C^2/B^2 = 1/2or B^2/C^2 = 2and therefore B is even.Therefore A and B have a common factor 2, contradictingthe original assertion that they don't.Hence A/B = sqrt(2) is impossible for straight integers,unless both A and B are equal to 0, which is fairly useless.This [ca] Integer mapping may address this but if so it'snot clear to me exactly how.> Basically, it's why, as one scales simple sine-> and cosine-based plots smaller and smaller, no> discontinuities will ever show up, even though> there're standard difficult numbers all over the> place, in-there.> It's part of the bridge, between simple Geom-> etry and higher-level Maths, that I discussed> while discussing my Proof of Fermat's Last> Theorem.> What I envision is that, through bi-directional> transformations, these new Integers can be> Mapped into the 'sticky-parts' of all manner of> Problems, Exactly, thereby, enabling folks to> proceed beyond the 'sticky-parts'.The problem above has a sticky-part -- namely, thecontradiction. Perhaps you can clarify with thisnew math how one resolves around this issue?> All I'm doing, at this 'time', is saying, Hey,> here's some new Integers. [I've been using> them for as long as I can remember, without> realizing that they were, in fact, Integers that> can, infact, transform to traditional Numbers.> In my Proof of Fermat's Last Theorem, for> instance, I Proved that, Given:> X^2 + Y^2 = Z^2> for all n > 2, either X[ca] or Y[ca] is al-> ways > 1 and [ca] is always > 0.]I see no N and am not sure what X[ca] means. You'llprobably want to define precisely how one adds X[ca]and Y[ca], subtracts, multiplies, and divides.#191, ewill3@earthlink.netIt's still legal to go .sigless. === Subject: Re: Finite Difference vs. Finite Element by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3HCkc009209;Hi all,I can see that most of the problem with finite difference is dealingwith geometries. But, the recent meshfree methods like Least Squarebase Finite Difference (LSFD) solves this problem. It is veryversatile that, I am trying to implement that in moving boundaryproblem. And the entire solver just takes in the grid points. We justhave to ensure the uniformity of grid distribution to some extent.And also, programming the solver took me just 4 hrs (for a non linearpoisson equation).Even some versions of Meshfree FEM is available. But, i donot havemuch knowledge in that.Shyam>Hallo John,>here some of my experiences regarding FD and FE from the users aswell>as from the programers viewpoint.>1. I started in 1990 with electromagnetic computations by using theFD>method>with a program we had bought for this purposes. The computations weretwo->dimensional. The results were fine, differences to experiments werearound>5%>in the most cases. Hardware resources to be used were rather low,maximum>number of degrees of freedom (DOF) was 5000. The grid was regularas it>was the case with all commercial FD programs I knew at this time. Soit was>a more or less intricate task to force irregular geometries in suchgrids,>sometimes one had to write separate programs to compute the grid fora>special system.beginnig>of>the 90s. The basic mathematical routines were not so difficult to be>developed.>The efforts to be spent regarding the user interface were muchhigher>(and much more boring). The results were accurate in the same manneras>described above, which means there were only differences around 5% in>comparison to experiments or analytical solutions.>This program too used regular grids. During this time I could notfind a way>to express the finite differences for the general case of anirregular grid.>E.g.>at nodes belonging to several triangles (may be four or five) I foundit>difficult to>express d/dx, d/dy, d/dz by any transformation of coordinates in away,>taking>all triangles into account and this by a general algorithm.>electromagnetic computations. With the irregular grid which can beapplied>easily in the FE method theoreticaly all geometries can be computednow.>Problems sometimes happend with meshing of 3D grids but thoseproblems>slowly vanished from update to update. Nearly everthing is fine now,the>results>show a good accuracy, being in the same range as with the FD method.Only>the hardware resources to be used , especially disk space, are huge.>www.max-baermann.de>John Hernlund schrieb in Nachricht ...> There,> I have been reading a lot of literature on the finite differenceand>finite>element methods recently and have found a great variety of opinions>regarding>which is best for this or that problem. In programming both types Ihave>found that both can produce nice results. I would like to hear more>opinions>on the matter however...>What do you think?href=http://www.public.asu.edu/~hernlund/>http:// www.public.asu.edu/~hernlund/ Hi all,> I can see that most of the problem with finite difference is dealing> with geometries. But, the recent meshfree methods like Least Square> base Finite Difference (LSFD) solves this problem. It is very> versatile that, I am trying to implement that in moving boundary> problem. And the entire solver just takes in the grid points. We just> have to ensure the uniformity of grid distribution to some extent.> And also, programming the solver took me just 4 hrs (for a non linear> poisson equation).> Even some versions of Meshfree FEM is available. But, i donot have> much knowledge in that.> Shyam>Hallo John,>here some of my experiences regarding FD and FE from the users as> well>as from the programers viewpoint.>1. I started in 1990 with electromagnetic computations by using the> FD>method>with a program we had bought for this purposes. The computations were> two->dimensional. The results were fine, differences to experiments were> around>5%>in the most cases. Hardware resources to be used were rather low,> maximum>number of degrees of freedom (DOF) was 5000. The grid was regular> as it>was the case with all commercial FD programs I knew at this time. So> it was>a more or less intricate task to force irregular geometries in such> grids,>sometimes one had to write separate programs to compute the grid for> a>special system.> beginnig>of>the 90s. The basic mathematical routines were not so difficult to be>developed.>The efforts to be spent regarding the user interface were much> higher>(and much more boring). The results were accurate in the same manner> as>described above, which means there were only differences around 5% in>comparison to experiments or analytical solutions.>This program too used regular grids. During this time I could not> find a way>to express the finite differences for the general case of an> irregular grid.>E.g.>at nodes belonging to several triangles (may be four or five) I found> it>difficult to>express d/dx, d/dy, d/dz by any transformation of coordinates in a> way,>taking>all triangles into account and this by a general algorithm.>electromagnetic computations. With the irregular grid which can be> applied>easily in the FE method theoreticaly all geometries can be computed> now.>Problems sometimes happend with meshing of 3D grids but those> problems>slowly vanished from update to update. Nearly everthing is fine now,> the>results>show a good accuracy, being in the same range as with the FD method.> Only>the hardware resources to be used , especially disk space, are huge.>www.max-baermann.de>John Hernlund schrieb in Nachricht ...> There,> I have been reading a lot of literature on the finite difference> and> finite>element methods recently and have found a great variety of opinions> regarding>which is best for this or that problem. In programming both types I> have>found that both can produce nice results. I would like to hear more> opinions>on the matter however...>What do you think?href=http://www.public.asu.edu/~hernlund/>http:// www.public.asu.edu/~hernlund/I am a little confused about the numerical stability issue. I think there>are at least two kinds of stability, one is D-Stable(or zero stable), the>other is absolute stable(or eigenvalue stable, time stable). Let's>restrict our mind to the absolute stable definition, which means that for>the fixed time step, as time goes to infinity, the numerical solution is>always bounded.>But this definition is only designed for the model equation >du/dt=c u>where the real part of c is negative: Re(c)<0.>I guess the idea is that the stability is a property of a scheme,>independent of the actual problem. If a scheme is stable for the model>equation, it should be also stable for other equations. But is this true?>How can we justify it?>Another thing is how to choose the proper time step for a given equation>using some scheme, say Euler scheme. The stability region is dependent on>the actual equation we are solving. We could find its Jacobian and then>the eigenvalues. But what if the eigenvalues are positive? Then all these>stability theories are no longer working.>For example, let's try to solve >du/dt=u>u(0)=1>using Euler scheme: u(n+1)=u(n)+dt*u(n).>How can I know whether my numerical solution is stable or not? Since the>actual solution is unbounded, the previous definition is meaningless.>How can I choose a proper time step?>I did some quick numerical experiments,it turned out that when the time>step is small enough, the numerical solutions are very close to the actual>solution. But when it is not that small, the numerical results are very>different from the exact one. But is this due to numerical accuracy or>stability? I am really confused here.>Shi Jin> The issue you mentioned (relative versus absolute stability) is one> couple of tutorials for a course in fluid-structure interaction I am> teaching. See> http://caswww.colorado.edu/courses.d/FSI.d/FSI.Ch03.d/FSI.Ch03 .pdf> http://caswww.colorado.edu/courses.d/FSI.d/FSI.Ch04.d/FSI.Ch04 .pdf> and recommended refs there.> One problem few books mention is overstabilty: the masking of physical> instability, eg airplane flutter, by the numerical dissipation of an> integrator like Backward Euler. Ideally the numerical scheme should> have exactly the same stability region as the differential equation> being integrated. Only a few integrators realize that goal.> As to the last point: if the time step is extremely small, roundoff> and truncation errors can dominate the discretization error. In my> experience that is not a very likely occurrence if one works in double> precision, unless you are dealing with a singular perturbation problem> (say, a differential difference system or a delayed differential> equation)Overstability is also common in circuit simulation, e.g. the Gear integrator in SPICE, which will often make an oscillator look like an amplifier. === Subject: Re: solution of polynomials with real coeffs by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3HKZ6728135;Routh test (also called Routh-Hurwitz criterion)forstability can be helpful in determining the locationof roots of a polynomial of whatever order.However, a polynomial of odd order will have at leastone real root. There is the Cardan's formulafor cubic polynomials. The quartic formula will dofor order four polynomials. So for other even orderpolynimials, the use of products of these formulas in the manner the quartic formula isdeveloped can be applied to determine the roots of whatever polynomial with real coefficients.>I would like to know if there is a condition using which I candetermine >if a given polynomial with real coeffs has real roots? For my case, I>know that the polynomial order never exceeds 15.>Ex:- 2nd order polynomial has real roots if b^2 - 4ac > 0. I amlooking >something in this line of thought for a general polynomial.>Any help is greatly appreciated. === Subject: Vector Norms and matrics norms in Numerical Computing by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i3HKZ5L28110;What are Norms .Of How many types and What about Vector Norms andmatrics norms in Numerical Computing