mm-255 === Subject: New book: Mathematical ConstantsHello!I am pleased to announce the publication of my book, Mathematical is an encyclopedia bursting with information and over 600 pages long. A table of contents and two sample essays are available (in PDF) at:http://us.cambridge.org/titles/catalogue.asp?isbn= 0521818052where one may also order a copy for $95. My website:http://pauillac.inria.fr/algo/bsolve/has three additional sample essays from the book, as well as supplementary materials (which will continue to expand as time passes).Cambridge has done a beautiful job in producing my book. It is far more encompassing and detailed than my website ever was. By the way, please note my new e-mail address. My old e-mail address (at MathSoft) should no longer be used.As far as I know, there is no book like this! The breadth and depth of the coverage, as well as many exotic topics (e.g., from statistical mechanics and theoretical computer science) and unsolved problems, should make Mathematical Constants a joy to === New book: Mathematical Constantsyipee!!! this book will be a true pleasure to read!!congratulations ... in the past we have === odium veritas paritSubject: Q: Norm of difference of fractional powers of operatorsand yet another norm inequality: $| A^{1/r} - B^{1/r} | le | A-B |^{1/r}$for $1 le r le infty$, where $A,B$ are positive operators ona Hilbert space and $| |$ is the operator norm.This is trivial for $r = 1$ and $r = infty$. For $r = 2$ I gota proof from Robert Israel some time ago. The case $r = 2$ alsofollows from Corollary 2 (with $p = infty$) in Fuad Kittaneh, Inequalities for the Schatten $p$-Norm. IV, Commun. Math. Phys. 106, 581-585 (1986)The case $r = 2^n$ can then be proved by induction.Does anybody know a proof for the other values of $r$? I don't seean easy way to generalise the === proofs for $r = 2$.MarkusSubject: Re: Number of cyclic digraphs without sources nor sinks> A cyclic digraph is a digraph in which every vertex is in a cycle. In> other words, cyclic digraps are non-acyclic digraphs.> I'm not sure this is correct. The paw (a graph with 4 vertices, 3 of> them forming a triangle and the fourth vertex adjacent to exactly one of> those three) can be oriented with a cycle. This oriented paw is now a> non-acyclic digraph, but it is not the case that every vertex is in a> cycle.> Now, if you are just looking for non-acyclic digraphs, since there is> much work done on acyclic graphs, you can just subtract those off the> number of total digraphs. But I assume you want every vertex to be in a> cycle. Is this not equivalent to the digraph being strongly connected ?No. For example take two directed triangles and add a directed edge from onevertex in one triangle to some vertex in the other triangle. Then everyvertex is contained in a directed cycle, but the graph is not stronglyconnected.The number of strongly connected digraphs on n nodes is a lower boundhowever and this is known at least partially. Here's the entry from Sloane'sOEIS:------------------------------------------------- -------------------------------ID Number: A003030 (Formerly M5064)URL: http://www.research.att.com/projects/OEIS?Anum=A003030Sequence : 1,1,18,1606,565080,734774776,3523091615568, 63519209389664176,4400410978376102609280, 1190433705317814685295399296, 1270463864957828799318424676767488Name: Strongly connected digraphs with n labeled nodes.-------------------------------------------------------- ----------------------If I have this right the number of digraphs on n vertices for which everyvertex has indegree at least one and outdegree at least one is the same asthe number of n x n 0-1 matrices with zeros on the main diagonal and havingno all zero row and no all zero column. I calculate these numbers for nfrom 1 to 4 to be: 0, 1, 18, 1699. This === sequence I did not find in theOEIS.--Edwin ClarkSubject: Re: Number of cyclic digraphs without sources nor sinks>A cyclic digraph is a digraph in which every vertex is in a cycle. In>other words, cyclic digraps are non-acyclic digraphs.>In a digraph, a source is a vertex with in-set of size zero; a sink is>a vertex with out-set of size zero.If a vertex is in a cycle, how could it be a source or sink?>I would like to know the number of labeled cyclic digraphs on n>vertices without sources nor sinks.I can't give you an exact count, but there are some obvious bounds.I assume you're talking about simple digraphs.For a lower bound, consider all digraphs containing the cycle 1->2->3->...->n->1 which contains all vertices. Then there are n(n-2) other possible directed edges, so thereare 2^(n(n-2)) labelled digraphs containing the given cycle, allof which are cyclic.For an upper bound, there are 2^(n(n-1)) labelled digraphs in all.The bounds are quite trivial, but the ratio (upper bound)/(lower bound)is only 2^n which is small compared to the number.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === Vancouver, BC, Canada V6T 1Z2Subject: Re: Number of cyclic digraphs without sources nor sinks>A cyclic digraph is a digraph in which every vertex is in a cycle. In>other words, cyclic digraps are non-acyclic digraphs.>In a digraph, a source is a vertex with in-set of size zero; a sink is>a vertex with out-set of size zero.> I can't give you an exact count, but there are some obvious bounds.> I assume you're talking about simple digraphs.Here's a better lower bound. Consider a random digraph on n vertices, whereeach possible directed edge has probability 1/2 of being present, and allare independent. The probability that a given vertex is in a 2-cycleis 1 - (3/4)^(n-1). Using the FKG inequality, the probability that all n vertices are in 2-cycles is at least (1 - (3/4)^(n-1))^n, whichis 1 - O(q^n) as n -> infinity if 3/4 < q < 1.Thus the number of cyclic digraphs is at least (1 - (3/4)^(n-1))^n 2^(n(n-1)).Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === Vancouver, BC, Canada V6T 1Z2Subject: This week in the mathematics arXiv (11 Aug - 15 Aug)Here are this week's titles in the mathematics arXiv, available at: http://front.math.ucdavis.edu/ http://front.math.ucdavis.edu/submissionsThis week in the mathematics arXiv may be freely redistributedwith attribution and without modification.Titles in the mathematics arXiv (11 Aug - 15 Aug)-------------------------------------------------AC: Commutative Algebra-----------------------math.AC/0308113 and norm || ||. LetT(t) be a strongly continuous semigroup of contractions generated byA, and suppose that the domain of A is the same as the domain of A^*(the adjoint operator of A). Suppose also that T(t) is normal(T(t)T(t)^*=T(t)^*T(t) for all t>0). Let B be a linear and boundedoperator. Then the semigroup S(t) generated by (A-BB^*) is alsonormal.T(t) is normal iff ||T(t)x||^2=||T(t)^*x||^2, t>0, x in H iffRe=Re, x in domain(A)=domain(A^*).Consequently, for S(t):Re<(A-BB^*)x,x>=Re<(A-BB^*)^*x,x>, x in domain(A).Hence S(t) is normal.I don't see how the identity Re=Re orRe<(A-BB^*)x,x>=Re<(A-BB^*)^*x,x> would imply the norms to be equal.Perhaps someone could help me with === the proof?-- Lauri Ylinenlauri.a.ylinen@tut.fiSubject: Re: perturbation of a normal semigroup>I am having problems with the following perturbation result. It is>Optimization, Vol. 17, No. 1, 1979).>Let H be a Hilbert space with inner product < , > and norm || ||. Let>T(t) be a strongly continuous semigroup of contractions generated by>A, and suppose that the domain of A is the same as the domain of A^*>(the adjoint operator of A). Suppose also that T(t) is normal>(T(t)T(t)^*=T(t)^*T(t) for all t>0). Let B be a linear and bounded>operator. Then the semigroup S(t) generated by (A-BB^*) is also>normal.Nonsense. If you're quoting correctly, there's something seriouslywrong with that paper.For a counterexample, consider the Hilbert space C^2 with [ 0 1 ] [ sqrt(2) 0 ]A = [ -1 0 ], B = [ 0 0 ] [ cos(t) sin(t) ]T(t) = [-sin(t) cos(t) ] is normal [ (1-s) e^(-s) s e^(-s) ]S(t) = [ -s e^(-s) (1+s) e^(-s) ]which is not normal since [ 0 4 s^2 e^(-2s) ]S(t) S(t)^* - S(t)^* S(t) = [ 4 s^2 e^(-2s) 0 ]>T(t) is normal iff ||T(t)x||^2=||T(t)^*x||^2, t>0, x in H iff>Re=Re, x in domain(A)=domain(A^*).What??? = = conjugate()so Re = Re always in all cases, normal or not.Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === Vancouver, BC, Canada V6T 1Z2Subject: Re: Euclid's BooksThe 'real' question was whether or I demand impunity from my brother.I perhaps overstated my case, and interpreted what my brother meant to theextreme.Does it matter...you better believe it!>Did Euclid himself actually, physically, write such books (or perhaps>dictate them to a slave)? Yes.>Did he do it without previous source material? No.>What is the real === question?Subject: Re: Euclid's Books> and this is why he thought the goal of the> Elements as a whole to be the construction of the so-called Platonic figures.I heard this too, long ago, but haven't seen any other reference to it at alluntil now. Is this a definite fact, in any sense? - or just an academicfolklore? I wonder, because an awful lot of Euclid (e.g. the Eudoxus stuff)doesn't seem in the slightest way connected to the Platonic solids.Anyone have any definite info on this?--------------------------------------------------------- --------------------- Bill Taylor W.Taylor@math.canterbury.ac.nz-------------------------------- ---------------------------------------------- Questions answered and answers questioned.--------------------------------------------------- === ---------------------------Subject: Assignment problemI came across the following question while working on a combinatoricsproblem. This is the very simplest version of the question.Given n sets of ordered pairs of integers, choose one from each pair so the resulting set of integers has the fewest members, i.e. the list of chosen integers has the fewest number of distinct values.Does anyone know if this is equivalent to one of the many types ofassignment problems out there? Is it NP-hard? I can write in as aninteger programming problem, but you.---------------------------------------------------------- --------Gary A. Ray voice: (253) 657-4165 (Desk)Boeing Company fax: (253) 657-4269P.O. Box 3999, MS 81-75 e-mail: === garyr@tahoma.ds.boeing.comSeattle, WA 98124-2499Subject: Re: Assignment problem>I came across the following question while working on a combinatorics>problem. This is the very simplest version of the question.>Given n sets of ordered pairs of integers, choose one from each pair so >the resulting set of integers has the fewest members, i.e. the list of >chosen integers has the fewest number of distinct values.>Does anyone know if this is equivalent to one of the many types of>assignment problems out there? Is it NP-hard? I can write in as an>integer programming problem, but I don't see that it really helps.Consider the pairs as edges in a graph, the integers corresponding tovertices. Then this is the Minimum Vertex Cover problem. It is NP-hard. Robert Israel israel@math.ubc.caDepartment of Mathematics http://www.math.ubc.ca/~israel University of British Columbia === Vancouver, BC, Canada V6T 1Z2Subject: Re: Assignment problem> Given n sets of ordered pairs of integers, choose one from each pair> so the resulting set of integers has the fewest members, i.e. the list> of chosen integers has the fewest number of distinct values.This is called Minimum Vertex Cover, and is certainly NP-hard ongeneral graphs. There are lots of results on approximating it, orcomputing it on various special classes of graphs.http://mathworld.wolfram.com/VertexCover.htmlhttp:// www.nada.kth.se/~viggo/wwwcompendium/node10.html David === desJardins2614Subject: Need help with steepest descentWould someone care to explain how to use the steepest descentalgorithm to minimize a function, say, F(x) = x1 * x1 + 25 * x2 * x2, -5 < x1,x2 < 5 :-). Links to web tutorials would also be appreciated.And no, this is not my homework, my homework is to minimizeRosenbrock's function :-). (I, uh, fell aspeep during the class :-(and now I can't find any references === steepest descent> Would someone care to explain how to use the steepest descent> algorithm to minimize a function, say, F(x) = x1 * x1 + 25 * x2 * x2, > -5 < x1,x2 < 5 :-). Links to web tutorials would also be appreciated.> And no, this is not my homework, my homework is to minimize> Rosenbrock's function :-). (I, uh, fell aspeep during the class :-(> and now I can't Andreycheck out mathworld. http://mathworld.wolfram.com/MethodofSteepestDescent.htmlhope you have a mathematica. if not, matlab. http://www-math.cudenver.edu/~aknyazev/teaching/98/4660/fp/r1/ the idea is to perturb a parameter by an increment i.( ratherarbitrary amount, you get to decide) by perturb, I mean add orsubtract. it doesn't matter whether you add first or subtract first.let's add incrememnt i to the x1 and x2, then look at what happens tothe F[x1, x2] that is look at it;s numerical derivative.so dfx1 = (f[x1+i,x2] - f[x1, x2])/ i and dfx2 = (f[x1,x2+i] - f[x1,x2])/ ithese are numerical derivatives. what happens if the derivative isnegative? this means you are moving downhill towards the local minima.you want to keepp this move.if this is positive than you're moving uphill then you want to makethe opposite move.flank the routine with while loop with conditions to stop it when itreaches a === certain minima criteria.good lucksean good luckSubject: Re: 8bit[[ This message was both posted and mailed: see the To, Cc, and Newsgroups headers for details. ]]> Would someone care to explain how to use the steepest descent> algorithm to minimize a function, say, F(x) = x1 * x1 + 25 * x2 * x2, > -5 < x1,x2 < 5 :-). Links to web tutorials would also be appreciated.> And no, this is not my homework, my homework is to minimize> Rosenbrock's function :-). (I, uh, fell aspeep during the class :-(> and now I can't find any references on the net.For a picture of a descent path down the Rosenbrock banana, see thepicture in the upper left corner of Would someone care to explain how to use the steepest descent> algorithm to minimize a function, say, F(x) = x1 * x1 + 25 * x2 * x2, > -5 < x1,x2 < 5 :-). Links to web tutorials would also be appreciated.Try adding the word gradient to your Google searches, and you'llprobably come up with better hits. The basic idea is that from somestarting point near your minimum (or from some arbitrary starting point ifyou have a nice convex function like the above) you repeatedly find thefunction's gradient and choose a new point in the opposite === direction.---Roy StognerSubject: Re: Small sample size and logit model>I have a sample of 6 observations and due to the conditions I cannot>make it bigger. Can I use any statistical models for such a sample? I>donot want to make a prediction, I just want to describe the past>period for wihch I have data. Can I use regresion? Can I use logit?>Perhaps you can recommend some books where this issue would be>discussed (small sample size, but I am NOT a mathematician, so I need> with 6 parameters (what is logit?)> you might desribe the six data points exactly, neglecting the data errors.> the rule of thumb is sqrt(of number of data points/2)>=number of parametrs.> here : 1, maybe 2.> software: see > http://plato.la.asu.edu/topics/problems/nlolsq.html> hth> dependentvariable takes only one of two possible values - zero or one. Youestimate the model using the maximum likelihood method.To ensure my understanding of your post I will ask one more question.Can I apply least squares method (preferably maximum likelihoodmethod) and just ignore t-test? Can I such === an estimation put into myph.d.?Subject: Help solving equation.E''(z) +(H/(z^2) - B).E(z)=(2*H)/(Z^3)* ... L S E(t).dt. (S is the integral sign). zWhere H and B are both constants.My Boundary conditions are E(1.1) = -0.45 and E'(500) = 0.Any assistance would be greatly appreciated. If you could e-mail me atandrewl at global.net.au I would === following equation.> E''(z) +(H/(z^2) - B).E(z)=(2*H)/(Z^3)* ...> L> S E(t).dt. (S is the integral sign).> zI'm assuming that your equation reads (in Mathematica notation) eqn = (h/z^2 - b) e[z] + z e''[z] == (2 h Integrate[e[t], {t, z, l}])/z^3By differentiating this equation, one can eliminate the integral from the resulting pair of equations leading to a _third_ order differential equations.> Where H and B are both constants.> My Boundary conditions are E(1.1) = -0.45 and E'(500) = 0.I assume that l = 500 then? You obtain a third boundary condition by evaluating your equation for z=l (where the integral vanishes): (h/l^2 - b) e[l] + l e''[l] == Physics, M013 Fax: +61 8 9380 1014The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling HighwayCrawley WA 6009 mailto:paul@physics.uwa.edu.au === AUSTRALIA http://physics.uwa.edu.au/~paulSubject: Re: Help following equation.> E''(z) +(H/(z^2) - B).E(z)=(2*H)/(Z^3)* ...> L> S E(t).dt. (S is the integral sign).> z> I'm assuming that your equation reads (in Mathematica notation)> eqn = (h/z^2 - b) e[z] + z e''[z] == > (2 h Integrate[e[t], {t, z, l}])/z^3That should have read eqn = e''[z] + (h/z^2 - b) e[z] == 2 h/z^3 Integrate[e[t], {t, z, l}]> By differentiating this equation, one can eliminate the integral from > the resulting pair of equations leading to a _third_ order differential > equations.> Where H and B are both constants.The equation reads b (3e[z] + z e'[z]) == (3h e[z])/z^2+(h e'[z])/z+ 3e''[z]+ z e'''[z]==0Mathematica returns a (formal) closed-form solution for this equation in terms of 1F2 hypergeometric functions involving roots of the cubic z^3 + h z - z + 3 h> My Boundary conditions are E(1.1) = -0.45 and E'(500) = 0.> I assume that l = 500 then? > You obtain a third boundary condition by evaluating your equation for > z=l (where the integral vanishes):> (h/l^2 - b) e[l] + l e''[l] == 0And that should have read e''[l] + (h/l^2 - b) e[l] == Physics, M013 Fax: +61 8 9380 1014The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling HighwayCrawley WA 6009 mailto:paul@physics.uwa.edu.au === AUSTRALIA http://physics.uwa.edu.au/~paulSubject: Grouping, non-repeatingLet's say you're going to have a dinner party group of 24 people (12couples). The group plans to get together monthly to have parties. The group has 2 rules: 1. Every couple must host at least once. 2. Any coulples dining together one month should not be paired again. For instance, in Januarycouple A hosts couples B, C, DE hosts F, G, HI hosts J, K, LIn February (and there after) couple A should never be in the samehouse as couples B or C, and A should never have to host again.So my question is this. Is there a generic equation (or relationship)that could tell me the minimum number of months and/or group sizesthat must exist in order for the === rules to stand up?Subject: Re: Grouping, non-repeatingRelated problems are the progressive party problem and the social golfer problem. Google will help you find paperson these problems. These are actually very difficultproblems to solve.-------------------------------------------------------- --------Erwin Kalvelagenerwin@gams.com, http://www.gams.com/~erwin------------------------------------ ----------------------------> Let's say you're going to have a dinner party group of 24 people (12> couples). The group plans to get together monthly to have parties. > The group has 2 rules: 1. Every couple must host at least once. 2. > Any coulples dining together one month should not be paired again. > For instance, in January> couple A hosts couples B, C, D> E hosts F, G, H> I hosts J, K, L> In February (and there after) couple A should never be in the same> house as couples B or C, and A should never have to host again.> So my question is this. Is there a generic equation (or relationship)> that could tell me the minimum number of months and/or group sizes> that must === exist in order for the rules to stand up?Subject: Re: Computational Evolution without velocitiesSorry, there was a typo! I should have squared DeltaTime.This is by no means cartoon physics. People are using this in researchsimulations and I'm wondering if there is a sound basis or spoon.************************************************Dr. Patrick Bangerthttp://www.knot-theory.orgResearch Instructor have heard the folklore that when one wants to simulate some structure> Newton's laws (i.e. non-relativistic) one can neglect the fact that the> update formula> x_new = x_old - Force DeltaTime / 2 Mass> holds for any simulation and we do not need to take care of velocities.What> I would like to know is does anyone know whether this holds only under> certain conditions (evolution at equilibrium, etc.) and/or knows ofresearch> papers that deal with this issue and perhaps give a proof of this claim asa> theorem. The strange thing is that apparently this is very well known and> widely practised but you very much! Best,> Pat> Free your mind. There is no spoon.> ************************************************> Dr. Patrick Bangert> http://www.knot-theory.org> Research Instructor for === Mathematics> International University BremenSubject: Re: Computational Evolution without velocitiesX-Enigmail-Version: 0.76.1.0X-Enigmail-Supports: pgp-inline, pgp-mimeThe underlying assumptions allowing to neglect the velocityterm is that it always remains negligible with respect to the forceterm. This situation occurs effectively in very viscous fluids,to random fluctuations. Dan> Sorry, there was a typo! I should have squared DeltaTime.> This is by no means cartoon physics. People are using this in research> simulations and Free your mind. There is no spoon.> ************************************************> Dr. Patrick Bangert> http://www.knot-theory.org> Research Instructor for have heard the folklore that when one wants to simulate some structure>>Newton's laws (i.e. non-relativistic) one can neglect the fact that the>>update formula>>x_new = x_old - Force DeltaTime / 2 Mass>>holds for any simulation and we do not need to take care of velocities.> What>>I would like to know is does anyone know whether this holds only under>>certain conditions (evolution at equilibrium, etc.) and/or knows of> research>>papers that deal with this issue and perhaps give a proof of this claim as> a>>theorem. The strange thing is that apparently this is very well known and>>widely practised but no one I've spoken to knew where to mind. There is no spoon.>>************************************************>>Dr. Patrick Bangert>>http://www.knot-theory.org>>Research Instructor for Mathematics>>International University === BremenSubject: Re: Computational Evolution without velocities> simulate some structure> Newton's laws (i.e. non-relativistic) one can neglect the fact that the> update formula> x_new = x_old - Force DeltaTime / 2 Mass> holds for any simulation and we do not need to take care of velocities. What> I would like to know is does anyone know whether this holds only under> certain conditions (evolution at equilibrium, etc.) and/or knows of research> papers that deal with this issue and perhaps give a proof of this claim as a> theorem. The strange thing is that apparently this is very well known and> widely practised but no one I've spoken to knew where to find the There is no spoon.> ************************************************> Dr. Patrick Bangert> http://www.knot-theory.org> Research Instructor for Mathematics> International University BremenIf F is constant, the above formula will give a constant incremental distance, i.e a constant velocity. In the absence of drag, that ain't newtonian. Perhaps there was an assumption of a viscous === retarding force.-- Joe LegrisSubject: Re: Computational folklore that when one wants to simulate some structure> Newton's laws (i.e. non-relativistic) one can neglect the fact that the> update formula> x_new = x_old - Force DeltaTime / 2 MassYou've got to stop reading those textbooks on cartoon === physics.Tom DavidsonRichmond, VASubject: Re: Computational All,> I have heard the folklore that when one wants to simulate some structure> Newton's laws (i.e. non-relativistic) one can neglect the fact that the> update formula> x_new = x_old - Force DeltaTime / 2 Mass> holds for any simulation and we do not need to take care of velocities. This equation is quite obviously wrong, for the simple fact that it does not even have consistent units. The left hand side has dimensions of [Length]; the right hand side is a length minus a velocity. You can'tsubtract a velocity from a length and get anything === meaningful.-- Gordon D. Pusch Subject: Re: Computational folklore that when one wants to simulate some structure> Newton's laws (i.e. non-relativistic) one can neglect the fact that the> update formula> x_new = x_old - Force DeltaTime / 2 Mass> holds for any simulation and we do not need to take care of velocities.> This equation is quite obviously wrong, for the simple fact that it> does not even have consistent units. The left hand side has dimensions> of [Length]; the right hand side is a length minus a velocity. You can't> subtract a velocity from a length and get anything meaningful.> -- Gordon D. PuschHe obviously meant x_new = x_old - Force * (DeltaTime)^2 / (2 * Mass)However, that does not seem right. Newton's second law is x = F/mIf you wish to eliminate velocities, then in an obvious notation, x(+) = 2x(0) - x(-) + (F/m) * (delta_t)^2will step forward. But you then have to have the values x(0) and x(-).I don't know what algorithm the OP could have been talking about sincedoesn't make sense physically.-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- === Bertolt Brecht, Galileo.Subject: Re: Computational Evolution without velocities* Gordon D. Pusch:>> Newton's laws (i.e. non-relativistic) one can neglect the fact that the>> update formula> x_new = x_old - Force DeltaTime / 2 Mass> holds for any simulation and we do not need to take care of velocities. > This equation is quite obviously wrong, [...]He may be talking about Stoermer-Verlet algorithms,which have update formulae that are indeed relativelysimple and don't contain the velocity. Also, they areused for n-body problems and hard sphere fluids and haveother interesting properties (look up symplectic algorithms).x_new = 2 x_old - x_older + DeltaTime^2 Force / Mass + O(DeltaTime^4)would be a correct formula. That is not far off from hisversion, and given that he heard it as folklore, it maywell be it.(Of course, the velocity is not neglected here. It is implicitlycontained in the ODE of 2nd order that is discretized directly.It could also be rewritten into two ODEs of 1st order and discretizedafterwards, which would lead to better known === schemes like Runge-Kutta.)-- RupertSubject: Wavelets package in JavaHas anyone ever used a wavelet (de)composition package under Java? Ora program that can be used with Java (looked at Jython, but I didn'tfind a wavelet package for that)?Please === Wavelets package in Java> Has anyone ever used a wavelet (de)composition package under Java? Or> a program that can be used with Java (looked at Jython, but I didn't> find a wavelet package for that)?> Please mail replies also to http://www.bearcave.com/software/java/wavelets/. Alsohttp://www.bearcave.com/misl/misl_tech/wavelets/index.html === (same site).Subject: Coupled system of non-linear PDEHi everybody,I'm trying to solve a non-linear PDE system in u(x,t) e v(x,t) ofthe form:--------------------------------------------------------- ---------------------------diff(u(x,t),t)-a(u,v)*diff(u(x,t),x ,x)-b(u,v)*diff(v(x,t),x,x);diff(v(x,t),t)-c(u,v)*diff(u(x,t), x,x)-d(u,v)*diff(v(x,t),x,x);with Neumann BC on x=0 e x=L. The initial conditions are:u(x,0) = {u_1 for x I have to implement a specific algorithm.> Just one question: if one talks about a Bernstein-B.8ezier coefficient> of a tensor-product surface one means a control point with x,y,z> -coordinates, right?I dont know what specific classification of a tensor product surface yourefer to in your mail : a Generic B-Spline or a NURBS or a BezierSurface (since you seem torefer to Bernstein polynomials and Bezier Basis Functions in your mail), and therefore,Starting with a generic definition, of a Tensor-Product Surface,S(u,v) = B i* Pi(x,y,z)where Bi is a generic Coefficient (a product of two curve coefficientsfor a surface)and therefore:[a] The Coefficient represents a term (of the polynomial function)that maps a point in R(3) space toR(2) (or parametric) space. The point is the control point of theTensor Product Surface for the specific u,vvalue in the parametric space where u -> [a,b] and v->[c,d][b] A Point on the tensor-product surface and a Control Point of thetensor-product surface are separate concepts and dont mean the same thing.hope this solves the problem.Anup R. === Joshiwww.me.mtu.edu/~arjoshiSubject: Re: Bernstein-B.8ezier coefficients?> I have to implement a specific algorithm.> Just one question: if one talks about a Bernstein-B.8ezier coefficient> of a tensor-product surface one means a control point with x,y,z> -coordinates, right?I think it means v = === fn(x,y,z)Subject: WORTH CHECKING OUT!!!!! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8MGMjk15740;Has anybody looked at the posting - prime number distribution, solved? Please give me some feedback, of any kind, even criticism could be helpful. If nobody is interested, I'll leave you all alone, but if you are interested, I have more to add. I'm 34 and I've worked on this problem most of my life, so I obviously have a lot invested in it. This is not a crackpot theory, so please === support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8MGMo515777;In the previous post, I should have said - any four ODD primes andindeed, any four odd numbers since they all share the samecharacteristic moduli's stated.Please read & respond to prime number === Numerical Library in Java by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8NFuOB11791;A new book with a numerical library in Java was published in August Title: A Numerical Library in Java for Scientists and EngineersAuthor: Hang T. LauISBN: 1584884304publisher: CRC Press email: orders@crcpress.com http://www.crcpress.com/The book contains the source code of a comprehensive numerical libraryin Java. The library can be used to solve a wide range of numericalproblems in linear algebra, the numerical solution of ordinary andpartial differential equations, Fast Fourier Transforms, Time SeriesAnalysis, optimization, parameter estimation and special functions === ofmathematical physics.Subject: Solving Nonlinear Equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8NDBC900577;Hi I am chemical engineer and I am solving some nonlinear equations F = F(x) I was using some solvers for nonlinear equation. But as thesesolvers calculate numerical Jacobian, it perturbs each variable andcalculates jacobian. This makes it very time consuming. Thats why Iused approximate jacobian (It is standard technique in many chemicalproblems) and used Newton raphson method to solve the equations. Now my problem is : Many time I don't have very good initial guess and it is sometimesfar away from the solution. So when i use Jacobian (either approximateor numerically calculated exact one), and solve for J * delX = -F for delX, the step delX is quite big. If I use X(k+1) = X(k) + delX then every time the soution goes somewhere and it fails..NEVERconverges. So i use some sort of relaxation factor X(k+1) = X(k) + w * delX where w = 0 to 1 So in the beginning iterations, i use small value of w (say 0.1 o0.2), and in later sateg increase it towarss 0.8, 0.9 when it is closeto convergence criteria. This works and i get solution. But value of w is manual. I am notusing any standard techniqus to get it. So is there any justificationfor this factor? is === Nonlinear Equation >Hi > I am chemical engineer and I am solving some nonlinear equations > F = F(x) > I was using some solvers for nonlinear equation. But as these >solvers calculate numerical Jacobian, it perturbs each variable and >calculates jacobian. This makes it very time consuming. Thats why I >used approximate jacobian (It is standard technique in many chemical >problems) and used Newton raphson method to solve the equations. > Now my problem is : > Many time I don't have very good initial guess and it is sometimes >far away from the solution. So when i use Jacobian (either approximate >or numerically calculated exact one), and solve for > J * delX = -F > for delX, the step delX is quite big. > If I use X(k+1) = X(k) + delX > then every time the soution goes somewhere and it fails..NEVER >converges. > So i use some sort of relaxation factor > X(k+1) = X(k) + w * delX where w = 0 to 1 > So in the beginning iterations, i use small value of w (say 0.1 o >0.2), and in later sateg increase it towarss 0.8, 0.9 when it is close >to convergence criteria. > This works and i get solution. But value of w is manual. I am not >using any standard techniqus to get it. So is there any justification >for this factor? is there any way (fast and reliable) to find the wheel. what you are doing is the damped Newton method with a handcrafted damping. there is a complete and sound theory for thisdeveloped in the seventies of the last century and even downloadable code,the nleq-series of codes from the codelib/elib.(deuflhards group). the essential idea is to choose w (in your notation) such that norm( F(X(k)+w*delx(k)) )^2 <= (1-factor*delx)*norm(F(X(k)))^2where 0 < factor < 1/2 for example factor =0.01 and norm(.)^2 is a scalar product, for example norm=euclidean lengthor norm(F)=euclidean length of (A*F) with a fixed matrix A for example A=approximate Jacobian. One can show that this converges to a solutionif on the set of x's with F-values norm(F(x)) <= norm(F(x(0))) the Jacobianbecomes never singular. for codes see http://plato.la.asu.edu/topics/problems/ === zero.htmlhthpeterSubject: Re: Solving Nonlinear Equation> Hi> I am chemical engineer and I am solving some nonlinear equations> F = F(x)To solve F(x)=0, apply an optimization routine to f(x) = sum F_i(x)^2.Many excellent routines are available on the web, see, e.g., http://www.mat.univie.ac.at/~neum/glopt/software_l.html[~ is a === tilde]Arnold NeumaierSubject: equations (not single equation) by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8NFuPU11799;hi actually in my previous posting I am using several equations and notsingle === equation. PrashantSubject: analysis-conneted by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8NFugr11872;Let p : [a,b] ---> R^3 be a continuous path and a < c < d < b. LetC={p(t) | c < t < d === }. Must p^-1 (C) be path-conneted? Subject: analysis-connected by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8NFubw11851;. Let A is a subset of R^2 be path- conneted. Regarding A as a subsetof the xy-plane in R^3, show that A is still path-conneted. Can youmake a === similar argument for A connected?Subject: suggest? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8NKTq430748;I don't know for sure, but the way I represented primes in theposting prime number distribution - solved may help you - I've been planning on trying to use it to attack the zeta fcn myself. I've kind of hit a roadblock, so please write something there when you check it out. It only uses (maybe) fourth grade math until I give a more rigorous treatment, and even that part isn't really that hard. Here's a taste of the way it represents primes:105-2^x=y ,(integer x, y<121)y=103,101,97,89,73,41,-23105+2^x=yy=107,109,113for larger y, the eqn is different. To get the rest of the === primes<121, need a bit more, see the posting.AaronSubject: Program code for finding eigenvector in Casio Calculator by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id h8OJQWL23555;hi,anyone know how to write or solve the eigenvector in Casio 9850GBmodel. i know hp and TI do. but not in casio 9850GB.if someone know,pls email to === me.that will be very helpful.thxSubject: Confidence interval for intra class correlation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id of a Continuous variable byusing intra-class correlation. Can somebody tell me how to calculatethe 95% confidence interval === programHow do we find angles in (-180 deg, +180 deg] domain?(If we use dot product, then +45 deg and -45 deg can not bedistinguished)Are there any particular functions in Matlab using which I can findthe angles in the (-180 deg, +180 === program>How do we find angles in (-180 deg, +180 deg] domain?>(If we use dot product, then +45 deg and -45 deg can not be>distinguished)>Are there any particular functions in Matlab using which I can find>the angles in the (-180 deg, simple method: for vectors a=(a1,a2) andb=(b1,b2), do the dot product first, then calculate the determinanta1b2 - a2b1. If the result is >0 then a is clockwise relative to b,if < 0 then a is ccw from b (and if =0 then a and b are aligned butthen the angle will be 0 anyway). You then have to adjust your angleaccordingly.(you will notice that this determinant is really the z component ofthe 3d cross-product).In 3d I think you would have to arbitrarily define an orientationrelative to === the plane that contains the 2 vectors. Subject: Information about Wavelet AnalysisI am looking for information (book name === internet source) about Wavelet analysis.Subject: Re: Information about Wavelet AnalysisS. Mallat, A Wavelet Tour of Signal Processing, Academic Pressis said being the bible of === wavelet analysis.AxelSubject: Re: analysis of a large ODE system. as per Dr. Sspellucci's suggestion.to someone in the forum and perhaps in my school. I went to the library and checked out the book Dr Spellucci hassuggested.( hahn's stability of motion in conjunction with thejacobians and eigenvalues) and guess what?it's been recalled by another person in my school. I wonder who elsefrom my school is reading these forums... I think I'll recall it whenI return it.what are the odds of two people from same school looking for the samebook on the same week right after the book was suggested in anewsgroup.specially when the suggestions for it came after eigenvalues comments.perhaps the book contains some information not found anywhere in theliterature. ( can't imagine what that will be) Though, if you need thebook like me, then you can't be all that great. LOL-----To Dr. Spellucci, that's a great book btw, Dr. Spelluci. (it's very mathematicallyformalized.) perhaps that's why you have suggested it.I have acces,now to some nice introductory materials into nonlineardynamics. I'm sure I would learn a lot based on your comments andthose of others. It really gives me what to focus on when reading thematerials.I really appreciated your comments as well as others. You have alwaysbeen so helpful. I wish people in academics were more like you. (Isound bitter. LOL)respectfully yours,sean from === UCIrvineSubject: Re: question about 3d plottin software data table limit> Hy, I have a quite noise problem, by a wavelet analysis I've obtained> a data table of frequency, time and energy, the problem is that the> table is very long (3 columns and somthing like 13 milion of rows) I> need to plot this table in a 3d graphic, but at now I don't find a> program (quite simple to use) that has a so high limit in is data> table lenght (usually are of 2^15 or 2^16 of rows). Do you know is> exist a program with a no limit data table lenght? Also a KioJust remark that a standart computer display has between1 and 1.5 millions of pixels. I think it's hopeless to tryto show 13 millions of informations on it.you can try xd3dhttp://www.cmap.polytechnique.fr/~jouve/xd3d/It will do === the job :)-- F.J.Subject: A Least Squares Problem latter is more appropriate,please don't redirect as I don't read it regularlyI'm trying to solve a least-squares problem:min{(Ax+b)(Ax+b)'} (1)whose explicit solution is:x = (A'*A)^-1 * A * b (2)Matrix A has form, e.g.: 1 0 0 0 1 -1 1 0 1 0 0 -1 1 -1 0 0 0 -1 0 -1i.e. it has exactly one pair of (1,-1) in each column. There won't beany all-zero rows. As you can see, it may contain two or more linearly dependent columns (two equal, two mutually negative, or other combination).Further, due to nature of the problem (I'd skip the physical backgroundfor now, I can repost if someone's interested) condition is thatsize(x) >= size(b)-1The problem is that equation (2) has explicit solution (A'*A has inversion) only ifsize(x) = size(b)-1 = rank(A)However, I have to solve the problem in general manner, i.e. I wantto obtain one of optimal solutions. Rank deficiency of A should bepreferredly solved by minimizing square sum of x, i.e. min{|x^2|},i.e. the task is to find such x that a) minimizes (1) and b) hasminimal sum(x*x).I plan to use Matlab as proving environment and Fortran/Lapack fordeployment (links to Fortran sources which solve that exact problem are also welcome). I probably have to find QR decomposition of A'A (GEQRF), but I'm lost at the moment what to do after that. I admit my linear algebra is a little rusty...There's an interesting example under Matlab's help on QR function,but it's not directly applicable -- in my case, (A'A) is (can be) singular. All suggestions/solutions are welcome. I suspect this is a well-knownproblem, but it's a little hard to find keywords to === Jugoslav___________www.geocities.com/jdujicSubject: Re: A Least Squares Problem >please don't redirect as I don't read it regularly >I'm trying to solve a least-squares problem: >min{(Ax+b)(Ax+b)'} (1) so Ax+b is a row for you? then: what is Ax ? >whose explicit solution is: >x = (A'*A)^-1 * A * b (2) >Matrix A has form, e.g.: > 1 0 0 0 1 > -1 1 0 1 0 > 0 -1 1 -1 0 > 0 0 -1 0 -1 if this is A then x should have 5 and b 4 components or otherwise your notation is faulty for this A A'*A is 5 times 5 and of rank 4 hence no inverse exists. snip obviously you are interested in the minimum norm least squares solution of norm(Ax+b,2) = min_x the best way to compute this is via the svd (available in matlab and also in lapack) it first computes (in the full -non economy version ) U , S and V which yield A=USV' here U is unitary and has as much rows as A, U'U=UU'=I V is also unitary and has as much columns as A, abd V'V=VV'=I (so U and V are quadratic) and S has the same shape as A, but has only some nonnegative elements along the diagonal, all other being zero . these elements are the singular values of A and their number is the rank of A. now you compute the minimum norm least squares solution x as x=V*S# *U'*b where S# has the same shape as S' (transpose of S) and along the diagonal the reciprocals of the nonzero singular values of A at the positions they occur in S: formally (S#)(i,i) = 1/S(i,i) if S(i,i) not= 0, and =0 otherwise. if you compute under roundoff, you must replace the decision S(i,i) not=0 by S(i,i)>alpha>0 for some reasonable alpha e.g. alpha=100*eps*norm(A) in matlab notation. you can get all his also from LAPACK and its derivatives (in other === languages). hth peter Subject: Re: A Least Squares Problem latter is more appropriate,| please don't redirect as I don't read it regularly| | I'm trying to solve a least-squares problem:| | min{(Ax+b)(Ax+b)'} (1)| | whose explicit solution is:| | x = (A'*A)^-1 * A * b (2)| | please don't redirect as I don't read it regularlyActually, sci.stat.math or sci.stat.consultwould be just as good.>I'd like>to know, however, what I am doing with this -- am I solving something>like>min{(Ax+b)(Ax+b)' + e||x||^2}Find some references on ridge regression (don't have any off the top of myhead, and my reference books are several miles away).A look at Rao's book (Linear Statistical Inference, Wiley) might help youin understanding the rank deficient case.Other possibilities:1) Use the SVD to determine the inverse. The usual deal there is to assumethat the singular values near zero are equal to zero. In exact arithmetic,this procedure leads to the minimum L2 norm solution you are looking for.2) If A is large, try conjugate gradients. It can do amazingly well onproblems of this sort.-- My real email address ismcintosh ##at## research ##dot## telcordia ##dot## === comSubject: Re: A Least Squares ProblemAn nxp matrix A can be factored into a product of 3 other matrices :A=U*L*Vt (where t==transpose) where n for n data and p for p parameters,using singular value decomposition SVD. U and V are data and parameter spaceeigenvectors and L a diagonal matrix containing at most r non-zeroeigenvalues of G with r<=p. So ifx = (A'*A)^-1 * A * bAt*A=V*L^2*Ut*U*L*Vt=V*L^2*Vt ,since Ut*U=Iand (At*A)^-1=V*L^-1*Ut and finally, since Vt*V=IThe least square solution is given byx=V*L^-1*Ut*bSVDCMP.f from numerical receips in fortran, and svd(A) from matlab givesimilar results.SVD is the best way to decompose a matrix since it provides extra infomationabout the resolution of the reconstruction (resolution matrices etc...)-----FiLiP Louis---> please don't redirect as I don't read it regularly> I'm trying to solve a least-squares problem:> min{(Ax+b)(Ax+b)'} (1)> whose explicit solution is:> x = (A'*A)^-1 * A * b (2)> Matrix A has form, e.g.:> 1 0 0 0 1> -1 1 0 1 0> 0 -1 1 -1 0> 0 0 -1 0 -1> i.e. it has exactly one pair of (1,-1) in each column. There won't be> any all-zero rows. As you can see, it may contain two or more linearly> dependent columns (two equal, two mutually negative, or othercombination).> Further, due to nature of the problem (I'd skip the physical background> for now, I can repost if someone's interested) condition is that> size(x) >= size(b)-1> The problem is that equation (2) has explicit solution (A'*A has> inversion) only if> size(x) = size(b)-1 = rank(A)> However, I have to solve the problem in general manner, i.e. I want> to obtain one of optimal solutions. Rank deficiency of A should be> preferredly solved by minimizing square sum of x, i.e. min{|x^2|},> i.e. the task is to find such x that a) minimizes (1) and b) has> minimal sum(x*x).> I plan to use Matlab as proving environment and Fortran/Lapack for> deployment (links to Fortran sources which solve that exact problem are> also welcome). I probably have to find QR decomposition of A'A (GEQRF),> but I'm lost at the moment what to do after that. I admit my linear> algebra is a little rusty...> There's an interesting example under Matlab's help on QR function,> but it's not directly applicable -- in my case, (A'A) is (can be)> singular.> All suggestions/solutions are welcome. I suspect this is a well-known> problem, but it's === ___________> www.geocities.com/jdujicSubject: Re: A Least Squares ProblemGoogle that reference.[38] S. M. Tan and C. Fox, Inverse Problems 453.707 classnotes, TheUniversity of Auckland, Aucland, New Zealand., 2001.What you are doing is referred to as regularized solution, or damped leastsquares.Read the first 3 chapters of the above reference, excellent and very briefreading on what you need to know when performing least squares, or damped(regularized) least squares.Alien+> An nxp matrix A can be factored into a product of 3 other matrices :> A=U*L*Vt (where t==transpose) where n for n data and p for p parameters,> using singular value decomposition SVD. U and V are data and parameterspace> eigenvectors and L a diagonal matrix containing at most r non-zero> eigenvalues of G with r<=p. So if> x = (A'*A)^-1 * A * b> At*A=V*L^2*Ut*U*L*Vt=V*L^2*Vt ,since Ut*U=I> and (At*A)^-1=V*L^-1*Ut and finally, since Vt*V=I> The least square solution is given by> x=V*L^-1*Ut*b> SVDCMP.f from numerical receips in fortran, and svd(A) from matlab give> similar results.> SVD is the best way to decompose a matrix since it provides extrainfomation> about the resolution of the reconstruction (resolution matrices etc...)> ---FiLiP Louis---> please don't redirect as I don't read it regularly> I'm trying to solve a least-squares problem:> min{(Ax+b)(Ax+b)'} (1)> whose explicit solution is:> x = (A'*A)^-1 * A * b (2)> Matrix A has form, e.g.:> 1 0 0 0 1> -1 1 0 1 0> 0 -1 1 -1 0> 0 0 -1 0 -1> i.e. it has exactly one pair of (1,-1) in each column. There won't be> any all-zero rows. As you can see, it may contain two or more linearly> dependent columns (two equal, two mutually negative, or other> combination).> Further, due to nature of the problem (I'd skip the physical background> for now, I can repost if someone's interested) condition is that> size(x) >= size(b)-1> The problem is that equation (2) has explicit solution (A'*A has> inversion) only if> size(x) = size(b)-1 = rank(A)> However, I have to solve the problem in general manner, i.e. I want> to obtain one of optimal solutions. Rank deficiency of A should be> preferredly solved by minimizing square sum of x, i.e. min{|x^2|},> i.e. the task is to find such x that a) minimizes (1) and b) has> minimal sum(x*x).> I plan to use Matlab as proving environment and Fortran/Lapack for> deployment (links to Fortran sources which solve that exact problem are> also welcome). I probably have to find QR decomposition of A'A (GEQRF),> but I'm lost at the moment what to do after that. I admit my linear> algebra is a little rusty...> There's an interesting example under Matlab's help on QR function,> but it's not directly applicable -- in my case, (A'A) is (can be)> singular.> All suggestions/solutions are welcome. I suspect this is a well-known> problem, but it's a little hard to find === www.geocities.com/jdujicSubject: Re: A Least Squares Problem problem:|| || min{(Ax+b)(Ax+b)'} (1)|| || whose explicit solution is:|| || x = (A'*A)^-1 * A * b (2)|| || The problem is that equation (2) has explicit solution (A'*A has|| inversion) only if|| size(x) = size(b)-1 = rank(A)|| || However, I have to solve the problem in general manner, | An nxp matrix A can be factored into a product of 3 other matrices :| A=U*L*Vt (where t==transpose) where n for n data and p for p parameters,| using singular value decomposition SVD. U and V are data and parameter space| eigenvectors and L a diagonal matrix containing at most r non-zero| eigenvalues of G with r<=p. So if| x = (A'*A)^-1 * A * b| At*A=V*L^2*Ut*U*L*Vt=V*L^2*Vt ,since Ut*U=I| and (At*A)^-1=V*L^-1*Ut and finally, since Vt*V=I| The least square solution is given by| x=V*L^-1*Ut*b...| SVD is the best way to decompose a matrix since it provides extra infomation| about the resolution of the reconstruction (resolution still don't see what to doif r | An nxp matrix A can be factored into a product of 3 other matrices :> | A=U*L*Vt (where t==transpose) where n for n data and p for p parameters,> | using singular value decomposition SVD. U and V are data and parameter space> | eigenvectors and L a diagonal matrix containing at most r non-zero> | eigenvalues of G with r<=p. So if> | x = (A'*A)^-1 * A * b> | At*A=V*L^2*Ut*U*L*Vt=V*L^2*Vt ,since Ut*U=I> | and (At*A)^-1=V*L^-1*Ut and finally, since Vt*V=I> | The least square solution is given by> | x=V*L^-1*Ut*b> ...> | SVD is the best way to decompose a matrix since it provides extra infomation> | about the resolution of the reconstruction ignorance, but I still don't see what to do> if r Remind you, the additional criterion for resolving rank deficiency should> be min{||x||}.There is a little mistake in the solution above: If you don not invertL itself, which is not possible in the rank deficient case, but invert onlyall nonzero elements of L, you get the uniqe x with minimum http://www.procoders.net schmitt@procoders.net A service to === open source is a service to mankind.Subject: Sinc Interpolation QuestionI have a vector y with 8 elements and a corresponding 8 x 8 (diagonal)covariance matrix Sy. I want to perform sinc interpolation on y andcompute the corresponding covariance.To sinc interpolate I perform:y = [224.2944 224.4289 230.3499 239.9113 251.0483 259.5003 260.7612 250.3568]';F = fft(eye(8));F2 = fft(eye(16));zf = [0 0 0 0]';y_interp = F2 * circshift([zf'; circshift(1/8 * F' * y, 4); zf'],8);In the above I can replace the circshift function fftshift to make iteasier to understand.How do I go about computing the corrected covariance Se_interp? Should it be === Regression Splines (MARS)Hi All,I am using MARS 3.5 for scattered data points approximation 2D, 3D, 5D. Whatminimum number of data points is required to start build approximation === help needed for runaway ODE?Hi All,I am solving a robotic control problem, envolving coupled nonlinear ODEs. Ifound the solution to the ODEs using relaxation method from the NumericReceipe in C for the two-point boundary value problem. But after I changedthe relaxation method to Runge-Kutta method, using the initial values solvedby relaxation method, I got a runaway type of solution with all solutiongrow to very large numbers. I also tried the stiff solver in the NRC, theresults were similar. I know my derivatives are correct in Range-Kuttabecause I computed derivative from the solution of relaxation method andthey matched the derivatives from my Range-Kutta routine. But thederivatives were wrong when I computed it from the solution of theRange-Kutta method. Any suggestions from numeric gurus here? I have beenstruggling on help.Everettsend your comments to everteq AT sbcglobal DOT net === runaway ODE?Hi Everett,try my ODEs from http://www.delphipages.com/result.cfm?ID=3482> Hi All,> I am solving a robotic control problem, envolving coupled nonlinear ODEs. I> found the solution to the ODEs using relaxation method from the Numeric> Receipe in C for the two-point boundary value problem. But after I changed> the relaxation method to Runge-Kutta method, using the initial values solved> by relaxation method, I got a runaway type of solution with all solution> grow to very large numbers. I also tried the stiff solver in the NRC, the> results were similar. I know my derivatives are correct in Range-Kutta> because I computed derivative from the solution of relaxation method and> they matched the derivatives from my Range-Kutta routine. But the> derivatives were wrong when I computed it from the solution of the> Range-Kutta method. Any suggestions from numeric gurus here? I have been> struggling on this simple ODE problem for comments to everteq AT sbcglobal DOT net or post it here. === All,> I am solving a robotic control problem, envolving coupled nonlinear ODEs.I> found the solution to the ODEs using relaxation method from the Numeric> Receipe in C for the two-point boundary value problem. But after I changed> the relaxation method to Runge-Kutta method, using the initial valuessolved> by relaxation method, I got a runaway type of solution with all solution> grow to very large numbers. I also tried the stiff solver in the NRC, the> results were similar. I know my derivatives are correct in Range-Kutta> because I computed derivative from the solution of relaxation method and> they matched the derivatives from my Range-Kutta routine. But the> derivatives were wrong when I computed it from the solution of the> Range-Kutta method. Any suggestions from numeric gurus here? I have been> struggling on this simple ODE problem for three weeks.It's hard to make a substantive suggestion without seeing the exactequations, but it sounds === chipSubject: Discrete Chebyshev (or other orthogonal) about discete Chebyshev polynomials of the first kind and their relation to Chebyshev polynomials of the second kind.In literature a lot of properties of the Chebyshev polynomials and the relations to other polynomials are known. But, afaik, all these nice propertie are derived under the assumption that we want to approach a continue function. I cannot find a list of properties in case we want to approach a discrete set of equidistant spaced points with these polynomials.The description in Numerical Recipes for example also assumes the the function is known at all locations.Can anyone help me === my research I want to find out more about discete Chebyshev >polynomials of the first kind and their relation to Chebyshev >polynomials of the second kind. >In literature a lot of properties of the Chebyshev polynomials and the >relations to other polynomials are known. But, afaik, all these nice >propertie are derived under the assumption that we want to approach a >continue function. I cannot find a list of properties in case we want to >approach a discrete set of equidistant spaced points with these polynomials. >The description in Numerical Recipes for example also assumes the the >function is known at all locations. >Can anyone help me discrete orthogonal for what scalar product? the chebyshev polynomials ofdegree <= m of the first kind are orthogonal on the discrete set {x_k} = zeros of T_{m+1} with the ordinary scalarproduct. for more information seeChebyshev Polynomialsby J.C. Mason (University of Huddersfield, UK)and D.C. Handscomb (lately of Oxford University)Chapman & Hall / CRC Press, 2002, xiii+341 pp.Hardback ISBN 0-8493-0355-9, $99.95or Pure and Applied Mathematics. New York: John Wiley & Sons, Inc. xvi, === 249 p. (1990)hthpeterSubject: Re: Discrete Chebyshev (or other find out more about discete Chebyshev> polynomials of the first kind and their relation to Chebyshev> polynomials of the second kind.> In literature a lot of properties of the Chebyshev polynomials and the> relations to other polynomials are known. But, afaik, all these nice> propertie are derived under the assumption that we want to approach a> continue function. I cannot find a list of properties in case we want to> approach a discrete set of equidistant spaced points with these polynomials.> The description in Numerical Recipes for example also assumes the the> function is known at all on a lot of possible> interesting words but nothing came out notes for a numerical analysis course: look on http://www.phys.virginia.edu/classes/551.jvn.fall01/ You want Chapter 2, Representation of Functions. 2. Abramowitz & Stegun, Handbook of Mathematical Functions (Dover PB) 3. Ralston, Introduction to Numerical Analysis-- Julian V. NobleProfessor Emeritus of ^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ Science knows only one commandment: contribute to science. -- === Bertolt Brecht, Galileo.Subject: Re: Discrete Chebyshev (or other orthogonal) polynomials.> 2. Abramowitz & Stegun, Handbook of Mathematical Functions> (Dover PB)Or download it === for free.http://jove.prohosting.com/~skripty/Subject: Re: Discrete Chebyshev (or other orthogonal) polynomials.HI,did u take a look at - Conte/DeBoor: Elementary Numerical Analysis, AnAlgorithmic Approach- ?I found it very useful, many times... IvanoMaurice ha scritto nel about discete Chebyshev> polynomials of the first kind and their relation to Chebyshev> polynomials of the second kind.> In literature a lot of properties of the Chebyshev polynomials and the> relations to other polynomials are known. But, afaik, all these nice> propertie are derived under the assumption that we want to approach a> continue function. I cannot find a list of properties in case we want to> approach a discrete set of equidistant spaced points with thesepolynomials.> The description in Numerical Recipes for example also assumes the the> function is known at all locations.> Can anyone help me === Help with adjustment to quadratic regression curve algorithmI'm trying to understand/get an algorithm to work which transforms aquadratic regression curve to be unimodal using Bezier control points. Thealgorithm works out the transform.Please understand that I am inexperienced in this field.I can post the whole algorithm if its needed to put things in context (itsnot very long), but what I'm particularly stuck on is this equation: mY0:=1/m SIGMA (Pi-(Y1)Si = (Y2)(Si)^2 ) i=1where: m=number of points (Si,Pi), Y1,Y2 are already transformed 'outer' points.What does the = mean in (Pi-(Y1)Si = (Y2)(Si)^2 ) ?The text states that:this statement evaluates the best least squares estimate of Y0 for theparabola with new values of Y2 and Y1 resulting from the first adjustment.The second adjustment, a vertical translation of the B.8ezier-reshapedparabola, occurs only when and Y1 and Y2 have been altered.In fact, I'm not sure that the steps in the algorithm I've done before thisequation are right either:(So ANY help appreciated.PS I have access to === matlabTIASubject: Books On Generalized Reduced Gradient Algorithm?Can any one recommend books or papers I can read that describe how to implement a Generalized Reduced Gradient Algorithm (such as that used in Excel's Solver tool)? Source code implementing this algorithm would be very useful as well. === Generalized Reduced Gradient Algorithm?I believe this is actually Leon Lasdon's GRG2 algorithm.Some references are:@misc{ fylstra98design, author = D. Fylstra and L. Lasdon and A. Warren and J. Watson, title = Design and use of the microsoft excel solvers, text = D. Fylstra, L. Lasdon, A. Warren, and J. Watson, Design and use of the microsoft excel solvers. To appear in Interfaces, 1998., year = 1998, url = citeseer.nj.nec.com/fylstra98design.html }L. S. Lasdon and A. D. Waren. Generalized reduced gradient software for linearly and nonlinearly constrained problems. In H. J. Greenberg, editor, Design and Implemetation of Optimization Software, pages 335---362. Sijthoff and Noordhoff, Netherlands, 1978.Lasdon L. S., Warren A. D., Jain A., and Ratner M. 1978. Design and Testing of a GRG Code for Nonlinear Optimization, ACM Trans. Mathematical Software, 4: 3450. -------------------------------------------------------------- --Erwin Kalvelagenerwin@gams.com, http://www.gams.com/~erwin------------------------------------ ----------------------------> Can any one recommend books or papers I can read that describe how to > implement a Generalized Reduced Gradient Algorithm (such as that used in > Excel's Solver tool)? Source code implementing this algorithm === info.> -VikSubject: skyline matrix solverPlease, I need a you,Cpplayer