mm-428 === Subject: : Re: Mutli-parameter border searchTen degrees of freedom in and two degrees of freedom out, with unknownrelationsps between input and output. To be able to move around the outputborder infers that the relationsps have been determined witn anacceptable error bounds. Seems like a Monte Carlo would help determine theinput vectors that yield points near the boundary, then some of inputdegrees of freedom could be constrained (limited). Further Monte Carloswould give sets of points fuzzy around the boundary. Another way to look atis to tnk of moving around the boundary, a two dimensional vectors, wchgenerates solution sets of 10 dimensional vectors. Typically many of thedimensions are not significant, can be ignored or minimized, or you havemultiple solutions of input vectors that yeild the same output vector.> I have a simulation that takes in quite a few parameters (say around> 10). The data output can be simplified to just one point on a> two-dimensional chart.> I am interested in is the border of that plot (i.e. the outlying> points), not the points in the middle. The relationsp between the> input parameters and the output points is not straightforward, though> not random. What is the best way to move around the border of the> plot when running the simulation (Ts will save much simulation time,> as I'd avoid running most simulations that> generate points around the center of the plot)?> What's the best approach? Is it using GAs - or a dimensional mapping> method such as Neldor-Mead? I'm a bit lost with ts, so any> suggestions will be gladly welcome.=== === Subject: : Re: Mutli-parameter border searchDo you have any pointers on how ts search should be implemented?> Ten degrees of freedom in and two degrees of freedom out, with unknown> relationsps between input and output. To be able to move around the output> border infers that the relationsps have been determined witn an> acceptable error bounds. Seems like a Monte Carlo would help determine the> input vectors that yield points near the boundary, then some of input> degrees of freedom could be constrained (limited). Further Monte Carlos> would give sets of points fuzzy around the boundary. Another way to look at> is to tnk of moving around the boundary, a two dimensional vectors, wch> generates solution sets of 10 dimensional vectors. Typically many of the> dimensions are not significant, can be ignored or minimized, or you have> multiple solutions of input vectors that yeild the same output vector.> I have a simulation that takes in quite a few parameters (say around> 10). The data output can be simplified to just one point on a> two-dimensional chart. I am interested in is the border of that plot (i.e. the outlying> points), not the points in the middle. The relationsp between the> input parameters and the output points is not straightforward, though> not random. What is the best way to move around the border of the> plot when running the simulation (Ts will save much simulation time,> as I'd avoid running most simulations that> generate points around the center of the plot)? What's the best approach? Is it using GAs - or a dimensional mapping> method such as Neldor-Mead? I'm a bit lost with ts, so any> suggestions will be gladly welcome.>=== === Subject: : Re: Mutli-parameter border searchsnip > I have a simulation that takes in quite a few parameters (say around > 10). The data output can be simplified to just one point on a > two-dimensional chart. > > I am interested in is the border of that plot (i.e. the outlying > points), not the points in the middle. The relationsp between the > input parameters and the output points is not straightforward, though > not random. What is the best way to move around the border of the > plot when running the simulation (Ts will save much simulation time, > as I'd avoid running most simulations that > generate points around the center of the plot)? > > What's the best approach? Is it using GAs - or a dimensional mapping > method such as Neldor-Mead? I'm a bit lost with ts, so any > suggestions will be gladly welcome. >if I underd ts right you have x in R^10 -> [ black box function taking x and producing ] -> y in R^2and you know in addition that y in Q, a box. you want to find (all????) xsuch that y is at the border of the box. say the box is [0,1] times [0,1].then y is at the border if and only if y(1)*(1-y(1))>=0 y(2)*(1-y(2))>=0 y(1)*(1-y(1))*y(2)*(1-y(2)) = 0 (and <= eps for near the border )so what you have is called a constraint satisfaction problem and for dimension<=10 it might be solvable completely. the problem I forsee is the black box nature of the function . the known methods for derivative freeoptimization typically detect (at best) one such x, not _all_and the CSP (constraint satisfaction problem) codes I know about need an explicit algorithm (in order to compute bounds, e.g. with intervalarithmetic means), not only a point evaluator. CSP codes can in principle cover the complete solution space. there is ongoing research on ts and for example the results of the COCONUTproject will be available as of march 1 to the public.http://www.mat.univie.ac.at/coconut/codes for derivativefree optimization can be found athttp://plato.la.asu.edu/topics/problems/nlounres.htmlandhttp ://plato.la.asu.edu/topics/problems/nlores.html (under the general nonlinear problem)the reverse problem (moving y on the boundary and finding a x such that F(x)=yis not well defined, in the sense of the implicit function theorem you have here 8 degrees of freedom in x )=== === Subject: : beyond Albert Einstein's general relativity by Rodgers's UNIFIED PHYSICS theory athttp://relativity.topcities.com/revolution.htmI hope Micko Wada isn't laugng at me!THANKS! IF YOU WANT TO PUT AUSTRALIAN DOLLARS INTO MY ACCOUNT: theswift code is METWAU4B, the beneficiary name is SUNCORP LTD,forwarding to D RODGERS, BSB484799, and accountnumber050772534........................=== === Subject: : How to check carefully that the inverse computed by a numerical method is correct ?Distribution: inetHow to check that the inverse computed by a numerical method (using LU factorisation) is correct ?Is it enough to check ||AA-1-I|| ?or/and should I just check the condition number of the matrix ? and if cond(A)*eps <<1 (where eps~10^-16 in double precision, eps~10^-31 in quadruple precision), it s safe.for lapack, they compute (||AA-1-I||/N*eps*||A||), why ?So I am a little confused ...Karim.=== === Subject: : Re: How to check carefully that the inverse computed by a numerical method is correct ?> How to check that the inverse computed by a numerical method (using LU > factorisation) is correct ?> Is it enough to check ||AA-1-I|| ?> or/and should I just check the condition number of the matrix ? and if > cond(A)*eps <<1 (where eps~10^-16 in double precision, eps~10^-31 in quadruple > precision), it s safe.for lapack, they compute (||AA-1-I||/N*eps*||A||), why ?Because c*N*eps*||A|| is the error error provided bybackward error analysis if partial pivoting (wch is themethod used in lapack) is used in the factorization.Typically c (the growth factor) is ignored since the bound isoften a severe overestimate.So I am a little confused ...> Karim.=== === Subject: : Re: sinc(x+y) relations?> I was wondering if there were any angle-sum relations for sinc(x+y) like> there are for sines and cosines (i.e. sin(x+y) => sin(x)cos(y)+cos(x)sin(y) )? I don't tnk so becasue of how the sinc> function is defined (sin(x)/x) - or it's not obvious if there are, but> thought I would ask anyway.The following copy (of a recent response of mine in the Mathematicanewsgroup) does not answer your question. But it does show how sinc of asum or difference can arise quite naturally, and so I thought it mightinterest you.It concerns the integral of sin(mx)sin(nx) from x = 0 to 2 pi.Mathematica's answer is, taken literally, invalid if m = +/-n. But sinc canbe used to write an answer wch is always valid.David-------------You have gotten several responses showing how, rather than merely replacingm or n by specific values, the results you desire can be obtained by takinglimits as m or n approach those values. However, your expectation that merereplacement should give the desired results is _not unjustified_. First,let's look atIn[1]:= Integrate[Sin[m*x]*Sin[n*x], {x, 0, 2*Pi}]Out[1]= (n*Cos[2*n*Pi]*Sin[2*m*Pi] - m*Cos[2*m*Pi]*Sin[2*n*Pi])/(m^2 - n^2)Ts expression may be rewritten as Sin[2*Pi*(m - n)]/(2(m - n)) - Sin[2*Pi*(m + n)]/(2(m + n)){Perhaps a clever person could show how to get Mathematica to do thatrewriting, but I did it by hand.}To avoid problems when we substitute using n = m or n = -m, we can againrewrite the expression in terms of a function wch is well known, but(unfortunately?) not implemented in Mathematica. It's the sine cardinalfunction, often abbreviated sinc.In[2]:= SinC[x_] := If[x == 0, 1, Sin[x]/x]Using it, the result of the integration may then be written nicely as# Pi*(SinC[2*Pi(m - n)] - SinC[2*Pi(m + n)])The above form is valid for all m and n; in other words, merely replacingm and n by specific values, we always get the correct result for theintegral. In an ideal world, I tnk that SinC would be implemented inMathematica, wch would then give #, rather than what it currently gives,for Out[1].David Cantrell=== === Subject: : Re: sinc(x+y) relations?> The foregoing can be rewritten assinc(x+y) = {sinc(x)cosc(y) + sinc(y)cosc(x)}(2/pi)Integrate[(1 -> z^2/(z^2+x^2))(1 - z^2/(z^2+y^2)),{z,0,Infinity}].wch can be rewritten as x y (sinc(x) cosc(y) + sinc(y) cosc(x)) / (x + y)wch is, of course, just (sin(x) cos(y) + sin(y) cos(x)) / (x + y) == sin(x+y) / (x+y)-- Abbott Phone: +61 8 9380 2734School of Physics, M013 Fax: +61 8 9380 1014The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling ghwayCrawley WA 6009 mailto:@physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~=== === Subject: : Re: sinc(x+y) relations?> The foregoing can be rewritten as sinc(x+y) = {sinc(x)cosc(y) + sinc(y)cosc(x)}(2/pi)Integrate[(1 -> z^2/(z^2+x^2))(1 - z^2/(z^2+y^2)),{z,0,Infinity}].> wch can be rewritten as> x y (sinc(x) cosc(y) + sinc(y) cosc(x)) / (x + y)> wch is, of course, just> (sin(x) cos(y) + sin(y) cos(x)) / (x + y) == sin(x+y) / (x+y)Naturally, but that's tautologically redundant. The tng is to express1/(x+y) as the product of two factors, one that depend on x only and theother that depends on y only, wch are intimately coupled in the complexplane with wchsinc(x+y) = (2/pi)Integrate[Sinc(z,x)Cosc(z,y) +Sinc(z,y)Cosc(z,x),{z,0,Infinity}]as I stated in my original post.-- Ciao,Gerry T.______A proof is a proof. What kind of a proof ? It's a proof. A proof is aproof. And when you have a good proof, it's because it's proven. -- JeanChr.8etien, ex Prime Minister of Canada.=== === Subject: : Re: sinc(x+y) relations?Originator: kblomste@cc.hut.fi (Kasimir Blomstedt)> The foregoing can be rewritten as> sinc(x+y) = {sinc(x)cosc(y) + sinc(y)cosc(x)}(2/pi)Integrate[(1 -> z^2/(z^2+x^2))(1 - z^2/(z^2+y^2)),{z,0,Infinity}].> wch can be rewritten as> x y (sinc(x) cosc(y) + sinc(y) cosc(x)) / (x + y)> wch is, of course, just> (sin(x) cos(y) + sin(y) cos(x)) / (x + y) == sin(x+y) / (x+y)>Naturally, but that's tautologically redundant. The tng is to express>1/(x+y) as the product of two factors, one that depend on x only and the>other that depends on y only, wch are intimately coupled in the complex>plane with wch>sinc(x+y) = (2/pi)Integrate[Sinc(z,x)Cosc(z,y) +>Sinc(z,y)Cosc(z,x),{z,0,Infinity}]>as I stated in my original post.To add to the choices, Abramowitz & Stegun (eq. 10.1.45) suggests thatsinc(x+y) = sin(x+y)/(x+y)= sum_{n = 0}^infty (2n+1) j_n(x) j_n(y) P_n(cos 0)= sum_{n = 0}^infty (2n+1) j_n(x) j_n(y)where j_n(z) is the spherical Bessel function of order n and P_n(z) is theLegendre polynomial of order n.-Kasimir Blomstedt, kblomste@vipunen.hut.fi=== === Subject: : Re: sinc(x+y) relations?Originator: kblomste@cc.hut.fi (Kasimir Blomstedt)>To add to the choices, Abramowitz & Stegun (eq. 10.1.45) suggests that>sinc(x+y) = sin(x+y)/(x+y)>= sum_{n = 0}^infty (2n+1) j_n(x) j_n(y) P_n(cos 0)>= sum_{n = 0}^infty (2n+1) j_n(x) j_n(y)>where j_n(z) is the spherical Bessel function of order n and P_n(z) is the>Legendre polynomial of order n.Also, there issinc(x+y)= sum_{m = -infty}^infty sinc(x+Pi m)sinc(y-Pi m),wch can be shown by extracting the sines from the sum and using residues toevaluate what's left.-Kasimir Blomstedt, kblomste@vipunen.hut.fi=== === Subject: : Re: sinc(x+y) relations?BTW, what is the title of the text by Abramowitz & Stegun?Pat>To add to the choices, Abramowitz & Stegun (eq. 10.1.45) suggests that>sinc(x+y) = sin(x+y)/(x+y)>= sum_{n = 0}^infty (2n+1) j_n(x) j_n(y) P_n(cos 0)>= sum_{n = 0}^infty (2n+1) j_n(x) j_n(y)>where j_n(z) is the spherical Bessel function of order n and P_n(z) isthe>Legendre polynomial of order n.> Also, there is> sinc(x+y)> = sum_{m = -infty}^infty sinc(x+Pi m)sinc(y-Pi m),> wch can be shown by extracting the sines from the sum and using residuesto> evaluate what's left.> -Kasimir Blomstedt, kblomste@vipunen.hut.fi=== === Subject: : Re: sinc(x+y) relations?>To add to the choices, Abramowitz & Stegun (eq. 10.1.45) suggests that>sinc(x+y) = sin(x+y)/(x+y)>= sum_{n = 0}^infty (2n+1) j_n(x) j_n(y) P_n(cos 0)>= sum_{n = 0}^infty (2n+1) j_n(x) j_n(y)>where j_n(z) is the spherical Bessel function of order n and P_n(z) isthe>Legendre polynomial of order n.> Also, there is> sinc(x+y)> = sum_{m = -infty}^infty sinc(x+Pi m)sinc(y-Pi m),> wch can be shown by extracting the sines from the sum and usingresidues to> evaluate what's left.> -Kasimir Blomstedt, kblomste@vipunen.hut.fi-- Ciao,Gerry T.=== === Subject: : Re: Multi-paramter border search by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1JDEMi11889;>I have a simulation that takes in quite a few parameters (say around>10). The data output can be simplified to just one point on a>two-dimensional chart. >I am interested in is the border of that plot (i.e. the outlying>points), not the points in the middle. The relationsp between the>input parameters and the output points is not straightforward, though>not random. What is the best way to move around the border of the>plot when running the simulation (Ts will save much simulation>time,>as I'd avoid running most simulations that>generate points around the center of the plot)?>What's the best approach? Is it using GAs - or a dimensional mapping>method such as Neldor-Mead? I'm a bit lost with ts, so any>suggestions will be gladly welcome.if your input parameters can be chosen independently from each other,you could couple your simulation with an optimization.For ts aim, constrain your input parameters (they will be the solution of the optimization procedure) witn reasonable bounds,fix one of the two values of your 2d-output and try to minimize resp.maximize the second one.Best wishesTorsten.=== === Subject: : US FlagI once saw a Matlab program to draw the American flag. Does anyone=== === Subject: : about fortran by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1JHQna02721;please i will like to know how fortran files can be printed and alsogive me an examp=== === Subject: : Re: about fortran> please i will like to know how fortran files can be printed and also> give me an exampAfter clarifying your question, as Spellucci suggested, youshould post it to the right newsgroup, comp.lang.fortran . To get youstarted,print*,hello,i,jwould print sometng likehello 2 3if i = 2 and j = 3.=== === Subject: : Re: about fortran> please i will like to know how fortran files can be printed and also> give me an exampAfter clarifying your question, as Spellucci suggested, you> should post it to the right newsgroup, comp.lang.fortran . To get you> started,> wle I agree that ts newsgroup isn't appropriate for ts kind ofquestions, I also wouldn't want to see it in comp.lang.fortran, wchis intended to discuss features of the Fortran language and alsoprovides valuable help to programmers. But for ts kind of questionsthe local advisor or any basic introduction is the proper source ofinformation. Alois-- Alois Steindl, Tel.: +43 (1) 58801 / 32558 Inst. for Mechanics II, Fax.: +43 (1) 58801 / 32598Vienna University of Technology,A-1040 Wiedner Hauptstr. 8-10 === === Subject: : Re: about fortran>please i will like to know how fortran files can be printed and also>give me an exampyour question is not clear.you mean : print a file say lu.f on your printer? surely ts is not your question (lpr lu.f in unix)you might mean: I want to write some file witn a fortran program to disc?for example: a table of data?wch fortran (because the expressive power of f90/f95 is much gher than f77 ?)I give you an example in f77: (compatible with f90) data : (x(i),y(i),z(i)) i=1,...,N you want a table with a heading , 60 lines a page, and every five lines a blankline, each value with 14 digits and exponential part: open(10,file='table.dat',status='new') write(10, fmt='((10x,''x'',22x,''y'',22x,''z'', 10(5(/3(e21.14,2x)),/)))') (x(i),y(i),z(i),i=1,N) close(10)ts is for free format. do not forget the continuation sign for fixed format.for integers the format is i10 for printing with 10 digits, 5x means 5 spaces, what is inside '' .. '' is printed literally. / means new linee21.14 or d21.14 is the float format first (21) number is length, second number of digits in mantissa === === Subject: : Re: L1 linear fit> Please point me to an efficient routine for linear L1 fit, that is,> finding a minimizer of x |-> norm(b-A*x, 1) with a given column vector b> and a (generally speaking) rectangular matrix A. Is there a routine about> as well-researched as least-squares routines?I am a MATLAB user, so a routine in ts system would be most welcome, but> a description in mathematical formulas would be satisfactory.I tried Google, but perhaps didn't set the key expressions right.> There is a Fortran 90 code to do ts, translated by Alan Miller. Seehttp://users.bigpond.net.au/amiller/toms/toms615.f90 . Here is anexcerpt from the documentation:! The purpose of ts program is to determine the best subset ofparameters! to fit a linear regression under an least absolute value criterion.! Ts program utilizes the simplex method of linear programmingwitn a! branch-and-bound algorithm to solve the best subset problem.! The algorithm is based on the publication:! Armstrong,R.D. and M.T. Kung An algorithm to select the bestsubset for! a least absolute value regression problem, Optimization inStatistics,! TIMS Studies of the Management Sciences, vol.33, 931-936 (1982).Another Fortran 90 code to do L1 and many other types of regression isby Burkardt -- seehttp://www.csit.fsu.edu/~burkardt/f_src/regression/ regression.html ,in particular algorithm A478_L1 .=== === Subject: : Re: L1 linear fit by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1JDEKJ11811;>Please point me to an efficient routine for linear L1 fit, that is,>finding a minimizer of x |-> norm(b-A*x, 1) with a given column>vector b>and a (generally speaking) rectangular matrix A. Is there a routine>about>as well-researched as least-squares routines?>I am a MATLAB user, so a routine in ts system would be most>welcome, but>a description in mathematical formulas would be satisfactory.>I tried Google, but perhaps didn't set the key expressions right.there are variants of the SIMPLEX algorithm especially suited for L1 fit athttp://www.netlib.org/toms/478http://www.netlib.org/toms/ 551If there is a routine for the solution of linear programming problemsin MATLAB, you can use ts routine, too, but most probably, it willnot be as efficient as the two mentioned above.Best wishesTorsten.=== === Subject: : Re: L1 linear fitPlease point me to an efficient routine for linear L1 fit, that is, >finding a minimizer of x |-> norm(b-A*x, 1) with a given column vector b >and a (generally speaking) rectangular matrix A. Is there a routine about >as well-researched as least-squares routines?I am a MATLAB user, so a routine in ts system would be most welcome, but >a description in mathematical formulas would be satisfactory.I tried Google, but perhaps didn't set the key expressions right. >the page http://plato.la.asu.edu/topics/problems/approx.htmlgives you several choices for that:APPROX (ths is netlib.ornl.gov/toms/458CL1RQNif you have the matlab optimization toolbox it might be simpler for you to uselinprog and the dard transformation of the problem to a LP problem minimize (w.r.t. eps(1),..,eps(m) and x(1),...,x(n) ) sum_{i=1 to m } eps(i) subject to -eps(i) <= sum_{j=1 to n} a(i,j)*x(j)-b(i) <= eps(i) i=1,..,m=== === Subject: : Re: Multivariate Statistical Process Control with PCA (T2 and SPE charts)> The group that I work for (see below) has done a lot of research in ts> area. I would say though that your best bet for reading about T2 and SPE in> general is the book by Jackson (very expensive book though!). As for> determining the variable(s) at fault, the paper by Miller, on contribution> plots looks at ts problem. A tutorial paper on how to actually use and> interpret all these plots is given by Kourti and MacGregor. The paper by> Nomikos and MacGregor gives an alternative method in the appendix to> calculate the SPE chart's limit.These books and papers should be in most university libraries. Finding the> contributing variables to SPE (or even T2) violations is the real purpose of> applying these statistical methods to one's data: if it is data from a> production process, then these would be the variables to target in order to> improve the process. One very simple way to compute the Q-chart's limits> from your given data is to take the Q-values that you have computed, and find> the point wch splits between 95% and the remaining 5% of your values.> Ts point is your 95% confidence limit, based on the distribution of your> data.Jackson, J.E. A User's Guide to Principal Components, Wiley, New York,> 1991.> P Miller, R.E. Swanson and C.F. Heckler, Contribution plots: a missing link> in multivariate quality control, Applied Math. and Comp. Science, vol 8(4),> p 775-792, 1998. [There is a version available from Paige Miller's website?]There is no longer a website with my paper. I can provide a Microsoft Word version of the paper (one or two iterations before publication, but the differences between the published version and my Word version are small, mostly grammar and punctuation changes).Just send me an e-mail.-- Paige MillerEastman Kodak Companypaige dot miller at kodak dot comhttp://www.kodak.comIt's notng until I call it! -- Klem, NL UmpireWhen you get the choice to sit it out or dance, I hope you dance -- Lee Ann Womack=== === Subject: : How many long primes are there?N can represent any integerL represents the length of the period of the expansion of 1/NThe residues at the same level have the same period lengthsFor each subsequent residue the period length halves until the length isodd. At that level the residues form a cycle.Can anyone tell me how many long primes there are to various bases?Can anyone tell me what was the deep work that Christopher Hooley has donethat Conway and Richard Guy alludes to in The Book of Numbers Why didn't they mention the correlation of period length to the quadraticgraphs below? They mentioned the Totient rule and Wilsons theorem. Is tstoo basic?Could someone enlighten me pleaseN L Graph of quadratic residues b where b also represents the base in wch the expansion of 1/n occurs2 1 1 3 2 2 1 1 4 2 3 1 1 5 4 |3|2| 2 4 1 1 7 6 |3|5| 3 | |2=4| 11 10 |7|6|8|2| 5 |5=3=9=4| 2 11 1 1 13 12 |2|11|6|7| 6 | 4 | 10| 3 | 3 = 9 | 4 |8|5| 2 |12 | 1 | 1 | 17 16 |6|11| 7|10|5|12|3|14| 8 | 2 | 15 | 8 | 9 | 4 | 4 | 13 | 2 | 16 | 1 | 1 | 19 18 |14|13|2|15|3|10| 9 | 6=17=4=16=9= 5| 6 |12|8| 3 |11=7| 2 |18| 1 |1| 23 22 |5|21|19|7|20|14|11|17|10|15| 11 |2= 4=16=3= 9=12= 6=13= 8=18| 2 |22| 1 | 1| 29 28 |2|27|10|19|11|18| 14 | 4 | 13 | 5 | 7 |16 = 24 =25 | 28 |3|26|8|21||14|15| 14 | 9 | 6 | 22 | 7 |23 = 7 = 20 | 4 |2|17| 2 | 28 | 1 | 1 | 31 30 |3|22|12|11| 15 |9=19=20=28| 30 |21|24|13|17| 15 |7=18=14= 10| 10 |23|29|27|15| 5 |2= 4=16= 8| 6 |6|26| 3 |5=25| 2 |30| 1 | 1| === === Subject: : solve linear systems in boost::numerics::ublas. I'm using boost::numerics::ublas to solve linear symmetric and hermitian systems (cholesky, qr, etc.). I'm aware that ublas only provide basic algebra operations (blas1, blas2 and blas3) so i've tried some other libraries to work with ublas, with no success so far. In order to do so i downloaded boost::numerics::bindings (wch aren't available in www.boost.org right now but in the list wch boost developers have in yahoo), those bindings enable you to use libraries developed in c or fortran to do more advanced algebra operations, but i'm stuck: + cLAPACK, when i tried to compile cLAPACK (a c library for wch there are bindings in boost), the build procedure in windows is quite tricky and seems to be out of date. I was unable to build the library and the prebuild libraries are outdated (1999). + Atlas, any of the prebuild libraries in sourceforge suits my microprocessor and i can't build them since i don't have cygwin (i have a 56k. modem and i can't download such a big package), evenmore, i intend to use the libraries in msvc and the object format differs. There's an alternative procedure to build in mscv, but i tnk the 'nmake' tool in msvc doesn't behave as the 'make' in the -ix world. am i wrong?. + Bayes++, ts library is intended to be used to calculate karman filters, but they provide uLAPACK compliance, when i compile it's ok, but when i try to link them, the functions wch provide LAPACK functionality are missing (in namespace bayesian_filters::LAPACK::rawLAPACK, dgetrs_, dgetrf_, etc.) I've also tried some other libraries like blitz++ and tnt (formerly lapack++), but i didn't suceed in building the libraries. So, i'm really stuck, i tnk i should look for prebuild libraries at ts point, but my micro is relatively odd (amd duron). Please, what step would you take next to perform cholesky factorization?. I don't have the expertise nor the time to write the algorithms hand-craft Any help would be appreciated, very much in advance and excuse my bad english. === === Subject: : Re: solve linear systems in boost::numerics::ublas> . I'm using boost::numerics::ublas to solve linear symmetric and> hermitian systems (cholesky, qr, etc.). I'm aware that ublas only provide> basic algebra operations (blas1, blas2 and blas3) so i've tried some> other libraries to work with ublas, with no success so far. In order to> do so i downloaded boost::numerics::bindings (wch aren't available in> www.boost.org right now but in the list wch boost developers have in> yahoo), those bindings enable you to use libraries developed in c or> fortran to do more advanced algebra operations, but i'm stuck: + cLAPACK, when i tried to compile cLAPACK (a c library for wch> there are bindings in boost), the build procedure in windows is quite> tricky and seems to be out of date. I was unable to build the library and> the prebuild libraries are outdated (1999). + Atlas, any of the prebuild libraries in sourceforge suits my> microprocessor and i can't build them since i don't have cygwin (i have a> 56k. modem and i can't download such a big package), evenmore, i intend> to use the libraries in msvc and the object format differs. There's an> alternative procedure to build in mscv, but i tnk the 'nmake' tool in> msvc doesn't behave as the 'make' in the -ix world. am i wrong?. + Bayes++, ts library is intended to be used to calculate karman> filters, but they provide uLAPACK compliance, when i compile it's ok, but> when i try to link them, the functions wch provide LAPACK functionality> are missing (in namespace bayesian_filters::LAPACK::rawLAPACK, dgetrs_,> dgetrf_, etc.) I've also tried some other libraries like blitz++ and tnt (formerly> lapack++), but i didn't suceed in building the libraries. So, i'm really stuck, i tnk i should look for prebuild libraries> at ts point, but my micro is relatively odd (amd duron). Please, what> step would you take next to perform cholesky factorization?. I don't have> the expertise nor the time to write the algorithms hand-craftYou should persist in trying to get [C]LAPACK to build, independently ofother libraries (ublas, etc.). [C]LAPACK does depend on [C]BLAS, but aversion of the latter is usually included in the former (see SRC/BLASdirectory). ATLAS is a replacement for [C]BLAS - you may improveperformance by linking with it, but it's not necessary. If you use the Fortran versions, you will also need G2C. See README and INSTALL files in LAPACK for info about exactly how toinstall. Oleg-- Oleg Trott === === Subject: : Re: solve linear systems in boost::numerics::ublas. I'm using boost::numerics::ublas to solve linear symmetric and> hermitian systems (cholesky, qr, etc.). I'm aware that ublas only provide> basic algebra operations (blas1, blas2 and blas3) so i've tried some> other libraries to work with ublas, with no success so far. In order to> do so i downloaded boost::numerics::bindings (wch aren't available in> www.boost.org right now but in the list wch boost developers have in> yahoo), those bindings enable you to use libraries developed in c or> fortran to do more advanced algebra operations, but i'm stuck: + cLAPACK, when i tried to compile cLAPACK (a c library for wch> there are bindings in boost), the build procedure in windows is quite> tricky and seems to be out of date. I was unable to build the library and> the prebuild libraries are outdated (1999). + Atlas, any of the prebuild libraries in sourceforge suits my> microprocessor and i can't build them since i don't have cygwin (i have a> 56k. modem and i can't download such a big package), evenmore, i intend> to use the libraries in msvc and the object format differs. There's an> alternative procedure to build in mscv, but i tnk the 'nmake' tool in> msvc doesn't behave as the 'make' in the -ix world. am i wrong?. + Bayes++, ts library is intended to be used to calculate karman> filters, but they provide uLAPACK compliance, when i compile it's ok, but> when i try to link them, the functions wch provide LAPACK functionality> are missing (in namespace bayesian_filters::LAPACK::rawLAPACK, dgetrs_,> dgetrf_, etc.) I've also tried some other libraries like blitz++ and tnt (formerly> lapack++), but i didn't suceed in building the libraries. So, i'm really stuck, i tnk i should look for prebuild libraries> at ts point, but my micro is relatively odd (amd duron). Please, what> step would you take next to perform cholesky factorization?. I don't have> the expertise nor the time to write the algorithms hand-craftYou should persist in trying to get [C]LAPACK to build, independently of> other libraries (ublas, etc.). [C]LAPACK does depend on [C]BLAS, but a> version of the latter is usually included in the former (see SRC/BLAS> directory). ATLAS is a replacement for [C]BLAS - you may improve> performance by linking with it, but it's not necessary. If you use the Fortran versions, you will also need G2C. Or you could just call the Fortran libraries directly fromC/C++...just remember in Fortran, everytng is passed by pointer. Ihave a class library that interfaces to LAPACK that I can show you asan example.TinoSee README and INSTALL files in LAPACK for info about exactly how to> install. Oleg=== === Subject: : Re: solve linear systems in boost::numerics::ublas> If you use the Fortran versions, you will also need G2C.Or you could just call the Fortran libraries directly from> C/C++...just remember in Fortran, everytng is passed by pointer. I> have a class library that interfaces to LAPACK that I can show you as> an example.It probably depends on the compiler actually, but none of the compilers Iuse (GCC and Intel) will link to LAPACK without G2C. I tnk the reason isthat some of the intrinsic Fortran functions (like lg10) do not exist in C,and so they have to be provided in libraries.Oleg-- Oleg Trott === === Subject: : Question about FFTW I have a question about difinition of angular frequency in fourier space. there are two option: 1) w=2*pi*n/(N*dt), n=0, .., N/2-1, -N/2, ..., -1 2) w=2*pi*n/ (N*dt), n=0,........ N-1 I tnk that both case should be right. But when I use fftw to calculatef''(x): f''(x)= IFFTW ( - w^2 FFTW(f(x)) ) ( I used the 3-d Real-Data DFT: dfftw-plan-dft-r2c-3d ) only the case (1) obtain correct result. Would you please explain a littleabout it? === === Subject: : Re: Question about FFTWAnd when I use Fourier Cosin Transform, both definitions work.I got confused here. I have a question about difinition of angular frequency in fourier space.> > there are two option:> 1) w=2*pi*n/(N*dt), n=0, .., N/2-1, -N/2, ..., -1> 2) w=2*pi*n/ (N*dt), n=0,........ N-1 I tnk that both case should be right. But when I use fftw to calculate> f''(x): f''(x)= IFFTW ( - w^2 FFTW(f(x)) ) ( I used the 3-d Real-Data DFT: dfftw-plan-dft-r2c-3d ) only the case (1) obtain correct result. Would you please explain a little> about it? > === === Subject: : Re: Question about FFTW I have a question about difinition of angular frequency in fourier space.> > there are two option:> 1) w=2*pi*n/(N*dt), n=0, .., N/2-1, -N/2, ..., -1> 2) w=2*pi*n/ (N*dt), n=0,........ N-1 I tnk that both case should be right. But when I use fftw to calculate> f''(x): f''(x)= IFFTW ( - w^2 FFTW(f(x)) ) ( I used the 3-d Real-Data DFT: dfftw-plan-dft-r2c-3d ) only the case (1) obtain correct result. Would you please explain a little> about it? > Google aliasing and Nyquist theorem. In a discrete system, the frequency ranges occur in repeating bands, of width 1/dt, because seen stroboscopically, frequencies separated by multiples of 1/dt are indistinguishable. Sampling once per second, a 100.2 Hz sine wave will produce exactly the same samples as a 0.2 Hz sine wave. Positive frequencies wrap round to negative ones at half the sampling rate--a 0.7 Hz sine wave will produce the same samples as a -0.3 Hz sine wave. Thus there are an infinity of possible choices, not just the two you cite. You can put your wraparound point anywhere you like, but not all definitions are equally useful.If you knew that your actual signal had only positive frequencies, for example, the 0...N-1 choice would be fine. If you knew that your signal was always between 100/dt and 101/dt, then the choice 100N...101N-1 would be appropriate (ts isn't necessarily silly, either--stroboscopic digitizers are becoming fairly widely used in real life).If your signals are band-limited to [-0.5/dt, 0.5.dt), your first choice is correct.Pl Hobbs=== === Subject: : Numerical Algorithms and Mathematical Programming SiteI would like to announce that my web site titled:Numerical Algorithms and Mathematical Programminghas MOVED to the following URL:http://www.numerical-algorithms.comI am presenting results on some interesting numerical problems.in PDF format.Functional Approximation using Normalized Two-Point HermitePolynomials:- Expansion of a function in terms of Two-Point Normalized Hermite Polynomials- Numerical Experiments [cos,sin] and [erf]; comparison with Brent's methodRoot Finding using Inverse Interpolation with NormalizedTwo-Point Hermite Polynomials:- Practical formulas for second and trd order interpolation- Numerical ExperimentsNote on solving the Cubic Equation (with Gary G. Ford):- Bi-Linear Transformation to single parameter canonical form- Series Expansion for the case of two almost identical rootsThe Constrained Geodesic and the Geometry of a Manifold definedby Algebraic Equality constraints; Examples:- The constrained geodesic for the case of holonomic constraints and for a combination of holonomic and non-holonomic constraints- Principal normal to the constrained geodesic; Riemann curvature tensor and Gauss curvature of the manifold- Taylor's series expansion of the Constrained Geodesic- Proposed algorithm for Constrained Optimization based on the Constrained Geodesic[Ts paper is long and has been split into four parts]Points on the Sphere Problem:- New Constrained Optimization Algorithm using the Reduced Hessian and the Block Reflector- List of known Optimum Configurations for the Geometric Objective Function for order to 96 [text format]- Lists of all known Meta-Stable configurations for the Geometric Objective Function for order to 16 [text format] for your interest,Jentje Goslinga=== === Subject: : Re: Modeling ground reaction forces (vecle dynamics)Ts is a very gh-level question, because I have yet to have an idea> of what I'm actually up to.I'm in the business of developing flight simulators, and one part of> that is to model the forces and moments produced by the landing gear> when the aircraft lands and rolls on the ground...You're up to *quite a bit*, fortunately, you won't have to start fromscratch or reinvent the wheel. So, make a search with, NASA TR landinggear, and the references will keep you busy for a long time. Or, youcan search directly on NASA's Technical Report server, http://ntrs.nasa.gov/btw, if you ever saw carrier landings (in rough seas & wet deck) thenyou'll get an idea of just how much they've perfected the art. --Dr.B.Voh---------------------------------------------------- --Applied Algorithms http://sdynamix.com=== === Subject: : Re: Modeling ground reaction forces (vecle dynamics)In message 19 FebI would very much appreciate discussion with experts in ts field and> any nts and references!I missed the original post, for reasons beyond me I only seem to getting aportion of ts ng, so hope I haven't misunderstood.I am doing some work on the subject, and particulary involved in a propertreatment of tyre forces at ts moment. I have a way to go yet, but somedisscussion on the subject would be welcome.I am a mechanical engineer with lots of vecle and simulation/modellingexperience. -- Terry Duell=== === Subject: : Re: Modeling ground reaction forces (vecle dynamics)Cc: tduell@planet.net.au> I am doing some work on the subject, and particulary involved in a proper> treatment of tyre forces at ts moment. I have a way to go yet, but some> disscussion on the subject would be welcome.How do you do it? Do you somehow model the actual tyre shapes, or justthe forces they produce? I *tnk* for my application it would beenough to reduce them to a massless contact point and (two-dimensional)tyre direction vector, and a velocity vector. Then the force is afunction of normal force, these two vectors, and the braking applied.The question is just, wch function! :-)And sometng needs to be conceived to force a contact point to rest atthe same position wrt ground once static friction governs and the forcesare below the dynamic friction threshold.I tnk we have three cases where there's simply a different number ofmodel Degrees of Freedom:1) The wheel is in the air.2) The wheel touches the ground and rolls.3) The wheel rests fixed on the ground.Perhaps ts is just a matter of cleanly defining the transitionsbetween these different model ``states''...-Gerhard[ posted and mailed ]-- Gerhard Wesp o o Tel.: +41 (0) 43 5347636Bachtobelstrasse 56 | http://www.cosy.sbg.ac.at/~gwesp/CH-8045 Zuerich _/=== === Subject: : Simple question regarding frequency detectionI am writing an application to detect a particular frequency(viz. 4.5Khz) in an input audio signal. I have written the code to capture thesound using a microphone; but I have to properly design a digital filterwch will return TRUE as soon as it detects sound of ts frequency inthe input signal. Also my project demands that I have to detect thesignal by looking at a very small number of samples e.g. 44. So I willapply the digital filter on a window of 44 samples of the input signalrepeatedly till I get a gh response indicating that I have got thereqd. frequency.I know DFT/FFT is an option but are there any better algorithms I canuse for my project. Please suggest the best algorithm that will do mywork. I am writing the code in C++ so if anyone can point me to existingcode on the net that will be really helpful.Atri.=== === Subject: : Re: Simple question regarding frequency detection> I am writing an application to detect a particular frequency (viz.> 4.5 Khz) in an input audio signal. I have written the code to> capture the sound using a microphone; but I have to properly design> a digital filter wch will return TRUE as soon as it detects sound> of ts frequency in the input signal. Also my project demands> that I have to detect the signal by looking at a very small number> of samples e.g. 44. So I will apply the digital filter on a window> of 44 samples of the input signal repeatedly till I get a gh> response indicating that I have got the reqd. frequency.> I know DFT/FFT is an option but are there any better algorithms> I can use for my project. Please suggest the best algorithm that> will do my work. I am writing the code in C++ so if anyone can> point me to existing code on the net that will be really helpful.> Atri.I would be inclined to use regression to fit the sampled signal to:c1 + c2 cos(t 2Pi 4500Hz) + c3 sin(t 2Pi 4500Hz)The amplitude of the 4.5 KHz is then sqrt(c2^2 + c3^2).For best results, the model should also include background signalsof known specific frequencies.The inverse matrix used in the regression can be calculated aheadof time since it doesn't change. / . | n Sum(cos) Sum(sin) | . A = | Sum(cos) Sum(cos^2) Sum(cos sin) | . | Sum(sin) Sum(cos sin) Sum(sin^2) | . / / / . | c1 | | Sum(y) | . C = | c2 | Y = | Sum(y cos) | . | c3 | | Sum(y sin) | . / /A C = YC = A^-1 Y=== === Subject: : Re: Simple question regarding frequency detection> I am writing an application to detect a particular frequency(viz. 4.5> Khz) in an input audio signal. I have written the code to capture the> sound using a microphone; but I have to properly design a digital filter> wch will return TRUE as soon as it detects sound of ts frequency in> the input signal. Also my project demands that I have to detect the> signal by looking at a very small number of samples e.g. 44. So I will> apply the digital filter on a window of 44 samples of the input signal> repeatedly till I get a gh response indicating that I have got the> reqd. frequency.I know DFT/FFT is an option but are there any better algorithms I can> use for my project. Please suggest the best algorithm that will do my> work. I am writing the code in C++ so if anyone can point me to existing> code on the net that will be really helpful.Atri.Here is Matlab code for the Goertzel algm if you just need aTRUE/FLASE indication. Set a threshold, and if y is greater than thethreshold, then the frequency is detected.x = input vectorf = frequency desiredN = number of samples to test overFs = sample frequencyfunction y = goert(x,f,N,Fs)k = fix(0.5 + N*f/Fs);w = (2*pi/N)*k;coeff = 2 * cos(w);Q1 = 0;Q2 = 0;for j = 1:N Q0 = coeff * Q1 - Q2 + x(j); Q2 = Q1; Q1 = Q0;endy = sqrt(Q1^2 + Q2^2 - Q1 * Q2 * coeff);Here is code to test the alghm. Plot F aganist y to see the detectoroutput as a function of frequency.fdet is the frequency desiredN = number of samples to test overFs = sample frequencyfunction [F,y] = testgoert(fdet,N,Fs)t = 0:1/Fs:((1/Fs)*N - (1/Fs));k = 1;for f = fix(fdet/2:(fdet/2)/15:fdet*2) x = sin(2*pi*f*t); y(k) = goert(x,fdet,N,Fs); F(k) = f; k = k + 1;endy = y(:);F = F(:);Maurice Givens=== === Subject: : Re: Simple question regarding frequency detection> I am writing an application to detect a particular frequency(viz. 4.5> Khz) in an input audio signal. I have written the code to capture the> sound using a microphone; but I have to properly design a digital filter> wch will return TRUE as soon as it detects sound of ts frequency in> the input signal. Also my project demands that I have to detect the> signal by looking at a very small number of samples e.g. 44. So I will> apply the digital filter on a window of 44 samples of the input signal> repeatedly till I get a gh response indicating that I have got the> reqd. frequency.I know DFT/FFT is an option but are there any better algorithms I can> use for my project. Please suggest the best algorithm that will do my> work. I am writing the code in C++ so if anyone can point me to existing> code on the net that will be really helpful.Atri.> Atri:Try the Goertzel algorithm.Good luck,OUP=== === Subject: : Re: Simple question regarding frequency detection> > I am writing an application to detect a particular frequency(viz. 4.5> Khz) in an input audio signal. I have written the code to capture the> sound using a microphone; but I have to properly design a digital filter> wch will return TRUE as soon as it detects sound of ts frequency in> the input signal. Also my project demands that I have to detect the> signal by looking at a very small number of samples e.g. 44. So I will> apply the digital filter on a window of 44 samples of the input signal> repeatedly till I get a gh response indicating that I have got the> reqd. frequency.> I know DFT/FFT is an option but are there any better algorithms I can> use for my project. Please suggest the best algorithm that will do my> work. I am writing the code in C++ so if anyone can point me to existing> code on the net that will be really helpful.> Atri.Atri:Try the Goertzel algorithm.Good luck,> OUP === === Subject: : Re: Simple question regarding frequency detection44 samples at 44.1K samples/sec.-Atri.> I am writing an application to detect a particular frequency(viz. 4.5> Khz) in an input audio signal. I have written the code to capture the> sound using a microphone; but I have to properly design a digital filter> wch will return TRUE as soon as it detects sound of ts frequency in> the input signal. Also my project demands that I have to detect the> signal by looking at a very small number of samples e.g. 44. So I will> apply the digital filter on a window of 44 samples of the input signal> repeatedly till I get a gh response indicating that I have got the> reqd. frequency.> I know DFT/FFT is an option but are there any better algorithms I can> use for my project. Please suggest the best algorithm that will do my> work. I am writing the code in C++ so if anyone can point me to existing> code on the net that will be really helpful.> Atri.> Atri: Try the Goertzel algorithm. Good luck,> OUP> 44 samples at what sample rate?> > --> Engineering is the art of making what you want from tngs you can get.> [OSlash][OSlash][OSlash][OSlash]=== === Subject: : Re: Simple question regarding frequency detectionThere is no detection bandwith specified. 1Khz, 100 Hz, 1 Hz, 0.1 Hz ?If you get a 4.8 KHz signal, should it be detected or not? The detectionbandwidth is directly related to the amount of time you have to observe thesignal. You are really trying to detect is about a 0.001 second burst ofa 4.500 KHz tone. That is 4.5 cycles of wanted tone, wch indicates thatthe detection bandwidth is very large. Noise and selectivity can beproblems with wide bandwidth. Checkout the signal spectrum or FT of a toneburst.> 44 samples at 44.1K samples/sec.> -Atri.> I am writing an application to detect a particular frequency(viz. 4.5> Khz) in an input audio signal. I have written the code to capture the> sound using a microphone; but I have to properly design a digitalfilter> wch will return TRUE as soon as it detects sound of ts frequencyin> the input signal. Also my project demands that I have to detect the> signal by looking at a very small number of samples e.g. 44. So Iwill> apply the digital filter on a window of 44 samples of the inputsignal> repeatedly till I get a gh response indicating that I have got the> reqd. frequency. I know DFT/FFT is an option but are there any better algorithms I can> use for my project. Please suggest the best algorithm that will do my> work. I am writing the code in C++ so if anyone can point me toexisting> code on the net that will be really helpful. Atri.> Atri: Try the Goertzel algorithm. Good luck, OUP 44 samples at what sample rate? > --> Engineering is the art of making what you want from tngs you can get.> [OSlash][OSlash][OSlash][OSlash]=== === Subject: : A difficult question of convolution by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i1KEBFX07024; Sir,I had a difficult time using a frequency domain convolution. The function is sin(2*pi*t/T) with period 0 I have an expression wch contain the term:> sinh(a+b)/(a+b) + sinh(a-b)/(a-b)> Although it looks like sometng that should reduce further, afterplaying> around with it for a wle (expanding sinh(a+b) and sinh(a-b) etc.) Iwasn't> able to find any simpler form. So I thought I would post it here in case> someone had some insights.It's arguable whether ts is a simplification butsinh(a+b)/(a+b) + sinh(a-b)/(a-b) := sinhc(a+b) + sinhc(a-b)= {exp[0 a + b;a + b 0] + exp[0 a - b;a - b 0]}[0 1]',where [...] is a matrix and [...]' is a matrix transpose.-- ,Gerry T.=== === Subject: : Re: Can ts be simplified? sinhc(a+b)+sinhc(a-b)Perhaps not; coming from Integral 0 to 1 of [2 cosh(a x) cosh(b x)]dx, could best be left as such.=== === Subject: : Re: Can ts be simplified? sinhc(a+b)+sinhc(a-b)>I have an expression wch contain the term:>sinh(a+b)/(a+b) + sinh(a-b)/(a-b)>Although it looks like sometng that should reduce further, after playing>around with it for a wle (expanding sinh(a+b) and sinh(a-b) etc.) I wasn't>able to find any simpler form. So I thought I would post it here in case>someone had some insights.What would you call simpler ? You presently have 3 add/substract ops., 2 divisions, 2 hyperbolic functionevaluations.It is unlikely that you could have only one hyperbolic function evaluation,you may replace one of the divisions with 3 multiplications, what moredo you expect ?=== === Subject: : Re: Can ts be simplified? sinhc(a+b)+sinhc(a-b)Ts very well may be as simple as it gets, and if so that's ok. But Ithought I would post ts just in case someone knew of a simpler form. Tsterm is actually part of a series solution that gets executed many manytimes, wch is why I'm interested.-Pat>I have an expression wch contain the term:sinh(a+b)/(a+b) + sinh(a-b)/(a-b)Although it looks like sometng that should reduce further, afterplaying>around with it for a wle (expanding sinh(a+b) and sinh(a-b) etc.) Iwasn't>able to find any simpler form. So I thought I would post it here in case>someone had some insights.> What would you call simpler ?> You presently have 3 add/substract ops., 2 divisions, 2 hyperbolicfunction> evaluations.> It is unlikely that you could have only one hyperbolic functionevaluation,> you may replace one of the divisions with 3 multiplications, what more> do you expect ?=== === Subject: : What's the computation time of ts Gradient optimization method, O(kN) or O(kN^2)?I am trying to figure out the complexity level of the gradient method forminimizing an objective function.Assume J is the scalar objection function of N vectors, each vector is of kdimension.Now we are trying to find a k-dimension vector x such that J is minimized,e.g. J is the sum of squares of the dices from N points in R^k space toa line segment,wchis connected by a known vector a and the unknown point x in R^k space.So given ts definition of J, the way we did is to calculate dJ/dx = 0 andfind the vector x.However, the expression dJ/dx is not usually in explicit form, i.e, x cannot directly be obtained andhave to be done by some iterative approach.Then for ts kind of optimization problem, how to define the computationalcomplexity?It is O(kN) or O(kN^2)?And how to verify ts?Fred=== === Subject: : Re: What's the computation time of ts Gradient optimization method, O(kN) or O(kN^2)?I am trying to figure out the complexity level of the gradient method for >minimizing an objective function.Assume J is the scalar objection function of N vectors, each vector is of k >dimension. >Now we are trying to find a k-dimension vector x such that J is minimized, >e.g. J is the sum of squares of the dices from N points in R^k space to >a line segment,wch >is connected by a known vector a and the unknown point x in R^k space.So given ts definition of J, the way we did is to calculate dJ/dx = 0 and >find the vector x. >However, the expression dJ/dx is not usually in explicit form, i.e, x can >not directly be obtained and >have to be done by some iterative approach.Then for ts kind of optimization problem, how to define the computational >complexity? >It is O(kN) or O(kN^2)?And how to verify ts?asking for the computational complexity in ts form requires exact solvability in finite time. if you underd under line segment L= {z : z=x+t*a , t arbitrary real }then J = sum_{i=1 to N} dist(x(i),L)^2 is a quadratic function in the unknown x (a given fixed), and the minimizationproblem can be solved by a finite algorithm (e.g. solving the linear system grad J=0). the solution of the problem is not well defined due to thearbitrariness of the chosen initial point x on the line and in order to make itunique you may add the constraint a'*x=0 (orthogonality). then the system becomes especially simple, the Hessian being a multiple of the unit matrix and hence the solution obtainable simply by evaluating the gradient at zero, wch involves O(kN) operations. If you restrict the possible values of t (wch would makesense to the wording segment) then J is piecewise quadratic in x and no finite expression for the solution exists. in ts case you may ask for a solution up to an error eps in norm, and then also the numerical method you choose comesinto play. since the gradient method is linearly convergent, the complexity would be O(log(eps)Nk). Observe that J is in ts case only once continuously differentiable, with the gradient Lipsctz but not smooth. === === Subject: : Microsoft and random numbersAnybody know whether MSFT has published the algorithm(s) used for thefollowing:1. Randomize and Rnd in VB 6.2. Randomize and Rnd in VB .NET3. Random class in .NET.-- http://www.dards.com/; See Howard Kaikow's web site.