mm-216 === Subject: method of moments in time domain mm-216 Good morning everyone, I was wondering if anyone had come across some open source method of moment (operating in the time domain) code? Is there any books or journal papers available that would help me to put together my own code? Can antennas be excited in method of moments code by ramped up sinusoids or gaussian pulses as they can in FDTD (finite difference time domain)? All comments are well appreciated, Alan === Subject: Re: method of moments in time domain > moment (operating in the time domain) code? Is there any books or journal > papers available that would help me to put together my own code? Have you read Eric Michielssen's papers on this subject? === Subject: Re: method of moments in time domain No I haven't read any of Michielssen's papers. So far I've been using Sergei Makarov's material found @ http://www2.ece.wpi.edu/books/aemm/index.shtml which is a frequency version of the MoM. To solve my current problem, I think I need to use MoMTD. I've read this book on the topic: S. M. Rao (Editor), Time Domain Electromagnetics, Academic Press, 1999. Which provides the necessary to write one's own code to investigate the induced current on wires illuminated by plane waves. I'd be more interested in exciting a dipole with a ramped up sinusoid, and then be able to compute the current @ the feed, the input impedance, near field values @ a point in space... Or can I get around the plane wave excitation problem by just setting the incident field @ the feedpoint to my ramped up Efield excitation? Alan > > moment (operating in the time domain) code? Is there any books or journal > > papers available that would help me to put together my own code? > Have you read Eric Michielssen's papers on this subject? === Subject: Re: method of moments in time domain > S. M. Rao (Editor), Time Domain Electromagnetics, Academic Press, > 1999. I recommend you look into Michielssen's work, as he has been active in the field of TD-MOM over the past 5 years, and his work is definitely more recent [1] than what is contained in a 1999 book (a book which is essentially a collection of even older papers). [1] Weile, D.S.; Pisharody, G.; Nan-Wei Chen; Shanker, B.; Michielssen, E.; A novel scheme for the solution of the time-domain integral equations of electromagnetics, Antennas and Propagation, IEEE Transactions on, === Subject: Re: method of moments in time domain Have a good evening, Alan >> S. M. Rao (Editor), Time Domain Electromagnetics, Academic Press, >> 1999. > I recommend you look into Michielssen's work, as he has been active > in the field of TD-MOM over the past 5 years, and his work is > definitely more recent [1] than what is contained in a 1999 book > (a book which is essentially a collection of even older papers). > [1] Weile, D.S.; Pisharody, G.; Nan-Wei Chen; Shanker, B.; Michielssen, > E.; > A novel scheme for the solution of the time-domain integral equations > of electromagnetics, Antennas and Propagation, IEEE Transactions on, === Subject: Re: method of moments in time domain Hi again, I've had a look @ Michielssen's paper & as you say R5, it's interesting. It just looks very complicated for a beginner in MoMTD! Apart from that, what's the best way to model the geometry? In Makarov's book that I mentionned earlier in the conversation, Delaunay triangulation was used. If I'm just wanting to model a dipole or a wire, what's the most simple meshing scheme that I can use? If the antenna is to be tilted in MoMTD, how can this be taken into account? Alan > Have a good evening, > Alan > >> S. M. Rao (Editor), Time Domain Electromagnetics, Academic Press, > >> 1999. > > I recommend you look into Michielssen's work, as he has been active > > in the field of TD-MOM over the past 5 years, and his work is > > definitely more recent [1] than what is contained in a 1999 book > > (a book which is essentially a collection of even older papers). > > [1] Weile, D.S.; Pisharody, G.; Nan-Wei Chen; Shanker, B.; Michielssen, > > E.; > > A novel scheme for the solution of the time-domain integral equations > > of electromagnetics, Antennas and Propagation, IEEE Transactions on, === Subject: Re: method of moments in time domain >Good morning everyone, >I was wondering if anyone had come across some open source method of >moment (operating in the time domain) code? Is there any books or journal >papers available that would help me to put together my own code? >Can antennas be excited in method of moments code by ramped up sinusoids or >gaussian pulses as they can in FDTD (finite difference time domain)? >All comments are well appreciated, > Alan yes. if you would look in Zentralblatt fuer Mathematik under method;moments you get 257 hits. for example this one 19. Zbl 1002.78531 Rius, Juan M.; beda, Eduard; Parr.97n, Josep On the testing of the magnetic field integral equation with RWG basis functions in method of moments. (English) IEEE Trans. Antennas Propag. 49, No.11, 1550-1553 (2001). MSC 2000: *78M05 78A45 hth peter === Subject: Re: method of moments in time domain Danke Schon Peter fur deine hilfe. Is this http://www.emis.de/ZMATH/en/full.html the Zentralblatt fuer Mathematik website? Alan > >Good morning everyone, > >I was wondering if anyone had come across some open source method of > >moment (operating in the time domain) code? Is there any books or > >journal > >papers available that would help me to put together my own code? > >Can antennas be excited in method of moments code by ramped up sinusoids > >or > >gaussian pulses as they can in FDTD (finite difference time domain)? > >All comments are well appreciated, > > Alan > yes. if you would look in > Zentralblatt fuer Mathematik > under method;moments > you get 257 hits. > for example this one > 19. Zbl 1002.78531 Rius, Juan M.; beda, Eduard; Parr.97n, Josep > On the testing of the magnetic field integral equation with RWG basis > functions in method of moments. (English) > IEEE Trans. Antennas Propag. 49, No.11, 1550-1553 (2001). MSC 2000: *78M05 > 78A45 > hth > peter === Subject: Re: method of moments in time domain Danke eine zweite mal! Unfortunately when I type my search in, only 3 hits come up that aren't relevant. It must be because my university hasn't subscribed to it! Alan > Danke Schon Peter fur deine hilfe. > Is this http://www.emis.de/ZMATH/en/full.html the Zentralblatt fuer > Mathematik website? > Alan > > >Good morning everyone, > > > >I was wondering if anyone had come across some open source method of > > >moment (operating in the time domain) code? Is there any books or > > >journal > > >papers available that would help me to put together my own code? > > > >Can antennas be excited in method of moments code by ramped up sinusoids > > >or > > >gaussian pulses as they can in FDTD (finite difference time domain)? > > > >All comments are well appreciated, > > > > > Alan > > > > yes. if you would look in > > Zentralblatt fuer Mathematik > > under method;moments > > you get 257 hits. > > for example this one > > 19. Zbl 1002.78531 Rius, Juan M.; beda, Eduard; Parr.97n, Josep > > On the testing of the magnetic field integral equation with RWG basis > > functions in method of moments. (English) > > IEEE Trans. Antennas Propag. 49, No.11, 1550-1553 (2001). MSC 2000: *78M05 > > 78A45 > > hth > > peter === Subject: Re: method of moments in time domain >Danke eine zweite mal! Unfortunately when I type my search in, only 3 >hits come up that aren't relevant. It must be because my university >hasn't subscribed to it! >Alan >> Danke Schon Peter fur deine hilfe. you searched with method ; moments in the title and have not excluded all sources (the 3 come if you select books?) hth peter === Subject: Re: method of moments in time domain get the hits. This is what I just noticed @ the top of the screen: Demo mode: there is a limit of max. 3 allowed answers for unregistered users. Alan > >Danke eine zweite mal! Unfortunately when I type my search in, only 3 > >hits come up that aren't relevant. It must be because my university > >hasn't subscribed to it! > >Alan > >> Danke Schon Peter fur deine hilfe. > you searched with > method ; moments > in the title and have not excluded all sources > (the 3 come if you select books?) > hth > peter === Subject: Re: method of moments in time domain >Danke Schon Peter fur deine hilfe. >Is this http://www.emis.de/ZMATH/en/full.html the Zentralblatt fuer >Mathematik website? >Alan sounds so. (for me it is http://www.emis.de/ZMATH/ cost free: because the university did pay it for me , fortunately and access to the above is refused. fortunately enough, almost all information is in English , so it is useful worldwide peter >> >Good morning everyone, >> > >I was wondering if anyone had come across some open source method of >> >moment (operating in the time domain) code? Is there any books or >> >journal >> >papers available that would help me to put together my own code? >> > >Can antennas be excited in method of moments code by ramped up sinusoids >> >or >> >gaussian pulses as they can in FDTD (finite difference time domain)? >> > >All comments are well appreciated, >> > > > Alan >> > > yes. if you would look in >> Zentralblatt fuer Mathematik >> under method;moments >> you get 257 hits. >> for example this one >> 19. Zbl 1002.78531 Rius, Juan M.; beda, Eduard; Parr.97n, Josep >> On the testing of the magnetic field integral equation with RWG basis >> functions in method of moments. (English) >> IEEE Trans. Antennas Propag. 49, No.11, 1550-1553 (2001). MSC 2000: *78M05 >> 78A45 >> hth >> peter === Subject: Re: 1-d navier stokes equation by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8PGGYP14425; function NavStokes(n) % Illustration of an application of BVP4C % This example takes the case of a steady-state ßuid, with only a % pressure gradient and ßow in the x-direction, therefore the % Navier-Stokes equation, which is % dv/dt + v * grad(v) = -grad(P) + laplacian(v) % becomes, with u as the x-ßuid speed % d^2u/dy^2 = (p1-p2)/L % The BCs are u(0)=u(1)=0, no-slip condition. % Where L is the length between the pressure gradients % n is the number of points used in the mesh if nargin~=1 || ~isnumeric(n) solinit = bvpinit(linspace(0,1,50),@ex1init); % initializing the mesh else solinit = bvpinit(linspace(0,1,n),@ex1init); % initializing the mesh end options = bvpset(ÔStats','on' ,'RelTol',1e-5); sol = bvp4c(@f,@ex1bc,solinit,options); % The solution at the mesh points x = sol.x; y = sol.y; figure; plot(x,y(1,:)') hold on a = linspace(0,1,50); analytical = (2-1)/0.735*0.5*a.*(1-a); % The analytical solution to the problem is (p1-p2)/L*0.5*y*(1-y) plot(a,analytical,'-r') legend(ÔBVP4C','Analytical Sol.'); xlabel(Ôy'); ylabel(ÔU');> title(ÔPressure-driven ßow in a Channel'); %------------------------------------------------------------- -- function dudy = f(y,u) dudy = zeros(2,1); p1 = 2; p2 = 1; L = 0.735; press_grad = (p1-p2)/L; dudy(1) = u(2); dudy(2) = -press_grad; %------------------------------------------------------------- -- function res = ex1bc(ua,ub) res = [ua(1) - 0; ub(1) - 0]; %------------------------------------------------------------- -- function v = ex1init(u) v = [0 0]; === Subject: Re: 1401 > 1401 div 3 = 467, > 467-360 = 107, > 107 / pi = 34.05915782, (34) > 34.05915782-34 = 0.05915782, > 0.05915872 * 88 = 5.205888292, (5) > 205 + 292 = 497, > 497 - 467 = 30. 1401 + 1234 = 2635, 1401 - 1234 = 167, 2635 - 167 = 2468. === Subject: Re: 1401 The resolution algorithm. Resolves all formulea. Resolves Pi to infinity. Reinvents the form and type of mathematics used. 1234 ... 1234 ... Any number subtracted and added to this sequence and the difference found can detect infinity of pi using a large sequencing database. Similar to it may be said the DNA code of just about any creature. Warning to consumer, you DNA sequence has only 23 distinctive threads. Rice has a much larger DNA sequence. Funny that not!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! === Subject: How to make a function that find the minimum value of a function with n variables in VB 6 ?? by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8Q0Nvl20995; Please I really want help. I have a function f(x1,x2,...,xn)=a*(b*(x1)+c*(x2)+...+d*(xn)+ e)^2. The restriction is: x1+x2+...+xi=1 with 0<(x1,x2,...xi)<1 (Example: x1=0.2; x2=0.4 and x3=0.4) and a,b,c,...,d,e real numbers. I need to find the minimum value of this function. An example of the function is: f(x1,x2,x3,x4)= 4,56(2*x1+3,54*x2+5*x3+8,1*x4+7,87)^2 I need a function to VB 6 that perform this. I need a library to VB6 too, that perform matrix calculations, like a*b=x. I am doing a job to possible answer this question to my e-mail. === Subject: Re: How to make a function that find the minimum value of a function with n variables in VB 6 ?? >Please I really want help. I have a function >f(x1,x2,...,xn)=a*(b*(x1)+c*(x2)+...+d*(xn)+ e)^2. The restriction is: >x1+x2+...+xi=1 with 0<(x1,x2,...xi)<1 (Example: x1=0.2; x2=0.4 and >x3=0.4) and a,b,c,...,d,e real numbers. I need to find the minimum >value of this function. An example of the function is: >f(x1,x2,x3,x4)= 4,56(2*x1+3,54*x2+5*x3+8,1*x4+7,87)^2 >I need a function to VB 6 that perform this. I need a library to VB6 >too, that perform matrix calculations, like a*b=x. I am doing a job to >possible answer this question to my e-mail. what about a=-infinity? bad joke _something is wrong with your model. don't know anything about VB6 but lots of optimization is here: http://plato.la.asu.edu/guide.html and linear algebra : look here http://www.netlib.org (lapack for example). vb this is visual basic? well f77 should be not too far hth peter === Subject: Re: How to make a function that find the minimum value of a function with n variables in VB 6 ?? > Please I really want help. I have a function > f(x1,x2,...,xn)=a*(b*(x1)+c*(x2)+...+d*(xn)+ e)^2. The restriction is: > x1+x2+...+xi=1 with 0<(x1,x2,...xi)<1 (Example: x1=0.2; x2=0.4 and > x3=0.4) and a,b,c,...,d,e real numbers. I need to find the minimum > value of this function. An example of the function is: > f(x1,x2,x3,x4)= 4,56(2*x1+3,54*x2+5*x3+8,1*x4+7,87)^2 > I need a function to VB 6 that perform this. I need a library to VB6 > too, that perform matrix calculations, like a*b=x. I am doing a job to > possible answer this question to my e-mail. Your problem is a typical nonlinear convex optimization problem which (fortunately) can be solved by semidefinite programming (SDP) using interior point method. First you need to remove the equality constrained. Then, you will need to rewrite objective function in a linear matrix inequality (LMI). Then solve this LMI with the inequality constraint using interior point method. That's the basic idea. If you do it once, you will find it very useful in many different areas. Good luck. Carson === Subject: Polynomial functions and equations discovered Polynomial functions and equations fully discovered and explained! Free to learn, study and use: linear, quadratic, cubic, quartic, quintic,...and the nth-degree polynomial. See http://www.nabla.hr === Subject: Re: Multiple Regression w/ Polynomial-in-Y? > Is what I'm tempted to call Polynomial Root Regression so obvious > Here's the model, which is readily solved with multiple regression > methods (also tried it successfully with shrinkage regression > technique such as Partial Least Squares) - note well, the coefficients > of the dependent-variable polynomial are unknown, and are estimated by > the (standard) regression (in addition to the usual weights for the > independent variables). > Here, a quadratic-in-Y polynomial (extension to higher-order > obvious): > x.w = y + c*y^2 + error > M indep. vars; N samples; (intercept term suppressed) > Both w vector and c coefficient unknown/to be estimated > |x(1,1), x(1,2)...x(1,M); -y(1)^2| |w(1)| |y(1)| > |x(2,1), x(2,2)...x(2,M); -y(2)^2| |w(2)| |y(2)| > |....... | |... | = |... | > |x(N,1), x(N,2)...x(N,M); -y(N)^2| |w(M)| |y(N)| > | c | > Solve above with appropriate linear least-squares solver (e.g., OLS, > or RR or PLS if multicollinearity). Yields w-vector and c coefficient > for quadratic term for y. > To predict a new y-value, just find the root y of the determined > polynomial > c*y^2 + y - x.w = 0 > I tried this successfully for a problem whose non-linearity was best > modeled on the dependent-variable side, rather than forcing polynomial > terms of independent variables (which I suspect would not be so good > for my application). One drawback is needing to decide which root is > the actual solution, but for many situations this is probably easy. > Thoughts? Does this have a name? Frank: Almost everyday I have to face this problem. In chemometrics there is something with the bastard name of quantitation limits between which the response is linear. the lower limit is upstream of the detection limit and the upper limit is where interactions at high concentrations cause the system to go non-linear. Have you considered using one or another data imputation procedures wherein you measure many variables in many samples, most ranging within the quantitation limits, along with your target variable that is biased at high concentrations? You take advantage of the fact that there usually usually is a web of correlations imposed by thermodynamics among the variables. The only problem is the time/cost of measuring, say, 20 samples to impute best estimates of response at high concentrations. === Subject: Re: Multiple Regression w/ Polynomial-in-Y? Sorry found the papers and the term is >>fixed-point regression<< or >>fixed-point solution<< !! here are some papers & sites:==> http://www.stat.wisc.edu/~wardrop/courses/302ch5.pdf and http://lec.ugr.es/~julyan/papers/rkpaper/node5.html (not sure of last but looked interesting) all the best Paul > Is what I'm tempted to call Polynomial Root Regression so obvious > Here's the model, which is readily solved with multiple regression > methods (also tried it successfully with shrinkage regression > technique such as Partial Least Squares) - note well, the coefficients > of the dependent-variable polynomial are unknown, and are estimated by > the (standard) regression (in addition to the usual weights for the > independent variables). > Here, a quadratic-in-Y polynomial (extension to higher-order > obvious): > x.w = y + c*y^2 + error > M indep. vars; N samples; (intercept term suppressed) > Both w vector and c coefficient unknown/to be estimated > |x(1,1), x(1,2)...x(1,M); -y(1)^2| |w(1)| |y(1)| > |x(2,1), x(2,2)...x(2,M); -y(2)^2| |w(2)| |y(2)| > |....... | |... | = |... | > |x(N,1), x(N,2)...x(N,M); -y(N)^2| |w(M)| |y(N)| > | c | > Solve above with appropriate linear least-squares solver (e.g., OLS, > or RR or PLS if multicollinearity). Yields w-vector and c coefficient > for quadratic term for y. > To predict a new y-value, just find the root y of the determined > polynomial > c*y^2 + y - x.w = 0 > I tried this successfully for a problem whose non-linearity was best > modeled on the dependent-variable side, rather than forcing polynomial > terms of independent variables (which I suspect would not be so good > for my application). One drawback is needing to decide which root is > the actual solution, but for many situations this is probably easy. > Thoughts? Does this have a name? === Subject: Re: Multiple Regression w/ Polynomial-in-Y? Doing more poking around and remembering the notion of a stationary value, it might be that your problem is related to Stationary Point Solutions of Multivariate Regression. Here is a paper where we are after the centre of y contours http://www.stat.rutgers.edu/~buyske/591/lect07.pdf Stationary Points are denoted as those points where differentials = zero but there is likely much more than that to this approach. I brießy looked at is some years ago in context of Neural Networks. Again one seems to get the dependent variable y on both sides of the equation. I will try to find the papers I know I have upstairs to look at this problem again wrt the Stationary Point. Paul > Is what I'm tempted to call Polynomial Root Regression so obvious > Here's the model, which is readily solved with multiple regression > methods (also tried it successfully with shrinkage regression > technique such as Partial Least Squares) - note well, the coefficients > of the dependent-variable polynomial are unknown, and are estimated by > the (standard) regression (in addition to the usual weights for the > independent variables). > Here, a quadratic-in-Y polynomial (extension to higher-order > obvious): > x.w = y + c*y^2 + error > M indep. vars; N samples; (intercept term suppressed) > Both w vector and c coefficient unknown/to be estimated > |x(1,1), x(1,2)...x(1,M); -y(1)^2| |w(1)| |y(1)| > |x(2,1), x(2,2)...x(2,M); -y(2)^2| |w(2)| |y(2)| > |....... | |... | = |... | > |x(N,1), x(N,2)...x(N,M); -y(N)^2| |w(M)| |y(N)| > | c | > Solve above with appropriate linear least-squares solver (e.g., OLS, > or RR or PLS if multicollinearity). Yields w-vector and c coefficient > for quadratic term for y. > To predict a new y-value, just find the root y of the determined > polynomial > c*y^2 + y - x.w = 0 > I tried this successfully for a problem whose non-linearity was best > modeled on the dependent-variable side, rather than forcing polynomial > terms of independent variables (which I suspect would not be so good > for my application). One drawback is needing to decide which root is > the actual solution, but for many situations this is probably easy. > Thoughts? Does this have a name? === Subject: Re: Multiple Regression w/ Polynomial-in-Y? If Y is truly the endogenous variable, then you are missing a Jacobian term for the model, if you want to get ML estimates. In econometrics, we call this FIML estimation. A similar situation which is nonlinear in the endogenous variable is a Box-Cox model. Clint Cummins === Subject: Re: Multiple Regression w/ Polynomial-in-Y? Paul, my motivation was a chemometric application, where I wanted to use PLS to estimate a regression between spectral intensity measurements (x) and chemical concentration (y). At higher concentrations, we have reason to believe there is self-absorption, thus a pragmatic way to model the emitted intensities versus concentration is: x.w = y - c*y^2 (c positive) that is, intensities will tend to be compressed at higher concentrations. I could have tried doing PLS using polynomial terms of x on the LHS, but my proposed approach seemed a more parsimonious model. Clint, I'm not sure I follow your point. I looked up FIML estimation, and discovered that two-stage least squares is related to that. But I don't really see the connection of either of these to what I'm Paul, indeed I hope others chime in. For what I was trying to do, the non-linear PLS (NLPLS), kernelized-PLS, etc seemed overkill, when I really simply needed to estimate the coefficient of my RHS quadratic term in Y (as well as w). Clint, I considered the Box-Cox transformation (Y^a), but was underwhelmed by the suggested methods of estimating a (e.g., iteratively, based on ML of presumed gaussian residuals). === Subject: Re: Multiple Regression w/ Polynomial-in-Y? > Paul, my motivation was a chemometric application, where I wanted to > use PLS to estimate a regression between spectral intensity > measurements (x) and chemical concentration (y). At higher > concentrations, we have reason to believe there is self-absorption, > thus a pragmatic way to model the emitted intensities versus > concentration is: > x.w = y - c*y^2 (c positive) > that is, intensities will tend to be compressed at higher > concentrations. > I could have tried doing PLS using polynomial terms of x on the LHS, > but my proposed approach seemed a more parsimonious model. > Clint, I'm not sure I follow your point. I looked up FIML estimation, > and discovered that two-stage least squares is related to that. But I > don't really see the connection of either of these to what I'm > Paul, indeed I hope others chime in. For what I was trying to do, the > non-linear PLS (NLPLS), kernelized-PLS, etc seemed overkill, when I > really simply needed to estimate the coefficient of my RHS quadratic > term in Y (as well as w). I am the author of a nonlinear regression program called NLREG (http://www.nlreg.com). NLREG has the ability to fit pretty much any function to data, but you have to specify the form of the function with parameters whose values are to be computed. In addition to normal function fitting of the form: y = f(x1,x2,...) NLREG also can handle general fitting of functions that don't have a simple variable on the left of the equal sign. This is done by minimizing a general function: f(y1, x2, x3, ...) An example of this type of fit can be seen at http://www.nlreg.com/circular.htm If you want to send me your data and function definition, I will be happy to try to run it through NLREG. Or you can download a free, demonstration copy of NLREG from http://www.nlreg.com/DownloadDemo.htm and try it yourself. -- Phil Sherrod (phil.sherrod Ôat' sandh.com) http://www.dtreg.com (decision tree modeling) http://www.nlreg.com (nonlinear regression) http://www.NewsRover.com (Usenet newsreader) http://www.LogRover.com (Web statistics analysis) === Subject: Re: Multiple Regression w/ Polynomial-in-Y? I have been trying to find more about Clint's suggestion re FIML there are a number but does this look good Clint? http://gsbwww.uchicago.edu/computing/research/SASManual/ets/ chap14/sect1.htm Paul > Paul, my motivation was a chemometric application, where I wanted to > use PLS to estimate a regression between spectral intensity > measurements (x) and chemical concentration (y). At higher > concentrations, we have reason to believe there is self-absorption, > thus a pragmatic way to model the emitted intensities versus > concentration is: > x.w = y - c*y^2 (c positive) > that is, intensities will tend to be compressed at higher > concentrations. > I could have tried doing PLS using polynomial terms of x on the LHS, > but my proposed approach seemed a more parsimonious model. > Clint, I'm not sure I follow your point. I looked up FIML estimation, > and discovered that two-stage least squares is related to that. But I > don't really see the connection of either of these to what I'm > Paul, indeed I hope others chime in. For what I was trying to do, the > non-linear PLS (NLPLS), kernelized-PLS, etc seemed overkill, when I > really simply needed to estimate the coefficient of my RHS quadratic > term in Y (as well as w). > Clint, I considered the Box-Cox transformation (Y^a), but was > underwhelmed by the suggested methods of estimating a (e.g., > iteratively, based on ML of presumed gaussian residuals). === Subject: Re: Multiple Regression w/ Polynomial-in-Y? Furthermore, something like MAPLE here would be good to find the roots. BTW MAPLE is lots of $ but handy to have around. Maybe you can download an older version from the net. Paul > Is what I'm tempted to call Polynomial Root Regression so obvious > Here's the model, which is readily solved with multiple regression > methods (also tried it successfully with shrinkage regression > technique such as Partial Least Squares) - note well, the coefficients > of the dependent-variable polynomial are unknown, and are estimated by > the (standard) regression (in addition to the usual weights for the > independent variables). > Here, a quadratic-in-Y polynomial (extension to higher-order > obvious): > x.w = y + c*y^2 + error > M indep. vars; N samples; (intercept term suppressed) > Both w vector and c coefficient unknown/to be estimated > |x(1,1), x(1,2)...x(1,M); -y(1)^2| |w(1)| |y(1)| > |x(2,1), x(2,2)...x(2,M); -y(2)^2| |w(2)| |y(2)| > |....... | |... | = |... | > |x(N,1), x(N,2)...x(N,M); -y(N)^2| |w(M)| |y(N)| > | c | > Solve above with appropriate linear least-squares solver (e.g., OLS, > or RR or PLS if multicollinearity). Yields w-vector and c coefficient > for quadratic term for y. > To predict a new y-value, just find the root y of the determined > polynomial > c*y^2 + y - x.w = 0 > I tried this successfully for a problem whose non-linearity was best > modeled on the dependent-variable side, rather than forcing polynomial > terms of independent variables (which I suspect would not be so good > for my application). One drawback is needing to decide which root is > the actual solution, but for many situations this is probably easy. > Thoughts? Does this have a name? === Subject: Re: Multiple Regression w/ Polynomial-in-Y? Firstly, I don't know what this is called. I was considering a model such as C1 * ln(y) + C2 * exp(y) just the other day so it was with some interest I saw your post. What did Karl Jung say about synchronicity??!! What I would like to know is why you were movivated to do this. You mentioned in your text the feeling or belief the RHS was nonlinear. polynomial formulation would be somehow better. Why did you suppress w0 the constant. Unless your formulation is a probability summation I think you should not do that. Then w0 estimates the error term I think Anyways, well done and I hope we hear from some math gurus here Paul Birke, P. Eng. > Is what I'm tempted to call Polynomial Root Regression so obvious > Here's the model, which is readily solved with multiple regression > methods (also tried it successfully with shrinkage regression > technique such as Partial Least Squares) - note well, the coefficients > of the dependent-variable polynomial are unknown, and are estimated by > the (standard) regression (in addition to the usual weights for the > independent variables). > Here, a quadratic-in-Y polynomial (extension to higher-order > obvious): > x.w = y + c*y^2 + error > M indep. vars; N samples; (intercept term suppressed) > Both w vector and c coefficient unknown/to be estimated > |x(1,1), x(1,2)...x(1,M); -y(1)^2| |w(1)| |y(1)| > |x(2,1), x(2,2)...x(2,M); -y(2)^2| |w(2)| |y(2)| > |....... | |... | = |... | > |x(N,1), x(N,2)...x(N,M); -y(N)^2| |w(M)| |y(N)| > | c | > Solve above with appropriate linear least-squares solver (e.g., OLS, > or RR or PLS if multicollinearity). Yields w-vector and c coefficient > for quadratic term for y. > To predict a new y-value, just find the root y of the determined > polynomial > c*y^2 + y - x.w = 0 > I tried this successfully for a problem whose non-linearity was best > modeled on the dependent-variable side, rather than forcing polynomial > terms of independent variables (which I suspect would not be so good > for my application). One drawback is needing to decide which root is > the actual solution, but for many situations this is probably easy. > Thoughts? Does this have a name? === Subject: Good Numerical Analysis Tutorial by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8R12ak06301; Does anyone know a good Numerical Analysis Tutorial site? Please help me if you know. === Subject: Re: Good Numerical Analysis Tutorial > Does anyone know a good Numerical Analysis Tutorial site? > Please help me if you know. You might find these notes useful. Enjoy. http://Galileo.phys.Virginia.EDU/classes/551.jvn.fall01/ 551Notes.htm -- Julian V. Noble Professor Emeritus of Physics ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: Good Numerical Analysis Tutorial >Does anyone know a good Numerical Analysis Tutorial site? The book Numerical Mathematics and Scientific Computation by Germund Dahlquist and .81ke Bj.9arck is online at http://www.mai.liu.se/~akbjo/NMbook.html . 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Eric Bourgade href=http://www.cs.wm.edu/~va/software/DirectSearch/wrappers/ direct.f>ht tp://www.cs.wm.edu/~va/software/DirectSearch/wrappers/direct.f href=http://www.netlib.org/opt/hooke.c>http://www.netlib.org/ opt/hooke.c href=http://ftp.math.ncsu.edu/FTP/kelley/optimization/matlab/ hooke.m>htt p://ftp.math.ncsu.edu/FTP/kelley/optimization/matlab/hooke.m I'm looking for the Hook_Jeeves algorithm.Actually I've found one,but >> it's not very clear.Can anyone help me in this respect? >> Padideh === Subject: Who konws how to use FFTW guru interface? I work with it for a few days, but it does not work even for a simple test, such as transfer use fftw_plan_guru_r2r for cos(x). There always causes cores when I excute the plan, and it also warnning me with required for this conversiono of constant. fftw_plan planf; int rank = 1; int howmany_rank = 1; double *f; fftw_iodim *dims, *howmany_dims; dims = (fftw_iodim *)malloc(rank*sizeof(fftw_iodim)); howmany_dims = (fftw_iodim *)malloc(howmany_rank*sizeof(fftw_iodim)); for (i=1; i<=rank; i++){ dims[i-1].n = M; dims[i-1].is = 0; dims[i-1].os = 0;} for (i=1; i<=howmany_rank; i++){ howmany_dims[i-1].n = M; howmany_dims[i-1].is = 1; howmany_dims[i-1].os = 1;} planf = fftw_plan_guru_r2r(rank, dims, howmany_rank, howmany_dims, f, f, FFTW_REDFT01, FFTW_EXHAUSTIVE); ...... fftw_execute_r2r(planf, f, f); ...... === Subject: Re: Who konws how to use FFTW guru interface? > There always causes cores when I excute the plan, and it also warnning > me with required for this conversiono of constant. That's because you're not passing the right argument types to fftw_plan_guru_r2r (the second-to-last argument should be an array, not a single constant). Also, your strides are incorrect (a stride of 0 means that the i-th element is stored at 0*i = 0, which is obviously not right). Steven === Subject: Re: solution of set of linear algebraic equation meet_brain@yahoo.com (Amit) dixit: >Hi Sylvain >If u find rank of the augmented matrix (A | b), it can not be more >than 2; it can either be 1 which will indicate system has multiple >solution or it can be 2 which will indicate system has a unique >solution. >But we know that system is inconsistent, it doesn't have a solution. >is there any way to infer inconsistency of a set of linear algebraic >equations. >Amit First of all, we're talking about inhomogenous systems (which have a non-zero right-hand side b). If the system is homogenous there is always of course the trivial solution of all zeros. For inhomogenous systems, what I meant and should have said is to compare rank(A) = rank(A|b) for a consistent system, and if so, then rank(A)=n for a unique solution. In your example of x+y=2 x+y=3 rank(A)=1 but rank(A|b)=2 so it's inconsistent. If you had x+y=2 x+y=2 rank(A)=1 and rank(A|b)=1 so it's consistent. But n=2 so there are infinitely many solutions. And with x+y=2 x+2y=3 rank(A)=2 and rank(A|b)=2 so it's consistent. And n=2 so there is only one solution x=1, y=1. >>meet_brain@yahoo.com (Amit) dixit: >I have a doubt. Suppose we are given with a set of linear algebraic >equations. how do we find out whether the given set of linear >algebraic equations have atleast one solution or not. In other words >how do we find out whether the given set of equations is a >consistent >or an inconsistent set. >for example- >x+y=2 >x+y=3 >is an inconsistent set but the rank of the A matrix (Ax=b) is 1. >Therefore I feel that finding out rank of the matrix and then >comparing it with the no. of variables is not the right approach. >second approach may be the degree of freedom (DOF)approach. But >then, >the present system has zero degrees of freedom even than the >solution >is not uniquely determined. >Both RANK & DOF approach does not seem to work here. Is there any >way >to find out whether a geven set of algebraic equations is a >consistent. >>Find the rank of the matrix A augmented with the right-hand side, >i.e. >>rank(A | b). === Subject: Re: solution of set of linear algebraic equation > Hi Sylvain > If u find rank of the augmented matrix (A | b), it can not be more > than 2; it can either be 1 which will indicate system has multiple > solution or it can be 2 which will indicate system has a unique > solution. > But we know that system is inconsistent, it doesn't have a solution. > is there any way to infer inconsistency of a set of linear algebraic > equations. > Amit See my other post for the math theory answer. However can't you infer inconsistency by comparing the rank or (A|b) with the rank of A? I haven't spent much time thinking about it, so you need to check but: it seems that if the ranks of these two quantities are *different* then the equations are going to have to be inconsistent.... If the ranks are the same then they are going to be solvable... Best wishes, andy. === Subject: prove existence of group multiplication modulo by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8RMk1b22716; Hi.. i m supposed to prove that a number n, let S_n denote the set of all numbers that are less than n and relatively prime to n. Show that S_n is a group for all n under multiplication modulo n.. now i cud convince myself to do this.. and know that it works.. i know that it satisfies the terms of closure, associativity, identity element, inverse element etc... can anyone help me detail a formal proof..?? === Subject: Re: prove existence of group multiplication modulo > Hi.. i m supposed to prove that a number n, let S_n denote the set of > all numbers that are less than n and relatively prime to n. Show that > S_n is a group for all n under multiplication modulo n.. > now i cud convince myself to do this.. and know that it works.. i know > that it satisfies the terms of closure, associativity, identity > element, inverse element etc... > can anyone help me detail a formal proof..?? That's the proof. If it's closed, multiplication is associative, it has an identity, and each element has an inverse, then it's a group. -- Julian V. Noble Professor Emeritus of Physics ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: Numerical Differentiation in C > Hi! > I'm writing a routine to calculate the n'th derivative of a function > numerically. > (snip) > { > df = (f(x+h) - f(x)) / h; > } Have a look at Numerical Recipes suggestions on http://www.library.cornell.edu/nr/bookcpdf/c5-7.pdf Might give you some insights and references to work on if you want to get further into it. === Subject: Re: Numerical Differentiation in C > I'm writing a routine to calculate the n'th derivative of a function > numerically. > df = (f(x+h) - f(x)) / h Numerical differentiation is a precarious operation and above is doomed to fail. You'll need to scale the increment, h = eps*abs(x) , where eps is related to machine epsilon. See, Jacobian fcns in the minpack dir for practical details. http://netlib.org/minpack === Subject: Re: Numerical Differentiation in C > I'm writing a routine > to calculate the n'th derivative of a function numerically. Try derivative arithmetic. I used Google to search for +derivative arithmetic and I found lots of stuff. === Subject: Re: Numerical Differentiation in C >>I'm writing a routine to calculate the n'th derivative of a function >>numerically. I took the standard difference equation from calculus and >>ßoat diff(ßoat(*f)(ßoat), ßoat x, ßoat n, ßoat h) >> //The difference equation from calc 1 >> df = (diff(f,x+h, n - 1, h) - diff(f,x, n - 1, h)) / h; >> df = (f(x+h) - f(x)) / h; > Peter Spellucci did cover the following point, but I want to emphasize > that it is one simple thing that will drastically improve the performance > of your program: Use > df = (f(x+h) - f(x-h))/(2*h) > as the difference equation. Look at it on a graph: The slope of a secant > line that goes on either side of the point at which you want the > derivative is usually much closer to the slope of the tangent line than is > the slope of a secant line that goes through the point at which you want > the derivative. Hmmmm, good point. > Also, minor point: Why use a ßoat instead of an int for n? *shrug* Cause. -paul === Subject: Re: Numerical Differentiation in C > I'm writing a routine to calculate the n'th derivative of a function Why not forward FFT your function, multiply by (i * omega) ^ n, and then inverse FFT back? === Subject: Re: Numerical Differentiation in C >> I'm writing a routine to calculate the n'th derivative of a function >Why not forward FFT your function, multiply by (i * omega) ^ n, >and then inverse FFT back? a good idea if your function has one (or say two, three ) variables and a lot of function values are known beforehand _not the situation where this kind of finite differencing is normally applied_ (lots of variables, function evaluation expensive, maybe result of a black box algorithm etc hth peter === Subject: Re: Numerical Differentiation in C > a lot of function values are known beforehand _not the situation where > this kind of finite differencing is normally applied_ (lots of variables, > function evaluation expensive, maybe result of a black box algorithm etc However, you have to pay the piper either through lots of fine samples for finite differencing or through uniform sampling for an FFT. An FFT approach has the added advantage of being more numerically stable than successive finite differencing at each derivative order. Yeah, the FFT might be expensive if the user cares only about the derivative at one point. Finally, FFT's can most certainly be connected to the outputs of black boxes for the purpose of analyzing or extracting information reported by the black boxes. Ever heard of SAR imagery? === Subject: Re: Numerical Differentiation in C >>I'm writing a routine to calculate the n'th derivative of a function > Why not forward FFT your function, multiply by (i * omega) ^ n, > and then inverse FFT back? I'm not familiar with FFTs; I know that they move from space to time domain and back again, but beyond that idea, I'm ignorant. So I'm unfamiliar with i and omega and how they factor into this matter. -paul === Subject: Re: Numerical Differentiation in C > domain and back again, but beyond that idea, I'm ignorant. > So I'm unfamiliar with i and omega and how they factor into this matter. Take a look at Eqn (33) at the Mathematica site: http://mathworld.wolfram.com/FourierTransform.html === Subject: Question about the modulo operator A few hours ago we discovered that Java allows the use of the modulo operator (%) for ßoating point numbers. My understanding is that modulo, from Gauss' time, was defined for natural numbers only (i.e. integers). The ßoating point modulo somehow doesn't seem right. I'm not arguing that the language should have it or not. What I'd like to know is if modulo is defined for any real number or only for natural numbers. If you have a web page with proof I'll be happy to continue the research myself. This seems wrong because, mathematically, things like: PI % 2 = 1.14159265.... would not allow reconstruction of PI by doing: (1.14159265...? + 2*1) == PI. (I think this means that modulo would not be defined for irrational numbers). P. === Subject: Re: Question about the modulo operator > A few hours ago we discovered that Java allows the use of the modulo > operator (%) for ßoating point numbers. My understanding is that > modulo, from Gauss' time, was defined for natural numbers only (i.e. > integers). The ßoating point modulo somehow doesn't seem right. I'm > not arguing that the language should have it or not. What I'd like to > know is if modulo is defined for any real number or only for natural > numbers. If you have a web page with proof I'll be happy to continue > the research myself. > This seems wrong because, mathematically, things like: > PI % 2 = 1.14159265.... > would not allow reconstruction of PI by doing: > (1.14159265...? + 2*1) == PI. > (I think this means that modulo would not be defined for irrational > numbers). What is the objection (other than you only just found out about this more general usage of the notion)? If one said remainder rather than modulo would anything be different? A common application is argument reduction for the complex exponential (aka sine and cosine). P.S. Many of the things you learned about modular arithmetic are special cases of interpolation theorems, so generalizations are nothing new. > P. === Subject: Re: Question about the modulo operator by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8SCO2Z26952; >A few hours ago we discovered that Java allows the use of the modulo operator >(%) for ßoating point numbers. My understanding is that modulo, from Gauss' >time, was defined for natural numbers only (i.e. integers). The ßoating point >modulo somehow doesn't seem right. I'm not arguing that the language should >have it or not. What I'd like to know is if modulo is defined for any real >number or only for natural numbers. If you have a web page with proof I'll be >happy to continue the research myself. >This seems wrong because, mathematically, things like: >PI % 2 = 1.14159265.... >would not allow reconstruction of PI by doing: >(1.14159265...? + 2*1) == PI. >(I think this means that modulo would not be defined for irrational numbers). I believe we must think and Ôfeel'at distance about mathematical concepts. For instance function modulo 2 is just NOT BIJECTIVE like many others :Abs(x),x^2,homographic ..and any periodical function: sin(x),frac(x/2) or constant f(x)=c. Second aspect|INFORMATION :with bijective operators we do not lose any information Ôcause we may get back ,... Can anyone please provide some guidance? YOU - C'est la vie,Alain. === Subject: Re: Solving non-linear system of n variable > you have some data say (xdata(i), ydata(i)) i=1,....,N > xdata as well as ydata can be vectors > in order to solve something with these data you must have a model > ydata=F(xdata, p) where p are the unknowns > and the true relationship might be > ydata(i)=F(xdata(i)+noisex(i),p)+noisey(i) this is precisely my system. a vector Yi is my output for my input vector Xi and i don't know the F(Xi) that gives Yi. all i can do is to get some observations for the values of Xi and Yi. it should be noted that Xi is n-tuple and Yi is m-tuple. > (the model may even change for different parts of the data set, this can also > be written in this form with F defined piecewise) > now you can try to identify p form these equations. > it depends on your assumptions on the nature of noise how to proceed > and nonlinear least squares is claerly the simplest way to go under the > usual Gaussian model. Yes, Gauss Siedel or Jacobi methods do speak about solving non-linear n-variable systems, provided we have Ômodels' for the observations for component results. What I mean is they give methods to solve equations in n-variable not really map observations for n-variable system to function. > but without such a model , which equations might you solve? Yes, you are right that I dont have equations defining my metrics/outputs. > a neural system is also nothing than such a model I am not sure, if I completely understand this statement. I believe that all we have to decide in modelling the neural system, is the learning function (one of the sigmoid functions?!!) based on the samples w.r.t the learning curve and the error factors and the layers and intermediate nodes -- by iterations. Correct me if I am wrong here. -sureshr === Subject: Re: Solving non-linear system of n variable >> you have some data say (xdata(i), ydata(i)) i=1,....,N >> xdata as well as ydata can be vectors >> in order to solve something with these data you must have a model >> ydata=F(xdata, p) where p are the unknowns >> and the true relationship might be >> ydata(i)=F(xdata(i)+noisex(i),p)+noisey(i) >this is precisely my system. a vector Yi is my output for my input >vector Xi and i don't know the F(Xi) that gives Yi. all i can do is to >get some observations for the values of Xi and Yi. it should be noted >that Xi is n-tuple and Yi is m-tuple. >> (the model may even change for different parts of the data set, this can also >> be written in this form with F defined piecewise) >> now you can try to identify p form these equations. >> it depends on your assumptions on the nature of noise how to proceed >> and nonlinear least squares is claerly the simplest way to go under the >> usual Gaussian model. >Yes, Gauss Siedel or Jacobi methods do speak about solving non-linear >n-variable systems, provided we have Ômodels' for the observations for >component results. What I mean is they give methods to solve equations >in n-variable not really map observations for n-variable system to >function. >> but without such a model , which equations might you solve? >Yes, you are right that I dont have equations defining my >metrics/outputs. >> a neural system is also nothing than such a model >I am not sure, if I completely understand this statement. I believe >that all we have to decide in modelling the neural system, is the >learning function (one of the sigmoid functions?!!) based on the >samples w.r.t the learning curve and the error factors and the layers >and intermediate nodes -- by iterations. Correct me if I am wrong yes but here you rely on the fact that the set of sigmoids is dense in the set of continuous functions in practice you must decide on the form you use and this in turn will inßuence your result. but without any model this would indeed be one way to go you will see that this kind of fitting is not that easy , though hth peter >here. >-sureshr === Subject: Re: curvature term for circular shapes >I have a computer program which comes up with Ôn' points representing >a closed polygon (contour) based on various factors. >The object which the contour or polygon represents is more or less >circular in nature. My program sometimes returns shapes which are >completely non-circular. >Question: >Is there a way to come up with a curvature term so that these Ôn' >points (x,y) would form more or less a circular shape? >The idea here is to insert this curvature term in my program so that >the result I get, the Ôn' points, would represent more or less a >circle and not any arbid shape. >Not sure if I have explained my problem well. >Tejas I assume you use some least squares fitting for the curve but unfortunately you said nothing concerning this, neither your model for the curve nor the model used. what means more or less circular? you could fit a circle (easy). you could fit an ellipse (not so easy, but not too hard with the right model) or you may fit by parametric splines with positivity condition on the curvature which gives in the parametric representation a nonlinear (quadratic) constraint from dot y dotdot y - dot dot x dot y >0 for the two splines (no assumption on the special form of curve beyond positive curvature) but this involves now , if done reasonably, least squares fitting by cubic splines with a free number of nodes and side constraints (nothing ready to use as far as I know) but there are sufficient software components out there to solve this (with some personal effort) see http://plato.la.asu.edu/topics/problems/nlores.html for optimization and http://www.netlib.org/dierckx for splines hth peter === Subject: Virtual community for the Simulation and the Numerical Modelling Finite Element Method - Analysis http://it.groups.yahoo.com/group/fem-analysis/ The best salutes, peace. Gabriele Martufi (ITALY) http://it.geocities.com/gabrielemartufi/ === Subject: Markov Chain Analysis by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8SEg5M08337; Does anyone know an algorithm for deriving expectations for infinite markov chains? Are there any tools? === Subject: interpolation question Question: For n=3, explain why L0(x)+L1(x)+L2(x)+L3(x)=1 for all x. Again thank you === Subject: Re: interpolation question >Question: For n=3, explain why > L0(x)+L1(x)+L2(x)+L3(x)=1 for all x. >Again thank you homework, clearly. didn't you learn about uniqueness of polynomial interpolation? and how reads the Lagrange interpolation formula? hth peter === Subject: Math can be entertaining While in high school in late 50's I came across _Mathematics for the Millions_ . It's long out of print. Is there something currently in print of similar nature aimed at general public? Is there anything similar aimed at lower elementary grades? === Subject: Re: Math can be entertaining > While in high school in late 50's I came across _Mathematics for the > Millions_ . > It's long out of print. Are you sure? This seems to be the book: http://www.amazon.com/exec/obidos/tg/detail/-/039331071X > Is there something currently in print of similar nature aimed at general > public? > Is there anything similar aimed at lower elementary grades? Not sure what the book above actually is, but if you're interested in recreational mathematics, I like the Dr. Ecco books by Dennis Shasha (the problems are nicely set up as stories) and the various books by Martin Gardner. cheers, Rick === Subject: Re: Math can be entertaining >> While in high school in late 50's I came across _Mathematics for the >> Millions_ . >> It's long out of print. > Are you sure? This seems to be the book: > http://www.amazon.com/exec/obidos/tg/detail/-/039331071X The sub-title, How to Master the Magic of Numbers is unfamiliar and associate with the book ( How to Lie with Statistics ). Then again my memory may be foggy after >45 years. The sample pages have right >> Is there something currently in print of similar nature aimed at >> general public? >> Is there anything similar aimed at lower elementary grades? > Not sure what the book above actually is, but if you're interested in > recreational mathematics, I like the Dr. Ecco books by Dennis Shasha > (the problems are nicely set up as stories) and the various books by > Martin Gardner. I'll check what books my local library has/can borrow by both authors. Mr. Gardener's name keeps popping up. A good sign. > cheers, > Rick === Subject: Re: Math can be entertaining >> While in high school in late 50's I came across _Mathematics for the >> Millions_ . It is NOT good. The level is far too low, and the author does not understand modern mathematics. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Math can be entertaining >While in high school in late 50's I came across _Mathematics for the >Millions_ . > It is NOT good. The level is far too low, and the author > does not understand modern mathematics. The level is too low for whom? I'm trying to attract attention of a 9 year old. I'm more interested in is the author accurate and comprehensible, than whether or not he covers topics of current interest. === Subject: Re: Math can be entertaining > While in high school in late 50's I came across _Mathematics for the > Millions_ . > It is NOT good. The level is far too low, and the author does not > understand modern mathematics. That's hardly surprising, it was written in the thirties IIRC. GH Hardy is scathing about Hogben in his A Mathematician's Apology, regarding him as concentrating too much on arithmetic rather than mathematics (I paraphrase). Perhaps I shouldn't suggest there is a distinction in this newsgroup. john === Subject: Re: Math can be entertaining >>While in high school in late 50's I came across _Mathematics for the >>Millions_ . >>It is NOT good. The level is far too low, and the author does not >>understand modern mathematics. > That's hardly surprising, it was written in the thirties IIRC. > GH Hardy is scathing about Hogben in his A Mathematician's Apology, > regarding him as concentrating too much on arithmetic rather than > mathematics (I paraphrase). Perhaps I shouldn't suggest there is a > distinction in this newsgroup. > john But my question is, Would it whet appetite? === Subject: Re: Math can be entertaining >While in high school in late 50's I came across _Mathematics for the >Millions_ . >It is NOT good. The level is far too low, and the author does not >understand modern mathematics. >> That's hardly surprising, it was written in the thirties IIRC. >> GH Hardy is scathing about Hogben in his A Mathematician's Apology, >> regarding him as concentrating too much on arithmetic rather than >> mathematics (I paraphrase). Perhaps I shouldn't suggest there is a >> distinction in this newsgroup. There is a huge distinction. Mathematics can find ways of doing arithmetic better, but learning arithmetic will not provide any understanding of the integers. I did read Hogben, but it gave me NO idea about the subject of mathematics as I understand it, and have been doing for decades. When I encountered a decent high school algebra book, I essentially learned algebra, except for some practice, in a few minutes. Even then, I had no idea of what is mathematics. >But my question is, Would it whet appetite? No; see the above. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: help with computation of a specific a definite integral via residue theorem Hi all, This is my first time posting here. I have the following integral which i am trying to evaluate via method of residues / infinity x*exp(i*x*a)*dx I = | --------------- / -infinity b^2 - x^2 from residue theorem, the lower path along the real axis can be found by subtracting an upper half-circle from a closed contour, C (the closed contour integration can readily be computed from the residues of the enclosed poles) / lim / pi I = O f(x)*dx - | f(R*exp(i*phi))*R*i*exp(i*phi)*dphi / C R->infinity / 0 the problem is that the integration on the half-circle goes to infinity as R->infinity; therefore the integral is indeterminant. however, i can use change of variables to write the integral as / infinity x*sin(x*a)*dx I = -2*i | ------------- = pi*i*cos(b*a) / 0 b^2 - x^2 where an identity in Table of Integrals, Series, and Products by Ryzhik and Gradshedyn was used to determine the value of the RHS. i however do not have a feel for how this solution was obtained and what limitations should be imposed on b (which pole should be chosen). for example, for most problems of this nature, the pole, b, with Im{b}>0 is chosen so that it lies in the upper half-space (above the real axis) and enclosed within the contour. however the 0 to infinity limits give a contour of integration only in the first quadrant (i think). i feel that there is no guarantee that this pole, b, will lie in the first quadrant (the real part of b could be negative). Michael Gilbert === Subject: Re: help with computation of a specific a definite integral via residue theorem earlier today. What I realized is that only one pole is needed because the other pole is redundant in the evaluation of the cosine function. What I mean is there are two poles, p=+b and p=-b, where p is the complex-valued pole. Now, say that I am limiting my analaysis to the pole that has imaginary part greater than zero, Im{p}>0 (from a condition elsewhere), then I only need that pole for this result because cos(a*p)=cos(a*b)=cos(a*-b) from the rule, cos(x)=cos(-x). so i can choose either of the poles (and the other pole is basically redundant and unnecessary). thus, i can evaluate the value of cos(a*p) with only the pole with condition Im{p}>0 Michael Gilbert > Hi all, > This is my first time posting here. I have the following integral which > i am trying to evaluate via method of residues > / infinity x*exp(i*x*a)*dx > I = | --------------- > / -infinity b^2 - x^2 > from residue theorem, the lower path along the real axis can be found by > subtracting an upper half-circle from a closed contour, C (the closed > contour integration can readily be computed from the residues of the > enclosed poles) > / lim / pi > I = O f(x)*dx - | f(R*exp(i*phi))*R*i*exp(i*phi)*dphi > / C R->infinity / 0 > the problem is that the integration on the half-circle goes to infinity > as R->infinity; therefore the integral is indeterminant. > however, i can use change of variables to write the integral as > / infinity x*sin(x*a)*dx > I = -2*i | ------------- = pi*i*cos(b*a) > / 0 b^2 - x^2 > where an identity in Table of Integrals, Series, and Products by Ryzhik > and Gradshedyn was used to determine the value of the RHS. i however do > not have a feel for how this solution was obtained and what limitations > should be imposed on b (which pole should be chosen). for example, for > most problems of this nature, the pole, b, with Im{b}>0 is chosen so > that it lies in the upper half-space (above the real axis) and enclosed > within the contour. however the 0 to infinity limits give a contour of > integration only in the first quadrant (i think). i feel that there is > no guarantee that this pole, b, will lie in the first quadrant (the real > part of b could be negative). > Michael Gilbert === Subject: Re: help with computation of a specific a definite integral via residuetheorem > Hi all, > This is my first time posting here. I have the following integral which > i am trying to evaluate via method of residues > / infinity x*exp(i*x*a)*dx > I = | --------------- > / -infinity b^2 - x^2 > from residue theorem, the lower path along the real axis can be found by > subtracting an upper half-circle from a closed contour, C (the closed > contour integration can readily be computed from the residues of the > enclosed poles) > / lim / pi > I = O f(x)*dx - | f(R*exp(i*phi))*R*i*exp(i*phi)*dphi > / C R->infinity / 0 > the problem is that the integration on the half-circle goes to infinity > as R->infinity; therefore the integral is indeterminant. > however, i can use change of variables to write the integral as > / infinity x*sin(x*a)*dx > I = -2*i | ------------- = pi*i*cos(b*a) > / 0 b^2 - x^2 > where an identity in Table of Integrals, Series, and Products by Ryzhik > and Gradshedyn was used to determine the value of the RHS. i however do > not have a feel for how this solution was obtained and what limitations > should be imposed on b (which pole should be chosen). for example, for > most problems of this nature, the pole, b, with Im{b}>0 is chosen so > that it lies in the upper half-space (above the real axis) and enclosed > within the contour. however the 0 to infinity limits give a contour of > integration only in the first quadrant (i think). i feel that there is > no guarantee that this pole, b, will lie in the first quadrant (the real > part of b could be negative). > Michael Gilbert Integrals like this are standard, and are done, for example, in Mathews & Walker, Mathematical Methods of Physics, or in E. Merzbacher's or A. Messiah's books on quantum mechanics. The idea is you convert to a contour integral around a closed contour by adding to the integral along the real axis from -R to +R, the integral along a semi-circle or radius R, closing the contour either in the upper or lower half-plane. Assume a > 0 : then |exp(iza)| < exp[-R a sin(theta)] . The latter -> 0 as R -> infty for 0 < theta < pi but it diverges for pi < theta < 2 pi . Hence for a > 0 you can only draw the circle in the upper half-plane. Now, in fact for large R you have | z dz exp(iaz) / (b^2 - z^2) | -> dtheta exp[-R a sin(theta)] and you have to prove that the integral of the latter, from 0 to pi, -> 0 when R -> infty . You can do this by noting that on that interval exp[-R a sin(theta)] is symmetric about pi /2 and so you can integrate from 0 to pi /2 and multiply the result by 2. But for 0 < theta < pi /2 , sin ( theta ) > 2 theta / pi (Jordan's lemma) so that exp[-R a sin(theta)] < exp[{-2Ra theta} / {pi} ] . The latter is easily integrated and goes to 0 as 1/R. Thus the integral on the big semicircle can be neglected (that is, when you do the contour integral over each piece of the contour, that piece gives nothing) and so you can evaluate the whole thing by the residue theorem. The problem is there are poles on the real axis, at z = +b and -b. You have to figure out how to handle these poles. Typically one gives b a slight positive imaginary part, which shifts the pole at +b up off the z-axis (that is, inside the contour) and the pole at -b gets shifted down (outside the contour). When the integral in question arises as the Green's function of the Helmholtz equation in 3D, the way we handle the poles is connected to how we impose boundary conditions on the solution. For example, the prescription above gives a Green's function with outgoing-wave boundary conditions, appropriate to a description where the scattered waves do not appear before the incident wave has reached the scatterer. A prescription where both poles remain on the real axis and you have to calculate the Cauchy principal value to give it a meaning corresponds to standing waves. Etc., etc. -- Julian V. Noble Professor Emeritus of Physics ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: help with computation of a specific a definite integral via residuetheorem by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8TCg8B27179; >Hi all, >This is my first time posting here. I have the following integral which >i am trying to evaluate via method of residues > / infinity x*exp(i*x*a)*dx >I = | --------------- > / -infinity b^2 - x^2 >from residue theorem, the lower path along the real axis can be found by >subtracting an upper half-circle from a closed contour, C (the closed >contour integration can readily be computed from the residues of the >enclosed poles) > / lim / pi >I = O f(x)*dx - | f(R*exp(i*phi))*R*i*exp(i*phi)*dphi > / C R->infinity / 0 >the problem is that the integration on the half-circle goes to infinity >as R->infinity; therefore the integral is indeterminant. >however, i can use change of variables to write the integral as > / infinity x*sin(x*a)*dx >I = -2*i | ------------- = pi*i*cos(b*a) > / 0 b^2 - x^2 >where an identity in Table of Integrals, Series, and Products by Ryzhik >and Gradshedyn was used to determine the value of the RHS. i however do >not have a feel for how this solution was obtained and what limitations >should be imposed on b (which pole should be chosen). for example, for >most problems of this nature, the pole, b, with Im{b}>0 is chosen so >that it lies in the upper half-space (above the real axis) and enclosed >within the contour. however the 0 to infinity limits give a contour of >integration only in the first quadrant (i think). i feel that there is >no guarantee that this pole, b, will lie in the first quadrant (the real >part of b could be negative). >Michael Gilbert Computing must be simpler with -1/2(1/(x-b)+1/(x+b)instead of x/(b^2-x^2). You have two parts -1/2*Int(exp(Iax)/(x-b)dx ) and -1/2*Int(exp(Iax)/(x+b)dx ) with limits -inf,+inf. 2 poles ! get on .. Check the result:(-1+I)*sign(a)*Pi* === Subject: Re: help with computation of a specific a definite integral via residue theorem > the problem is that the integration on the half-circle goes to infinity > as R->infinity; therefore the integral is indeterminant. Did you include the effect of the R-squared in the denominator? === Subject: Help on properties of derivatives of concave functions by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8TDw9a00975; By developing an economic model, I'm facing the following problem: a function f(x) is increasing and concave in x, x>=0 (strict concavity or not does not matter), therefore f'(x) is positive (or non negative) and f''(x) is negative (or non positive). In a formula, we have a term defined by -f'(x)/(x.f''(x)). For which additional property on f(x), we can have this last term lesser or equal to one? I suspect that an assumption on f'''(x) is needed but I can't prove it. === Subject: Re: Help on properties of derivatives of concave functions >By developing an economic model, I'm facing the following problem: a >function f(x) is increasing and concave in x, x>=0 (strict concavity >or not does not matter), therefore f'(x) is positive (or non negative) >and f''(x) is negative (or non positive). In a formula, we have a term >defined by -f'(x)/(x.f''(x)). For which additional property on f(x), >we can have this last term lesser or equal to one? >I suspect that an assumption on f'''(x) is needed but I can't prove >it. using a Maclaurin expansion at x=0 as a first step I got f'(0)=0 and for f''(0)<0 f'''(0)<0 but for large values of x this does not apply and I see nothing beyond a differential inequality hth peter === Subject: Object oriented C++ in numerical analysis posting-account=7NK6QgwAAAA2HEio3VYYkWgWPfUia5JV Is there a book of numerical recipes which makes use of object oriented programming in C++? The classic Numerical Recipes in C++ hardly makes any use of C++'s OO capabilities, and other books I've come across only make very limited use of them. Marco === Subject: Re: Object oriented C++ in numerical analysis >Is there a book of numerical recipes which makes use of object >oriented programming in C++? The classic Numerical Recipes in >C++ hardly makes any use of C++'s OO capabilities, and other >books I've come across only make very limited use of them. >Marco maybe Daoqi Yang: C++ and object oriented numeric computing springer publisher 2001, ISBN 0-387-98990-0 is of the intended type hth peter === Subject: Re: Object oriented C++ in numerical analysis > >Is there a book of numerical recipes which makes use of object > >oriented programming in C++? The classic Numerical Recipes in > >C++ hardly makes any use of C++'s OO capabilities, and other > >books I've come across only make very limited use of them. > >Marco > maybe Daoqi Yang: C++ and object oriented numeric computing > springer publisher 2001, ISBN 0-387-98990-0 > is of the intended type > hth > peter I endorse Peter's suggestion. Additionally, if you come from a F77 background, Yang's book will promote your transition from FORTRAN to C++ nothing of significance to OO numerics. -- You're Welcome, Gerry T. ______ Object-oriented programming is an exceptionally bad idea which could only have originated in California. -- E. Dijkstra. === Subject: Re: Object oriented C++ in numerical analysis >> >Is there a book of numerical recipes which makes use of object >> >oriented programming in C++? The classic Numerical Recipes in >> >C++ hardly makes any use of C++'s OO capabilities, and other >> >books I've come across only make very limited use of them. >> > >Marco >> >maybe Daoqi Yang: C++ and object oriented numeric computing >>springer publisher 2001, ISBN 0-387-98990-0 >>is of the intended type >>hth >>peter > I endorse Peter's suggestion. Additionally, if you come from a F77 > background, Yang's book will promote your transition from FORTRAN to C++ > nothing of significance to OO numerics. Note to the original poster: I'd be cautious about accepting any advice from Gerry Thomas, he's malicous, dishonest, and none too bright to boot. Hang around c.l.f or s.m.n-a for a while, and you'll see what I mean -- pretty much everyone in these ng's detests the pathetic little troll... Aardpig === Subject: Re: Object oriented C++ in numerical analysis in > >> >Is there a book of numerical recipes which makes use of object > >> >oriented programming in C++? The classic Numerical Recipes in > >> >C++ hardly makes any use of C++'s OO capabilities, and other > >> >books I've come across only make very limited use of them. > >> >> >Marco > >> > >>maybe Daoqi Yang: C++ and object oriented numeric computing > >>springer publisher 2001, ISBN 0-387-98990-0 > >>is of the intended type > >>hth > >>peter > > I endorse Peter's suggestion. Additionally, if you come from a F77 > > background, Yang's book will promote your transition from FORTRAN to C++ add > > nothing of significance to OO numerics. > Note to the original poster: I'd be cautious about accepting any advice > from Gerry Thomas, he's malicous, dishonest, and none too bright to > boot. Hang around c.l.f or s.m.n-a for a while, and you'll see what I > mean -- pretty much everyone in these ng's detests the pathetic little > troll... > Aardpig Read Daoqi Yang book for yourself and form your own judgment instead of looking to c.l.f.'s cretinious police for propaganda on matters Fortran or numerics, they being worth less than a pig's turd which is why Fortran has long been ßushed down the toilet -- You're Welcome, Gerry T. ______ Ah, Klinger, my constant reminder that Darwin was right!. -- Maj Charles Winchester III, the 4077th M*A*S*H . === Subject: Re: Object oriented C++ in numerical analysis ... > Read Daoqi Yang book for yourself and form your own judgment instead of > looking to c.l.f.'s cretinious police for propaganda on matters Fortran or > numerics, they being worth less than a pig's turd which is why Fortran has > long been ßushed down the toilet ... For sure only a person lacking of serious arguments can use such a language. Using such a counterproductive language also tells a lot about the author intellectual level. Dan === Subject: Re: Object oriented C++ in numerical analysis > >Is there a book of numerical recipes which makes use of object > >oriented programming in C++? The classic Numerical Recipes in > >C++ hardly makes any use of C++'s OO capabilities, and other > >books I've come across only make very limited use of them. > >Marco > maybe Daoqi Yang: C++ and object oriented numeric computing > springer publisher 2001, ISBN 0-387-98990-0 > is of the intended type > hth > peter I realize that this is an analysis forum ... is OOP really much of an improvement? I've had a little exposure to OOP and been unimpressed. === Subject: Re: Object oriented C++ in numerical analysis >> >Is there a book of numerical recipes which makes use of object >> >oriented programming in C++? The classic Numerical Recipes in >> >C++ hardly makes any use of C++'s OO capabilities, and other >> >books I've come across only make very limited use of them. >> > >Marco >> >maybe Daoqi Yang: C++ and object oriented numeric computing >>springer publisher 2001, ISBN 0-387-98990-0 >>is of the intended type >>hth >>peter > I realize that this is an analysis forum ... is OOP really much of an > improvement? I've had a little exposure to OOP and been unimpressed. This depends entirely on what you want to do with a computer. If your objective is to perform complicated calculations that no one else understands, producing masses of data whose main destiny is to ride around on a disk for years, then, no, you don't need OOP. On the other hand, if the objective is to deliver useable code to other technologists, who want to specify the details of some system in either a textual or graphic manner, then see the results presented graphically, and interactively modify the parameters of the calculation, then you certainly need an object-oriented architecture. To address the original question, the obvious place to apply C++ in numerical calculations is in matrix/vector manipulations. You can do that, and you will essentially replicate the capabilities of Matlab. The problem is that when you accept the notion of abstracting a purely mathematical problem before you solve it, you also suffer the weakness of mathematical abstraction: a lot of information that was relevant to the original problelm has been thrown away. This is where OO architecture can save a great deal of work. One does not merely define a column or row vector class, but you create classes for vectors of abscissas, which are always monotonically ordered, and vectors of things that are naturally ordinates, which contain a reference to the vector of abcissas. All these classes might contain string descriptions that can be axis labels, and information on the physical units employed. This then lets you automate tasks like plotting your results. In summary, object-oriented programming can be quite useful in numerical computation, but it requires that you re-factor your task in a way that is generally orthogonal to the (application domain) / (pure mathematical problem) separation. - Bill Frensley === Subject: Re: Object oriented C++ in numerical analysis > >Is there a book of numerical recipes which makes use of object > >oriented programming in C++? The classic Numerical Recipes in > >C++ hardly makes any use of C++'s OO capabilities, and other > >books I've come across only make very limited use of them. > >Marco >maybe Daoqi Yang: C++ and object oriented numeric computing >springer publisher 2001, ISBN 0-387-98990-0 >is of the intended type >hth >peter >> I realize that this is an analysis forum ... is OOP really much of an >> improvement? I've had a little exposure to OOP and been unimpressed. >This depends entirely on what you want to do with a computer. If your >objective is to perform complicated calculations that no one else understands, >producing masses of data whose main destiny is to ride around on a disk >for years, then, no, you don't need OOP. Utter baloney. I am not quite sure what OOP is, but there is nothing which mathematicians would not have added to the existing programming languages if there had been a call to mathematicians as to what computations should be made. To mathematicians, there are already an infinite number of types, and there are lots of natural operations. Which ones happen to be clumsily implemented in hardware is due to the practice of language designers leaving out important possibilities, and also attempting to keep fools from doing foolish things. This keeps intelligent people from doing intelligent things, and also, as Einstein has stated, nature is winning in the war of producing better fools. No matter which ones are implemented, it is always possible to do the others, including the supposedly prohibited ones, by working hard enough. Any computer-language combination can be simulated on any other with ONE program, if there is enough memory. It may not be very efficient, but it is quite easy to come up with a method of doing it; just hit it over the head with a hammer. > On the other hand, if the objective is to deliver useable code to other >technologists, who want to specify the details of some system in either a >textual or graphic manner, then see the results presented graphically, and >interactively modify the parameters of the calculation, then you certainly >need an object-oriented architecture. See the above. You need a ßexible architecture to do it well, not an object-oriented one. The current architectures do not have the ßexibility of the old von Neumann ones, to some extent deliberately. > To address the original question, the obvious place to apply C++ in >numerical calculations is in matrix/vector manipulations. See the above. ALL languages after the early days should have had essentially unlimited types. They also should have had strings of values to the left of the equal sign, which I do not believe C++ has. You can do that, >and you will essentially replicate the capabilities of Matlab. I am unfamiliar with Matlab, but my impression of the packages such as Mathematica and Maple is that they try to automate too much, and consequently make things difficult. The intelligent programmer should be in charge, and we even need interactive compilation. The problem is >that when you accept the notion of abstracting a purely mathematical problem >before you solve it, you also suffer the weakness of mathematical abstraction: >a lot of information that was relevant to the original problelm has been >thrown away. This is where OO architecture can save a great deal of work. >One does not merely define a column or row vector class, but you create >classes for vectors of abscissas, which are always monotonically ordered, and >vectors of things that are naturally ordinates, which contain a reference to >the vector of abcissas. All these classes might contain string descriptions >that can be axis labels, and information on the physical units employed. >This then lets you automate tasks like plotting your results. You have even overly limited what should be done in a formulation. All of this can be done using abstract mathematics. It is those who do not understand the power of mathematics, but limit it to a few types, who have this problem. Automation may or may not work, and can be very limiting. > In summary, object-oriented programming can be quite useful in numerical >computation, but it requires that you re-factor your task in a way that is >generally orthogonal to the (application domain) / (pure mathematical problem) >separation. No, you are limiting the pure mathematical problem. Any properly formulated problem has all of what you claimed would be left out included. The main part of mathematics for the user is as a very general language, not a means of calculation. This comes later, after the problem is formulated. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: Object oriented C++ in numerical analysis > I realize that this is an analysis forum ... is OOP really much of an > improvement? I've had a little exposure to OOP and been unimpressed. I'm not very experienced with OOP either, but I gather there are two main purposes: 1) the conceptual part, to encourage appropriate program/data structure 2) the practical part, to ease maintenance/expandability, software engineering I think most good programmers do 1) no matter the language, but I think 2) becomes important for big projects. I have no experience in multi-generation team written software, so can't say how much OOP software engineering concerns matter. Matt Wolinsky === Subject: Re: Object oriented C++ in numerical analysis > I think most good programmers do 1) no matter the language, but I > think 2) becomes important for big projects. I have no experience in One thing to keep in mind is that numerical recipes is mostly concerned with laying out algorithms for fairly simple, stateless functions, such as calculating a Bessel function. About the most you can do for a Bessel function method is to overload the function name. So the main thing to be on the look out for with numerical recipes is to make sure that the functions you are interested in are properly encapsulated, and if you want, you can add a little polymorphism such that you can use a single function name for ßoat vs double vs complex arguments. Even in the worst case, making a numerical recipe function OO can probably be achieved by writing a C++ wrapper function that guarantees that encapsulation is satisfied. More important from an OO point of view is the architecture of the higher level objects and methods that will be calling these low level numerical recipe functions. In other words, concentrate the effort on putting OO in the code where it is needed -- managing objects and their lifetimes and keeping them loosely coupled from other objects. Practically speaking, if you find that you have function calls consisting of 25 parameters, then that is probably a good indication of an area where a few objects could be used to represent the parameters and control their lines of communication. === Subject: Question about LU algorithm in NRiC Hi I am not an expert, but I have a simple question regarding the LU decomposition algorithm as implemented in the NRiC. The algorithm outputs an array Ôindx' which contains the permutation of the rows for partial pivoting. My question is: what exactly does each element of the array represent? This is quite unclear from the descriptions in the book (or I have not done enough RTFM, which may be the case as well :) ) In any case, the algorithm yields the following indx output on the matrix [1 2 3] [2 -1 -1] [3 4 -1] indx ==> [2 3 3] I do not understand the semantics of the indx result. -- Marco Antoniotti === Subject: Re: Question about LU algorithm in NRiC > indx ==> [2 3 3] 2 in position 1: exchange column (or row?) 1 and 2 3 in position 2: *then* exchange 2 and 3 3 : well, that the last one, so you can't exchange anything anymore V. -- email: lastname at cs utk edu homepage: www cs utk edu tilde lastname === Subject: Re: Parameter estimation with maximum likelihood method using Matlab 6 by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8TKIOj04067; I am investigating procedures to estimate missile aerodynamic parameters including bending. I have a similar question. Is MMLE3 or the methods used in the MATLAB system identification toolbox more approiate. John Knox, Lockheed-Martin Missiles and Fire Control. >I'm using Matlab 6.0 to perform Maximum Likelihood estimation >for identifying parameters. >The problem is that i am dealing with time varying parameters (which >are to be identified). and along with that other terms in matrices >(A,B,C,D,K) are also time varying. I am elaborating this as follows: >my system is of state space form: >dx/dt=f(x(t),aux(t),exi(t)) >z=g(x(t))+eta(t) >I contacted Ljung who is the author of system identification toolbox. >but he could not satisfy me. He suggest that i should change the file >gnnew.m, where Ôltti' function is used. but with that i could not >understand the algorithm behind such estimation. >so my main questions are: >. how can we use Matlab system identification toolbox to estimate >parameters of this kind. >. there is toolbox MMLE3 in Matlab. I want to know about this toolbox. >. what could be the suitable algorithm for this time varying problem. >. How can we change the matlab file gnnew.m for our case. >. nmerical algorithm for the minimization of cost function. >I also want to evaluate the Cramer-Rao bounds as a measure of the >parameter >accuracy. Does anyone know where I can get an algorithm to do this? >If anyone doing this kind of job plz contact with me. >Saqib Yousaf === Subject: Re: polar grid to cartesian grid by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8TNken21010; it also depends on how many angles you have . for limited number of angles , you need to have a very high sampling frequncy (not necessarily , but preferably) - this means it would increase the isoropic bandwidth of the fourier domain of the object ... this in turn means your interpolated points in the polar grid is actually going to be far in the cartesian grid. Vaidyanathan Ravi Shankar === Subject: Re: matrices - solving for a by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8TNt5I21730; >I have to solve a 4x4 matrix for a so that the matrix is singular. >The matrix reads: [1 a a a; a 1 a a; a a 1 a; a a a 1]. I have tried >numerous methods, but I can't figure it out!! a cannot equal 1 - any >suggestions how to go about this?? >~Jill Hi Jill, Sounds like you have an Identity Matix, ID Matrices always equal One. A=1. === Subject: Re: matrices - solving for a If 2 lines or 2 columns of a matrix are linearly dependent, the matrix is singular. with a=1, all lines and colums are =. You have 4 times the same equation , hence 1 equation, with 4 unknowns. Roland > >I have to solve a 4x4 matrix for a so that the matrix is singular. > >The matrix reads: [1 a a a; a 1 a a; a a 1 a; a a a 1]. I have tried > >numerous methods, but I can't figure it out!! a cannot equal 1 - any > >suggestions how to go about this?? > >~Jill > Hi Jill, Sounds like you have an Identity Matix, ID Matrices always > equal One. A=1. === Subject: Re: matrices - solving for a >The matrix reads: [1 a a a; a 1 a a; a a 1 a; a a a 1]. I have tried You need to solve for the determinant and then find what values of ``a'' cause the determinant to go to zero. Clearly a=1 can be seen to cause the matrix to go singular. Check out a site such as http://mcraefamily.com/MathHelp/MatrixDeterminant.htm to see the full formula for the determinant of a 4x4 matrix. I think the equation for the determinant in terms of a is: 1 - 6*a*a + 8*a*a*a - 3*a*a*a*a Factor the above polynomial and see if any roots other than a = 1 exist. Hint: there is another root lying somewhere between a=-0.5 and a=0.0. Also, this polynomial is very easy to factor because some roots occur more than once. === Subject: Solving 4x4 Matrices by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8U1X5L30254; What is the procedure for solving a 4x4 matrix === Subject: Re: Solving 4x4 Matrices >What is the procedure for solving a 4x4 matrix in water or in sulfuric acid? no joke; you mean linear systems solvers or inversion: the same as for general n , dense code: lu-decomposition hth peter === Subject: Re: Solving 4x4 Matrices Am 30.09.04 03:33 schrieb Steve: > What is the procedure for solving a 4x4 matrix numerical recipes online. (Don't have the matrix-inversion by gauss-jordan, LU-decomposition and others in detail and with commented excerpts from their program-code. HTH Gottfried Helms === Subject: Re: Solving 4x4 Matrices > What is the procedure for solving a 4x4 matrix There isn't one. You can solve an equation (or a problem), but the statement solve a 4x4 matrix is meaningless. === Subject: Gain of Simplex Allo, I want to implement a simplex method in my firm. It is a Aluminium combinate that produces folium,and other products of Aluminiu it have approxcimatly 1800 employers. I think tahat we have approximately a 250 equations and that problem is 80-85% linear. My question is how much we could gain if I set a problem correctly, I have all data. I know that this is not posibile to ansver correctly on my question but I think if this is posible ansver approximately. robert.bralic@si.htnet.hr === Subject: Re: Gain of Simplex : Allo, : : : I want to implement a simplex method in my firm. : It is a Aluminium combinate that produces folium,and : other products of Aluminiu it have approxcimatly : 1800 employers. : I think tahat we have approximately a 250 equations : and that problem is 80-85% linear. : My question is how much we could gain if I set : a problem correctly, I have all data. : I know that this is not posibile to ansver correctly : on my question but I think if this is posible : ansver approximately. : : : robert.bralic@si.htnet.hr Think of about 1% - 5% increase in net profit. D. Baruth === Subject: Best fitting plane I have a set of 3D data points and try to fit a plane through it. For data points without errors the SVD -> eigenvector method works fine. But what about data with errors? (And what about correlated errors?) I think something like orthogonal data regression (ODR) must be the solution of this problem, but I don't know how to exactly do this. Can I do this with netlib/odrpack ? But how .... Can anybody provide some examples? Manuel M. === Subject: Re: Best fitting plane Hello one very simple option is to use standard linear regression for a function of the type z=f(x,y) = a + b*x + c*y and determine the optimal values for the parameters a, b and c Send mail if you can not locate a readable description, i have lecture notes (in german) that explain how to proceed. Hope this helps Andreas > I have a set of 3D data points and try to fit a plane through it. For > data points without errors the SVD -> eigenvector method works fine. > But what about data with errors? (And what about correlated errors?) I > think something like orthogonal data regression (ODR) must be the > solution of this problem, but I don't know how to exactly do this. Can > I do this with netlib/odrpack ? But how .... > Can anybody provide some examples? > Manuel M. -- Mathematics, HTI Phone: ++41 +32 32 16 258 Quellgasse 21 Fax: ++41 +32 321 500 CH-2501 Biel WWW: www.hta-bi.bfh.ch/~sha Switzerland === Subject: Re: Best fitting plane >I have a set of 3D data points and try to fit a plane through it. For >data points without errors the SVD -> eigenvector method works fine. >But what about data with errors? (And what about correlated errors?) I >think something like orthogonal data regression (ODR) must be the >solution of this problem, but I don't know how to exactly do this. Can I >do this with netlib/odrpack ? But how .... >Can anybody provide some examples? >Manuel M. the svd method can also be applied in the noisy case, known as total least squares. the essential trick consists in subtracting the mean of the x,y,z-data and here is a snippet of code from tom davis which I found in my annotations: (of course you culd also use odrpack, but this is much more compute intense. besides, it has a very fine userguide , so if you are in need of a nonlinear fitting, this is a good way to go) % Article: 151969 of comp.soft-sys.matlab % TLS Hyperplane % P is a point on the fitted plane % and N its normal vector. % error-free data a=1; b=2; c=-2; n=20; x=10*rand(n,1); y=20*rand(n,1); z=a*x+b*y+c; P=[mean(x),mean(y),mean(z)]; [U,S,V]=svd([x-P(1),y-P(2),z-P(3)],0); N=-1/V(end,end)*V(:,end); % Z = AX + BY + C A=N(1), B=N(2), C=-P*N % A = 1.0000; B = 2.0000; C = -2.0000; % data with random error sigma=3; x=x+sigma*(2*rand(n,1)-1); y=y+sigma*(2*rand(n,1)-1); z=z+sigma*(2*rand(n,1)-1); P=[mean(x),mean(y),mean(z)]; [U,S,V]=svd([x-P(1),y-P(2),z-P(3)],0); N=-1/V(end,end)*V(:,end); % Z = AX + BY + C A=N(1), B=N(2), C=-P*N figure(1) plot3(x,y,z,'bo'), hold on plot3(P(1),P(2),P(3),'ro') [X,Y]=meshgrid(linspace(min(x),max(x),10),... linspace(min(y),max(y),10)); mesh(X,Y,A*X+B*Y+C) grid on, hold off %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% hth peter === Subject: Best fitting plane > I have a set of 3D data points and try to fit a plane through it. For > data points without errors the SVD -> eigenvector method works fine. > But what about data with errors? (And what about correlated errors?) I > think something like orthogonal data regression (ODR) must be the > solution of this problem, but I don't know how to exactly do this. Can I > do this with netlib/odrpack ? But how .... If you describe your plane P by normal vector w and bias b, that is P = { x | - b = 0 } you have for an arbitrary point y dist(y, P) = | - b | as the orthogonal distance to the plane. So if you have points y_i and you want to fit such a plane, you have to construct a matrix A with rows [ (y_i)_x | (y_i)_y | (y_i)_z | -1 ] and minimize || A u ||^2 with u = [ wx, wy, wz, b ]^T under the constraint wx*wx + wy*wx + wz*wz = 1. This can be done by using QR decomposition of A followed by singular value decomposition of a submatrix of R. Write if you are interested in these details. === Subject: local solution of a polynomial I am interested in finding a solution of a polynomial, say a quintic, in a neighborhood of a value of x, say 0. I could write a programs to solve this numerically. However I am wondering if there is a simple, more elegance way of doing this? packat === Subject: Re: local solution of a polynomial > I am interested in finding a solution of a polynomial, say a > quintic, in a neighborhood of a value of x, say 0. > I could write a programs to solve this numerically. However > I am wondering if there is a simple, more elegance way of > doing this? > packat With a general quintic, you (and everyone else) are out of luck for simple, elegant ways (presumably with elementary formulas in a finite number of steps). To read more about it, search under Galois Theory. There are formulas involving functions considered standard by many mathematicians: the theta functions, if I remember correctly. But in practice, it is an overkill: one would be done with a numerical procedure faster than using those functions. Programs can be, as you may well know, written with various levels of efficiency. There must be well-tested programs publicly available for that. At some time, Laguerre's method was considered quite efficient (again, conduct a search). === Subject: Reference for Integro-Differential Equations Methods? Hi everyone I'd like to seek a good reference for numerical methods of solving integro-differential equations, in particular for solving Boltzmann-like ones. Any recommendations? Yi-Zen === Subject: Re: Reference for Integro-Differential Equations Methods? > Hi everyone > I'd like to seek a good reference for numerical methods of solving > integro-differential equations, in particular for solving Boltzmann-like > ones. Any recommendations? > Yi-Zen If they are nonlinear, such equations are usually solved iteratively from some good initial guess. In principle the nonlinear mapping must be compact for this process to converge. Physicists solve Hartree-Fock integro-differential equations routinely. Davis, Nonlinear Differential and Integral Equations (Dover) is not bad. -- Julian V. Noble Professor Emeritus of Physics ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: Reference for Integro-Differential Equations Methods? >Hi everyone >I'd like to seek a good reference for numerical methods of solving >integro-differential equations, in particular for solving Boltzmann-like >ones. Any recommendations? >Yi-Zen search in zentralblatt fuer mathematik with titel boltzmann and global index integro differential gives 270 hits for example this one: 2. Zbl 1038.65004 Matheis, Ingo; Wagner, Wolfgang 5. Zbl 1019.65100 Akesbi, S.; Maitre, E. Theoretical and numerical analysis of a minimal residual solver for 2D Boltzmann transport equation. (English) 16. Zbl 0982.65145 Pareschi, Lorenzo; Russo, Giovanni An introduction to Monte Carlo methods for the Boltzmann equation. (English) ESAIM, Proc. 10, 35-75, electronic only (2001). MSC 2000: *65R20 82C40 82C80 20. Zbl 0986.65145 Pareschi, L.; Russo, G. Fast spectral methods for Boltzmann and Landau integral operators of gas and plasma kinetic theory. (English) Ann. Univ. Ferrara, Nuova Ser., Sez. VII 46, Suppl., 329-341 (2000). MSC 2000: *65R20 76M22 82C40, Reviewer: Nikolai Pleshchinskii hth peter === Subject: Re: finding long primary numbers by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8UFSf401911; What is a long primary number? === Subject: Re: finding long primary numbers > What is a long primary number? What do you mean by primary number? Google finds http://mathworld.wolfram.com/PrimaryPseudoperfectNumber.html If you mean prime numbers, my current recommendation is: 2^24036583-1, or if your pocket calculator's display is too narrow to display the result: http://mersenne.org/prime7.txt Hugo === Subject: hierarchical cluster analysis I have some questions regarding hierarchical cluster analysis. (1) What exactly is a linkage distance (when I have a dendrogram)? I kind of have an idea that it is the coefficient of the distance at which different objects form a cluster, and different lower level clusters form higher level clusters. But, I don't really know how to explain to people. (2) How do I decide at which point I should cut the dendrogram tree (how do I decide how many clusters to retain)? Is it just a subjective process? Ô`'`'`'`[CapitalO Tilde]`'`'`'`'`[ CapitalOTilde]`'`'`'`[CapitalOT ilde]'`'`'`'`[Ca pitalOTilde]`'`'`'`[CapitalOTil de]`'`'`'`'`[Cap italOTilde]`'`'`'`[CapitalOTild e] `'`'`'`'`[Capit alOTilde]` sci.psychology.research is a moderated newsgroup. here bimonthly or the charter on the web at http://psychcentral.com/spr/ Submissions are acknowledged automatically. === Subject: Re: hierarchical cluster analysis >I have some questions regarding hierarchical cluster analysis. > (1) What exactly is a linkage distance (when I have a dendrogram)? > kind of have an idea that it is the coefficient of the distance at > which different objects form a cluster, and different lower level > clusters form higher level clusters. But, I don't really know how > to explain to people. The concept of linkage distance is needed to define what a distance between two clusters. The distance between two elements is a simple metric, but not between two sets of elements. It's possible to define the notion of a multimetric (D. Wolpert Metrics for more than two points at once arXiv:nlin.AO/0404032) in a rigorous way, but linkage distances are simple heuristics that try to do the same. There are a few: 1. Average linkage: how different on average are all pairs of elements, the first element is from the first, and the second from the second cluster. 2. Single linkage: how different is the closest pair of neighboring elements 3. Complete linkage: how different are the elements from the most different pair of two clusters 4. Ward's method: Ward's minimum variance linkage method attempts to minimize the increase in the total sum of squared deviations from the mean of a cluster. 5. Weighted linkage method: it is a derivative of average linkage method, but where both clusters are weighted equally in order to remove the inßuence of different cluster size. > (2) How do I decide at which point I should cut the dendrogram tree > (how do I decide how many clusters to retain)? Is it just a > subjective process? The subjective guideline is that the longer the branches you cut, the better the cut. For example, this is a better place to cut: +------------| | +----------------| than this one: +---| | +-----| You can do a rudimentary kind of a test by creating several bootstrap replications of the data, doing the splitting for each replication independently, and verifying if the resulting cluster assignment is identical across the replications. If it is not, the splitting has not been significantly obvious. It rarely is, in fact, except for a subset of typical elements. -- mag. Aleks Jakulin http://www.ailab.si/aleks/ Artificial Intelligence Laboratory, Faculty of Computer and Information Science, University of Ljubljana, Slovenia. Ô`'`'`'`[CapitalO Tilde]`'`'`'`'`[ CapitalOTilde]`'`'`'`[CapitalOT ilde]'`'`'`'`[Ca pitalOTilde]`'`'`'`[CapitalOTil de]`'`'`'`'`[Cap italOTilde]`'`'`'`[CapitalOTild e] `'`'`'`'`[Capit alOTilde]` sci.psychology.research is a moderated newsgroup. here bimonthly or the charter on the web at http://psychcentral.com/spr/ Submissions are acknowledged automatically. === Subject: Storage of Input and Outputdata or numerical analysis Hello all, I'm interested in an approach to store input- and outputdata for numerical computer simulations, e.g. FEM, in an structured way, e.g. in a DBMS. There should be a possibility to store the e.g. geometry, material properties, dofs, forces for several models and the resulting data of the model run. There should be a way to query the data to compare the input/output data. The output-data could be of a size of several megabytes. I heard something about HDF5 or netCDF. Woody === Subject: Fast Hessian diagonal or trace - is it possible? I have a real valued function f(x) of a vector x (x has dimension n) that can be evaluated in time O(n). It's derivative df/dx (which is also a vector of dimension n) can also be evaluated in time O(n). (This is a neural network problem, and the fast derivative finding algorithm is backpropagation). I want to find the trace of the hessian of this function f(x). Is this possible in O(n) time? Or equivalently, can I get the diagonal of the hessian in O(n) time (as this would give me the trace)? I've seen one neat speeding up trick you can do with the hessian (ref: Pearlmutter, Fast exact multiplication by the hessian). Using this trick it is possible to form the inner product of the hessian matrix H with a vector V simply by evaluating (df/dx(x+eV)-df/dx(x-eV)) /2e where e is a small real number (and df/dx(x+eV) means the gradient evaluated at (x+eV)). This achieves the answer in O(w) time. Can anyone think of perhaps a similar trick to get what I want, i.e. the diagonal or trace? Mike Fairbank. === Subject: what is this called? Suppose you have an integer number x, and you calculate what is left after substracting mod(x,b), so you actually calculate f(x) = x - mod(x,b) Does this f(x) has a special name? Bart -- Share what you know. Learn what you don't. === Subject: Re: what is this called? >Suppose you have an integer number x, and you calculate what is left >after substracting mod(x,b), so you actually calculate >f(x) = x - mod(x,b) >Does this f(x) has a special name? For nonnegative x and positive b, this is the largest multiple of b not exceeding x. When the operands have other signs, it depends upon how you define mod. -- During a wedding ceremony, members of the audience cried. Scientists examined the tears. They determined the liquid was eye dew. pmontgom@cwi.nl Microsoft Research and CWI Home: Bellevue, WA === Subject: Re: what is this called? >Suppose you have an integer number x, and you calculate what is left >after substracting mod(x,b), so you actually calculate >f(x) = x - mod(x,b) >Does this f(x) has a special name? The largest multiple of b which is less than or equal to x. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 === Subject: Re: what is this called? Doesn't that just leave, in computing jargon, x div b. Or in maths jargan, f(x) = integer part of (x/b) > Suppose you have an integer number x, and you calculate what is left > after substracting mod(x,b), so you actually calculate > f(x) = x - mod(x,b) > Does this f(x) has a special name? > Bart > -- > Share what you know. Learn what you don't. === Subject: Re: what is this called? > Doesn't that just leave, in computing jargon, x div b. > Or in maths jargan, f(x) = integer part of (x/b) > > Suppose you have an integer number x, and you calculate what is left > > after substracting mod(x,b), so you actually calculate > > f(x) = x - mod(x,b) > > Does this f(x) has a special name? > > Bart > > -- > > Share what you know. Learn what you don't. Clearly it is not the result of integer division. It also isn't the remainder of integer division. Example: 37 - mod(37,6) = 36 37 - mod(37,5) = 35 37 - mod(37,13) = 26 I have never come across this function and, AFIK, it has no special name. -- Julian V. Noble Professor Emeritus of Physics ^^^^^^^^^^^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ For there was never yet philosopher that could endure the toothache patiently. -- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1. === Subject: Re: I need some help with this one, please! > 1, 4, 2, 16, 5, 36, 7, ____, ____, ____ 1, 4, 2, 16, 5, 36, 7, 64, 13, 100 === Subject: Re: I need some help with this one, please! by support1.mathforum.org (8.11.6/8.11.6/The Math Forum, $Revision: 1.9 primary) id i8TCg8d27156; >> 1, 4, 2, 16, 5, 36, 7, ____, ____, ____ >How about 1, 4, 2, 16, 5, 36, 7, 0, 0, 0? >There are two things you must never attempt to prove: the >unprovable -- and the obvious. >Democracy: The triumph of popularity over principle. >http://www.crbond.com Are sure about all terms of Ôyour' suite : 1, 4, 2*, 16, 5, 36, 7, ____, ____, ____ The following one: 1, 4 , 3, 16, 5, 36, 7, has a nice solution: 1,2^2, 3,4^2, 5,6^2, 7,8^2 good game ! Try to write down a formula A(n)= ... for integral n values. Friendly,Alain. === Subject: International Conference on Computational Science (2nd CFP) Please excuse us if you receive this announcement more than once. ************************************************************** ******* * ICCS 2005: 5th International Conference on Computational Science * * Atlanta, May 22-25, 2005 http://www.iccs-meeting.org/ * ************************************************************** ******* You are invited to submit a paper and/or a proposal to organize a workshop at ICCS 2005, Altanta, USA, May 22-25, 2005. Please see http://www.iccs-meeting.org/ for more information. The theme for ICCS 2005 in Atlanta, USA, is Advancing Science through Computation, to mark several decades of progress in computational science theory and practice, leading to greatly improved applications science. Original contributions not exceeding 8 pages are invited for publication and oral presentation. All accepted papers will be printed in the conference proceedings published by Springer-Verlag in the Lecture Notes in Computer Science series. Selected papers will also be published as special issues of appropriate journals. Topics of Interest ------------------ ICCS 2005 invites original contributions on all topics related to computational science, including, but not limited to: * Scientific Computing * Problem Solving Environments * Advanced Numerical Algorithms * Complex Systems: Modeling and Simulation * Hybrid Computational Methods * Computational Science in Data Mining/Information Retrieval * Web- and Grid-based Simulation and Computing * Parallel and Distributed Computing * Visualization in Computational Science * Applications of Computation as a Scientific Paradigm * New Algorithms for Computational Kernels and Applications * Education in Computational Science Important deadlines: -------------------- Notification of acceptance of papers: .............. January 31, 2005 Camera ready papers: .............................. February 14, 2005 Early registration: .................................. March 30, 2005 ------------------------- iccs2005@mathcs.emory.edu Scientific Chair ..................................... Vaidy Sunderam Workshops Chair ..................................... Dick van Albada Overall Co-chair ...................................... Jack Dongarra Overall Chair ...................................... Peter M.A. Sloot