mm-2099 === Subject: Re: A new kind of Calculus theory <1119117828.984597@athnrd02> You expect us to read!? === Subject: integration of nyquist rate sampled data Hi all, I have to cope with the following problem: Consider X(f), the Fourier transform of x(t). X(f) exists only in the interval f1<=f<=f2 and -f1<=f<=-f2, so according to the sampling theorem it can be reconstructed from its samples if it is sampled with the proper period. x(t) is nyquist sampled and known only on certain samples t1,t2,...,tn, so X(f) is known on certain samples f1,f2,...,fn. I have to calculate the following definite integral: Y(f) = int{ int { H(a,b-a,f-b) * X(a) * X(b-a) * X(f-b) da} db } (Where a,b=-inf..inf and H(x1,x2,x3) is an analytically known function) only on f1,f2,...,fn Can i use the trapezoidal rule on f1,f2,..,fn or I have to use an explicit interpolation formula and reconstruct X(f) and then integrate? What is the connection with the sampling theorem (does it proves that if the trapezoidal rule is used, the result of f1,f2,..,fn is exact?) === Subject: Re: complexity of determining spectral radius > >I am interested in determining the complexity of finding the spectral > >radius (largest eigenvalue) of a positive nxn matrix A. Actually, I only > >need to know whether the spectral radius exceeds a certain given > >threshold t. All the entries of A, as well as t, are positive rational > >numbers. > >I can compute in O(n^3) the determinant of (tI-A) and see if it is zero, > >which tells me whether t is an eigenvalue of A. If it is not, then I can > >form the characteristic polynomial of A, f(x)=det(xI-a), and look for > >its largest root. What is the complexity of finding an interval about > >the largest root of f(x) that is small enough that t is not in it? > >Steve Borbash from the perron frobenius theorem you know that the spectral radius is itself > a simple eigenvalue. hence it suffices to show a sign change of the determinant. > this can be done by tabulating the sign and the log of abs of the determinant > (the determinant itself will soon produce overflow or underflow) near > t. if you would be happy and your matrix symmetric, then one symmetric > L-D-LT decomposition would suffice to check whether the spectralradius > exceeds t using the sylvester inertia theorem. > anyway, the complexity will be O(n^3). > hth > peter Peter, det(tI-A) is positive or negative. If positive, then t > spectral radius, if negative then t < spectral radius. Computing the determinant of the matrix tI-A, which has rational entries, takes time O(n^3). What effect do the sizes of the rational numbers (numerators & denominators) in the matrix have on the running time of computing the determinant? By the way, my matrix is not symmetric. --Steve === Subject: Re: complexity of determining spectral radius >> > >I am interested in determining the complexity of finding the spectral >> >radius (largest eigenvalue) of a positive nxn matrix A. Actually, I only >> >need to know whether the spectral radius exceeds a certain given >> >threshold t. All the entries of A, as well as t, are positive rational >> >numbers. >> >I can compute in O(n^3) the determinant of (tI-A) and see if it is zero, >> >which tells me whether t is an eigenvalue of A. If it is not, then I can >> >form the characteristic polynomial of A, f(x)=det(xI-a), and look for >> >its largest root. What is the complexity of finding an interval about >> >the largest root of f(x) that is small enough that t is not in it? >> >Steve Borbash >> from the perron frobenius theorem you know that the spectral radius is itself >> a simple eigenvalue. hence it suffices to show a sign change of the determinant. >> this can be done by tabulating the sign and the log of abs of the determinant >> (the determinant itself will soon produce overflow or underflow) near >> t. if you would be happy and your matrix symmetric, then one symmetric >> L-D-LT decomposition would suffice to check whether the spectralradius >> exceeds t using the sylvester inertia theorem. >> anyway, the complexity will be O(n^3). >> hth >> peter >det(tI-A) is positive or negative. If positive, then t > spectral >radius, if negative then t < spectral radius. Computing the determinant >of the matrix tI-A, which has rational entries, takes time O(n^3). What >effect do the sizes of the rational numbers (numerators & denominators) >in the matrix have on the running time of computing the determinant? >By the way, my matrix is not symmetric. This is not correct. A positive matrix, symmetric or not, may have other simple positive eigenvalues besides the Perron-Frobenius one, so the sign may change many times. You need to know that det(rI-A) has no roots >= t. Sturm's Theorem can be used. I don't know the complexity. But I think you may need all coefficients of the characteristic polynomial. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada === Subject: Re: complexity of determining spectral radius > > > >I am interested in determining the complexity of finding the spectral > >radius (largest eigenvalue) of a positive nxn matrix A. Actually, I only > >need to know whether the spectral radius exceeds a certain given > >threshold t. All the entries of A, as well as t, are positive rational > >numbers. > > >I can compute in O(n^3) the determinant of (tI-A) and see if it is zero, > >which tells me whether t is an eigenvalue of A. If it is not, then I can > >form the characteristic polynomial of A, f(x)=det(xI-a), and look for > >its largest root. What is the complexity of finding an interval about > >the largest root of f(x) that is small enough that t is not in it? > > >Steve Borbash > > from the perron frobenius theorem you know that the spectral radius is itself > a simple eigenvalue. hence it suffices to show a sign change of the determinant. > this can be done by tabulating the sign and the log of abs of the determinant > (the determinant itself will soon produce overflow or underflow) near > t. if you would be happy and your matrix symmetric, then one symmetric > L-D-LT decomposition would suffice to check whether the spectralradius > exceeds t using the sylvester inertia theorem. > anyway, the complexity will be O(n^3). > hth > peter > >>det(tI-A) is positive or negative. If positive, then t > spectral >>radius, if negative then t < spectral radius. Computing the determinant >>of the matrix tI-A, which has rational entries, takes time O(n^3). What >>effect do the sizes of the rational numbers (numerators & denominators) >>in the matrix have on the running time of computing the determinant? > >>By the way, my matrix is not symmetric. > >This is not correct. A positive matrix, symmetric or not, may have other >simple positive eigenvalues besides the Perron-Frobenius one, so the sign >may change many times. You need to know that det(rI-A) has no roots >= t. >Sturm's Theorem can be used. I don't know the complexity. But I think >you may need all coefficients of the characteristic polynomial. > >Robert Israel israel@math.ubc.ca >Department of Mathematics http://www.math.ubc.ca/~israel >University of British Columbia Vancouver, BC, Canada arithmetic operations with t-digit numbers can be done in t log t time using the fft. the information length of a determinant is linear in the information length of the original entries and quadratic in the dimension. and the coefficients of the characteristic polynomial can be expressed as determinants. using sturm sequences (euclids algorithm) requires exact computation since it is notoriously illconditioned. hence this limits the dimension of the tractable problem strictly. I thought of numerical tabulating using Gaussian elimination on an interval containing t and a (easily obtainable) upper bound for the spectral radius. here the choice of the stepsize is a problem of course. but you could do the following: if you apply Newtons method to the characteristic polynomial with an upper bound for the spectral radius (e.g. from gerschgorins theorem) then you know it will converge monotonically to the spectral radius which is the dominant eigenvalue. hence if the iteration sequence falls below t, you know that the spectral radius does not exceed t. in order to apply Newtons method, you do not need the standard (mclaurin series) representation of the characteristic polynomial but only n+1 of its values (using the uniqueness of polynomial interpolation) you can then evaluate a value of it and its derivatives using the Newton interpolation scheme (divided differences) this should make this tractable for larger n too. hth peter === Subject: New mathematics/physical sciences positions at http://jobs.phds.org, June 20, 2005 New job listings at http://jobs.phds.org - Jobs for PhDs List your job at no cost! http://jobs.phds.org/jobs/post * Fixed Income Derivatives Quantitative Developer NY $200+: Huxley Associates, New York. 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PhD Studentship Applications are invited for a PhD Studentship within the Space and Solar Plasma Physics Group of the Institute for Materials Research and School of Computing, Science and Engineering at the University of Salford. The project... http://jobs.phds.org/jobs/tsikd/listing_2005_06_16 === Subject: Re: Error on kurtosis and skewness > There is a way to compute the standard error of those quantities for > non-normal distributions. You do the integral to compute the variance > and take the square root. ... alternatively you go to text-book sources and find extensive results there, but perhaps only in the one book ... see Chapters 12,13 of Kendall MG and Stuart A (1969) The Advanced Theory of Statistics Vol 1 (3rd Edition), Griffin,London. Later editions may have different section numbering. David Jones === Subject: Re: numerical errors in primal-dual methods > > In primal-dual methods for linear programming, a suitable elimination > of variables yields an equation of the form > >A S^{-1/2} X^{1/2} A^T Delta_lambda = right_hand_side > >where X is the diagonal matrix of primal variables, S is the diagonal >matrix of slack variables, and delta_lambda is the unknown update >vector of dual variables (look e.g. for normal equations > in S. J. Wright's book, 1997). > >In finite precision arithmetic, some components of S eventually go > to zero, and the normal matrix A S^{-1/2} X^{1/2} A^T becomes >ill-defined. > > Do you know of a way to deal with this issue? >Is there a standard way? > >PA. > > at least if you have strict complementarity then all variables approach theit limit at the same rate, hence you have more a problem of scaling than of illconditioning (given the standard assumption of full rank in A) Zbl 0994.90139 Wright, Stephen J. Effects of finite-precision arithmetic on interior-point methods for nonlinear programming. (English) and also of Margaret Wright: Zbl 0957.65056 Wright, Margaret H. Ill-conditioning and computational error in interior methods for nonlinear programming. (English) hth peter === Subject: Re: Uses of a very good bidimensional nonlinear equation solver >Peter, > >Please, could you give me some references >about those fitting problems involving two nonlinear parameters? > > we had just one heer in the thread on fitting a @D parabola. fitting with two exponentials (a0+a1*exp(b1*x)+a2*exp(b2*x) ) given b1,b2, you can express the optimal a0,a1,a2 by a linear system and it remains a nonlinear system for the b1,b2. fitting rationals with denominator degree 2. fitting a Gaussian hth peter >Peter Spellucci escreveu: >> >Hi! >> > >> >Suppose you have a very good bidimensional numerical >> >nonlinear equation solver, that is, suppose you have >> >a very good numerical routine that computes efficiently >> >F^(-1)(c) (including degenerated cases), where F is a >> >(nonlinear) function from the plane to the plane. >> > >> >I=B4m looking for pratical uses of such routine, that is, >> >are there particular problems in optics, mechanics, >> >biology, etc that would go to benefit of such numerical >> >bidimensional inversor? >> > >> >Any pointers are welcome! >> > >> > >> >> any fitting problem involving two nonlinear parameters (there are many such) >> would be a good test case. but two variables is indeed a bit special. >> hth >> peter > === Subject: Problems with condition number I am currently writing a program to work out an engineering system. The program boils down to the linear combination of a set of eigenvalues Ax = b using a non-linear least squares fit. Currently the condition number of matrix A is in the region of 10^100 (the exact number depends on system properties). I think that the problem lies in that A is made up of a function [exp(g*2*(L+x)/h)+exp(g*2*(L-x)/h)]*[other stuff]. At the point where the boundary conditions are being evaluated L = x, L/h = 10 (this can't be made smaller due to the geometric conditions in the system) and g is the eigenvalues which have already calcualted by the program. Due to this the equation becomes [very large number +1]*[~1]. Is there any way that i can manipulate the first exp term to give smaller numbers? The engenvalues g are already calcualted from the system properties. Is there any way that i can manipulate them after they have been calcualted. Chris === Subject: Re: Problems with condition number >I am currently writing a program to work out an engineering system. >The program boils down to the linear combination of a set of >eigenvalues Ax = b using a non-linear least squares fit. >Currently the condition number of matrix A is in the region of 10^100 >(the exact number depends on system properties). I think that the >problem lies in that A is made up of a function >[exp(g*2*(L+x)/h)+exp(g*2*(L-x)/h)]*[other stuff]. At the point where >the boundary conditions are being evaluated L = x, L/h = 10 (this can't >be made smaller due to the geometric conditions in the system) and g is >the eigenvalues which have already calcualted by the program. Due to >this the equation becomes [very large number +1]*[~1]. >Is there any way that i can manipulate the first exp term to give >smaller numbers? The engenvalues g are already calcualted from the >system properties. Is there any way that i can manipulate them after >they have been calcualted. > >Chris > if all this should make sense, then you must compute with considerably more than 100 decimal (330 binary) digits and of course also the eigenvalues must be computed so precise. or, 10^100 simply means that indeed the matrix is singular? if all your matrix elements have the form you indicate then you could factor out exp(g*2*L/h) leaving 2*cosh(g*2*x/h) inside but this alone will not help with the illconditioning, it makes the entries a bit smaller or larger, depending on the sign of g. maybe you reformulate your problem a bit? but for getting help here, more details must be given hth peter === Subject: Re: Problems with condition number This is cross posted to sci.math.num-analysis Chris === Subject: Re: Problems with condition number This is cross posted to sci.math === Subject: Re: matrix transpose in DGEMM and DGEMV in BLAS > Since this restriction has been here forever, am I > misunderstanding something basic here? Is it *supposed* > to act like this? Is this a bug so old that is now > a feature? It turns out that I was, of course, misunderstanding something basic. My mental model of DGEMV was from looking at DGEMM. In DGEMM, the m, n, and k arguments are listed once. DGEMM must choose which ones specify the sizes of the A and B arrays depending on the transpose flags. Here m and n specify the size of the output C array. DGEMV doesn't have size arguments that need to be shared . . . m and n specify the original size of the A array regardless of whether it is transposed. Here m or n specifies the size of the output C array depending on the transpose flag. So in the test program the following DO 20 COL = 1, 7 CALL DGEMV('t', 5, 3, ONE, AT, 3, B(1,COL), 1, ZERO, > C(1,COL), 1) 20 CONTINUE should be changed to DO 20 COL = 1, 7 CALL DGEMV('t', 3, 5, ONE, AT, 3, B(1,COL), 1, ZERO, > C(1,COL), 1) 20 CONTINUE === Subject: Stochastic differential equations : Ito vs Stratonovich, Weak scheme vs Strong scheme, for additive noise I am integrating numerically the following SDE : dîx/dtî + g dx/dt = F(x,t) + b*L(t) dependant), g is a damping constant, L(t) is a Langevin force (gaussian white noise), and F(x,t) = a*(sin(x) + i*t) I am interrested in the distribution of times t for which the value of x is greater than some x_max. I first used a simple Euler Stratonovich scheme and it gives good results. What I mean by good results is that the distribution of times t has the correct shape according to experimental results. I then wanted to use a second order scheme. So, I found a book (Numerical Solution of Stochastic Differential Equations, by Peter E. Kloeden), in which I found a second order weak Ito-Taylor scheme. It allows me to integrate the equation for a wider set of parameters (for example, if a is too big, I needed to reduce the integration step too much for the Euler scheme), but the distribution I get has a different shape. According to what I know, since this equation has additive noise, it shouldn't matter if I take an Ito scheme or a Stratonovich scheme, am I right ? If it is right, can the difference in the shape of the distribution happens because I used a weak scheme instead of a strong scheme ? I thought a weak scheme was sufficient, because I am not interrested in value. I am not very used to SDE, so maybe there is something I don't understand correctly. Also, are there C++ libraries that can be used to solve SDE ? By the way, if there is a better newsgroup in which I should post, just tell me. Maxime Boissonneault === Subject: Re: Matlab 7.1 Benchmark Mr. Steinhaus created an excellent comparison and we used to reference this directly but many of the benchmark results and feature comparisons are so far out of date it is misrepresentative of many of the compared products. If you have questions on certain items from the report related to the current state of O-Matrix, please let us know. Beau Paisley, Harmonic Software www.omatrix.com >> We have updated our benchmark results at >> http://www.omatrix.com/bench.html >> for Matlab version 7.1 that's nice, if your intent is to offer cursory comparison between > Matlab and Omatrix. if your intent is to offer something that is more > serious, you should have a look at this: http://www.scientificweb.de/ncrunch/ some of the software versions in this benchmark are out of date, > by now. nevertheless, the effort is far more comprehensive and there > is a more complete description of tests, OS, and platform. > -- > 'No one' === Subject: Re: Matlab 7.1 Benchmark so it seemed fairest. Beau Paisley, Harmonic Software www.omatrix.com >> I downloaded your Omatrix and tested one benchmark you listed there: >> Inverse of 500x500 random matrix Omatrix=0.109 & MATLAB=0.198 >> On my system (R13 3GHz P4) I get Omatrix=0.109 & MATLAB=0.109. I >> replaced the default MATLAB ATLAS/BLAS with GOTO BLAS, which probably >> explains it. Going to 1000x1000 matrix the timings are Omatrix=0.672 >> MATLAB=0.609. Interesting. So this kind of begs the question...why doesn't Matlab use > the > GOTO BLAS for the default?