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  • Best open source Mixed Integer Optimization Solver

    - by Mark
    I am using CPLEX for solving huge optimization models (more than 100k variables) now I'd like to see if I can find an open source alternative, I solve mixed integer problems (MILP) and CPLEX works great but it is very expensive if we want to scale so I really need to find an alternative or start writing our own ad-hoc optimization library (which will be painful) Any suggestion/insight would be much appreciated

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  • Optimizing mathematics on arrays of floats in Ada 95 with GNATC

    - by mat_geek
    Consider the bellow code. This code is supposed to be processing data at a fixed rate, in one second batches, It is part of an overal system and can't take up too much time. When running over 100 lots of 1 seconds worth of data the program takes 35 seconds; or 35%. How do I improce the code to get the processing time down to a minimum? The code will be running on an Intel Pentium-M which is a P3 with SSE2. package FF is new Ada.Numerics.Generic_Elementary_Functions(Float); N : constant Integer := 820; type A is array(1 .. N) of Float; type A3 is array(1 .. 3) of A; procedure F(state : in out A3; result : out A3; l : in A; r : in A) is s : Float; t : Float; begin for i in 1 .. N loop t := l(i) + r(i); t := t / 2.0; state(1)(i) := t; state(2)(i) := t * 0.25 + state(2)(i) * 0.75; state(3)(i) := t * 1.0 /64.0 + state(2)(i) * 63.0 /64.0; for r in 1 .. 3 loop s := state(r)(i); t := FF."**"(s, 6.0) + 14.0; if t > MAX then t := MAX; elsif t < MIN then t := MIN; end if; result(r)(i) := FF.Log(t, 2.0); end loop; end loop; end;

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  • Collision Attacks, Message Digests and a Possible solution

    - by Dominar
    I've been doing some preliminary research in the area of message digests. Specifically collision attacks of cryptographic hash functions such as MD5 and SHA-1, such as the Postscript example and X.509 certificate duplicate. From what I can tell in the case of the postscript attack, specific data was generated and embedded within the header of the postscript (which is ignored during rendering) which brought about the internal state of the md5 to a state such that the modified wording of the document would lead to a final MD equivalent to the original. The X.509 took a similar approach where by data was injected within the comment/whitespace of the certificate. Ok so here is my question, and I can't seem to find anyone asking this question: Why isn't the length of ONLY the data being consumed added as a final block to the MD calculation? In the case of X.509 - Why is the whitespace and comments being taken into account as part of the MD? Wouldn't a simple processes such as one of the following be enough to resolve the proposed collision attacks: MD(M + |M|) = xyz MD(M + |M| + |M| * magicseed_0 +...+ |M| * magicseed_n) = xyz where : M : is the message |M| : size of the message MD : is the message digest function (eg: md5, sha, whirlpool etc) xyz : is the acutal message digest value for the message M magicseed_{i}: Is a set random values generated with seed based on the internal-state prior to the size being added. This technqiue should work, as to date all such collision attacks rely on adding more data to the original message. In short, the level of difficulty involved in generating a collision message such that: It not only generates the same MD But is also comprehensible/parsible/compliant and is also the same size as the original message, is immensely difficult if not near impossible. Has this approach ever been discussed? Any links to papers etc would be nice.

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  • Typesetting LaTeX fraction terms to be larger in an equation

    - by gotgenes
    I have the following formula in LaTeX, based on Fisher's Exact Test. (NOTE: requires the use of the amsmath package for \binom.) \begin{equation} P(i,j) = \sum_{x=|N(V_i) \cap V_j|}^{\min\{|V_j|, |N(V_i)|} \frac{ \binom{|V_j|}{x} \binom{|V - V_j|}{|N(V_i)| - x}} {\binom{|V|}{|N(V_i)|}} \end{equation} This renders the fraction portion with very small, difficult to read text: I would like my text more readable, as in the following example: What trickery can I use to get LaTeX to render my equation similarly?

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  • Rendering LaTeX on third-party websites?

    - by A. Rex
    There are some sites on the web that render LaTeX into some more readable form, such as Wikipedia, some Wordpress blogs, and MathOverflow. They may use images, MathML, jsMath, or something like that. There are other sites on the web where LaTeX appears inline and is not rendered, such as the arXiv, various math forums, or my email. In fact, it is quite common to see an arXiv paper's abstract with raw LaTeX in it, e.g. this paper. Is there a plugin available for Firefox, or would it be possible to write one, that renders LaTeX within pages that do not provide a rendering mechanism themselves? (The LaTeX would be enclosed within dollar signs, e.g. $\pi$. See the arXiv link above.) Some notes: It may be impossible to render some of the code, because authors often copy-paste code directly from their source TeX files, which may contain things like "\cite{foo}" or undefined commands. These should be left alone. This question is a repost of a question from MathOverflow that was closed for not being related to math. There is one answer there, which is helpful, but perhaps Stack Overflow can provide better answers. I program a lot, but Javascript is not my specialty, so comments along the lines of "look at this library" are not particularly helpful to me (but may be to others).

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  • Optimizing mathematics on arrays of floats in Ada 95 with GNAT

    - by mat_geek
    Consider the bellow code. This code is supposed to be processing data at a fixed rate, in one second batches, It is part of an overal system and can't take up too much time. When running over 100 lots of 1 seconds worth of data the program takes 35 seconds (or 35%), executing this function in a loop. The test loop is timed specifically with Ada.RealTime. The data is pregenerated so the majority of the execution time is definatetly in this loop. How do I improce the code to get the processing time down to a minimum? The code will be running on an Intel Pentium-M which is a P3 with SSE2. package FF is new Ada.Numerics.Generic_Elementary_Functions(Float); N : constant Integer := 820; type A is array(1 .. N) of Float; type A3 is array(1 .. 3) of A; procedure F(state : in out A3; result : out A3; l : in A; r : in A) is s : Float; t : Float; begin for i in 1 .. N loop t := l(i) + r(i); t := t / 2.0; state(1)(i) := t; state(2)(i) := t * 0.25 + state(2)(i) * 0.75; state(3)(i) := t * 1.0 /64.0 + state(2)(i) * 63.0 /64.0; for r in 1 .. 3 loop s := state(r)(i); t := FF."**"(s, 6.0) + 14.0; if t > MAX then t := MAX; elsif t < MIN then t := MIN; end if; result(r)(i) := FF.Log(t, 2.0); end loop; end loop; end; psuedocode for testing create two arrays of 80 random A3 arrays, called ls and rs; init the state and result A3 array record the realtime time now, called last for i in 1 .. 100 loop for j in 1 .. 80 loop F(state, result, ls(j), rs(j)); end loop; end loop; record the realtime time now, called curr output the duration between curr and last

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  • Code optimization on minutes pr hour calculation

    - by corger
    Hi All, The following code takes a timeframe in minutes since midnight and creates an array with minutes pr hour. But, it's slow. Any better suggestions out there? (no, changing language is not an option :-) ) Const clDeparture As Long = 123 Const clArrival As Long = 233 Dim lHour As Long Dim lMinute As Long Dim alHour(25) As Long For lMinute = 0 To 1440 If lMinute >= clDeparture And lMinute < clArrival Then alHour(Int(lMinute / 60)) = alHour(Int(lMinute / 60)) + 1 End If Next The array should now contain: (0,0) (1,0) (2,57) (3,53) (4,0) ..... Regards

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  • What is a data structure for quickly finding non-empty intersections of a list of sets?

    - by Andrey Fedorov
    I have a set of N items, which are sets of integers, let's assume it's ordered and call it I[1..N]. Given a candidate set, I need to find the subset of I which have non-empty intersections with the candidate. So, for example, if: I = [{1,2}, {2,3}, {4,5}] I'm looking to define valid_items(items, candidate), such that: valid_items(I, {1}) == {1} valid_items(I, {2}) == {1, 2} valid_items(I, {3,4}) == {2, 3} I'm trying to optimize for one given set I and a variable candidate sets. Currently I am doing this by caching items_containing[n] = {the sets which contain n}. In the above example, that would be: items_containing = [{}, {1}, {1,2}, {2}, {3}, {3}] That is, 0 is contained in no items, 1 is contained in item 1, 2 is contained in itmes 1 and 2, 2 is contained in item 2, 3 is contained in item 2, and 4 and 5 are contained in item 3. That way, I can define valid_items(I, candidate) = union(items_containing[n] for n in candidate). Is there any more efficient data structure (of a reasonable size) for caching the result of this union? The obvious example of space 2^N is not acceptable, but N or N*log(N) would be.

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  • Mathematics - Why is Differential Calculus (MVP) in PHP a tabu?

    - by Email
    Hi I want to do a Mean-Variance-Optimization (Markowitz) but i never found anything written in php that does this. MVP needs differential calculus. Can it be done in php and why arent there any classes/works from universities? For a webapplication (regarding performance) would another language be the better choice to handle heavy calculations? Thanks so much for any help/answer on this

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  • Adding an equation or formula to a figure caption in LaTeX

    - by gotgenes
    I have a figure in LaTeX with a caption to which I need to add a formula (equation* or displaymath environments). For example: \documentclass[12pt]{article} \begin{document} \begin{figure}[tbph] \begin{center} %... \end{center} \caption{As you can see \begin{displaymath}4 \ne 5\end{displaymath} } \label{fig:somefig} \end{figure} \end{document} This makes pdflatex angry, though it will produce a PDF. ! Argument of \@caption has an extra }. <inserted text> \par l.9 } What's the right way to go about adding an equation to a figure caption? NOTE: Please do not suggest simply using the $ ... $ math environment; the equation shown is a toy example; my real equation is much more intricate. See also: Adding a caption to an equation in LaTeX (the reverse of this question)

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  • How to use constraint programming for optimizing shopping baskets?

    - by tangens
    I have a list of items I want to buy. The items are offered by different shops and different prices. The shops have individual delivery costs. I'm looking for an optimal shopping strategy (and a java library supporting it) to purchase all of the items with a minimal total price. Example: Item1 is offered at Shop1 for $100, at Shop2 for $111. Item2 is offered at Shop1 for $90, at Shop2 for $85. Delivery cost of Shop1: $10 if total order < $150; $0 otherwise Delivery cost of Shop2: $5 if total order < $50; $0 otherwise If I buy Item1 and Item2 at Shop1 the total cost is $100 + $90 +$0 = $190. If I buy Item1 and Item2 at Shop2 the total cost is $111 + $85 +$0 = $196. If I buy Item1 at Shop1 and Item2 at Shop2 the total cost is $100 + $10 + $85 + $0 = 195. I get the minimal price if I order Item1 and Item2 at Shop1: $190 What I tried so far I asked another question before that led me to the field of constraint programming. I had a look at cream and choco, but I did not figure out how to create a model to solve my problem. | shop1 | shop2 | shop3 | ... ----------------------------------------- item1 | p11 | p12 | p13 | item2 | p21 | p22 | p23 | . | | | | . | | | | ----------------------------------------- shipping | s1 | s2 | s3 | limit | l1 | l2 | l3 | ----------------------------------------- total | t1 | t2 | t3 | ----------------------------------------- My idea was to define these constraints: each price "p xy" is defined in the domain (0, c) where c is the price of the item in this shop only one price in a line should be non zero if one or more items are bought from one shop and the sum of the prices is lower than limit, then add shipping cost to the total cost shop total cost is the sum of the prices of all items in a shop total cost is the sum of all shop totals The objective is "total cost". I want to minimize this. In cream I wasn't able to express the "if then" constraint for conditional shipping costs. In choco these constraints exist, but even for 5 items and 10 shops the program was running for 10 minutes without finding a solution. Question How should I express my constraints to make this problem solvable for a constraint programming solver?

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  • why do we need advanced knowledge of mathematics & physics for programming?

    - by Sumeet
    Guys, I have been very good in mathematics and physics in my schools and colleges. Right now I am a programmer. Even in the colleges I have to engrossed my self into computers and programming things all the time. As I used to like it very much. But I have always felt the lack of advanced mathematics and physics in all the work I have done (Programs). Programming never asked me any advanced mathematics and physics knowledge in what I was very good. It always ask u some optimized loops, and different programming technologies which has never been covered in advanced mathematics and physics. Even at the time of selection in big College , such a kind of advanced knowledge is required. Time by time I got out of touch of all that facts and concepts (advanced mathematics and physics). And now after, 5 years in job I found it hard to resolve Differentiations and integrations from Trigonometry. Which sometimes make me feel like I have wasted time in those concepts because they are never used. (At that time I knew that I am going to be a programmer) If one need to be a programmer why do all this advanced knowledge is required. One can go with elementry knowledge a bit more. You never got to think like scientists and R&D person in your Schols and colleges for being a programmer? Just think and let me know your thoughts. I must be wrong somewhere in what I think , but not able to figure that out..? Regards Sumeet

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  • optimized grid for rectangular items

    - by peterchen
    I have N rectangular items with an aspect ratio Aitem (X:Y). I have a rectangular display area with an aspect ratio Aview The items should be arranged in a table-like layout (i.e. r rows, c columns). what is the ideal grid rows x columns, so that individual items are largest? (rows * colums = N, of course - i.e. there may be "unused" grid places). A simple algorithm could iterate over rows = 1..N, calculate the required number of columns, and keep the row/column pair with the largest items. I wonder if there's a non-iterative algorithm, though (e.g. for Aitem = Aview = 1, rows / cols can be approximated by sqrt(N)).

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  • Efficient implementation of natural logarithm (ln) and exponentiation

    - by Donotalo
    Basically, I'm looking for implementation of log() and exp() functions provided in C library <math.h>. I'm working with 8 bit microcontrollers (OKI 411 and 431). I need to calculate Mean Kinetic Temperature. The requirement is that we should be able to calculate MKT as fast as possible and with as little code memory as possible. The compiler comes with log() and exp() functions in <math.h>. But calling either function and linking with the library causes the code size to increase by 5 Kilobytes, which will not fit in one of the micro we work with (OKI 411), because our code already consumed ~12K of available ~15K code memory. The implementation I'm looking for should not use any other C library functions (like pow(), sqrt() etc). This is because all library functions are packed in one library and even if one function is called, the linker will bring whole 5K library to code memory.

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  • How computer multiplies 2 numbers?

    - by ckv
    How does a computer perform a multiplication on 2 numbers say 100 * 55. My guess was that the computer did repeated addition to achieve multiplication. Of course this could be the case for integer numbers. However for floating point numbers there must be some other logic. Note: This was asked in an interview.

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  • Merging and splitting overlapping rectangles to produce non-overlapping ones

    - by uj
    I am looking for an algorithm as follows: Given a set of possibly overlapping rectangles (All of which are "not rotated", can be uniformly represented as (left,top,right,bottom) tuplets, etc...), it returns a minimal set of (non-rotated) non-overlapping rectangles, that occupy the same area. It seems simple enough at first glance, but prooves to be tricky (at least to be done efficiently). Are there some known methods for this/ideas/pointers? Methods for not necessarily minimal, but heuristicly small, sets, are interesting as well, so are methods that produce any valid output set at all.

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  • Speeding up Math calculations in Java

    - by Simon
    I have a neural network written in Java which uses a sigmoid transfer function defined as follows: private static double sigmoid(double x) { return 1 / (1 + Math.exp(-x)); } and this is called many times during training and computation using the network. Is there any way of speeding this up? It's not that it's slow, it's just that it is used a lot, so a small optimisation here would be a big overall gain.

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  • Getting confused why i dont get expected amount ?

    - by Stackfan
    I have 1 result and which i will receive in Bank account, Based on that account i have to Put a balance to user account. How can you find the Handling cost from total tried 491.50 / 0.95 = 517.36 which is wrong ? It should be 500.00 (to my expectation) User balance requires 500.00 When 500.00 selected he gets 5% discount There is a handling cost for this ex: 1) Discount: 500.00 - 5% = 475.00 2) Handling cost: (475.00 x 0.034) + 0.35 = 16.50 3) Total: 475.00 + 16.50 = 491.50 So problem is from 491.50, i have to find atleast handling cost to get promised Balance. Any solution ? Cant figure it out myself...

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  • Wpf. Chart optimization. More than million points

    - by Evgeny
    I have custom control - chart with size, for example, 300x300 pixels and more than one million points (maybe less) in it. And its clear that now he works very slowly. I am searching for algoritm which will show only few points with minimal visual difference. I have link to component which have functionallity exactly what i need (2 million points demo): http://www.mindscape.co.nz/demo/SilverlightElements/demopage.html#/ChartOverviewPage I will be grateful for any matherials, links or thoughts how to realize such functionallity.

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  • Netlogo programming question - is it possible to put balanced chemical equations in a model?

    - by user286190
    hi I was wondering if it was possible to put balanced chemical equations, and if possible including state symbols, in the existing netlogo model that i am using, i havenot seen any examples in the models library so was not sure if it was possible. I wanted the model to be able to allow the user to input a balanced chemical equilibrium equation, or the model displays the the equation so then the user can select from them if they do not want to enter their own any help will be greatly appreciated thanks for example ethane + oxygen -- carbon dioxide + steam C2H6 + O2 -- CO2 + H2O

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  • difference equations in MATLAB - why the need to switch signs?

    - by jefflovejapan
    Perhaps this is more of a math question than a MATLAB one, not really sure. I'm using MATLAB to compute an economic model - the New Hybrid ISLM model - and there's a confusing step where the author switches the sign of the solution. First, the author declares symbolic variables and sets up a system of difference equations. Note that the suffixes "a" and "2t" both mean "time t+1", "2a" means "time t+2" and "t" means "time t": %% --------------------------[2] MODEL proc-----------------------------%% % Define endogenous vars ('a' denotes t+1 values) syms y2a pi2a ya pia va y2t pi2t yt pit vt ; % Monetary policy rule ia = q1*ya+q2*pia; % ia = q1*(ya-yt)+q2*pia; %%option speed limit policy % Model equations IS = rho*y2a+(1-rho)yt-sigma(ia-pi2a)-ya; AS = beta*pi2a+(1-beta)*pit+alpha*ya-pia+va; dum1 = ya-y2t; dum2 = pia-pi2t; MPs = phi*vt-va; optcon = [IS ; AS ; dum1 ; dum2; MPs]; He then computes the matrix A: %% ------------------ [3] Linearization proc ------------------------%% % Differentiation xx = [y2a pi2a ya pia va y2t pi2t yt pit vt] ; % define vars jopt = jacobian(optcon,xx); % Define Linear Coefficients coef = eval(jopt); B = [ -coef(:,1:5) ] ; C = [ coef(:,6:10) ] ; % B[c(t+1) l(t+1) k(t+1) z(t+1)] = C[c(t) l(t) k(t) z(t)] A = inv(C)*B ; %(Linearized reduced form ) As far as I understand, this A is the solution to the system. It's the matrix that turns time t+1 and t+2 variables into t and t+1 variables (it's a forward-looking model). My question is essentially why is it necessary to reverse the signs of all the partial derivatives in B in order to get this solution? I'm talking about this step: B = [ -coef(:,1:5) ] ; Reversing the sign here obviously reverses the sign of every component of A, but I don't have a clear understanding of why it's necessary. My apologies if the question is unclear or if this isn't the best place to ask.

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  • NET Math Libraries

    - by JoshReuben
    NET Mathematical Libraries   .NET Builder for Matlab The MathWorks Inc. - http://www.mathworks.com/products/netbuilder/ MATLAB Builder NE generates MATLAB based .NET and COM components royalty-free deployment creates the components by encrypting MATLAB functions and generating either a .NET or COM wrapper around them. .NET/Link for Mathematica www.wolfram.com a product that 2-way integrates Mathematica and Microsoft's .NET platform call .NET from Mathematica - use arbitrary .NET types directly from the Mathematica language. use and control the Mathematica kernel from a .NET program. turns Mathematica into a scripting shell to leverage the computational services of Mathematica. write custom front ends for Mathematica or use Mathematica as a computational engine for another program comes with full source code. Leverages MathLink - a Wolfram Research's protocol for sending data and commands back and forth between Mathematica and other programs. .NET/Link abstracts the low-level details of the MathLink C API. Extreme Optimization http://www.extremeoptimization.com/ a collection of general-purpose mathematical and statistical classes built for the.NET framework. It combines a math library, a vector and matrix library, and a statistics library in one package. download the trial of version 4.0 to try it out. Multi-core ready - Full support for Task Parallel Library features including cancellation. Broad base of algorithms covering a wide range of numerical techniques, including: linear algebra (BLAS and LAPACK routines), numerical analysis (integration and differentiation), equation solvers. Mathematics leverages parallelism using .NET 4.0's Task Parallel Library. Basic math: Complex numbers, 'special functions' like Gamma and Bessel functions, numerical differentiation. Solving equations: Solve equations in one variable, or solve systems of linear or nonlinear equations. Curve fitting: Linear and nonlinear curve fitting, cubic splines, polynomials, orthogonal polynomials. Optimization: find the minimum or maximum of a function in one or more variables, linear programming and mixed integer programming. Numerical integration: Compute integrals over finite or infinite intervals, over 2D and higher dimensional regions. Integrate systems of ordinary differential equations (ODE's). Fast Fourier Transforms: 1D and 2D FFT's using managed or fast native code (32 and 64 bit) BigInteger, BigRational, and BigFloat: Perform operations with arbitrary precision. Vector and Matrix Library Real and complex vectors and matrices. Single and double precision for elements. Structured matrix types: including triangular, symmetrical and band matrices. Sparse matrices. Matrix factorizations: LU decomposition, QR decomposition, singular value decomposition, Cholesky decomposition, eigenvalue decomposition. Portability and performance: Calculations can be done in 100% managed code, or in hand-optimized processor-specific native code (32 and 64 bit). Statistics Data manipulation: Sort and filter data, process missing values, remove outliers, etc. Supports .NET data binding. Statistical Models: Simple, multiple, nonlinear, logistic, Poisson regression. Generalized Linear Models. One and two-way ANOVA. Hypothesis Tests: 12 14 hypothesis tests, including the z-test, t-test, F-test, runs test, and more advanced tests, such as the Anderson-Darling test for normality, one and two-sample Kolmogorov-Smirnov test, and Levene's test for homogeneity of variances. Multivariate Statistics: K-means cluster analysis, hierarchical cluster analysis, principal component analysis (PCA), multivariate probability distributions. Statistical Distributions: 25 29 continuous and discrete statistical distributions, including uniform, Poisson, normal, lognormal, Weibull and Gumbel (extreme value) distributions. Random numbers: Random variates from any distribution, 4 high-quality random number generators, low discrepancy sequences, shufflers. New in version 4.0 (November, 2010) Support for .NET Framework Version 4.0 and Visual Studio 2010 TPL Parallellized – multicore ready sparse linear program solver - can solve problems with more than 1 million variables. Mixed integer linear programming using a branch and bound algorithm. special functions: hypergeometric, Riemann zeta, elliptic integrals, Frensel functions, Dawson's integral. Full set of window functions for FFT's. Product  Price Update subscription Single Developer License $999  $399  Team License (3 developers) $1999  $799  Department License (8 developers) $3999  $1599  Site License (Unlimited developers in one physical location) $7999  $3199    NMath http://www.centerspace.net .NET math and statistics libraries matrix and vector classes random number generators Fast Fourier Transforms (FFTs) numerical integration linear programming linear regression curve and surface fitting optimization hypothesis tests analysis of variance (ANOVA) probability distributions principal component analysis cluster analysis built on the Intel Math Kernel Library (MKL), which contains highly-optimized, extensively-threaded versions of BLAS (Basic Linear Algebra Subroutines) and LAPACK (Linear Algebra PACKage). Product  Price Update subscription Single Developer License $1295 $388 Team License (5 developers) $5180 $1554   DotNumerics http://www.dotnumerics.com/NumericalLibraries/Default.aspx free DotNumerics is a website dedicated to numerical computing for .NET that includes a C# Numerical Library for .NET containing algorithms for Linear Algebra, Differential Equations and Optimization problems. The Linear Algebra library includes CSLapack, CSBlas and CSEispack, ports from Fortran to C# of LAPACK, BLAS and EISPACK, respectively. Linear Algebra (CSLapack, CSBlas and CSEispack). Systems of linear equations, eigenvalue problems, least-squares solutions of linear systems and singular value problems. Differential Equations. Initial-value problem for nonstiff and stiff ordinary differential equations ODEs (explicit Runge-Kutta, implicit Runge-Kutta, Gear's BDF and Adams-Moulton). Optimization. Unconstrained and bounded constrained optimization of multivariate functions (L-BFGS-B, Truncated Newton and Simplex methods).   Math.NET Numerics http://numerics.mathdotnet.com/ free an open source numerical library - includes special functions, linear algebra, probability models, random numbers, interpolation, integral transforms. A merger of dnAnalytics with Math.NET Iridium in addition to a purely managed implementation will also support native hardware optimization. constants & special functions complex type support real and complex, dense and sparse linear algebra (with LU, QR, eigenvalues, ... decompositions) non-uniform probability distributions, multivariate distributions, sample generation alternative uniform random number generators descriptive statistics, including order statistics various interpolation methods, including barycentric approaches and splines numerical function integration (quadrature) routines integral transforms, like fourier transform (FFT) with arbitrary lengths support, and hartley spectral-space aware sequence manipulation (signal processing) combinatorics, polynomials, quaternions, basic number theory. parallelized where appropriate, to leverage multi-core and multi-processor systems fully managed or (if available) using native libraries (Intel MKL, ACMS, CUDA, FFTW) provides a native facade for F# developers

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