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  • A Taxonomy of Numerical Methods v1

    - by JoshReuben
    Numerical Analysis – When, What, (but not how) Once you understand the Math & know C++, Numerical Methods are basically blocks of iterative & conditional math code. I found the real trick was seeing the forest for the trees – knowing which method to use for which situation. Its pretty easy to get lost in the details – so I’ve tried to organize these methods in a way that I can quickly look this up. I’ve included links to detailed explanations and to C++ code examples. I’ve tried to classify Numerical methods in the following broad categories: Solving Systems of Linear Equations Solving Non-Linear Equations Iteratively Interpolation Curve Fitting Optimization Numerical Differentiation & Integration Solving ODEs Boundary Problems Solving EigenValue problems Enjoy – I did ! Solving Systems of Linear Equations Overview Solve sets of algebraic equations with x unknowns The set is commonly in matrix form Gauss-Jordan Elimination http://en.wikipedia.org/wiki/Gauss%E2%80%93Jordan_elimination C++: http://www.codekeep.net/snippets/623f1923-e03c-4636-8c92-c9dc7aa0d3c0.aspx Produces solution of the equations & the coefficient matrix Efficient, stable 2 steps: · Forward Elimination – matrix decomposition: reduce set to triangular form (0s below the diagonal) or row echelon form. If degenerate, then there is no solution · Backward Elimination –write the original matrix as the product of ints inverse matrix & its reduced row-echelon matrix à reduce set to row canonical form & use back-substitution to find the solution to the set Elementary ops for matrix decomposition: · Row multiplication · Row switching · Add multiples of rows to other rows Use pivoting to ensure rows are ordered for achieving triangular form LU Decomposition http://en.wikipedia.org/wiki/LU_decomposition C++: http://ganeshtiwaridotcomdotnp.blogspot.co.il/2009/12/c-c-code-lu-decomposition-for-solving.html Represent the matrix as a product of lower & upper triangular matrices A modified version of GJ Elimination Advantage – can easily apply forward & backward elimination to solve triangular matrices Techniques: · Doolittle Method – sets the L matrix diagonal to unity · Crout Method - sets the U matrix diagonal to unity Note: both the L & U matrices share the same unity diagonal & can be stored compactly in the same matrix Gauss-Seidel Iteration http://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method C++: http://www.nr.com/forum/showthread.php?t=722 Transform the linear set of equations into a single equation & then use numerical integration (as integration formulas have Sums, it is implemented iteratively). an optimization of Gauss-Jacobi: 1.5 times faster, requires 0.25 iterations to achieve the same tolerance Solving Non-Linear Equations Iteratively find roots of polynomials – there may be 0, 1 or n solutions for an n order polynomial use iterative techniques Iterative methods · used when there are no known analytical techniques · Requires set functions to be continuous & differentiable · Requires an initial seed value – choice is critical to convergence à conduct multiple runs with different starting points & then select best result · Systematic - iterate until diminishing returns, tolerance or max iteration conditions are met · bracketing techniques will always yield convergent solutions, non-bracketing methods may fail to converge Incremental method if a nonlinear function has opposite signs at 2 ends of a small interval x1 & x2, then there is likely to be a solution in their interval – solutions are detected by evaluating a function over interval steps, for a change in sign, adjusting the step size dynamically. Limitations – can miss closely spaced solutions in large intervals, cannot detect degenerate (coinciding) solutions, limited to functions that cross the x-axis, gives false positives for singularities Fixed point method http://en.wikipedia.org/wiki/Fixed-point_iteration C++: http://books.google.co.il/books?id=weYj75E_t6MC&pg=PA79&lpg=PA79&dq=fixed+point+method++c%2B%2B&source=bl&ots=LQ-5P_taoC&sig=lENUUIYBK53tZtTwNfHLy5PEWDk&hl=en&sa=X&ei=wezDUPW1J5DptQaMsIHQCw&redir_esc=y#v=onepage&q=fixed%20point%20method%20%20c%2B%2B&f=false Algebraically rearrange a solution to isolate a variable then apply incremental method Bisection method http://en.wikipedia.org/wiki/Bisection_method C++: http://numericalcomputing.wordpress.com/category/algorithms/ Bracketed - Select an initial interval, keep bisecting it ad midpoint into sub-intervals and then apply incremental method on smaller & smaller intervals – zoom in Adv: unaffected by function gradient à reliable Disadv: slow convergence False Position Method http://en.wikipedia.org/wiki/False_position_method C++: http://www.dreamincode.net/forums/topic/126100-bisection-and-false-position-methods/ Bracketed - Select an initial interval , & use the relative value of function at interval end points to select next sub-intervals (estimate how far between the end points the solution might be & subdivide based on this) Newton-Raphson method http://en.wikipedia.org/wiki/Newton's_method C++: http://www-users.cselabs.umn.edu/classes/Summer-2012/csci1113/index.php?page=./newt3 Also known as Newton's method Convenient, efficient Not bracketed – only a single initial guess is required to start iteration – requires an analytical expression for the first derivative of the function as input. Evaluates the function & its derivative at each step. Can be extended to the Newton MutiRoot method for solving multiple roots Can be easily applied to an of n-coupled set of non-linear equations – conduct a Taylor Series expansion of a function, dropping terms of order n, rewrite as a Jacobian matrix of PDs & convert to simultaneous linear equations !!! Secant Method http://en.wikipedia.org/wiki/Secant_method C++: http://forum.vcoderz.com/showthread.php?p=205230 Unlike N-R, can estimate first derivative from an initial interval (does not require root to be bracketed) instead of inputting it Since derivative is approximated, may converge slower. Is fast in practice as it does not have to evaluate the derivative at each step. Similar implementation to False Positive method Birge-Vieta Method http://mat.iitm.ac.in/home/sryedida/public_html/caimna/transcendental/polynomial%20methods/bv%20method.html C++: http://books.google.co.il/books?id=cL1boM2uyQwC&pg=SA3-PA51&lpg=SA3-PA51&dq=Birge-Vieta+Method+c%2B%2B&source=bl&ots=QZmnDTK3rC&sig=BPNcHHbpR_DKVoZXrLi4nVXD-gg&hl=en&sa=X&ei=R-_DUK2iNIjzsgbE5ID4Dg&redir_esc=y#v=onepage&q=Birge-Vieta%20Method%20c%2B%2B&f=false combines Horner's method of polynomial evaluation (transforming into lesser degree polynomials that are more computationally efficient to process) with Newton-Raphson to provide a computational speed-up Interpolation Overview Construct new data points for as close as possible fit within range of a discrete set of known points (that were obtained via sampling, experimentation) Use Taylor Series Expansion of a function f(x) around a specific value for x Linear Interpolation http://en.wikipedia.org/wiki/Linear_interpolation C++: http://www.hamaluik.com/?p=289 Straight line between 2 points à concatenate interpolants between each pair of data points Bilinear Interpolation http://en.wikipedia.org/wiki/Bilinear_interpolation C++: http://supercomputingblog.com/graphics/coding-bilinear-interpolation/2/ Extension of the linear function for interpolating functions of 2 variables – perform linear interpolation first in 1 direction, then in another. Used in image processing – e.g. texture mapping filter. Uses 4 vertices to interpolate a value within a unit cell. Lagrange Interpolation http://en.wikipedia.org/wiki/Lagrange_polynomial C++: http://www.codecogs.com/code/maths/approximation/interpolation/lagrange.php For polynomials Requires recomputation for all terms for each distinct x value – can only be applied for small number of nodes Numerically unstable Barycentric Interpolation http://epubs.siam.org/doi/pdf/10.1137/S0036144502417715 C++: http://www.gamedev.net/topic/621445-barycentric-coordinates-c-code-check/ Rearrange the terms in the equation of the Legrange interpolation by defining weight functions that are independent of the interpolated value of x Newton Divided Difference Interpolation http://en.wikipedia.org/wiki/Newton_polynomial C++: http://jee-appy.blogspot.co.il/2011/12/newton-divided-difference-interpolation.html Hermite Divided Differences: Interpolation polynomial approximation for a given set of data points in the NR form - divided differences are used to approximately calculate the various differences. For a given set of 3 data points , fit a quadratic interpolant through the data Bracketed functions allow Newton divided differences to be calculated recursively Difference table Cubic Spline Interpolation http://en.wikipedia.org/wiki/Spline_interpolation C++: https://www.marcusbannerman.co.uk/index.php/home/latestarticles/42-articles/96-cubic-spline-class.html Spline is a piecewise polynomial Provides smoothness – for interpolations with significantly varying data Use weighted coefficients to bend the function to be smooth & its 1st & 2nd derivatives are continuous through the edge points in the interval Curve Fitting A generalization of interpolating whereby given data points may contain noise à the curve does not necessarily pass through all the points Least Squares Fit http://en.wikipedia.org/wiki/Least_squares C++: http://www.ccas.ru/mmes/educat/lab04k/02/least-squares.c Residual – difference between observed value & expected value Model function is often chosen as a linear combination of the specified functions Determines: A) The model instance in which the sum of squared residuals has the least value B) param values for which model best fits data Straight Line Fit Linear correlation between independent variable and dependent variable Linear Regression http://en.wikipedia.org/wiki/Linear_regression C++: http://www.oocities.org/david_swaim/cpp/linregc.htm Special case of statistically exact extrapolation Leverage least squares Given a basis function, the sum of the residuals is determined and the corresponding gradient equation is expressed as a set of normal linear equations in matrix form that can be solved (e.g. using LU Decomposition) Can be weighted - Drop the assumption that all errors have the same significance –-> confidence of accuracy is different for each data point. Fit the function closer to points with higher weights Polynomial Fit - use a polynomial basis function Moving Average http://en.wikipedia.org/wiki/Moving_average C++: http://www.codeproject.com/Articles/17860/A-Simple-Moving-Average-Algorithm Used for smoothing (cancel fluctuations to highlight longer-term trends & cycles), time series data analysis, signal processing filters Replace each data point with average of neighbors. Can be simple (SMA), weighted (WMA), exponential (EMA). Lags behind latest data points – extra weight can be given to more recent data points. Weights can decrease arithmetically or exponentially according to distance from point. Parameters: smoothing factor, period, weight basis Optimization Overview Given function with multiple variables, find Min (or max by minimizing –f(x)) Iterative approach Efficient, but not necessarily reliable Conditions: noisy data, constraints, non-linear models Detection via sign of first derivative - Derivative of saddle points will be 0 Local minima Bisection method Similar method for finding a root for a non-linear equation Start with an interval that contains a minimum Golden Search method http://en.wikipedia.org/wiki/Golden_section_search C++: http://www.codecogs.com/code/maths/optimization/golden.php Bisect intervals according to golden ratio 0.618.. Achieves reduction by evaluating a single function instead of 2 Newton-Raphson Method Brent method http://en.wikipedia.org/wiki/Brent's_method C++: http://people.sc.fsu.edu/~jburkardt/cpp_src/brent/brent.cpp Based on quadratic or parabolic interpolation – if the function is smooth & parabolic near to the minimum, then a parabola fitted through any 3 points should approximate the minima – fails when the 3 points are collinear , in which case the denominator is 0 Simplex Method http://en.wikipedia.org/wiki/Simplex_algorithm C++: http://www.codeguru.com/cpp/article.php/c17505/Simplex-Optimization-Algorithm-and-Implemetation-in-C-Programming.htm Find the global minima of any multi-variable function Direct search – no derivatives required At each step it maintains a non-degenerative simplex – a convex hull of n+1 vertices. Obtains the minimum for a function with n variables by evaluating the function at n-1 points, iteratively replacing the point of worst result with the point of best result, shrinking the multidimensional simplex around the best point. Point replacement involves expanding & contracting the simplex near the worst value point to determine a better replacement point Oscillation can be avoided by choosing the 2nd worst result Restart if it gets stuck Parameters: contraction & expansion factors Simulated Annealing http://en.wikipedia.org/wiki/Simulated_annealing C++: http://code.google.com/p/cppsimulatedannealing/ Analogy to heating & cooling metal to strengthen its structure Stochastic method – apply random permutation search for global minima - Avoid entrapment in local minima via hill climbing Heating schedule - Annealing schedule params: temperature, iterations at each temp, temperature delta Cooling schedule – can be linear, step-wise or exponential Differential Evolution http://en.wikipedia.org/wiki/Differential_evolution C++: http://www.amichel.com/de/doc/html/ More advanced stochastic methods analogous to biological processes: Genetic algorithms, evolution strategies Parallel direct search method against multiple discrete or continuous variables Initial population of variable vectors chosen randomly – if weighted difference vector of 2 vectors yields a lower objective function value then it replaces the comparison vector Many params: #parents, #variables, step size, crossover constant etc Convergence is slow – many more function evaluations than simulated annealing Numerical Differentiation Overview 2 approaches to finite difference methods: · A) approximate function via polynomial interpolation then differentiate · B) Taylor series approximation – additionally provides error estimate Finite Difference methods http://en.wikipedia.org/wiki/Finite_difference_method C++: http://www.wpi.edu/Pubs/ETD/Available/etd-051807-164436/unrestricted/EAMPADU.pdf Find differences between high order derivative values - Approximate differential equations by finite differences at evenly spaced data points Based on forward & backward Taylor series expansion of f(x) about x plus or minus multiples of delta h. Forward / backward difference - the sums of the series contains even derivatives and the difference of the series contains odd derivatives – coupled equations that can be solved. Provide an approximation of the derivative within a O(h^2) accuracy There is also central difference & extended central difference which has a O(h^4) accuracy Richardson Extrapolation http://en.wikipedia.org/wiki/Richardson_extrapolation C++: http://mathscoding.blogspot.co.il/2012/02/introduction-richardson-extrapolation.html A sequence acceleration method applied to finite differences Fast convergence, high accuracy O(h^4) Derivatives via Interpolation Cannot apply Finite Difference method to discrete data points at uneven intervals – so need to approximate the derivative of f(x) using the derivative of the interpolant via 3 point Lagrange Interpolation Note: the higher the order of the derivative, the lower the approximation precision Numerical Integration Estimate finite & infinite integrals of functions More accurate procedure than numerical differentiation Use when it is not possible to obtain an integral of a function analytically or when the function is not given, only the data points are Newton Cotes Methods http://en.wikipedia.org/wiki/Newton%E2%80%93Cotes_formulas C++: http://www.siafoo.net/snippet/324 For equally spaced data points Computationally easy – based on local interpolation of n rectangular strip areas that is piecewise fitted to a polynomial to get the sum total area Evaluate the integrand at n+1 evenly spaced points – approximate definite integral by Sum Weights are derived from Lagrange Basis polynomials Leverage Trapezoidal Rule for default 2nd formulas, Simpson 1/3 Rule for substituting 3 point formulas, Simpson 3/8 Rule for 4 point formulas. For 4 point formulas use Bodes Rule. Higher orders obtain more accurate results Trapezoidal Rule uses simple area, Simpsons Rule replaces the integrand f(x) with a quadratic polynomial p(x) that uses the same values as f(x) for its end points, but adds a midpoint Romberg Integration http://en.wikipedia.org/wiki/Romberg's_method C++: http://code.google.com/p/romberg-integration/downloads/detail?name=romberg.cpp&can=2&q= Combines trapezoidal rule with Richardson Extrapolation Evaluates the integrand at equally spaced points The integrand must have continuous derivatives Each R(n,m) extrapolation uses a higher order integrand polynomial replacement rule (zeroth starts with trapezoidal) à a lower triangular matrix set of equation coefficients where the bottom right term has the most accurate approximation. The process continues until the difference between 2 successive diagonal terms becomes sufficiently small. Gaussian Quadrature http://en.wikipedia.org/wiki/Gaussian_quadrature C++: http://www.alglib.net/integration/gaussianquadratures.php Data points are chosen to yield best possible accuracy – requires fewer evaluations Ability to handle singularities, functions that are difficult to evaluate The integrand can include a weighting function determined by a set of orthogonal polynomials. Points & weights are selected so that the integrand yields the exact integral if f(x) is a polynomial of degree <= 2n+1 Techniques (basically different weighting functions): · Gauss-Legendre Integration w(x)=1 · Gauss-Laguerre Integration w(x)=e^-x · Gauss-Hermite Integration w(x)=e^-x^2 · Gauss-Chebyshev Integration w(x)= 1 / Sqrt(1-x^2) Solving ODEs Use when high order differential equations cannot be solved analytically Evaluated under boundary conditions RK for systems – a high order differential equation can always be transformed into a coupled first order system of equations Euler method http://en.wikipedia.org/wiki/Euler_method C++: http://rosettacode.org/wiki/Euler_method First order Runge–Kutta method. Simple recursive method – given an initial value, calculate derivative deltas. Unstable & not very accurate (O(h) error) – not used in practice A first-order method - the local error (truncation error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size In evolving solution between data points xn & xn+1, only evaluates derivatives at beginning of interval xn à asymmetric at boundaries Higher order Runge Kutta http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods C++: http://www.dreamincode.net/code/snippet1441.htm 2nd & 4th order RK - Introduces parameterized midpoints for more symmetric solutions à accuracy at higher computational cost Adaptive RK – RK-Fehlberg – estimate the truncation at each integration step & automatically adjust the step size to keep error within prescribed limits. At each step 2 approximations are compared – if in disagreement to a specific accuracy, the step size is reduced Boundary Value Problems Where solution of differential equations are located at 2 different values of the independent variable x à more difficult, because cannot just start at point of initial value – there may not be enough starting conditions available at the end points to produce a unique solution An n-order equation will require n boundary conditions – need to determine the missing n-1 conditions which cause the given conditions at the other boundary to be satisfied Shooting Method http://en.wikipedia.org/wiki/Shooting_method C++: http://ganeshtiwaridotcomdotnp.blogspot.co.il/2009/12/c-c-code-shooting-method-for-solving.html Iteratively guess the missing values for one end & integrate, then inspect the discrepancy with the boundary values of the other end to adjust the estimate Given the starting boundary values u1 & u2 which contain the root u, solve u given the false position method (solving the differential equation as an initial value problem via 4th order RK), then use u to solve the differential equations. Finite Difference Method For linear & non-linear systems Higher order derivatives require more computational steps – some combinations for boundary conditions may not work though Improve the accuracy by increasing the number of mesh points Solving EigenValue Problems An eigenvalue can substitute a matrix when doing matrix multiplication à convert matrix multiplication into a polynomial EigenValue For a given set of equations in matrix form, determine what are the solution eigenvalue & eigenvectors Similar Matrices - have same eigenvalues. Use orthogonal similarity transforms to reduce a matrix to diagonal form from which eigenvalue(s) & eigenvectors can be computed iteratively Jacobi method http://en.wikipedia.org/wiki/Jacobi_method C++: http://people.sc.fsu.edu/~jburkardt/classes/acs2_2008/openmp/jacobi/jacobi.html Robust but Computationally intense – use for small matrices < 10x10 Power Iteration http://en.wikipedia.org/wiki/Power_iteration For any given real symmetric matrix, generate the largest single eigenvalue & its eigenvectors Simplest method – does not compute matrix decomposition à suitable for large, sparse matrices Inverse Iteration Variation of power iteration method – generates the smallest eigenvalue from the inverse matrix Rayleigh Method http://en.wikipedia.org/wiki/Rayleigh's_method_of_dimensional_analysis Variation of power iteration method Rayleigh Quotient Method Variation of inverse iteration method Matrix Tri-diagonalization Method Use householder algorithm to reduce an NxN symmetric matrix to a tridiagonal real symmetric matrix vua N-2 orthogonal transforms     Whats Next Outside of Numerical Methods there are lots of different types of algorithms that I’ve learned over the decades: Data Mining – (I covered this briefly in a previous post: http://geekswithblogs.net/JoshReuben/archive/2007/12/31/ssas-dm-algorithms.aspx ) Search & Sort Routing Problem Solving Logical Theorem Proving Planning Probabilistic Reasoning Machine Learning Solvers (eg MIP) Bioinformatics (Sequence Alignment, Protein Folding) Quant Finance (I read Wilmott’s books – interesting) Sooner or later, I’ll cover the above topics as well.

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  • West Palm Beach Developers&rsquo; Group June 2013 Meeting Recap &ndash; ASP.NET Web API and Web Sockets with Shervin Shakibi

    - by Sam Abraham
    Originally posted on: http://geekswithblogs.net/wildturtle/archive/2013/07/02/west-palm-beach-developersrsquo-group-june-2013-meeting-recap-ndash.aspxOur West Palm Beach Developers’ Group June 2013 meeting featured Shervin Shakibi, Microsoft Regional Director and Certified Trainer. Shervin spoke on the ASP.NET Web API and Web Sockets, two new features shipped along with ASP.NET MVC4. Talk was simply awesome and very interactive as Shervin answered many audience questions. Our event was sponsored by Steven Douglas Associates and hosted by PC Professor. Below are some photos of our event (Pardon my flash malfunction):   Shervin Presenting on the Web API A partial view of the standing-room only meeting.

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  • Itzik Ben-Gan is in town

    - by Dave Ballantyne
    Not that you would know it from the page below,  but Itzik Ben-Gan is back in London to do a 5 day training course, start 03october.  http://www.qa.com/training-courses/technical-it-training/microsoft/microsoft-sql-server/microsoft-sql-server-2008-and-r2/advanced-t-sql-querying,-programming-and-tuning-for-sql-server-2005--2008Why QA are not screaming this from the rafters, I will never be able to fathom.  Its kind of like going for a physics course and finding that Steven Hawking is taking the class. Training budgets are tight at the moment and £2500+ is a fair amount to pay but ,as the saying goes,  but if you pay peanuts you get monkeys. Looks like you will need to be quick , the site is saying "Fewer than 5 places available".

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  • ASP.NET MVC HandleError Attribute

    - by Ben Griswold
    Last Wednesday, I took a whopping 15 minutes out of my day and added ELMAH (Error Logging Modules and Handlers) to my ASP.NET MVC application.  If you haven’t heard the news (I hadn’t until recently), ELMAH does a killer job of logging and reporting nearly all unhandled exceptions.  As for handled exceptions, I’ve been using NLog but since I was already playing with the ELMAH bits I thought I’d see if I couldn’t replace it. Atif Aziz provided a quick solution in his answer to a Stack Overflow question.  I’ll let you consult his answer to see how one can subclass the HandleErrorAttribute and override the OnException method in order to get the bits working.  I pretty much took rolled the recommended logic into my application and it worked like a charm.  Along the way, I did uncover a few HandleError fact to which I wasn’t already privy.  Most of my learning came from Steven Sanderson’s book, Pro ASP.NET MVC Framework.  I’ve flipped through a bunch of the book and spent time on specific sections.  It’s a really good read if you’re looking to pick up an ASP.NET MVC reference. Anyway, my notes are found a comments in the following code snippet.  I hope my notes clarify a few things for you too. public class LogAndHandleErrorAttribute : HandleErrorAttribute {     public override void OnException(ExceptionContext context)     {         // A word from our sponsors:         //      http://stackoverflow.com/questions/766610/how-to-get-elmah-to-work-with-asp-net-mvc-handleerror-attribute         //      and Pro ASP.NET MVC Framework by Steven Sanderson         //         // Invoke the base implementation first. This should mark context.ExceptionHandled = true         // which stops ASP.NET from producing a "yellow screen of death." This also sets the         // Http StatusCode to 500 (internal server error.)         //         // Assuming Custom Errors aren't off, the base implementation will trigger the application         // to ultimately render the "Error" view from one of the following locations:         //         //      1. ~/Views/Controller/Error.aspx         //      2. ~/Views/Controller/Error.ascx         //      3. ~/Views/Shared/Error.aspx         //      4. ~/Views/Shared/Error.ascx         //         // "Error" is the default view, however, a specific view may be provided as an Attribute property.         // A notable point is the Custom Errors defaultRedirect is not considered in the redirection plan.         base.OnException(context);           var e = context.Exception;                  // If the exception is unhandled, simply return and let Elmah handle the unhandled exception.         // Otherwise, try to use error signaling which involves the fully configured pipeline like logging,         // mailing, filtering and what have you). Failing that, see if the error should be filtered.         // If not, the error simply logged the exception.         if (!context.ExceptionHandled                || RaiseErrorSignal(e)                   || IsFiltered(context))                  return;           LogException(e); // FYI. Simple Elmah logging doesn't handle mail notifications.     }

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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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  • Internet Explorer 9 Cannot open file on download CTRL + J doesn’t work can’t open list of downloads

    - by simonsabin
    If any of the above symptoms are causing a problem, i.e. 1. You download a file and the download dialog disappears. 2. You select open when you download a file and nothing happens. 3. The View Downloads doesn’t work 4. CTRL + J doesn’t work (view downloads) The solution is to clear your download history See IE9 - View downloads / Ctrl+J do not open. I cannot open any file. But SAVE function still work fine. 64 bit version. for details the answer is provided by Steven. S on June 20th. I hope that...(read more)

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  • What discipline does Computer Science belong to?

    - by Macneil
    Is Computer Science science, applied mathematics, engineering, art, philosophy? "Other"? To provide background, here is Steven Wartik's blog posting for Scientific American titled "I'm not a real scientist, and that's okay." The article covers some good topics for this question, but it leaves open more than it answers. If you can think of the discipline, how would computer science fit into its definition? Should the discipline for Computer Science be based on what programmers do, or what academics do? What kind of answers do you get from people who've seemed to think deeply about this? What reasons do they give?

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  • Chainload boot of Ubuntu installed on 32GB SD card from legacy Grub boot on USB

    - by Gary Darsey
    I have Ubuntu installed on a 32 GB SD card (in the Storage Expansion slot on an Acer Aspire One) with Grub2 installed in the same partition. I boot into legacy Grub on a USB drive and would like to boot by chainloading Grub2 from Grub (kernel/initrd or symlink booting would also be fine), but I haven't figured out how to do this from legacy Grub CLI. Output from blkid for this partition is /dev/mmcblk0p1: LABEL="Ubuntu" UUID="7ceb9fa7-238c-4c5d-bb8e-2c655652ddec" TYPE='ext4" / fdisk -lu information Boot indicator ID 83. Related entries in grub.cfg: search --no-floppy --fs-uuid --set-root 7ceb9fa7-238c-4c5d-bb8e-2c655652ddec linux /boot/vmlinuz-3.5.0-17-generic root=UUID=7ceb9fa7-238c-4c5d-bb8e-2c655652ddec... initrd /boot/initrd.img-3.5.0-17-generic I can't seem to replicate this in legacy Grub. Is there any way get Grub2 to chainload? How do I set root with UUID in legacy Grub? I prefer to boot from USB. Would Grub2 on USB (copying the grub.cfg generated during installation) be an option?

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  • NightHacking demo: Java in the Internet of Things

    - by terrencebarr
    The NightHacking session with Steven Chin was good fun. Check out the video on “Java in the Internet of Things” and a live demo of the Smart Solar Tracking System with Java ME Embedded 3.2. Real hardware and demo flakiness included See here. While you are at, have a look at some of the other NightHacking sessions and a number of other videos on the YouTube Java Channel. Cheers, – Terrence Filed under: Mobile & Embedded Tagged: "Oracle Java ME Embedded", demo, embedded, iot, Java Embedded, nighthacking, video, webcast

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  • NightHacking demo: Java in the Internet of Things

    - by terrencebarr
    The NightHacking session with Steven Chin was good fun. Check out the video on “Java in the Internet of Things” and a live demo of the Smart Solar Tracking System with Java ME Embedded 3.2. Real hardware and demo flakiness included See here. While you are at, have a look at some of the other NightHacking sessions and a number of other videos on the YouTube Java Channel. Cheers, – Terrence Filed under: Mobile & Embedded Tagged: "Oracle Java ME Embedded", demo, embedded, iot, Java Embedded, nighthacking, video, webcast

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  • Q&amp;A: What is the UK pricing for the Windows Azure CDN?

    - by Eric Nelson
    The pricing for Windows Azure Content Delivery Network (CDN) was announced last week. The prices are: £0.091 per GB transferred from North America & Europe locations £0.1213 per GB transferred from other locations £0.0061 per 10,000 transactions CDN rates are effective for all billing periods that begin subsequent to June 30, 2010. All usage for billing periods beginning prior to July 1, 2010 will not be charged. To help you determine which pricing plan best suits your needs, please review the comparison table, which includes the CDN information. Steven Nagy has also done an interesting follow up post on CDN. Related Links: Q&A- How can I calculate the TCO and ROI when considering the Windows Azure Platform? Q&A- When do I get charged for compute hours on Windows Azure? Q&A- What are the UK prices for the Windows Azure Platform

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  • Principes universels du design de William Lidwell , Kritina Holden , Jill Butler, critique par Benwit

    Je viens de lire un livre intitulé "Principes universels du design" [IMG]http://images-eu.amazon.com/images/P/2212128622.08.LZZZZZZZ.jpg[/IMG] Sur la couverture recto/verso, ce qui ressemble à des traits jaunes verticaux, ce sont les noms des 125 principes de design présentés dans ce livre. Entendons nous bien, il ne s'agit pas de Design Pattern (modèle de conception pour votre modèle de données) mais des principes de design utilisé lors de la conception d'objets (IHM comprise). Quels principes de design utilisez vous dans la conception de vos IHM ? Avez vous lu ce livre, pensez vous le lire ?...

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  • How to mount a hidden NTFS WinRE which are on an external HDD

    - by annabinna
    A friend have given me her external hard drive which contains a backup of his Windows data. The disk has two NTFS partitions, once of them tagged as WinRe. When I do fdisk -lu I get Disk /dev/sdc: 120.0 GB, 120034123776 bytes 255 heads, 63 sectors/track, 14593 cylinders, total 234441648 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x59725972 Dispositiu Arrenc. Inici Final Blocs Id Sistema /dev/sdc1 2048 3074047 1536000 27 Hidden NTFS WinRE /dev/sdc2 * 3074048 234438655 115682304 7 HPFS/NTFS/exFAT I never fought against this type of partitions and I haven't any idea of how to mount this and recover the data. Can someone help me?

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  • Développement d'applications professionnelles avec Android 2 de Reto Meier, critique par verdvaine yan

    Je viens de lire "Développement d'applications professionnelles avec Android 2" de Reto Meier, ingénieur chez Google. [IMG]http://images-eu.amazon.com/images/P/274402452X.08.LZZZZZZZ.jpg[/IMG] Je le trouve très complémentaire aux tutoriaux qu'on trouve sur Internet. Il aborde beaucoup de sujets et le nombre de pages n'est pas dù à des captures d'écrans ! Ce que j'ai particulièremen apprécié, ce sont toutes les petites informations tirées de son expérience qu'il distille au fil des pages. L'avez vous lu ? Si oui, par rapport à d'autres livres sur le sujet ? Allez vous le lire ?...

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  • NightHacking Tour Continues - Don't Miss It!

    - by Tori Wieldt
    Java Evangelist Steven Chin (@steveonjava) has been motorcycling across Europe, dropping in on developers and Java User Groups to do some hacking. The visits he has already made are up on the Youtube/Java channel (including James Gosling, Ben Evans, Stephen Colebourne and Trisha Gee).  Steve will be at J-Fall in the Netherlands all day Wednesday, Oct 31. You can watch streaming live and join in on the conversation. (You mean you missed the discussion about long variable names?) Watch for #nighthacking on Twitter. Some upcoming stops on the tour include: Adam Bien (Java Champion and Author) - Friday Nov 2 at 11AM CEST (2AM PST) Andres Almiray (Griffon Founder and Author) - Sunday Nov 4 at 8PM CEST (11AM PST) In total, there will be over 20 different interviews, several JUG visits, and special coverage of J-Fall and Devoxx conferences.You can view the full schedule and watch streaming video at nighthacking.com.

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  • Développement d'applications professionnelles avec Android 2 de Reto Meier, critique par Benwit

    Je viens de lire "Développement d'applications professionnelles avec Android 2" de Reto Meier, ingénieur chez Google. [IMG]http://images-eu.amazon.com/images/P/274402452X.08.LZZZZZZZ.jpg[/IMG] Je le trouve très complémentaire aux tutoriaux qu'on trouve sur Internet. Il aborde beaucoup de sujets et le nombre de pages n'est pas dù à des captures d'écrans ! Ce que j'ai particulièremen apprécié, ce sont toutes les petites informations tirées de son expérience qu'il distille au fil des pages. L'avez vous lu ? Si oui, par rapport à d'autres livres sur le sujet ? Allez vous le lire ?...

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  • Hyper-V Windows Guest Isletim Sistemleri için E-Business Suite R12 Sertifikasi Yayinlandi

    - by TUFEKCIOGLU,FATIH
    Windows Server 2012 Hyper-V sanal makinalarinda guest isletim sistemi olarak çalisan Windows Server 2008 (32-bit) ve Windows Server 2008 R2 için Oracle E-Business Suite Release 12 (12.1) sertifikasi yayinlandi. Hyper-V, Microsoft Windows sunucularda bulunan dahili bir özelliktir ve sanallastirilmis ortamlar olusturmaya ve yönetmeye olanak saglar. Bu sertifika ile E-Business Suite, yukarida belirtilen Windows sanallastirilmis guest isletim sistemleri üzerinde desteklenmektedir. Referanslar : •Note 761567.1 - Oracle E-Business Suite Installation and Upgrade Notes Release 12 (12.1.1) for Microsoft Windows Server (32-bit)•Note 1188535.1 - Migrating Oracle E-Business Suite R12 to Microsoft Windows Server 2008 R2•Note 1563794.1 - Certified Software on Microsoft Windows Server 2012 Hyper-V•Windows Server Hyper-V Overview Orjinal Kaynak (Original Source) : Steven Chan Oracle Blog : https://blogs.oracle.com/stevenChan/entry/e_business_suite_r12_certified

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  • Draggable & Resizable Editors

    - by Geertjan
    Thanks to a cool tip from Steven Yi (here in the comments to a blog entry), I was able to make a totally pointless but fun set of draggable and resizable editors: What you see above are two JEditorPanes within JPanels. The JPanels are within ComponentWidgets provided by the NetBeans Visual Library, which is also where the special border comes from. The ComponentWidgets are within a Visual Library Scene, which is within a JScrollPane in a TopComponent. Each editor has this, which means the NetBeans Java Editor is bound to the JEditorPane: jEditorPane1.setContentType("text/x-java"); EditorKit kit = CloneableEditorSupport.getEditorKit("text/x-java"); jEditorPane1.setEditorKit(kit); jEditorPane1.getDocument().putProperty("mimeType", "text/x-java"); A similar thing is done in the other JEditorPane, i.e., it is bound to the XML Editor. While the XML Editor also has code completion, in addition to syntax coloring, as can be seen above, this is not the case for the JEditorPane bound to the Java Editor, since the JEditorPane doesn't have a Java classpath, which is needed for Java code completion to work.

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  • Justice : Apple s'attaque à Android 4.1 dans un deuxième procès contre Samsung

    Justice : Apple s'attaque directement à Android 4.1 Dans un deuxième procès qui l'oppose à Samsung Combien de fois avons-nous lu, dans certains commentaires de nos articles qui traitent des procès qu'Apple intente aux constructeurs qui lancent des modèles sous Android, que Google et son OS mobile n'étaient pas visés ? Nous avions beau expliquer que c'était bien Android qui était dans la ligne de mire d'Apple (ce qui est d'ailleurs l'analyse de Google), pour certains, nous ne faisions qu'extrapoler pour jeter le doute dans l'esprit des développeurs de cette plateforme. Que ceux-là veuillent bien nous excuser. Apple assigne aujourd'hui Android 4.1 en justice. ...

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  • Java JRE 7 Certified with Oracle E-Business Suite

    - by LuciaC
    Java Runtime Environment 7u10 (a.k.a. JRE 1.7.0_10 build 18) and later updates on the JRE 7 codeline are now certified with Oracle E-Business Suite Release 11i and 12 Windows-based desktop clients. What do you need to do and where are the official patch requirements? EBS customers should ensure that they are running JRE 7u10, at minimum, on Windows desktop clients.  There are also official patch requirements to avoid compatibility issues of E-Business Suite with JRE 7.  Please refer to Steven Chan's Oracle E-Business Suite Technology blog for all the detailed information regarding this certification.

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  • Bing disponible comme plateforme pour développeurs avec l'imagerie 3D et la recherche vocale contextuelle

    Bing disponible comme plateforme pour développeurs avec l'imagerie 3D et la recherche vocale contextuelleMicrosoft a annoncé pendant la Build 2013 à Seattle qu'il a ouvert son outil de recherche Bing comme plateforme pour développeurs. Gurdeep Singh Pall, Vice-Président de l'entreprise, a fait une démonstration pour exposer les capacités dont les développeurs pourront désormais bénéficier. Pall a montré comment ajouter un élément à la to-do liste d'une application tiers en faisant usage du contrôle vocal de Bing. Dans le cas d'espèce, il a créé un mémento sur ses intentions de voyage en direction de l'Espagne.La démonstration est allée un peu plus loin. Pall a obtenu des informations lu...

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  • L'homme vs. l'ordinateur : une machine peut-elle faire votre métier ? Dans quelles tâches un PC peut-il remplacer l'humain ?

    L'homme vs. l'ordinateur : une machine peut-elle faire votre métier ? Dans quelles tâches un PC peut-il remplacer l'humain ? Steven Hsu est physicien, et il a énoncé la phrase suivante, à propos des nouveaux admis dans les Universités : "Certains sont moins bons pour prédire les UG GPA qu'un algorithme tout simple". Une constatation cinglante qui réveille la bonne vieille problématique man versus machine. Dans beaucoup de métiers en rapport avec les sciences, la logique ou les chiffres, des travailleurs effectuent des calculs et des opérations qui semblent extrêmement complexes, ne serait-ce -dans une banque- que pour déterminer si une personne peut se voir accorder un prêt. Pourtant, dans c...

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  • sorting two tables (full join)

    - by Ruslan
    i'm joining tables like: select * from tableA a full join tableB b on a.id = b.id But the output should be: row without null fields row with null fields in tableB row with null fields in tableA Like: a.id a.name b.id b.name 5 Peter 5 Jones 2 Steven 2 Pareker 6 Paul null null 4 Ivan null null null null 1 Smith null null 3 Parker

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