Search Results

Search found 8 results on 1 pages for 'eigenvalue'.

Page 1/1 | 1 

• get eigenvalue pca with java

  - by Muhamad Burhanudin

  I try use PCA to reduce dimention, and i use jama for help me using matrix. but, i got problem when get eigenvalue with jama. for example i hava 2 image dimention 100x100, then i create single matrix 2 image x (100x100). there is 20.000 pixel. and how to get reduction with eigenvalue? this is sample my code : public static void main(String[] args) { BufferedImage bi; int[] rgb; int R, G, B; // int[] jum; double[][] gray = new double[500][500] ; String[] baris = new String[1000]; try { //bi = ImageIO.read(new File("D:\\c.jpg")); int[][] pixelData = new int[bi.getHeight() * bi.getWidth()][3]; int counter = 0; for (int i = 0; i < bi.getHeight(); i++) { for (int j = 0; j < bi.getWidth(); j++) { gray[i][j] = getPixelData(bi, i, j); // R = getR(bi, i, j); //G = getG(bi, i, j); //B = getB(bi, i, j); //jum = R + G + B; // gray[counter] = Double.toString(R + G + B / 3); // System.out.println("Gray " +gray); //for (int k = 0; k < rgb.length; k++) { // pixelData[counter][k] = rgb[k]; // } counter++; } } } catch (IOException e) { e.printStackTrace(); } Matrix matrix = new Matrix(gray); PCA pca = new PCA(matrix); pca.getEigenvalue(6); String n = pca.toString(); System.err.println("nilai n "+n); //double dete = pcadete(matrix,3600); } private static int getPixelData(BufferedImage bi, int x, int y) { int argb = bi.getRGB(y, x); int r, g, b; int gray; int rgb[] = new int[]{ (argb >> 16) & 0xff, //red (argb >> 8) & 0xff, //green (argb) & 0xff //blue }; r = rgb[0]; g = rgb[1]; b = rgb[2]; gray = (r + g + b) / 3; System.out.println("gray: " + gray); return gray; } when i show eigenvalue in this code : PCA pca = new PCA(matrix); pca.getEigenvalue(6); String n = pca.toString(); System.err.println("nilai n "+n); Result is : nilai n PCA@c3e9e9 Can, u tell me what way to get eigenvalue and reduction dimension.

  Read the article

• Sorting eigenvectors by their eigenvalues (associated sorting)

  - by fbrereto

  I have an unsorted vector of eigenvalues and a related matrix of eigenvectors. I'd like to sort the columns of the matrix with respect to the sorted set of eigenvalues. (e.g., if eigenvalue[3] moves to eigenvalue[2], I want column 3 of the eigenvector matrix to move over to column 2.) I know I can sort the eigenvalues in O(N log N) via std::sort. Without rolling my own sorting algorithm, how do I make sure the matrix's columns (the associated eigenvectors) follow along with their eigenvalues as the latter are sorted?

  Read the article

• Tool to diagonalize large matrices

  - by Xodarap

  I want to compute a diffusion kernel, which involves taking exp(b*A) where A is a large matrix. In order to play with values of b, I'd like to diagonalize A (so that exp(A) runs quickly). My matrix is about 25k x 25k, but is very sparse - only about 60k values are non-zero. Matlab's "eigs" function runs of out memory, as does octave's "eig" and R's "eigen." Is there a tool to find the decomposition of large, sparse matrices? Dunno if this is relevant, but A is an adjacency matrix, so it's symmetric, and it is full rank.

  Read the article

• 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.

  Read the article

• 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

  Read the article

• linear algebra libraries for clusters

  - by Abruzzo Forte e Gentile

  Hi all I need to develop applications doing linear algebra + eigenvalue + linear equation solutions over a cluster of pcs ( I have a lot of machines available ). I discovered Scalapack libraries but they seem to me developed long time ago. Do you know if these are other libs available that I should learn doing math & linear algebra in a cluster? My language is C++ and off course I am newbie to this topic. Kind Regards to everybody AFG

  Read the article

• How to store generated eigen faces for future face recognition?

  - by user3237134

  My code works in the following manner: 1.First, it obtains several images from the training set 2.After loading these images, we find the normalized faces,mean face and perform several calculation. 3.Next, we ask for the name of an image we want to recognize 4.We then project the input image into the eigenspace, and based on the difference from the eigenfaces we make a decision. 5.Depending on eigen weight vector for each input image we make clusters using kmeans command. Source code i tried: clear all close all clc % number of images on your training set. M=1200; %Chosen std and mean. %It can be any number that it is close to the std and mean of most of the images. um=60; ustd=32; %read and show images(bmp); S=[]; %img matrix for i=1:M str=strcat(int2str(i),'.jpg'); %concatenates two strings that form the name of the image eval('img=imread(str);'); [irow icol d]=size(img); % get the number of rows (N1) and columns (N2) temp=reshape(permute(img,[2,1,3]),[irow*icol,d]); %creates a (N1*N2)x1 matrix S=[S temp]; %X is a N1*N2xM matrix after finishing the sequence %this is our S end %Here we change the mean and std of all images. We normalize all images. %This is done to reduce the error due to lighting conditions. for i=1:size(S,2) temp=double(S(:,i)); m=mean(temp); st=std(temp); S(:,i)=(temp-m)*ustd/st+um; end %show normalized images for i=1:M str=strcat(int2str(i),'.jpg'); img=reshape(S(:,i),icol,irow); img=img'; end %mean image; m=mean(S,2); %obtains the mean of each row instead of each column tmimg=uint8(m); %converts to unsigned 8-bit integer. Values range from 0 to 255 img=reshape(tmimg,icol,irow); %takes the N1*N2x1 vector and creates a N2xN1 matrix img=img'; %creates a N1xN2 matrix by transposing the image. % Change image for manipulation dbx=[]; % A matrix for i=1:M temp=double(S(:,i)); dbx=[dbx temp]; end %Covariance matrix C=A'A, L=AA' A=dbx'; L=A*A'; % vv are the eigenvector for L % dd are the eigenvalue for both L=dbx'*dbx and C=dbx*dbx'; [vv dd]=eig(L); % Sort and eliminate those whose eigenvalue is zero v=[]; d=[]; for i=1:size(vv,2) if(dd(i,i)>1e-4) v=[v vv(:,i)]; d=[d dd(i,i)]; end end %sort, will return an ascending sequence [B index]=sort(d); ind=zeros(size(index)); dtemp=zeros(size(index)); vtemp=zeros(size(v)); len=length(index); for i=1:len dtemp(i)=B(len+1-i); ind(i)=len+1-index(i); vtemp(:,ind(i))=v(:,i); end d=dtemp; v=vtemp; %Normalization of eigenvectors for i=1:size(v,2) %access each column kk=v(:,i); temp=sqrt(sum(kk.^2)); v(:,i)=v(:,i)./temp; end %Eigenvectors of C matrix u=[]; for i=1:size(v,2) temp=sqrt(d(i)); u=[u (dbx*v(:,i))./temp]; end %Normalization of eigenvectors for i=1:size(u,2) kk=u(:,i); temp=sqrt(sum(kk.^2)); u(:,i)=u(:,i)./temp; end % show eigenfaces; for i=1:size(u,2) img=reshape(u(:,i),icol,irow); img=img'; img=histeq(img,255); end % Find the weight of each face in the training set. omega = []; for h=1:size(dbx,2) WW=[]; for i=1:size(u,2) t = u(:,i)'; WeightOfImage = dot(t,dbx(:,h)'); WW = [WW; WeightOfImage]; end omega = [omega WW]; end % Acquire new image % Note: the input image must have a bmp or jpg extension. % It should have the same size as the ones in your training set. % It should be placed on your desktop ed_min=[]; srcFiles = dir('G:\newdatabase\*.jpg'); % the folder in which ur images exists for b = 1 : length(srcFiles) filename = strcat('G:\newdatabase\',srcFiles(b).name); Imgdata = imread(filename); InputImage=Imgdata; InImage=reshape(permute((double(InputImage)),[2,1,3]),[irow*icol,1]); temp=InImage; me=mean(temp); st=std(temp); temp=(temp-me)*ustd/st+um; NormImage = temp; Difference = temp-m; p = []; aa=size(u,2); for i = 1:aa pare = dot(NormImage,u(:,i)); p = [p; pare]; end InImWeight = []; for i=1:size(u,2) t = u(:,i)'; WeightOfInputImage = dot(t,Difference'); InImWeight = [InImWeight; WeightOfInputImage]; end noe=numel(InImWeight); % Find Euclidean distance e=[]; for i=1:size(omega,2) q = omega(:,i); DiffWeight = InImWeight-q; mag = norm(DiffWeight); e = [e mag]; end ed_min=[ed_min MinimumValue]; theta=6.0e+03; %disp(e) z(b,:)=InImWeight; end IDX = kmeans(z,5); clustercount=accumarray(IDX, ones(size(IDX))); disp(clustercount); QUESTIONS: 1.It is working fine for M=50(i.e Training set contains 50 images) but not for M=1200(i.e Training set contains 1200 images).It is not showing any error.There is no output.I waited for 10 min still there is no output. I think it is going infinite loop.What is the problem?Where i was wrong? 2.Instead of running the training set everytime how eigen faces generated are stored so that stored eigen faces are used for future face recoginition for a new input image.So it reduces wastage of time.

  Read the article

• Vectorization of matlab code for faster execution

  - by user3237134

  My code works in the following manner: 1.First, it obtains several images from the training set 2.After loading these images, we find the normalized faces,mean face and perform several calculation. 3.Next, we ask for the name of an image we want to recognize 4.We then project the input image into the eigenspace, and based on the difference from the eigenfaces we make a decision. 5.Depending on eigen weight vector for each input image we make clusters using kmeans command. Source code i tried: clear all close all clc % number of images on your training set. M=1200; %Chosen std and mean. %It can be any number that it is close to the std and mean of most of the images. um=60; ustd=32; %read and show images(bmp); S=[]; %img matrix for i=1:M str=strcat(int2str(i),'.jpg'); %concatenates two strings that form the name of the image eval('img=imread(str);'); [irow icol d]=size(img); % get the number of rows (N1) and columns (N2) temp=reshape(permute(img,[2,1,3]),[irow*icol,d]); %creates a (N1*N2)x1 matrix S=[S temp]; %X is a N1*N2xM matrix after finishing the sequence %this is our S end %Here we change the mean and std of all images. We normalize all images. %This is done to reduce the error due to lighting conditions. for i=1:size(S,2) temp=double(S(:,i)); m=mean(temp); st=std(temp); S(:,i)=(temp-m)*ustd/st+um; end %show normalized images for i=1:M str=strcat(int2str(i),'.jpg'); img=reshape(S(:,i),icol,irow); img=img'; end %mean image; m=mean(S,2); %obtains the mean of each row instead of each column tmimg=uint8(m); %converts to unsigned 8-bit integer. Values range from 0 to 255 img=reshape(tmimg,icol,irow); %takes the N1*N2x1 vector and creates a N2xN1 matrix img=img'; %creates a N1xN2 matrix by transposing the image. % Change image for manipulation dbx=[]; % A matrix for i=1:M temp=double(S(:,i)); dbx=[dbx temp]; end %Covariance matrix C=A'A, L=AA' A=dbx'; L=A*A'; % vv are the eigenvector for L % dd are the eigenvalue for both L=dbx'*dbx and C=dbx*dbx'; [vv dd]=eig(L); % Sort and eliminate those whose eigenvalue is zero v=[]; d=[]; for i=1:size(vv,2) if(dd(i,i)>1e-4) v=[v vv(:,i)]; d=[d dd(i,i)]; end end %sort, will return an ascending sequence [B index]=sort(d); ind=zeros(size(index)); dtemp=zeros(size(index)); vtemp=zeros(size(v)); len=length(index); for i=1:len dtemp(i)=B(len+1-i); ind(i)=len+1-index(i); vtemp(:,ind(i))=v(:,i); end d=dtemp; v=vtemp; %Normalization of eigenvectors for i=1:size(v,2) %access each column kk=v(:,i); temp=sqrt(sum(kk.^2)); v(:,i)=v(:,i)./temp; end %Eigenvectors of C matrix u=[]; for i=1:size(v,2) temp=sqrt(d(i)); u=[u (dbx*v(:,i))./temp]; end %Normalization of eigenvectors for i=1:size(u,2) kk=u(:,i); temp=sqrt(sum(kk.^2)); u(:,i)=u(:,i)./temp; end % show eigenfaces; for i=1:size(u,2) img=reshape(u(:,i),icol,irow); img=img'; img=histeq(img,255); end % Find the weight of each face in the training set. omega = []; for h=1:size(dbx,2) WW=[]; for i=1:size(u,2) t = u(:,i)'; WeightOfImage = dot(t,dbx(:,h)'); WW = [WW; WeightOfImage]; end omega = [omega WW]; end % Acquire new image % Note: the input image must have a bmp or jpg extension. % It should have the same size as the ones in your training set. % It should be placed on your desktop ed_min=[]; srcFiles = dir('G:\newdatabase\*.jpg'); % the folder in which ur images exists for b = 1 : length(srcFiles) filename = strcat('G:\newdatabase\',srcFiles(b).name); Imgdata = imread(filename); InputImage=Imgdata; InImage=reshape(permute((double(InputImage)),[2,1,3]),[irow*icol,1]); temp=InImage; me=mean(temp); st=std(temp); temp=(temp-me)*ustd/st+um; NormImage = temp; Difference = temp-m; p = []; aa=size(u,2); for i = 1:aa pare = dot(NormImage,u(:,i)); p = [p; pare]; end InImWeight = []; for i=1:size(u,2) t = u(:,i)'; WeightOfInputImage = dot(t,Difference'); InImWeight = [InImWeight; WeightOfInputImage]; end noe=numel(InImWeight); % Find Euclidean distance e=[]; for i=1:size(omega,2) q = omega(:,i); DiffWeight = InImWeight-q; mag = norm(DiffWeight); e = [e mag]; end ed_min=[ed_min MinimumValue]; theta=6.0e+03; %disp(e) z(b,:)=InImWeight; end IDX = kmeans(z,5); clustercount=accumarray(IDX, ones(size(IDX))); disp(clustercount); Running time for 50 images:Elapsed time is 103.947573 seconds. QUESTIONS: 1.It is working fine for M=50(i.e Training set contains 50 images) but not for M=1200(i.e Training set contains 1200 images).It is not showing any error.There is no output.I waited for 10 min still there is no output. I think it is going infinite loop.What is the problem?Where i was wrong?

  Read the article

1