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  • Extreme Optimization – Numerical Algorithm Support

    - by JoshReuben
    Function Delegates Many calculations involve the repeated evaluation of one or more user-supplied functions eg Numerical integration. The EO MathLib provides delegate types for common function signatures and the FunctionFactory class can generate new delegates from existing ones. RealFunction delegate - takes one Double parameter – can encapsulate most of the static methods of the System.Math class, as well as the classes in the Extreme.Mathematics.SpecialFunctions namespace: var sin = new RealFunction(Math.Sin); var result = sin(1); BivariateRealFunction delegate - takes two Double parameters: var atan2 = new BivariateRealFunction (Math.Atan2); var result = atan2(1, 2); TrivariateRealFunction delegate – represents a function takes three Double arguments ParameterizedRealFunction delegate - represents a function taking one Integer and one Double argument that returns a real number. The Pow method implements such a function, but the arguments need order re-arrangement: static double Power(int exponent, double x) { return ElementaryFunctions.Pow(x, exponent); } ... var power = new ParameterizedRealFunction(Power); var result = power(6, 3.2); A ComplexFunction delegate - represents a function that takes an Extreme.Mathematics.DoubleComplex argument and also returns a complex number. MultivariateRealFunction delegate - represents a function that takes an Extreme.Mathematics.LinearAlgebra.Vector argument and returns a real number. MultivariateVectorFunction delegate - represents a function that takes a Vector argument and returns a Vector. FastMultivariateVectorFunction delegate - represents a function that takes an input Vector argument and an output Matrix argument – avoiding object construction  The FunctionFactory class RealFromBivariateRealFunction and RealFromParameterizedRealFunction helper methods - transform BivariateRealFunction or a ParameterizedRealFunction into a RealFunction delegate by fixing one of the arguments, and treating this as a new function of a single argument. var tenthPower = FunctionFactory.RealFromParameterizedRealFunction(power, 10); var result = tenthPower(x); Note: There is no direct way to do this programmatically in C# - in F# you have partial value functions where you supply a subset of the arguments (as a travelling closure) that the function expects. When you omit arguments, F# generates a new function that holds onto/remembers the arguments you passed in and "waits" for the other parameters to be supplied. let sumVals x y = x + y     let sumX = sumVals 10     // Note: no 2nd param supplied.     // sumX is a new function generated from partially applied sumVals.     // ie "sumX is a partial application of sumVals." let sum = sumX 20     // Invokes sumX, passing in expected int (parameter y from original)  val sumVals : int -> int -> int val sumX : (int -> int) val sum : int = 30 RealFunctionsToVectorFunction and RealFunctionsToFastVectorFunction helper methods - combines an array of delegates returning a real number or a vector into vector or matrix functions. The resulting vector function returns a vector whose components are the function values of the delegates in the array. var funcVector = FunctionFactory.RealFunctionsToVectorFunction(     new MultivariateRealFunction(myFunc1),     new MultivariateRealFunction(myFunc2));  The IterativeAlgorithm<T> abstract base class Iterative algorithms are common in numerical computing - a method is executed repeatedly until a certain condition is reached, approximating the result of a calculation with increasing accuracy until a certain threshold is reached. If the desired accuracy is achieved, the algorithm is said to converge. This base class is derived by many classes in the Extreme.Mathematics.EquationSolvers and Extreme.Mathematics.Optimization namespaces, as well as the ManagedIterativeAlgorithm class which contains a driver method that manages the iteration process.  The ConvergenceTest abstract base class This class is used to specify algorithm Termination , convergence and results - calculates an estimate for the error, and signals termination of the algorithm when the error is below a specified tolerance. Termination Criteria - specify the success condition as the difference between some quantity and its actual value is within a certain tolerance – 2 ways: absolute error - difference between the result and the actual value. relative error is the difference between the result and the actual value relative to the size of the result. Tolerance property - specify trade-off between accuracy and execution time. The lower the tolerance, the longer it will take for the algorithm to obtain a result within that tolerance. Most algorithms in the EO NumLib have a default value of MachineConstants.SqrtEpsilon - gives slightly less than 8 digits of accuracy. ConvergenceCriterion property - specify under what condition the algorithm is assumed to converge. Using the ConvergenceCriterion enum: WithinAbsoluteTolerance / WithinRelativeTolerance / WithinAnyTolerance / NumberOfIterations Active property - selectively ignore certain convergence tests Error property - returns the estimated error after a run MaxIterations / MaxEvaluations properties - Other Termination Criteria - If the algorithm cannot achieve the desired accuracy, the algorithm still has to end – according to an absolute boundary. Status property - indicates how the algorithm terminated - the AlgorithmStatus enum values:NoResult / Busy / Converged (ended normally - The desired accuracy has been achieved) / IterationLimitExceeded / EvaluationLimitExceeded / RoundOffError / BadFunction / Divergent / ConvergedToFalseSolution. After the iteration terminates, the Status should be inspected to verify that the algorithm terminated normally. Alternatively, you can set the ThrowExceptionOnFailure to true. Result property - returns the result of the algorithm. This property contains the best available estimate, even if the desired accuracy was not obtained. IterationsNeeded / EvaluationsNeeded properties - returns the number of iterations required to obtain the result, number of function evaluations.  Concrete Types of Convergence Test classes SimpleConvergenceTest class - test if a value is close to zero or very small compared to another value. VectorConvergenceTest class - test convergence of vectors. This class has two additional properties. The Norm property specifies which norm is to be used when calculating the size of the vector - the VectorConvergenceNorm enum values: EuclidianNorm / Maximum / SumOfAbsoluteValues. The ErrorMeasure property specifies how the error is to be measured – VectorConvergenceErrorMeasure enum values: Norm / Componentwise ConvergenceTestCollection class - represent a combination of tests. The Quantifier property is a ConvergenceTestQuantifier enum that specifies how the tests in the collection are to be combined: Any / All  The AlgorithmHelper Class inherits from IterativeAlgorithm<T> and exposes two methods for convergence testing. IsValueWithinTolerance<T> method - determines whether a value is close to another value to within an algorithm's requested tolerance. IsIntervalWithinTolerance<T> method - determines whether an interval is within an algorithm's requested tolerance.

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  • Indexing with pointer C/C++

    - by Leavenotrace
    Hey I'm trying to write a program to carry out newtons method and find the roots of the equation exp(-x)-(x^2)+3. It works in so far as finding the root, but I also want it to print out the root after each iteration but I can't get it to work, Could anyone point out my mistake I think its something to do with my indexing? Thanks a million :) #include <stdio.h> #include <math.h> #include <malloc.h> //Define Functions: double evalf(double x) { double answer=exp(-x)-(x*x)+3; return(answer); } double evalfprime(double x) { double answer=-exp(-x)-2*x; return(answer); } double *newton(double initialrt,double accuracy,double *data) { double root[102]; data=root; int maxit = 0; root[0] = initialrt; for (int i=1;i<102;i++) { *(data+i)=*(data+i-1)-evalf(*(data+i-1))/evalfprime(*(data+i-1)); if(fabs(*(data+i)-*(data+i-1))<accuracy) { maxit=i; break; } maxit=i; } if((maxit+1==102)&&(fabs(*(data+maxit)-*(data+maxit-1))>accuracy)) { printf("\nMax iteration reached, method terminated"); } else { printf("\nMethod successful"); printf("\nNumber of iterations: %d\nRoot Estimate: %lf\n",maxit+1,*(data+maxit)); } return(data); } int main() { double root,accuracy; double *data=(double*)malloc(sizeof(double)*102); printf("NEWTONS METHOD PROGRAMME:\nEquation: f(x)=exp(-x)-x^2+3=0\nMax No iterations=100\n\nEnter initial root estimate\n>> "); scanf("%lf",&root); _flushall(); printf("\nEnter accuracy required:\n>>"); scanf("%lf",&accuracy); *data= *newton(root,accuracy,data); printf("Iteration Root Error\n "); printf("%d %lf \n", 0,*(data)); for(int i=1;i<102;i++) { printf("%d %5.5lf %5.5lf\n", i,*(data+i),*(data+i)-*(data+i-1)); if(*(data+i*sizeof(double))-*(data+i*sizeof(double)-1)==0) { break; } } getchar(); getchar(); free(data); return(0); }

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  • whats the diference between train, validation and test set, in neural networks?

    - by Daniel
    Im using this library http://pastebin.com/raw.php?i=aMtVv4RZ to implement a learning agent. I have generated the train cases, but i dont know for sure what are the validation and test sets, the teacher says: 70% should be train cases, 10% will be test cases and the rest 20% should be validation cases. Thanks. edit i have this code, for training.. but i have no ideia when to stop training.. def train(self, train, validation, N=0.3, M=0.1): # N: learning rate # M: momentum factor accuracy = list() while(True): error = 0.0 for p in train: input, target = p self.update(input) error = error + self.backPropagate(target, N, M) print "validation" total = 0 for p in validation: input, target = p output = self.update(input) total += sum([abs(target - output) for target, output in zip(target, output)]) #calculates sum of absolute diference between target and output accuracy.append(total) print min(accuracy) print sum(accuracy[-5:])/5 #if i % 100 == 0: print 'error %-14f' % error if ? < ?: break

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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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  • Which of these algorithms is best for my goal?

    - by JonathonG
    I have created a program that restricts the mouse to a certain region based on a black/white bitmap. The program is 100% functional as-is, but uses an inaccurate, albeit fast, algorithm for repositioning the mouse when it strays outside the area. Currently, when the mouse moves outside the area, basically what happens is this: A line is drawn between a pre-defined static point inside the region and the mouse's new position. The point where that line intersects the edge of the allowed area is found. The mouse is moved to that point. This works, but only works perfectly for a perfect circle with the pre-defined point set in the exact center. Unfortunately, this will never be the case. The application will be used with a variety of rectangles and irregular, amorphous shapes. On such shapes, the point where the line drawn intersects the edge will usually not be the closest point on the shape to the mouse. I need to create a new algorithm that finds the closest point to the mouse's new position on the edge of the allowed area. I have several ideas about this, but I am not sure of their validity, in that they may have far too much overhead. While I am not asking for code, it might help to know that I am using Objective C / Cocoa, developing for OS X, as I feel the language being used might affect the efficiency of potential methods. My ideas are: Using a bit of trigonometry to project lines would work, but that would require some kind of intense algorithm to test every point on every line until it found the edge of the region... That seems too resource intensive since there could be something like 200 lines that would have each have to have as many as 200 pixels checked for black/white.... Using something like an A* pathing algorithm to find the shortest path to a black pixel; however, A* seems resource intensive, even though I could probably restrict it to only checking roughly in one direction. It also seems like it will take more time and effort than I have available to spend on this small portion of the much larger project I am working on, correct me if I am wrong and it would not be a significant amount of code (100 lines or around there). Mapping the border of the region before the application begins running the event tap loop. I think I could accomplish this by using my current line-based algorithm to find an edge point and then initiating an algorithm that checks all 8 pixels around that pixel, finds the next border pixel in one direction, and continues to do this until it comes back to the starting pixel. I could then store that data in an array to be used for the entire duration of the program, and have the mouse re-positioning method check the array for the closest pixel on the border to the mouse target position. That last method would presumably execute it's initial border mapping fairly quickly. (It would only have to map between 2,000 and 8,000 pixels, which means 8,000 to 64,000 checked, and I could even permanently store the data to make launching faster.) However, I am uncertain as to how much overhead it would take to scan through that array for the shortest distance for every single mouse move event... I suppose there could be a shortcut to restrict the number of elements in the array that will be checked to a variable number starting with the intersecting point on the line (from my original algorithm), and raise/lower that number to experiment with the overhead/accuracy tradeoff. Please let me know if I am over thinking this and there is an easier way that will work just fine, or which of these methods would be able to execute something like 30 times per second to keep mouse movement smooth, or if you have a better/faster method. I've posted relevant parts of my code below for reference, and included an example of what the area might look like. (I check for color value against a loaded bitmap that is black/white.) // // This part of my code runs every single time the mouse moves. // CGPoint point = CGEventGetLocation(event); float tX = point.x; float tY = point.y; if( is_in_area(tX,tY, mouse_mask)){ // target is inside O.K. area, do nothing }else{ CGPoint target; //point inside restricted region: float iX = 600; // inside x float iY = 500; // inside y // delta to midpoint between iX,iY and tX,tY float dX; float dY; float accuracy = .5; //accuracy to loop until reached do { dX = (tX-iX)/2; dY = (tY-iY)/2; if(is_in_area((tX-dX),(tY-dY),mouse_mask)){ iX += dX; iY += dY; } else { tX -= dX; tY -= dY; } } while (abs(dX)>accuracy || abs(dY)>accuracy); target = CGPointMake(roundf(tX), roundf(tY)); CGDisplayMoveCursorToPoint(CGMainDisplayID(),target); } Here is "is_in_area(int x, int y)" : bool is_in_area(NSInteger x, NSInteger y, NSBitmapImageRep *mouse_mask){ NSAutoreleasePool * pool = [[NSAutoreleasePool alloc] init]; NSUInteger pixel[4]; [mouse_mask getPixel:pixel atX:x y:y]; if(pixel[0]!= 0){ [pool release]; return false; } [pool release]; return true; }

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  • 5 Ways to Determine Mobile Location

    - by David Dorf
    In my previous post, I mentioned the importance of determining the location of a consumer using their mobile phone.  Retailers can track anonymous mobile phones to determine traffic patterns both inside and outside their stores.  And with consumers' permission, retailers can send location-aware offers to mobile phones; for example, a coupon for cereal as you walk down that aisle.  When paying with Square, your location is matched with the transaction.  So there are lots of reasons for retailers to want to know the location of their customers.  But how is it done? I thought I'd dive a little deeper on that topic and consider the approaches to determining location. 1. Tower Triangulation By comparing the relative signal strength from multiple antenna towers, a general location of a phone can be roughly determined to an accuracy of 200-1000 meters.  The more towers involved, the more accurate the location. 2. GPS Using Global Positioning Satellites is more accurate than using cell towers, but it takes longer to find the satellites, it uses more battery, and it won't well indoors.  For geo-fencing applications, like those provided by Placecast and Digby, cell towers are often used to determine if the consumer is nearing a "fence" then switches to GPS to determine the actual crossing of the fence. 3. WiFi Triangulation WiFi triangulation is usually more accurate than using towers just because there are so many more WiFi access points (i.e. radios in routers) around. The position of each WiFi AP needs to be recorded in a database and used in the calculations, which is what Skyhook has been doing since 2008.  Another advantage to this method is that works well indoors, although it usually requires additional WiFi beacons to get the accuracy down to 5-10 meters.  Companies like ZuluTime, Aisle411, and PointInside have been perfecting this approach for retailers like Meijer, Walgreens, and HomeDepot. Keep in mind that a mobile phone doesn't have to connect to the WiFi network in order for it to be located.  The WiFi radio in the phone only needs to be on.  Even when not connected, WiFi radios talk to each other to prepare for a possible connection. 4. Hybrid Approaches Naturally the most accurate approach is to combine the approaches described above.  The more available data points, the greater the accuracy.  Companies like ShopKick like to add in acoustic triangulation using the phone's microphone, and NearBuy can use video analytics to increase accuracy. 5. Magnetic Fields The latest approach, and this one is really new, takes a page from the animal kingdom.  As you've probably learned from guys like Marlin Perkins, some animals use the Earth's magnetic fields to navigate.  By recording magnetic variations within a store, then matching those readings with ones from a consumer's phone, location can be accurately determined.  At least that's the approach IndoorAtlas is taking, and the science seems to bear out.  It works well indoors, and doesn't require retailers to purchase any additional hardware.  Keep an eye on this one.

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  • Issues in Convergence of Sequential minimal optimization for SVM

    - by Amol Joshi
    I have been working on Support Vector Machine for about 2 months now. I have coded SVM myself and for the optimization problem of SVM, I have used Sequential Minimal Optimization(SMO) by Mr. John Platt. Right now I am in the phase where I am going to grid search to find optimal C value for my dataset. ( Please find details of my project application and dataset details here http://stackoverflow.com/questions/2284059/svm-classification-minimum-number-of-input-sets-for-each-class) I have successfully checked my custom implemented SVM`s accuracy for C values ranging from 2^0 to 2^6. But now I am having some issues regarding the convergence of the SMO for C 128. Like I have tried to find the alpha values for C=128 and it is taking long time before it actually converges and successfully gives alpha values. Time taken for the SMO to converge is about 5 hours for C=100. This huge I think ( because SMO is supposed to be fast. ) though I`m getting good accuracy? I am screwed right not because I can not test the accuracy for higher values of C. I am actually displaying number of alphas changed in every pass of SMO and getting 10, 13, 8... alphas changing continuously. The KKT conditions assures convergence so what is so weird happening here? Please note that my implementation is working fine for C<=100 with good accuracy though the execution time is long. Please give me inputs on this issue. Thank You and Cheers.

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  • Function approximation with Maclaurin series

    - by marines
    I need to approx (1-x)^0.25 with given accuracy (0.0001 e.g.). I'm using expansion found on Wikipedia for (1+x)^0.25. I need to stop approximating when current expression is less than the accuracy. long double s(long double x, long double d) { long double w = 1; long double n = 1; // nth expression in series long double tmp = 1; // sum while last expression is greater than accuracy while (fabsl(tmp) >= d) { tmp *= (1.25 / n - 1) * (-x); // the next expression w += tmp; // is added to approximation n++; } return w; } Don't mind long double n. :P This works well when I'm not checking value of current expression but when I'm computing 1000 or more expressions. Domain of the function is <-1;1 and s() calculates approximation well for x in <-1;~0.6. The bigger the argument is the bigger is the error of calculation. From 0.6 it exceeds the accuracy. I'm not sure if the problem is clear enough because I don't know English math language well. The thing is what's the matter with while condition and why the function s() doesn't approximate correctly.

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  • WebCenter Customer Spotlight: Alberta Agriculture and Rural Developmen

    - by me
    Author: Peter Reiser - Social Business Evangelist, Oracle WebCenter  Solution SummaryAlberta Agriculture and Rural Development is a government ministry that works with producers and consumers to create a strong, competitive, and sustainable agriculture and food industry in the province of Alberta, Canada The primary business challenge faced by the Alberta Ministry of Agriculture was that of managing the rapid growth of their information.  They needed to incorporate a system that would work across 22 different divisions within the ministry and deliver an improved and more efficient experience for Desktop, Web and Mobile users, while addressing their regulatory compliance needs as part of the Canadian government. The customer implemented a centralized Enterprise Content Management solution based on Oracle WebCenter Content and developed a strong and repeatable information life cycle management methodology across all their 22 divisions and agencies. With the implemented solution, Alberta Agriculture and Rural Development  centrally manages over 20 million documents for 22 divisions and agencies and they have improved time required to find records,  reliability of information, improved speed and accuracy of reporting and data security. Company OverviewAlberta Agriculture and Rural Development is a government ministry that works with producers and consumers to create a strong, competitive, and sustainable agriculture and food industry in the province of Alberta, Canada.  Business ChallengesThe business users were overwhelmed by growth in documents (over 20 million files across 22 divisions and agencies) and it was difficult to find and manage documents and versions. There was a strong need for a personalized easy-to-use, secure and dependable method of managing and consuming content via desktop, Web, and mobile, while improving efficiency and maintaining regulatory compliance by removing the risk of non-uniform approaches to retention and disposition. Solution DeployedAs a first step Alberta Agriculture and Rural Development developed a business case with clear defined business drivers: Reduce time required to find records Locate “lost” records Capture knowledge lost through attrition Increase the ease of retrieval Reduce personal copies Increase reliability of information Improve speed and accuracy of reporting Improve data security The customer implemented a centralized Enterprise Content Management solution based on Oracle WebCenter Content. They used an incremental implementation approach aligned with their divisional and agency structure which allowed continuous process improvement. This led to a very strong and repeatable information life cycle management methodology across all their 22 divisions and agencies. Business ResultsAlberta Agriculture and Rural Development achieved impressive business results: Centrally managing over 20 million files for 22 divisions and agencies Federated model to manage documents in SharePoint and other applications Doing records management for both paper and electronic records Reduced time required to find records Increased the ease of retrieval Increased reliability of information Improved speed and accuracy of reporting Improved data security Additional Information Oracle Open World 2012 Presentation Oracle WebCenter Content

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  • Getting started with object detection - Image segmentation algorithm

    - by Dev Kanchen
    Just getting started on a hobby object-detection project. My aim is to understand the underlying algorithms and to this end the overall accuracy of the results is (currently) more important than actual run-time. I'm starting with trying to find a good image segmentation algorithm that provide a good jump-off point for the object detection phase. The target images would be "real-world" scenes. I found two techniques which mirrored my thoughts on how to go about this: Graph-based Image Segmentation: http://www.cs.cornell.edu/~dph/papers/seg-ijcv.pdf Contour and Texture Analysis for Image Segmentation: http://www.eng.utah.edu/~bresee/compvision/files/MalikBLS.pdf The first one was really intuitive to understand and seems simple enough to implement, while the second was closer to my initial thoughts on how to go about this (combine color/intensity and texture information to find regions). But it's an order of magnitude more complex (at least for me). My question is - are there any other algorithms I should be looking at that provide the kind of results that these two, specific papers have arrived at. Are there updated versions of these techniques already floating around. Like I mentioned earlier, the goal is relative accuracy of image segmentation (with an eventual aim to achieve a degree of accuracy of object detection) over runtime, with the algorithm being able to segment an image into "naturally" or perceptually important components, as these two algorithms do (each to varying extents). Thanks! P.S.1: I found these two papers after a couple of days of refining my search terms and learning new ones relevant to the exact kind of techniques I was looking for. :) I have just about reached the end of my personal Google creativity, which is why I am finally here! Thanks for the help. P.S.2: I couldn't find good tags for this question. If some relevant ones exist, @mods please add them. P.S.3: I do not know if this is a better fit for cstheory.stackexchange (or even cs.stackexchange). I looked but cstheory seems more appropriate for intricate algorithmic discussions than a broad question like this. Also, I couldn't find any relevant tags there either! But please do move if appropriate.

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  • Geolocation through Android's GPS Provider on a website?

    - by Corey Ogburn
    I'm trying to get the geolocation of the mobile device in a regular website, not a webview of an application or anything native like that. I'm getting a location, but it's highly inaccurate, the accuracy comes back as 3230 or some other outrageous number. I'm assuming that's in meters, either way it's not nearly accurate enough. By comparison, the same webpage on a laptop gets an accuracy of 30-40. My first thought was that it was using the Network Provider instead of the GPS Provider, telling me where I am based on tower location and reach. A little research later I found enableHighAccuracy and set it true in the options that I pass. After including that, I still notice no difference. Here's the test page's HTML/javascript: <html> <head> <script type="text/javascript" src="http://ecn.dev.virtualearth.net/mapcontrol/mapcontrol.ashx?v=7.0"></script> <script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.4.3/jquery.min.js"></script> <script type="text/javascript"> function OnLoad() { $("#Status").text("Init"); if (navigator.geolocation) { $("#Status").text("Supports Geolocation"); navigator.geolocation.getCurrentPosition(HandleLocation, LocationError, { enableHighAccuracy: true }); $("#Status").text("Sent position request..."); } else { $("#Status").text("Doesn't support geolocation"); } } function HandleLocation(position) { $("#Status").text("Received response:"); $("#Position").text("(" + position.coords.latitude + ", " + position.coords.longitude + ") accuracy: " + position.coords.accuracy); var loc = new Microsoft.Maps.Location(position.coords.latitude, position.coords.longitude); GetMap(loc); } function LocationError(error) { switch(error.code) { case error.PERMISSION_DENIED: alert("Location not provided"); break; case error.POSITION_UNAVAILABLE: alert("Current location not available"); break; case error.TIMEOUT: alert("Timeout"); break; default: alert("unknown error"); break; } } function GetMap(loc) { var map = new Microsoft.Maps.Map(document.getElementById("mapDiv"), {credentials: "Aj59meaCR1e7rNgkfQy7j08Pd3mzfP1r04hGesGmLe2a3ZwZ3iGecwPX2SNPWq5a", center: loc, mapTypeId: Microsoft.Maps.MapTypeId.road, zoom: 15}); } </script> </head> <body onload="javascript:OnLoad()"> <div id="Status"></div> <div id="Position"></div><br/> <div id='mapDiv' style="position:relative; width:600px; height:400px;"></div> </body> </html> I'm testing this on a rooted MyTouch 3G running Cyanogen 6.1 stable, Android 2.2 and GPS is enabled. In case rooting was a problem, I have also had various friends and coworkers try the webpage on their non-rooted 2.0+ Android devices. Each phone had various effects on the accuracy, but none were better than 1000, I attribute this to the different carriers. I have not (but eventually will) tested with iPhone or other location-aware cell phones.

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  • Can I have multiple colors in a single TextBlock in WPF?

    - by Siracuse
    I have a line of text in a textblock that reads: "Detected [gesture] with an accuracy of [accuracy]" In WPF, is it possible for me to be able to change the color of the elements within a textblock? Can I have a textblock be multiple colors? For example, I would like the whole TextBlock to be black except the gesture name, which I would like to be red. Is this possible in WPF?

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  • Cepstral Analysis for pitch detection

    - by Ohmu
    Hi! I'm looking to extract pitches from a sound signal. Someone on IRC just explain to me how taking a double FFT achieves this. Specifically: take FFT take log of square of absolute value (can be done with lookup table) take another FFT take absolute value I am attempting this using vDSP I can't understand how I didn't come across this technique earlier. I did a lot of hunting and asking questions; several weeks worth. More to the point, I can't understand why I didn't think of it. I am attempting to achieve this with vDSP library. it looks as though it has functions to handle all of these tasks. However, I'm wondering about the accuracy of the final result. I have previously used a technique which scours the frequency bins of a single FFT for local maxima. when it encounters one, it uses a cunning technique (the change in phase since the last FFT) to more accurately place the actual peak within the bin. I am worried that this precision will be lost with this technique I'm presenting here. I guess the technique could be used after the second FFT to get the fundamental accurately. But it kind of looks like the information is lost in step 2. as this is a potentially tricky process, could someone with some experience just look over what I'm doing and check it for sanity? also, I've heard there is an alternative technique involving fitting a quadratic over neighbouring bins. Is this of comparable accuracy? if so, I would favour it, as it doesn't involve remembering bin phases. so questions: does this approach makes sense? Can it be improved? I'm a bit worried about And the log square component; there seems to be a vDSP function to do exactly that: vDSP_vdbcon however, there is no indication it precalculates a log-table -- I assume it doesn't, as the FFT function requires an explicit pre-calculation function to be called and passed into it. and this function doesn't. Is there some danger of harmonics being picked up? is there any cunning way of making vDSP pull out the maxima, biggest first? Can anyone point me towards some research or literature on this technique? the main question: is it accurate enough? Can the accuracy be improved? I have just been told by an expert that the accuracy IS INDEED not sufficient. Is this the end of the line? Pi PS I get SO annoyed (npi) when I want to create tags, but cannot. :| I have suggested to the maintainers that SO keep track of attempted tags, but I'm sure I was ignored. we need tags for vDSP, accelerate framework, cepstral analysis

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  • MTN WMS Implementation Story

    - by aditya.agarkar
    MTN is Africa's largest cellular phone company serving millions of customers across 21 countries. MTN uses Oracle WMS to manage its distribution activities and its sizzling growth. Just for perspective, since 2004, Africa has been the fastest growing mobile phone market in the world. If you want to know more about MTN and the WMS Project at MTN, a summarized view of MTN WMS project is here. The WMS Project at MTN was presented at Oracle Open World in 2007. The extensive automation at MTN includes interface with Conveyor for item transport, High Speed Sorter for item routing, Put to Light for packing accuracy, ASRS Carousel/Lift for inventory Security and Storage Optimization, Check Weight Scale for shipping accuracy, Automated Carton Erectors for package creation and Automated Carton Labeling. Subsequent to this presentation and their go-live in 2007, the MTN warehouse has scaled new heights. The volume has grown manifolds (as can be expected in a fast growing cellular market). Oracle WMS has been able to scale very well to the increase in volume, just as it was designed to do. Here are a couple of videos that highlight the WMS operations at MTN:  1) Video Interview with Margaretha Theart (Warehouse Manager at MTN) 2) Automation Video at MTN (Hat tip: Syed Imran) Enjoy!

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  • How do I choose the scaling factor of a 3D game world?

    - by concept3d
    I am making a 3D tank game prototype with some physics simulation, am using C++. One of the decisions I need to make is the scale of the game world in relation to reality. For example, I could consider 1 in-game unit of measurement to correspond to 1 meter in reality. This feels intuitive, but I feel like I might be missing something. I can think of the following as potential problems: 3D modelling program compatibility. (?) Numerical accuracy. (Does this matter?) Especially at large scales, how games like Battlefield have huge maps: How don't they lose numerical accuracy if they use 1:1 mapping with real world scale, since floating point representation tend to lose more precision with larger numbers (e.g. with ray casting, physics simulation)? Gameplay. I don't want the movement of units to feel slow or fast while using almost real world values like -9.8 m/s^2 for gravity. (This might be subjective.) Is it ok to scale up/down imported assets or it's best fit with a world with its original scale? Rendering performance. Are large meshes with the same vertex count slower to render? I'm wondering if I should split this into multiple questions...

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  • Javascript: Machine Constants Applicable?

    - by DavidB2013
    I write numerical routines for students of science and engineering (although they are freely available for use by anybody else as well) and am wondering how to properly use machine constants in a JavaScript program, or if they are even applicable. For example, say I am writing a program in C++ that numerically computes the roots of the following equation: exp(-0.7x) + sin(3x) - 1.2x + 0.3546 = 0 A root-finding routine should be able to compute roots to within the machine epsilon. In C++, this value is specified by the language: DBL_EPSILON. C++ also specifies the smallest and largest values that can be held by a float or double variable. However, how does this convert to JavaScript? Since a Javascript program runs in a web browser, and I don't know what kind of computer will run the program, and JavaScript does not have corresponding predefined values for these quantities, how can I implement my own version of these constants so that my programs compute results to as much accuracy as allowed on the computer running the web browser? My first draft is to simply copy over the literal constants from C++: FLT_MIN: 1.17549435082229e-038 FLT_MAX: 3.40282346638529e+038 DBL_EPSILON: 2.2204460492503131e-16 I am also willing to write small code blocks that could compute these values for each machine on which the program is run. That way, a supercomputer might compute results to a higher accuracy than an old, low-level, PC. BUT, I don't know if such a routine would actually reach the computer, in which case, I would be wasting my time. Anybody here know how to compute and use (in Javascript) values that correspond to machine constants in a compiled language? Is it worth my time to write small programs in Javascript that compute DBL_EPSILON, FLT_MIN, FLT_MIN, etc. for use in numerical routines? Or am I better off simply assigning literal constants that come straight from C++ on a standard Windows PC?

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  • Creating Gun objects with upgrades?

    - by zardon
    I have a series of guns in my game. I use the Gun class/object like this: (Just an example) @interface Gun : NSObject { NSString *name; // Six-shooter NSNumber *cost; NSNumber *clipPrice; // ie: 700 NSNumber *clipCapacity; // 6 NSNumber *ammoCapacity; // 6 NSNumber *damage; // 0-10 NSNumber *accuracy; // 0-10 NSNumber *fireRate; // 0-10 NSNumber *range; // 0-10 // Not sure if I have all the stats, but this is fine for now } Lets say I want to have 3 upgrades per gun. My problem is I am not sure how to do this. Examples: increase fire-rate increase range increase accuracy silencer double ammo capacity (ie: Drum) double clip capacity (ie: Taped magazine) Thus my question is, I'd like to implement an upgrade system to guns but I am not sure how to do it. Would there be an Upgrade object which is a child to the Gun class, or would it be seperate class altogether. Thanks for your time.

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  • Join Us!! Live Webinar: Using UPK for Testing

    - by Di Seghposs
    Create Manual Test Scripts 50% Faster with Oracle User Productivity Kit  Thursday, March 29, 2012 11:00 am – 12:00 pm ET Click here to register now for this informative webinar. Oracle UPK enhances the testing phase of the implementation lifecycle by reducing test plan creation time, improving accuracy, and providing the foundation for reusable training documentation, application simulations, and end-user performance support—all critical assets to support an enterprise application implementation. With Oracle UPK: Reduce manual test plan development time - Accelerate the testing cycle by significantly reducing the time required to create the test plan. Improve test plan accuracy - Capture test steps automatically using Oracle UPK and import those steps directly to any of these testing suites eliminating many of the errors that occur when writing manual tests. Create the foundation for reusable assets - Recorded simulations can be used for other lifecycle phases of the project, such as knowledge transfer for training and support. With its integration to Oracle Application Testing Suite, IBM Rational, and HP Quality Center, Oracle UPK allows you to deploy high-quality applications quickly and effectively by providing a consistent, repeatable process for gathering requirements, planning and scheduling tests, analyzing results, and managing  issues. Join this live webinar and learn how to decrease your time to deployment and enhance your testing plans today! 

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  • What is the recommended MongoDB schema for this quiz-engine scenario?

    - by hughesdan
    I'm working on a quiz engine for learning a foreign language. The engine shows users four images simultaneously and then plays an audio file. The user has to match the audio to the correct image. Below is my MongoDB document structure. Each document consists of an image file reference and an array of references to audio files that match that image. To generate a quiz instance I select four documents at random, show the images and then play one audio file from the four documents at random. The next step in my application development is to decide on the best document schema for storing user guesses. There are several requirements to consider: I need to be able to report statistics at a user level. For example, total correct answers, total guesses, mean accuracy, etc) I need to be able to query images based on the user's learning progress. For example, select 4 documents where guess count is 10 and accuracy is <=0.50. The schema needs to be optimized for fast quiz generation. The schema must not cause future scaling issues vis a vis document size. Assume 1mm users who make an average of 1000 guesses. Given all of this as background information, what would be the recommended schema? For example, would you store each guess in the Image document or perhaps in a User document (not shown) or a new document collection created for logging guesses? Would you recommend logging the raw guess data or would you pre-compute statistics by incrementing counters within the relevant document? Schema for Image Collection: _id "505bcc7a45c978be24000005" date 2012-09-21 02:10:02 UTC imageFileName "BD3E134A-C7B3-4405-9004-ED573DF477FE-29879-0000395CF1091601" random 0.26997075392864645 user "2A8761E4-C13A-470E-A759-91432D61B6AF-25982-0000352D853511AF" audioFiles [ 0 { audioFileName "C3669719-9F0A-4EB5-A791-2C00486665ED-30305-000039A3FDA7DCD2" user "2A8761E4-C13A-470E-A759-91432D61B6AF-25982-0000352D853511AF" audioLanguage "English" date 2012-09-22 01:15:04 UTC } 1 { audioFileName "C3669719-9F0A-4EB5-A791-2C00486665ED-30305-000039A3FDA7DCD2" user "2A8761E4-C13A-470E-A759-91432D61B6AF-25982-0000352D853511AF" audioLanguage "Spanish" date 2012-09-22 01:17:04 UTC } ]

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  • How to prevent overlapping of gunshot sounds when using fast-firing weapons

    - by G3tinmybelly
    So I am now trying to find sounds for my guns but when I grab a gun sound effect and play it in my game a lot of the sounds are either terrible sounding or have this horrible echoing effect because as a gun shoots sometimes the previous sound is playing still. public void shoot(float x, float y, float direction){ if(empty){ PlayHUD.message = "No more bullets!"; return; } if(reloading){ return; } if(System.currentTimeMillis() - lastShot < fireRate){ //AssetsLoader.lmgSound.stop(); return; } float dx = (float) (-13 * Math.cos(direction) + 75 * Math.sin(direction)); float dy = (float) (-14 * -Math.sin(direction) + 75 * Math.cos(direction)); float dx1 = (float) (-13 * Math.cos(direction) + 75 * Math.sin(direction)); float dy1 = (float) (-14 * -Math.sin(direction) + 75 * Math.cos(direction)); PlayState.effects.add(new MuzzleFlashEffect(x + dx1, y + dy1, (float) Math.toDegrees(-direction))); PlayState.projectiles.add(new Bullet(this, x + dx, y + dy, (float) (direction + (Math.toRadians(MathUtils.random(-accuracy, accuracy)))))); if(OptionState.soundOn){ AssetsLoader.lmgSound.play(OptionState.volume); } bulletsInClip--; lastShot = System.currentTimeMillis(); } Here is the code for where the sound plays. Every time this method is called the sound is called but it happens so often in this case that there is this terrible echoing. Any idea on how to fix this?

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  • How to gun shots sounds right in game development?

    - by G3tinmybelly
    So I am now trying to find sounds for my guns but when I grab a gun sound effect and play it in my game a lot of the sounds are either terrible sounding or have this horrible echoing effect because as a gun shoots sometimes the previous sound is playing still. public void shoot(float x, float y, float direction){ if(empty){ PlayHUD.message = "No more bullets!"; return; } if(reloading){ return; } if(System.currentTimeMillis() - lastShot < fireRate){ //AssetsLoader.lmgSound.stop(); return; } float dx = (float) (-13 * Math.cos(direction) + 75 * Math.sin(direction)); float dy = (float) (-14 * -Math.sin(direction) + 75 * Math.cos(direction)); float dx1 = (float) (-13 * Math.cos(direction) + 75 * Math.sin(direction)); float dy1 = (float) (-14 * -Math.sin(direction) + 75 * Math.cos(direction)); PlayState.effects.add(new MuzzleFlashEffect(x + dx1, y + dy1, (float) Math.toDegrees(-direction))); PlayState.projectiles.add(new Bullet(this, x + dx, y + dy, (float) (direction + (Math.toRadians(MathUtils.random(-accuracy, accuracy)))))); if(OptionState.soundOn){ AssetsLoader.lmgSound.play(OptionState.volume); } bulletsInClip--; lastShot = System.currentTimeMillis(); } Here is the code for where the sound plays. Every time this method is called the sound is called but it happens so often in this case that there is this terrible echoing. Any idea on how to fix this?

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  • Milliseconds in DateTime.Now on .NET Compact Framework always zero?

    - by Marcel
    Hi all, i want to have a time stamp for logs on a Windows Mobile project. The accuracy must be in the range a hundred milliseconds at least. However my call to DateTime.Now returns a DateTime object with the Millisecond property set to zero. Also the Ticks property is rounded accordingly. How to get better time accuracy? Remember, that my code runs on on the Compact Framework, version 3.5. I use a HTC touch Pro 2 device.

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  • iPhone SDK: CLocationAccuracy. What constants map to what positioning technology?

    - by buzzappsoftware
    With respect to CLocationManager docs.... Constant values you can use to specify the accuracy of a location. extern const CLLocationAccuracy kCLLocationAccuracyBestForNavigation; extern const CLLocationAccuracy kCLLocationAccuracyBest; extern const CLLocationAccuracy kCLLocationAccuracyNearestTenMeters; extern const CLLocationAccuracy kCLLocationAccuracyHundredMeters; extern const CLLocationAccuracy kCLLocationAccuracyKilometer; extern const CLLocationAccuracy kCLLocationAccuracyThreeKilometers; Given that, I have the following questions. What triangulation method (GPS, cell tower or wi-fi) corresponds to each accuracy level? Does iPhone SDK utilize Skyhook Wireless API? For kCLLocationAccuracyBestForNavigation, there is note stating the phone must be plugged in. Is this enforced or is it just warning the developer the battery is likely to drain quick from using the GPS receiver. Thanks in advance.

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  • How was non-decimal money represented in software?

    - by dan04
    A lot of the answers to the questions about the accuracy of float and double recommend the use of decimal for monetary amounts. This works because today all currencies are decimal except MGA and MRO, and those have subunits of 1/5 so are still decimal-friendly. But what about the software used in U.S. stock markets when prices were in 1/16ths of dollar? The accuracy of binary data types wouldn't have been an issue, right? Going further back, how did pre-1971 British accounting software deal with pounds, shillings, and pence? Did their versions of COBOL have a special PIC clause for it? Were all amounts stored in pence? How was decimalisation handled?

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