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  • Point in polygon OR point on polygon using LINQ

    - by wageoghe
    As noted in an earlier question, How to Zip enumerable with itself, I am working on some math algorithms based on lists of points. I am currently working on point in polygon. I have the code for how to do that and have found several good references here on SO, such as this link Hit test. So, I can figure out whether or not a point is in a polygon. As part of determining that, I want to determine if the point is actually on the polygon. This I can also do. If I can do all of that, what is my question you might ask? Can I do it efficiently using LINQ? I can already do something like the following (assuming a Pairwise extension method as described in my earlier question as well as in links to which my question/answers links, and assuming a Position type that has X and Y members). I have not tested much, so the lambda might not be 100% correct. Also, it does not take very small differences into account. public static PointInPolygonLocation PointInPolygon(IEnumerable<Position> pts, Position pt) { int numIntersections = pts.Pairwise( (p1, p2) => { if (p1.Y != p2.Y) { if ((p1.Y >= pt.Y && p2.Y < pt.Y) || (p1.Y < pt.Y && p2.Y >= pt.Y)) { if (p1.X < p1.X && p2.X < pt.X) { return 1; } if (p1.X < pt.X || p2.X < pt.X) { if (((pt.Y - p1.Y) * ((p1.X - p2.X) / (p1.Y - p2.Y)) * p1.X) < pt.X) { return 1; } } } } return 0; }).Sum(); if (numIntersections % 2 == 0) { return PointInPolygonLocation.Outside; } else { return PointInPolygonLocation.Inside; } } This function, PointInPolygon, takes the input Position, pt, iterates over the input sequence of position values, and uses the Jordan Curve method to determine how many times a ray extended from pt to the left intersects the polygon. The lambda expression will yield, into the "zipped" list, 1 for every segment that is crossed, and 0 for the rest. The sum of these values determines if pt is inside or outside of the polygon (odd == inside, even == outside). So far, so good. Now, for any consecutive pairs of position values in the sequence (i.e. in any execution of the lambda), we can also determine if pt is ON the segment p1, p2. If that is the case, we can stop the calculation because we have our answer. Ultimately, my question is this: Can I perform this calculation (maybe using Aggregate?) such that we will only iterate over the sequence no more than 1 time AND can we stop the iteration if we encounter a segment that pt is ON? In other words, if pt is ON the very first segment, there is no need to examine the rest of the segments because we have the answer. It might very well be that this operation (particularly the requirement/desire to possibly stop the iteration early) does not really lend itself well to the LINQ approach. It just occurred to me that maybe the lambda expression could yield a tuple, the intersection value (1 or 0 or maybe true or false) and the "on" value (true or false). Maybe then I could use TakeWhile(anontype.PointOnPolygon == false). If I Sum the tuples and if ON == 1, then the point is ON the polygon. Otherwise, the oddness or evenness of the sum of the other part of the tuple tells if the point is inside or outside.

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  • Speeding up a search .net 4.0

    - by user231465
    Wondering if I can speed up the search. I need to build a functionality that has to be used by many UI screens The one I have got works but I need to make sure I am implementing a fast algoritim if you like It's like an incremental search. User types a word to search for eg const string searchFor = "Guinea"; const char nextLetter = ' ' It looks in the list and returns 2 records "Guinea and Guinea Bissau " User types a word to search for eg const string searchFor = "Gu"; const char nextLetter = 'i' returns 3 results. This is the function but I would like to speed it up. Is there a pattern for this kind of search? class Program { static void Main() { //Find all countries that begin with string + a possible letter added to it //const string searchFor = "Guinea"; //const char nextLetter = ' '; //returns 2 results const string searchFor = "Gu"; const char nextLetter = 'i'; List<string> result = FindPossibleMatches(searchFor, nextLetter); result.ForEach(x=>Console.WriteLine(x)); //returns 3 results Console.Read(); } /// <summary> /// Find all possible matches /// </summary> /// <param name="searchFor">string to search for</param> /// <param name="nextLetter">pretend user as just typed a letter</param> /// <returns></returns> public static List<string> FindPossibleMatches (string searchFor, char nextLetter) { var hashedCountryList = new HashSet<string>(CountriesList()); var result=new List<string>(); IEnumerable<string> tempCountryList = hashedCountryList.Where(x => x.StartsWith(searchFor)); foreach (string item in tempCountryList) { string tempSearchItem; if (nextLetter == ' ') { tempSearchItem = searchFor; } else { tempSearchItem = searchFor + nextLetter; } if(item.StartsWith(tempSearchItem)) { result.Add(item); } } return result; } /// <summary> /// Returns list of countries. /// </summary> public static string[] CountriesList() { return new[] { "Afghanistan", "Albania", "Algeria", "American Samoa", "Andorra", "Angola", "Anguilla", "Antarctica", "Antigua And Barbuda", "Argentina", "Armenia", "Aruba", "Australia", "Austria", "Azerbaijan", "Bahamas", "Bahrain", "Bangladesh", "Barbados", "Belarus", "Belgium", "Belize", "Benin", "Bermuda", "Bhutan", "Bolivia", "Bosnia Hercegovina", "Botswana", "Bouvet Island", "Brazil", "Brunei Darussalam", "Bulgaria", "Burkina Faso", "Burundi", "Byelorussian SSR", "Cambodia", "Cameroon", "Canada", "Cape Verde", "Cayman Islands", "Central African Republic", "Chad", "Chile", "China", "Christmas Island", "Cocos (Keeling) Islands", "Colombia", "Comoros", "Congo", "Cook Islands", "Costa Rica", "Cote D'Ivoire", "Croatia", "Cuba", "Cyprus", "Czech Republic", "Czechoslovakia", "Denmark", "Djibouti", "Dominica", "Dominican Republic", "East Timor", "Ecuador", "Egypt", "El Salvador", "England", "Equatorial Guinea", "Eritrea", "Estonia", "Ethiopia", "Falkland Islands", "Faroe Islands", "Fiji", "Finland", "France", "Gabon", "Gambia", "Georgia", "Germany", "Ghana", "Gibraltar", "Great Britain", "Greece", "Greenland", "Grenada", "Guadeloupe", "Guam", "Guatemela", "Guernsey", "Guiana", "Guinea", "Guinea Bissau", "Guyana", "Haiti", "Heard Islands", "Honduras", "Hong Kong", "Hungary", "Iceland", "India", "Indonesia", "Iran", "Iraq", "Ireland", "Isle Of Man", "Israel", "Italy", "Jamaica", "Japan", "Jersey", "Jordan", "Kazakhstan", "Kenya", "Kiribati", "Korea, South", "Korea, North", "Kuwait", "Kyrgyzstan", "Lao People's Dem. Rep.", "Latvia", "Lebanon", "Lesotho", "Liberia", "Libya", "Liechtenstein", "Lithuania", "Luxembourg", "Macau", "Macedonia", "Madagascar", "Malawi", "Malaysia", "Maldives", "Mali", "Malta", "Mariana Islands", "Marshall Islands", "Martinique", "Mauritania", "Mauritius", "Mayotte", "Mexico", "Micronesia", "Moldova", "Monaco", "Mongolia", "Montserrat", "Morocco", "Mozambique", "Myanmar", "Namibia", "Nauru", "Nepal", "Netherlands", "Netherlands Antilles", "Neutral Zone", "New Caledonia", "New Zealand", "Nicaragua", "Niger", "Nigeria", "Niue", "Norfolk Island", "Northern Ireland", "Norway", "Oman", "Pakistan", "Palau", "Panama", "Papua New Guinea", "Paraguay", "Peru", "Philippines", "Pitcairn", "Poland", "Polynesia", "Portugal", "Puerto Rico", "Qatar", "Reunion", "Romania", "Russian Federation", "Rwanda", "Saint Helena", "Saint Kitts", "Saint Lucia", "Saint Pierre", "Saint Vincent", "Samoa", "San Marino", "Sao Tome and Principe", "Saudi Arabia", "Scotland", "Senegal", "Seychelles", "Sierra Leone", "Singapore", "Slovakia", "Slovenia", "Solomon Islands", "Somalia", "South Africa", "South Georgia", "Spain", "Sri Lanka", "Sudan", "Suriname", "Svalbard", "Swaziland", "Sweden", "Switzerland", "Syrian Arab Republic", "Taiwan", "Tajikista", "Tanzania", "Thailand", "Togo", "Tokelau", "Tonga", "Trinidad and Tobago", "Tunisia", "Turkey", "Turkmenistan", "Turks and Caicos Islands", "Tuvalu", "Uganda", "Ukraine", "United Arab Emirates", "United Kingdom", "United States", "Uruguay", "Uzbekistan", "Vanuatu", "Vatican City State", "Venezuela", "Vietnam", "Virgin Islands", "Wales", "Western Sahara", "Yemen", "Yugoslavia", "Zaire", "Zambia", "Zimbabwe" }; } } } Any suggestions? Thanks

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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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  • Using a mounted NTFS share with nginx

    - by Hoff
    I have set up a local testing VM with Ubuntu Server 12.04 LTS and the LEMP stack. It's kind of an unconventional setup because instead of having all my PHP scripts on the local machine, I've mounted an NTFS share as the document root because I do my development on Windows. I had everything working perfectly up until this morning, now I keep getting a dreaded 'File not found.' error. I am almost certain this must be somehow permission related, because if I copy my site over to /var/www, nginx and php-fpm have no problems serving my PHP scripts. What I can't figure out is why all of a sudden (after a reboot of the server), no PHP files will be served but instead just the 'File not found.' error. Static files work fine, so I think it's PHP that is causing the headache. Both nginx and php-fpm are configured to run as the user www-data: root@ubuntu-server:~# ps aux | grep 'nginx\|php-fpm' root 1095 0.0 0.0 5816 792 ? Ss 11:11 0:00 nginx: master process /opt/nginx/sbin/nginx -c /etc/nginx/nginx.conf www-data 1096 0.0 0.1 6016 1172 ? S 11:11 0:00 nginx: worker process www-data 1098 0.0 0.1 6016 1172 ? S 11:11 0:00 nginx: worker process root 1130 0.0 0.4 175560 4212 ? Ss 11:11 0:00 php-fpm: master process (/etc/php5/php-fpm.conf) www-data 1131 0.0 0.3 175560 3216 ? S 11:11 0:00 php-fpm: pool www www-data 1132 0.0 0.3 175560 3216 ? S 11:11 0:00 php-fpm: pool www www-data 1133 0.0 0.3 175560 3216 ? S 11:11 0:00 php-fpm: pool www root 1686 0.0 0.0 4368 816 pts/1 S+ 11:11 0:00 grep --color=auto nginx\|php-fpm I have mounted the NTFS share at /mnt/webfiles by editing /etc/fstab and adding the following line: //192.168.0.199/c$/Websites/ /mnt/webfiles cifs username=Jordan,password=mypasswordhere,gid=33,uid=33 0 0 Where gid 33 is the www-data group and uid 33 is the user www-data. If I list the contents of one of my sites you can in fact see that they belong to the user www-data: root@ubuntu-server:~# ls -l /mnt/webfiles/nTv5-2.0 total 8 drwxr-xr-x 0 www-data www-data 0 Jun 6 19:12 app drwxr-xr-x 0 www-data www-data 0 Aug 22 19:00 assets -rwxr-xr-x 0 www-data www-data 1150 Jan 4 2012 favicon.ico -rwxr-xr-x 0 www-data www-data 1412 Dec 28 2011 index.php drwxr-xr-x 0 www-data www-data 0 Jun 3 16:44 lib drwxr-xr-x 0 www-data www-data 0 Jan 3 2012 plugins drwxr-xr-x 0 www-data www-data 0 Jun 3 16:45 vendors If I switch to the www-data user, I have no problem creating a new file on the share: root@ubuntu-server:~# su www-data $ > /mnt/webfiles/test.txt $ ls -l /mnt/webfiles | grep test\.txt -rwxr-xr-x 0 www-data www-data 0 Sep 8 11:19 test.txt There should be no problem reading or writing to the share with php-fpm running as the user www-data. When I examine the error log of nginx, it's filled with a bunch of lines that look like the following: 2012/09/08 11:22:36 [error] 1096#0: *1 FastCGI sent in stderr: "Primary script unknown" while reading response header from upstream, client: 192.168.0.199, server: , request: "GET / HTTP/1.1", upstream: "fastcgi://unix:/var/run/php5-fpm.sock:", host: "192.168.0.123" 2012/09/08 11:22:39 [error] 1096#0: *1 FastCGI sent in stderr: "Primary script unknown" while reading response header from upstream, client: 192.168.0.199, server: , request: "GET /apc.php HTTP/1.1", upstream: "fastcgi://unix:/var/run/php5-fpm.sock:", host: "192.168.0.123" It's bizarre that this was working previously and now all of sudden PHP is complaining that it can't "find" the scripts on the share. Does anybody know why this is happening? EDIT I tried editing php-fpm.conf and changing chdir to the following: chdir = /mnt/webfiles When I try and restart the php-fpm service, I get the error: Starting php-fpm [08-Sep-2012 14:20:55] ERROR: [pool www] the chdir path '/mnt/webfiles' does not exist or is not a directory This is a total load of bullshit because this directory DOES exist and is mounted! Any ls commands to list that directory work perfectly. Why the hell can't PHP-FPM see this directory?! Here are my configuration files for reference: nginx.conf user www-data; worker_processes 2; error_log /var/log/nginx/nginx.log info; pid /var/run/nginx.pid; events { worker_connections 1024; multi_accept on; } http { include fastcgi.conf; include mime.types; default_type application/octet-stream; set_real_ip_from 127.0.0.1; real_ip_header X-Forwarded-For; ## Proxy proxy_redirect off; proxy_set_header Host $host; proxy_set_header X-Real-IP $remote_addr; proxy_set_header X-Forwarded-For $proxy_add_x_forwarded_for; client_max_body_size 32m; client_body_buffer_size 128k; proxy_connect_timeout 90; proxy_send_timeout 90; proxy_read_timeout 90; proxy_buffers 32 4k; ## Compression gzip on; gzip_types text/plain text/css application/x-javascript text/xml application/xml application/xml+rss text/javascript; gzip_disable "MSIE [1-6]\.(?!.*SV1)"; ### TCP options tcp_nodelay on; tcp_nopush on; keepalive_timeout 65; sendfile on; include /etc/nginx/sites-enabled/*; } my site config server { listen 80; access_log /var/log/nginx/$host.access.log; error_log /var/log/nginx/error.log; root /mnt/webfiles/nTv5-2.0/app/webroot; index index.php; ## Block bad bots if ($http_user_agent ~* (HTTrack|HTMLParser|libcurl|discobot|Exabot|Casper|kmccrew|plaNETWORK|RPT-HTTPClient)) { return 444; } ## Block certain Referers (case insensitive) if ($http_referer ~* (sex|vigra|viagra) ) { return 444; } ## Deny dot files: location ~ /\. { deny all; } ## Favicon Not Found location = /favicon.ico { access_log off; log_not_found off; } ## Robots.txt Not Found location = /robots.txt { access_log off; log_not_found off; } if (-f $document_root/maintenance.html) { rewrite ^(.*)$ /maintenance.html last; } location ~* \.(?:ico|css|js|gif|jpe?g|png)$ { # Some basic cache-control for static files to be sent to the browser expires max; add_header Pragma public; add_header Cache-Control "max-age=2678400, public, must-revalidate"; } location / { try_files $uri $uri/ index.php; if (-f $request_filename) { break; } rewrite ^(.+)$ /index.php?url=$1 last; } location ~ \.php$ { include /etc/nginx/fastcgi.conf; fastcgi_pass unix:/var/run/php5-fpm.sock; } } php-fpm.conf ;;;;;;;;;;;;;;;;;;;;; ; FPM Configuration ; ;;;;;;;;;;;;;;;;;;;;; ; All relative paths in this configuration file are relative to PHP's install ; prefix (/opt/php5). This prefix can be dynamicaly changed by using the ; '-p' argument from the command line. ; Include one or more files. If glob(3) exists, it is used to include a bunch of ; files from a glob(3) pattern. This directive can be used everywhere in the ; file. ; Relative path can also be used. They will be prefixed by: ; - the global prefix if it's been set (-p arguement) ; - /opt/php5 otherwise ;include=etc/fpm.d/*.conf ;;;;;;;;;;;;;;;;;; ; Global Options ; ;;;;;;;;;;;;;;;;;; [global] ; Pid file ; Note: the default prefix is /opt/php5/var ; Default Value: none pid = /var/run/php-fpm.pid ; Error log file ; Note: the default prefix is /opt/php5/var ; Default Value: log/php-fpm.log error_log = /var/log/php5-fpm/php-fpm.log ; Log level ; Possible Values: alert, error, warning, notice, debug ; Default Value: notice ;log_level = notice ; If this number of child processes exit with SIGSEGV or SIGBUS within the time ; interval set by emergency_restart_interval then FPM will restart. A value ; of '0' means 'Off'. ; Default Value: 0 ;emergency_restart_threshold = 0 ; Interval of time used by emergency_restart_interval to determine when ; a graceful restart will be initiated. This can be useful to work around ; accidental corruptions in an accelerator's shared memory. ; Available Units: s(econds), m(inutes), h(ours), or d(ays) ; Default Unit: seconds ; Default Value: 0 ;emergency_restart_interval = 0 ; Time limit for child processes to wait for a reaction on signals from master. ; Available units: s(econds), m(inutes), h(ours), or d(ays) ; Default Unit: seconds ; Default Value: 0 ;process_control_timeout = 0 ; Send FPM to background. Set to 'no' to keep FPM in foreground for debugging. ; Default Value: yes ;daemonize = yes ;;;;;;;;;;;;;;;;;;;; ; Pool Definitions ; ;;;;;;;;;;;;;;;;;;;; ; Multiple pools of child processes may be started with different listening ; ports and different management options. The name of the pool will be ; used in logs and stats. There is no limitation on the number of pools which ; FPM can handle. Your system will tell you anyway :) ; Start a new pool named 'www'. ; the variable $pool can we used in any directive and will be replaced by the ; pool name ('www' here) [www] ; Per pool prefix ; It only applies on the following directives: ; - 'slowlog' ; - 'listen' (unixsocket) ; - 'chroot' ; - 'chdir' ; - 'php_values' ; - 'php_admin_values' ; When not set, the global prefix (or /opt/php5) applies instead. ; Note: This directive can also be relative to the global prefix. ; Default Value: none ;prefix = /path/to/pools/$pool ; The address on which to accept FastCGI requests. ; Valid syntaxes are: ; 'ip.add.re.ss:port' - to listen on a TCP socket to a specific address on ; a specific port; ; 'port' - to listen on a TCP socket to all addresses on a ; specific port; ; '/path/to/unix/socket' - to listen on a unix socket. ; Note: This value is mandatory. ;listen = 127.0.0.1:9000 listen = /var/run/php5-fpm.sock ; Set listen(2) backlog. A value of '-1' means unlimited. ; Default Value: 128 (-1 on FreeBSD and OpenBSD) ;listen.backlog = -1 ; List of ipv4 addresses of FastCGI clients which are allowed to connect. ; Equivalent to the FCGI_WEB_SERVER_ADDRS environment variable in the original ; PHP FCGI (5.2.2+). Makes sense only with a tcp listening socket. Each address ; must be separated by a comma. If this value is left blank, connections will be ; accepted from any ip address. ; Default Value: any ;listen.allowed_clients = 127.0.0.1 ; Set permissions for unix socket, if one is used. In Linux, read/write ; permissions must be set in order to allow connections from a web server. Many ; BSD-derived systems allow connections regardless of permissions. ; Default Values: user and group are set as the running user ; mode is set to 0666 ;listen.owner = www-data ;listen.group = www-data ;listen.mode = 0666 ; Unix user/group of processes ; Note: The user is mandatory. If the group is not set, the default user's group ; will be used. user = www-data group = www-data ; Choose how the process manager will control the number of child processes. ; Possible Values: ; static - a fixed number (pm.max_children) of child processes; ; dynamic - the number of child processes are set dynamically based on the ; following directives: ; pm.max_children - the maximum number of children that can ; be alive at the same time. ; pm.start_servers - the number of children created on startup. ; pm.min_spare_servers - the minimum number of children in 'idle' ; state (waiting to process). If the number ; of 'idle' processes is less than this ; number then some children will be created. ; pm.max_spare_servers - the maximum number of children in 'idle' ; state (waiting to process). If the number ; of 'idle' processes is greater than this ; number then some children will be killed. ; Note: This value is mandatory. pm = dynamic ; The number of child processes to be created when pm is set to 'static' and the ; maximum number of child processes to be created when pm is set to 'dynamic'. ; This value sets the limit on the number of simultaneous requests that will be ; served. Equivalent to the ApacheMaxClients directive with mpm_prefork. ; Equivalent to the PHP_FCGI_CHILDREN environment variable in the original PHP ; CGI. ; Note: Used when pm is set to either 'static' or 'dynamic' ; Note: This value is mandatory. pm.max_children = 50 ; The number of child processes created on startup. ; Note: Used only when pm is set to 'dynamic' ; Default Value: min_spare_servers + (max_spare_servers - min_spare_servers) / 2 pm.start_servers = 20 ; The desired minimum number of idle server processes. ; Note: Used only when pm is set to 'dynamic' ; Note: Mandatory when pm is set to 'dynamic' pm.min_spare_servers = 5 ; The desired maximum number of idle server processes. ; Note: Used only when pm is set to 'dynamic' ; Note: Mandatory when pm is set to 'dynamic' pm.max_spare_servers = 35 ; The number of requests each child process should execute before respawning. ; This can be useful to work around memory leaks in 3rd party libraries. For ; endless request processing specify '0'. Equivalent to PHP_FCGI_MAX_REQUESTS. ; Default Value: 0 pm.max_requests = 500 ; The URI to view the FPM status page. If this value is not set, no URI will be ; recognized as a status page. By default, the status page shows the following ; information: ; accepted conn - the number of request accepted by the pool; ; pool - the name of the pool; ; process manager - static or dynamic; ; idle processes - the number of idle processes; ; active processes - the number of active processes; ; total processes - the number of idle + active processes. ; max children reached - number of times, the process limit has been reached, ; when pm tries to start more children (works only for ; pm 'dynamic') ; The values of 'idle processes', 'active processes' and 'total processes' are ; updated each second. The value of 'accepted conn' is updated in real time. ; Example output: ; accepted conn: 12073 ; pool: www ; process manager: static ; idle processes: 35 ; active processes: 65 ; total processes: 100 ; max children reached: 1 ; By default the status page output is formatted as text/plain. Passing either ; 'html' or 'json' as a query string will return the corresponding output ; syntax. Example: ; http://www.foo.bar/status ; http://www.foo.bar/status?json ; http://www.foo.bar/status?html ; Note: The value must start with a leading slash (/). The value can be ; anything, but it may not be a good idea to use the .php extension or it ; may conflict with a real PHP file. ; Default Value: not set pm.status_path = /status ; The ping URI to call the monitoring page of FPM. If this value is not set, no ; URI will be recognized as a ping page. This could be used to test from outside ; that FPM is alive and responding, or to ; - create a graph of FPM availability (rrd or such); ; - remove a server from a group if it is not responding (load balancing); ; - trigger alerts for the operating team (24/7). ; Note: The value must start with a leading slash (/). The value can be ; anything, but it may not be a good idea to use the .php extension or it ; may conflict with a real PHP file. ; Default Value: not set ping.path = /ping ; This directive may be used to customize the response of a ping request. The ; response is formatted as text/plain with a 200 response code. ; Default Value: pong ping.response = pong ; The timeout for serving a single request after which the worker process will ; be killed. This option should be used when the 'max_execution_time' ini option ; does not stop script execution for some reason. A value of '0' means 'off'. ; Available units: s(econds)(default), m(inutes), h(ours), or d(ays) ; Default Value: 0 ;request_terminate_timeout = 0 ; The timeout for serving a single request after which a PHP backtrace will be ; dumped to the 'slowlog' file. A value of '0s' means 'off'. ; Available units: s(econds)(default), m(inutes), h(ours), or d(ays) ; Default Value: 0 ;request_slowlog_timeout = 0 ; The log file for slow requests ; Default Value: not set ; Note: slowlog is mandatory if request_slowlog_timeout is set ;slowlog = log/$pool.log.slow ; Set open file descriptor rlimit. ; Default Value: system defined value ;rlimit_files = 1024 ; Set max core size rlimit. ; Possible Values: 'unlimited' or an integer greater or equal to 0 ; Default Value: system defined value ;rlimit_core = 0 ; Chroot to this directory at the start. This value must be defined as an ; absolute path. When this value is not set, chroot is not used. ; Note: you can prefix with '$prefix' to chroot to the pool prefix or one ; of its subdirectories. If the pool prefix is not set, the global prefix ; will be used instead. ; Note: chrooting is a great security feature and should be used whenever ; possible. However, all PHP paths will be relative to the chroot ; (error_log, sessions.save_path, ...). ; Default Value: not set ;chroot = ; Chdir to this directory at the start. ; Note: relative path can be used. ; Default Value: current directory or / when chroot ;chdir = /var/www ; Redirect worker stdout and stderr into main error log. If not set, stdout and ; stderr will be redirected to /dev/null according to FastCGI specs. ; Note: on highloaded environement, this can cause some delay in the page ; process time (several ms). ; Default Value: no ;catch_workers_output = yes ; Pass environment variables like LD_LIBRARY_PATH. All $VARIABLEs are taken from ; the current environment. ; Default Value: clean env ;env[HOSTNAME] = $HOSTNAME ;env[PATH] = /usr/local/bin:/usr/bin:/bin ;env[TMP] = /tmp ;env[TMPDIR] = /tmp ;env[TEMP] = /tmp ; Additional php.ini defines, specific to this pool of workers. These settings ; overwrite the values previously defined in the php.ini. The directives are the ; same as the PHP SAPI: ; php_value/php_flag - you can set classic ini defines which can ; be overwritten from PHP call 'ini_set'. ; php_admin_value/php_admin_flag - these directives won't be overwritten by ; PHP call 'ini_set' ; For php_*flag, valid values are on, off, 1, 0, true, false, yes or no. ; Defining 'extension' will load the corresponding shared extension from ; extension_dir. Defining 'disable_functions' or 'disable_classes' will not ; overwrite previously defined php.ini values, but will append the new value ; instead. ; Note: path INI options can be relative and will be expanded with the prefix ; (pool, global or /opt/php5) ; Default Value: nothing is defined by default except the values in php.ini and ; specified at startup with the -d argument ;php_admin_value[sendmail_path] = /usr/sbin/sendmail -t -i -f [email protected] ;php_flag[display_errors] = off ;php_admin_value[error_log] = /var/log/fpm-php.www.log ;php_admin_flag[log_errors] = on ;php_admin_value[memory_limit] = 32M php_admin_value[sendmail_path] = /usr/sbin/sendmail -t -i

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