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  • Multiplying Block Matrices in Numpy

    - by Ada Xu
    Hi Everyone I am python newbie I have to implement lasso L1 regression for a class assignment. This involves solving a quadratic equation involving block matrices. minimize x^t * H * x + f^t * x where x 0 Where H is a 2 X 2 block matrix with each element being a k dimensional matrix and x and f being a 2 X 1 vectors each element being a k dimension vector. I was thinking of using nd arrays. such that np.shape(H) = (2, 2, k, k) np.shape(x) = (2, k) But I figured out that np.dot(X, H) doesn't work here. Is there an easy way to solve this problem? Thanks in advance.

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  • Which is the best Linux C/C++ debugger (or front-end to gdb) to help teaching programming?

    - by omer.gimenez
    I teach a sort of "lite" C++ programming course to novices ("lite" meaning no pointers, no classes, just plain old C, plus references and STL string and vectors). Students have no previous experience in programming, so I believe that using an interactive debugger would help them understand program flow, variables, and recursion. The course is taught in Linux. Teaching them to use gdb is just overkill (they will not use nor understand most features). I just need something simple but easy to use: to see at which line the program is now, what is in the stack (local variables, previous calls, etc.). I look something similar to old Turbo Pascal or Turbo C++ Borland's debugger, or Visual Studio debugger. Thank you,

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  • loading Data in VBA from a text file

    - by omegayen
    I am not very familiar with VBA but need to use it for a new software program I am using (not Microsoft related) I have a text file that has columns of data I would like to read into VBA. Specifically the text file has 4 entries per row. Thus I would like to load in the column vectors (N by 1). The text file is separated by a space between each entry. So for example I want to load in column one and save it as array A, then column two and save as array B, then column three and save as array C, and then column four and save as array D. This code snippet found below from http://www.tek-tips.com/faqs.cfm?fid=482 is something I found that can load in text to an array, but I need to adapt it to be able to save the columns as different arrays as specified above... Open "MyFile.txt" For Input As #1 ReDim Txt$(0) Do While Not EOF(1) ReDim Preserve Txt$(UBound(Txt$) + 1) Input #1, Txt$(UBound(Txt$)) Loop Close #1

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  • Nonstatic conversion functions; Casting different types, e.g. DirectX vector to OpenGL vector

    - by Markus
    I am currently working on a game "engine" that needs to move values between a 3D engine, a physics engine and a scripting language. Since I need to apply vectors from the physics engine to 3D objects very often and want to be able to control both the 3D, as well as the physics objects through the scripting system, I need a mechanism to convert a vector of one type (e.g. vector3d<float>) to a vector of the other type (e.g. btVector3). Unfortunately I can make no assumptions on how the classes/structs are laid out, so a simple reinterpret_cast probably won't do. So the question is: Is there some sort of 'static'/non-member casting method to achieve basically this: vector3d<float> operator vector3d<float>(btVector3 vector) { // convert and return } btVector3 operator btVector3(vector3d<float> vector) { // convert and return } Right now this won't compile since casting operators need to be member methods. (error C2801: 'operator foo' must be a non-static member)

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  • Switching from C++ (with a lot of STL use) to C for interpreter building

    - by wndsr
    I'm switching from C++ to C because I'm rebuilding my toy interpreter. I was used to vectors for dynamic allocation of objects like tokens or instructions of my programs, stacks and mainly strings with all their aspects. Now, in C I'm not going to have all these anymore. I know that I will have to use a lot of memory management, too. I'm completely new to C, I only know the high-level easy-life data structures from the STL, how can I get started with strings and dynamic memory allocation?

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  • c++ Array passing dilemma

    - by Thomas
    Hi, I am writing a function that takes a string, string pointer and an int. The function splits the string based on a set of rules and puts each token into an array. I need to return the array out of the function with the number of elements in the int variable etc. I am stuck as to how I return the array as I can not use auto other wise it is destroyed and I am reluctant to use new as I feel this is patchy. I have other ideas on how to go about this but would like to see how other people go about this first. I could also be wrong and it could be possible to pass an auto out of an array. I can also not use vectors so there goes a copy constructor.

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  • How can you deflect a direction/magnitude vector based on a direction/magnitude vector and a collided triangle?

    - by JeanOTF
    So, I have a Triangle-AABB collision algorithm and I have it returning the triangle that the AABB collided with. I was hoping with the 3 vectors of the triangle and the direction/magnitude of the movement would let me determine a deflected vector so that when you run against the wall at an angle you move slower, depending on the angle of collision, but along side the wall. This would remove the sticky collision problem with only moving when there is not a collision. Any suggestions or references would be greatly appreciated! Thanks.

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  • Getting object coordinates from camera

    - by user566757
    I've implemented a camera in Java using a position vector and three direction vectors so I can use gluLookAt(); moving around in `ghost mode' works fine enough, but I want to add collision detection. I can't seem to figure out how to transform my position vector to coordinates in which OpenGL draws my objects. A rough sketch of my drawing loop is this: glClear(GL_COLOR_BUFFER_BIT | GL_DEPTH_BUFFER_BIT); glLoadIdentity(); camera.setView(); drawer.drawTheScene(); I'm at a loss of how to proceed; looking at the ModelView matrix between calls and my position vector, I haven't found any kind of correlation.

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  • Insert at specific location of a 2d vector

    - by Elgoog
    I have a 2d vector which represents a 2d grid; so grid[0][2] for example. I am needing to 'insert' -might not be the right word here. a vector at a specific location say grid[3][2] there will definitely be a grid[0][0] but when im needing to insert into grid[3][2] there may be nothing before it other than grid[0][0] and there needs to be the space in between for later on. Is there any way to do this? Thank you for your help. ps: I should note that the size of the vectors are not known (they will grow over time)

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  • Duplicates in a sorted java array

    - by Max Frazier
    I have to write a method that takes an array of ints that is already sorted in numerical order then remove all the duplicate numbers and return an array of just the numbers that have no duplicates. That array must then be printed out so I can't have any null pointer exceptions. The method has to be in O(n) time, can't use vectors or hashes. This is what I have so far but it only has the first couple numbers in order without duplicates and then just puts the duplicates in the back of the array. I can't create a temporary array because it gives me null pointer exceptions. public static int[] noDups(int[] myArray) { int j = 0; for (int i = 1; i < myArray.length; i++) { if (myArray[i] != myArray[j]) { j++; myArray[j] = myArray[i]; } } return myArray; }

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  • Why the valid looking statement gives error in MATLAB?

    - by user198729
    It's from this question? Why the two solutions doesn't work, though it looks very valid for me: >> t = -pi:0.1:pi; >> r = ((sin(t)*sqrt(cos(t)))*(sin(t) + (7/5))^(-1)) - 2*sin(t) + 2 ; ??? Error using ==> mtimes Inner matrix dimensions must agree. >> t = -pi:0.1:pi; >> r = ((sin(t).*sqrt(cos(t))).*(sin(t) + (7/5)).^(-1)) - 2*sin(t) + 2 ; >> plot(r,t) ??? Error using ==> plot Vectors must be the same lengths. What's wrong with the above?

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  • How to interpret situations where Math.Acos() reports invalid input?

    - by Sean Ochoa
    Hey all. I'm computing the angle between two vectors, and sometimes Math.Acos() returns NaN when it's input is out of bounds (-1 input && input 1) for a cosine. What does that mean, exactly? Would someone be able to explain what's happening? Any help is appreciated! Here's me method: public double AngleBetween(vector b) { var dotProd = this.Dot(b); var lenProd = this.Len*b.Len; var divOperation = dotProd/lenProd; // http://msdn.microsoft.com/en-us/library/system.math.acos.aspx return Math.Acos(divOperation) * (180.0 / Math.PI); }

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  • About curse of dimensionality

    - by Dan
    My question is about this topic I've been reading about a bit. Basically my understanding is that in higher dimensions all points end up being very close to each other. The doubt I have is whether this means that calculating distances the usual way (euclidean for instance) is valid or not. If it were still valid, this would mean that when comparing vectors in high dimensions, the two most similar wouldn't differ much from a third one even when this third one could be completely unrelated. Is this correct? Then in this case, how would you be able to tell whether you have a match or not?

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  • How can I neatly clean my R workspace while preserving certain objects?

    - by briandk
    Suppose I'm messing about with some data by binding vectors together, as I'm wont to do on a lazy sunday afternoon. x <- rnorm(25, mean = 65, sd = 10) y <- rnorm(25, mean = 75, sd = 7) z <- 1:25 dd <- data.frame(mscore = x, vscore = y, caseid = z) I've now got my new dataframe dd, which is wonderful. But there's also still the detritus from my prior slicings and dicings: > ls() [1] "dd" "x" "y" "z" What's a simple way to clean up my workspace if I no longer need my "source" columns, but I want to keep the dataframe? That is, now that I'm done manipulating data I'd like to just have dd and none of the smaller variables that might inadvertently mask further analysis: > ls() [1] "dd" I feel like the solution must be of the form rm(ls[ -(dd) ]) or something, but I can't quite figure out how to say "please clean up everything BUT the following objects."

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  • c++ Sorting a vector based on values of other vector, or what's faster?

    - by pollux
    Hi, There are a couple of other posts about sorting a vector A based on values in another vector B. Most of the other answers tell to create a struct or a class to combine the values into one object and use std::sort. Though I'm curious about the performance of such solutions as I need to optimize code which implements bubble sort to sort these two vectors. I'm thinking to use a vector<pair<int,int>> and sort that. I'm working on a blob-tracking application (image analysis) where I try to match previously tracked blobs against newly detected blobs in video frames where I check each of the frames against a couple of previously tracked frames and of course the blobs I found in previous frames. I'm doing this at 60 times per second (speed of my webcam). Any advice on optimizing this is appreciated. The code I'm trying to optimize can be shown here: http://code.google.com/p/projectknave/source/browse/trunk/knaveAddons/ofxBlobTracker/ofCvBlobTracker.cpp?spec=svn313&r=313 Thanks

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  • Pushing an array into a vector.

    - by Sunil
    I've a 2d array, say A[2][3]={{1,2,3},{4,5,6}}; and I want to push it into a 2D vector(vector of vectors). I know you can use two for loops to push the elements one by on on to the first vector and then push that into the another vector which makes it 2d vector but I was wondering if there is any way in C++ to do this in a single loop. For example I want to do something like this: myvector.pushback(A[1]+3); // where 3 is the size or number of columns in the array. I understand this is not a correct code but I put this just for understanding purpose. Thanks

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  • C++ vector insights

    - by Sunscreen
    Hi, I am a little bit frustrated of how to use vectors in C++. I use them widely though I am not exactly certail of how I use them. Below are teh questions? If I have a vector lets say: std::vector<CString> v_strMyVector, with (int)v_strMyVector.size > i can I access the i member: v_strMyVector[i] == "xxxx"; ? (it works, though why?) Do i always need to define an iterator to acces to go to the beginning of the vector, and lop on its members ? What is the purpose of an iterator if I have access to all members of the vector directly (see 1)? Thanks in advance, Sun

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  • apply function using expand.grid in R

    - by kolonel
    I have two vectors x and y. I create a grid using the following function: v = expand.grid(x, y) I have a function defined as follows N <- function(a, b , dat){ m = ncol(Filter(function(z) a*max(z)*min(z) < b , dat[1:ncol(dat)])) return(m) } and then I need to maximize N over a grid of x,y: Maximize <- function(x , y ,dat){ v = as.matrix(expand.grid(x,y)) # Here is where I want to map the values of v and get the maximum element and # get the tuple in v that maximized N temp1 <- max(apply(v , 1 , N(v[[1]] , v[[2]] , dat))) } Thanks

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  • How to update a vector in method

    - by gurpinars
    I'm new to C++ and trying to understand vectors. My goal is to update a vector in method: #include <vector> #include <iostream> using namespace std; void test(vector<int>& array){ for(int i=0;i<10;i++){ array.push_back(i); } } int main(){ // some integer value vector<int> array(10); test(array); for(int i=0;i<array.size();++i) cout<<array.at(i)<<endl; cout<<"array size:"<<array.size()<<endl; return 0; } output: 0 0 0 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 array size:20 I haven't figure out why 10 zeros add vector at first?

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  • Best neural network for certain type of pattern analysis?

    - by fred basset
    Hi All, I'm working on a system that will send telemetry data on machine operation back to a central server for analysis. One of the machine parameters we're measuring is motor current drawn vs time. After an operation is finished we plan to send back an array of currents vs time to the server. A successful operation would have a pattern like a trapezoid, problematic operations would have a pattern completely different, more like a large spike in values. Can anyone recommend a type of neural network that would be good at classifying these 1D vectors of current values into a pass/fail type output? Thanks, Fred

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  • C++ methods which take templated classes as argument.

    - by Nils
    I have a templated class Vector<class T, int N> Where T is the type of the components (double for example) and n the number of components (so N=3 for a 3D vector) Now I want to write a method like double findStepsize(Vector<double,2> v) {..} I want to do this also for three and higher dimensional vectors. Of course I could just introduce further methods for higher dimensions, but the methods would have a lot of redundant code, so I want a more generic solution. Is there a way to create a method which takes a templated class without further specializing it (in this case without specifying T or N)? Like double findStepsize(Vector<T,N> v) ?

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  • read integers from a file into a vector in C++

    - by user2922063
    I am trying to read an unknown number of double values stored on separate lines from a text file into a vector called rainfall. My code won't compile; I am getting the error no match for 'operator>>' in 'inputFile >> rainfall' for the while loop line. I understand how to read in from a file into an array, but we are required to use vectors for this project and I'm not getting it. I appreciate any tips you can give on my partial code below. vector<double> rainfall; // a vector to hold rainfall data // open file ifstream inputFile("/home/shared/data4.txt"); // test file open if (inputFile) { int count = 0; // count number of items in the file // read the elements in the file into a vector while ( inputFile >> rainfall ) { rainfall.push_back(count); ++count; } // close the file

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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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  • OpenGL - Calculating camera view matrix

    - by Karle
    Problem I am calculating the model, view and projection matrices independently to be used in my shader as follows: gl_Position = projection * view * model * vec4(in_Position, 1.0); When I try to calculate my camera's view matrix the Z axis is flipped and my camera seems like it is looking backwards. My program is written in C# using the OpenTK library. Translation (Working) I've created a test scene as follows: From my understanding of the OpenGL coordinate system they are positioned correctly. The model matrix is created using: Matrix4 translation = Matrix4.CreateTranslation(modelPosition); Matrix4 model = translation; The view matrix is created using: Matrix4 translation = Matrix4.CreateTranslation(-cameraPosition); Matrix4 view = translation; Rotation (Not-Working) I now want to create the camera's rotation matrix. To do this I use the camera's right, up and forward vectors: // Hard coded example orientation: // Normally calculated from up and forward // Similar to look-at camera. Vector3 r = Vector.UnitX; Vector3 u = Vector3.UnitY; Vector3 f = -Vector3.UnitZ; Matrix4 rot = new Matrix4( r.X, r.Y, r.Z, 0, u.X, u.Y, u.Z, 0, f.X, f.Y, f.Z, 0, 0.0f, 0.0f, 0.0f, 1.0f); This results in the following matrix being created: I know that multiplying by the identity matrix would produce no rotation. This is clearly not the identity matrix and therefore will apply some rotation. I thought that because this is aligned with the OpenGL coordinate system is should produce no rotation. Is this the wrong way to calculate the rotation matrix? I then create my view matrix as: // OpenTK is row-major so the order of operations is reversed: Matrix4 view = translation * rot; Rotation almost works now but the -Z/+Z axis has been flipped, with the green cube now appearing closer to the camera. It seems like the camera is looking backwards, especially if I move it around. My goal is to store the position and orientation of all objects (including the camera) as: Vector3 position; Vector3 up; Vector3 forward; Apologies for writing such a long question and thank you in advance. I've tried following tutorials/guides from many sites but I keep ending up with something wrong. Edit: Projection Matrix Set-up Matrix4 projection = Matrix4.CreatePerspectiveFieldOfView( (float)(0.5 * Math.PI), (float)display.Width / display.Height, 0.1f, 1000.0f);

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  • The View-Matrix and Alternative Calculations

    - by P. Avery
    I'm working on a radiosity processor in DirectX 9. The process requires that the camera be placed at the center of a mesh face and a 'screenshot' be taken facing 5 different directions...forward...up...down...left...right... ...The problem is that when the mesh face is facing up( look vector: 0, 1, 0 )...a view matrix cannot be determined using standard trigonometry functions: Matrix4 LookAt( Vector3 eye, Vector3 target, Vector3 up ) { // The "look-at" vector. Vector3 zaxis = normal(target - eye); // The "right" vector. Vector3 xaxis = normal(cross(up, zaxis)); // The "up" vector. Vector3 yaxis = cross(zaxis, xaxis); // Create a 4x4 orientation matrix from the right, up, and at vectors Matrix4 orientation = { xaxis.x, yaxis.x, zaxis.x, 0, xaxis.y, yaxis.y, zaxis.y, 0, xaxis.z, yaxis.z, zaxis.z, 0, 0, 0, 0, 1 }; // Create a 4x4 translation matrix by negating the eye position. Matrix4 translation = { 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, -eye.x, -eye.y, -eye.z, 1 }; // Combine the orientation and translation to compute the view matrix return ( translation * orientation ); } The above function comes from http://3dgep.com/?p=1700... ...Is there a mathematical approach to this problem? Edit: A problem occurs when setting the view matrix to up or down directions, here is an example of the problem when facing down: D3DXVECTOR4 vPos( 3, 3, 3, 1 ), vEye( 1.5, 3, 3, 1 ), vLook( 0, -1, 0, 1 ), vRight( 1, 0, 0, 1 ), vUp( 0, 0, 1, 1 ); D3DXMATRIX mV, mP; D3DXMatrixPerspectiveFovLH( &mP, D3DX_PI / 2, 1, 0.5f, 2000.0f ); D3DXMatrixIdentity( &mV ); memcpy( ( void* )&mV._11, ( void* )&vRight, sizeof( D3DXVECTOR3 ) ); memcpy( ( void* )&mV._21, ( void* )&vUp, sizeof( D3DXVECTOR3 ) ); memcpy( ( void* )&mV._31, ( void* )&vLook, sizeof( D3DXVECTOR3 ) ); memcpy( ( void* )&mV._41, ( void* )&(-vEye), sizeof( D3DXVECTOR3 ) ); D3DXVec4Transform( &vPos, &vPos, &( mV * mP ) ); Results: vPos = D3DXVECTOR3( 1.5, -6, -0.5, 0 ) - this vertex is not properly processed by shader as the homogenous w value is 0 it cannot be normalized to a position within device space...

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