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  • Information on Rojiani's Numerical methods C textbook

    - by yCalleecharan
    Hi, having taken a look at a few textbooks that discuss numerical methods and C programming, I was gladly surprised when browsing through "programming in C with numerical methods for engineers" by Rojiani. I understand of course it's important that one need to have a solid background in numerical methods prior to try implementing them on a computer. I would like to know if someone here has been using this book and if possible point out strengths and weaknesses of this textbook. Thanks a lot...

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  • good books on numerical computation with C

    - by yCalleecharan
    Hi, I've read the post "What is the best book on numerical methods?" and I wish to ask more or less the same question but in relation to C programming. Most of the time, C programming books on numerical methods are just another version of the author's previous Fortran book on the same subject. I've seen Applied numerical methods in C by Nakamura, Shoichiro and the C codes are not good programming practice. I've heard bad comments about Numerical Recipes by Press. Do you know good books on C that discusses numerical methods. It's seem better for me to ask about good books on C discussing numerical methods than rather asking books on numerical methods that discusses C. I've heard about Numerical Algorithms with C by Giesela Engeln-Müllges and A Numerical Library in C for Scientists and Engineers bu Lau but haven't read them. Good books will always have algorithms implemented in the programming language in a smart way. Thanks a lot...

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  • F# Extention Methods on Lists, IEnumberable, etc

    - by flevine100
    I have searched StackOverflow (and other sources) for this answer, but can't seem to find anything. In C#, if I had a widget definition, say: class widget { public string PrettyName() { ... do stuff here } } and I wanted to allow for easy printing of a list of Widgets, I might do this: namespace ExtensionMethods { public static PrintAll( this IEnumerable<Widget> widgets, TextWriter writer ) { foreach(var w in widgets) { writer.WriteLine( w.PrettyName() ) } } } How would I accomplish something similar with a record type and a collection (List or Seq preferrably in F#). I'd love to have a list of Widgest and be able to call a function right on the collection that did something like this. Assume (since it's F#) that the function would not be changing the state of the collection that it's attached to, but returning some new value.

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  • Are long methods always bad?

    - by wobbily_col
    So looking around earlier I noticed some comments about long methods being bad practice. I am not sure I always agree that long methods are bad (and would like opinions from others). For example I have some Django views that do a bit of processing of the objects before sending them to the view, a long method being 350 lines of code. I have my code written so that it deals with the paramaters - sorting / filtering the queryset, then bit by bit does some processing on the objects my query has returned. So the processing is mainly conditional aggregation, that has complex enough rules it can't easily be done in the database, so I have some variables declared outside the main loop then get altered during the loop. varaible_1 = 0 variable_2 = 0 for object in queryset : if object.condition_condition_a and variable_2 > 0 : variable 1+= 1 ..... ... . more conditions to alter the variables return queryset, and context So according to the theory I should factor out all the code into smaller methods, so That I have the view method as being maximum one page long. However having worked on various code bases in the past, I sometimes find it makes the code less readable, when you need to constantly jump from one method to the next figuring out all the parts of it, while keeping the outermost method in your head. I find that having a long method that is well formatted, you can see the logic more easily, as it isn't getting hidden away in inner methods. I could factor out the code into smaller methods, but often there is is an inner loop being used for two or three things, so it would result in more complex code, or methods that don't do one thing but two or three (alternatively I could repeat inner loops for each task, but then there will be a performance hit). So is there a case that long methods are not always bad? Is there always a case for writing methods, when they will only be used in one place?

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  • Read XML Files using LINQ to XML and Extension Methods

    - by psheriff
    In previous blog posts I have discussed how to use XML files to store data in your applications. I showed you how to read those XML files from your project and get XML from a WCF service. One of the problems with reading XML files is when elements or attributes are missing. If you try to read that missing data, then a null value is returned. This can cause a problem if you are trying to load that data into an object and a null is read. This blog post will show you how to create extension methods to detect null values and return valid values to load into your object. The XML Data An XML data file called Product.xml is located in the \Xml folder of the Silverlight sample project for this blog post. This XML file contains several rows of product data that will be used in each of the samples for this post. Each row has 4 attributes; namely ProductId, ProductName, IntroductionDate and Price. <Products>  <Product ProductId="1"           ProductName="Haystack Code Generator for .NET"           IntroductionDate="07/01/2010"  Price="799" />  <Product ProductId="2"           ProductName="ASP.Net Jumpstart Samples"           IntroductionDate="05/24/2005"  Price="0" />  ...  ...</Products> The Product Class Just as you create an Entity class to map each column in a table to a property in a class, you should do the same for an XML file too. In this case you will create a Product class with properties for each of the attributes in each element of product data. The following code listing shows the Product class. public class Product : CommonBase{  public const string XmlFile = @"Xml/Product.xml";   private string _ProductName;  private int _ProductId;  private DateTime _IntroductionDate;  private decimal _Price;   public string ProductName  {    get { return _ProductName; }    set {      if (_ProductName != value) {        _ProductName = value;        RaisePropertyChanged("ProductName");      }    }  }   public int ProductId  {    get { return _ProductId; }    set {      if (_ProductId != value) {        _ProductId = value;        RaisePropertyChanged("ProductId");      }    }  }   public DateTime IntroductionDate  {    get { return _IntroductionDate; }    set {      if (_IntroductionDate != value) {        _IntroductionDate = value;        RaisePropertyChanged("IntroductionDate");      }    }  }   public decimal Price  {    get { return _Price; }    set {      if (_Price != value) {        _Price = value;        RaisePropertyChanged("Price");      }    }  }} NOTE: The CommonBase class that the Product class inherits from simply implements the INotifyPropertyChanged event in order to inform your XAML UI of any property changes. You can see this class in the sample you download for this blog post. Reading Data When using LINQ to XML you call the Load method of the XElement class to load the XML file. Once the XML file has been loaded, you write a LINQ query to iterate over the “Product” Descendants in the XML file. The “select” portion of the LINQ query creates a new Product object for each row in the XML file. You retrieve each attribute by passing each attribute name to the Attribute() method and retrieving the data from the “Value” property. The Value property will return a null if there is no data, or will return the string value of the attribute. The Convert class is used to convert the value retrieved into the appropriate data type required by the Product class. private void LoadProducts(){  XElement xElem = null;   try  {    xElem = XElement.Load(Product.XmlFile);     // The following will NOT work if you have missing attributes    var products =         from elem in xElem.Descendants("Product")        orderby elem.Attribute("ProductName").Value        select new Product        {          ProductId = Convert.ToInt32(            elem.Attribute("ProductId").Value),          ProductName = Convert.ToString(            elem.Attribute("ProductName").Value),          IntroductionDate = Convert.ToDateTime(            elem.Attribute("IntroductionDate").Value),          Price = Convert.ToDecimal(elem.Attribute("Price").Value)        };     lstData.DataContext = products;  }  catch (Exception ex)  {    MessageBox.Show(ex.Message);  }} This is where the problem comes in. If you have any missing attributes in any of the rows in the XML file, or if the data in the ProductId or IntroductionDate is not of the appropriate type, then this code will fail! The reason? There is no built-in check to ensure that the correct type of data is contained in the XML file. This is where extension methods can come in real handy. Using Extension Methods Instead of using the Convert class to perform type conversions as you just saw, create a set of extension methods attached to the XAttribute class. These extension methods will perform null-checking and ensure that a valid value is passed back instead of an exception being thrown if there is invalid data in your XML file. private void LoadProducts(){  var xElem = XElement.Load(Product.XmlFile);   var products =       from elem in xElem.Descendants("Product")      orderby elem.Attribute("ProductName").Value      select new Product      {        ProductId = elem.Attribute("ProductId").GetAsInteger(),        ProductName = elem.Attribute("ProductName").GetAsString(),        IntroductionDate =            elem.Attribute("IntroductionDate").GetAsDateTime(),        Price = elem.Attribute("Price").GetAsDecimal()      };   lstData.DataContext = products;} Writing Extension Methods To create an extension method you will create a class with any name you like. In the code listing below is a class named XmlExtensionMethods. This listing just shows a couple of the available methods such as GetAsString and GetAsInteger. These methods are just like any other method you would write except when you pass in the parameter you prefix the type with the keyword “this”. This lets the compiler know that it should add this method to the class specified in the parameter. public static class XmlExtensionMethods{  public static string GetAsString(this XAttribute attr)  {    string ret = string.Empty;     if (attr != null && !string.IsNullOrEmpty(attr.Value))    {      ret = attr.Value;    }     return ret;  }   public static int GetAsInteger(this XAttribute attr)  {    int ret = 0;    int value = 0;     if (attr != null && !string.IsNullOrEmpty(attr.Value))    {      if(int.TryParse(attr.Value, out value))        ret = value;    }     return ret;  }   ...  ...} Each of the methods in the XmlExtensionMethods class should inspect the XAttribute to ensure it is not null and that the value in the attribute is not null. If the value is null, then a default value will be returned such as an empty string or a 0 for a numeric value. Summary Extension methods are a great way to simplify your code and provide protection to ensure problems do not occur when reading data. You will probably want to create more extension methods to handle XElement objects as well for when you use element-based XML. Feel free to extend these extension methods to accept a parameter which would be the default value if a null value is detected, or any other parameters you wish. NOTE: You can download the complete sample code at my website. http://www.pdsa.com/downloads. Choose “Tips & Tricks”, then "Read XML Files using LINQ to XML and Extension Methods" from the drop-down. Good Luck with your Coding,Paul D. Sheriff  

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  • Naming methods that do the same thing but return different types

    - by Konstantin Ð.
    Let's assume that I'm extending a graphical file chooser class (JFileChooser). This class has methods which display the file chooser dialog and return a status signature in the form of an int: APPROVE_OPTION if the user selects a file and hits Open /Save, CANCEL_OPTION if the user hits Cancel, and ERROR_OPTION if something goes wrong. These methods are called showDialog(). I find this cumbersome, so I decide to make another method that returns a File object: in the case of APPROVE_OPTION, it returns the file selected by the user; otherwise, it returns null. This is where I run into a problem: would it be okay for me to keep the showDialog() name, even though methods with that name — and a different return type — already exist? To top it off, my method takes an additional parameter: a File which denotes in which directory the file chooser should start. My question to you: Is it okay to call a method the same name as a superclass method if they return different types? Or would that be confusing to API users? (If so, what other name could I use?) Alternatively, should I keep the name and change the return type so it matches that of the other methods? public int showDialog(Component parent, String approveButtonText) // Superclass method public File showDialog(Component parent, File location) // My method

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  • Extension methods conflict

    - by Yochai Timmer
    Lets say I have 2 extension methods to string, in 2 different namespaces: namespace test1 { public static class MyExtensions { public static int TestMethod(this String str) { return 1; } } } namespace test2 { public static class MyExtensions2 { public static int TestMethod(this String str) { return 2; } } } These methods are just for example, they don't really do anything. Now lets consider this piece of code: using System; using test1; using test2; namespace blah { public static class Blah { public Blah() { string a = "test"; int i = a.TestMethod(); //Which one is chosen ? } } } I know that only one of the extension methods will be chosen. Which one will it be ? and why ? How can I choose a certain method from a certain namespace ? Edit: Usually I'd use Namespace.ClassNAME.Method() ... But that just beats the whole idea of extension methods. And I don't think you can use Variable.Namespace.Method()

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  • Do private static methods in C# hurt anything?

    - by fish
    I created a private validation method for a certain validation that happens multiple times in my class (I can't store the validated data for various reasons). Now, ReSharper suggests that the function could be made static. I'm a little reluctant to do so due known problems with static methods. It would be a private static method. My question is, can private static methods cause similar coupling and testing problems like public static methods? Is it a bad practice? I would guess not, but I'm not sure if there is a pitfall here.

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  • Can't I just use all static methods?

    - by Reddy S R
    What's the difference between the two UpdateSubject methods below? I felt using static methods is better if you just want to operate on the entities. In which situations should I go with non-static methods? public class Subject { public int Id {get; set;} public string Name { get; set; } public static bool UpdateSubject(Subject subject) { //Do something and return result return true; } public bool UpdateSubject() { //Do something on 'this' and return result return true; } } I know I will be getting many kicks from the community for this really annoying question but I could not stop myself asking it. Does this become impractical when dealing with inheritance?

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  • Naming methods that perform HTTP GET/POST calls?

    - by antonpug
    In the application I am currently working on, there are generally 3 types of HTTP calls: pure GETs pure POSTs (updating the model with new data) "GET" POSTs (posting down an object to get some data back, no updates to the model) In the integration service, generally we name methods that post "postSomething()", and methods that get, "getSomething()". So my question is, if we have a "GET" POST, should the method be called: getSomething - seeing as the purpose is to obtain data postSomething - since we are technically using POST performSomeAction - arbitrary name that's more relevant to the action What are everyone's thoughts?

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  • What is the best Java numerical method package?

    - by Bob Cross
    I am looking for a Java-based numerical method package that provides functionality including: Solving systems of equations using different numerical analysis algorithms. Matrix methods (e.g., inversion). Spline approximations. Probability distributions and statistical methods. In this case, "best" is defined as a package with a mature and usable API, solid performance and numerical accuracy. Edit: derick van brought up a good point in that cost is a factor. I am heavily biased in favor of free packages but others may have a different emphasis.

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  • The best way to have a pointer to several methods - critique requested

    - by user827992
    I'm starting with a short introduction of what i know from the C language: a pointer is a type that stores an adress or a NULL the * operator reads the left value of the variable on its right and use this value as address and reads the value of the variable at that address the & operator generate a pointer to the variable on its right so i was thinking that in C++ the pointers can work this way too, but i was wrong, to generate a pointer to a static method i have to do this: #include <iostream> class Foo{ public: static void dummy(void){ std::cout << "I'm dummy" << std::endl; }; }; int main(){ void (*p)(); p = Foo::dummy; // step 1 p(); p = &(Foo::dummy); // step 2 p(); p = Foo; // step 3 p->dummy(); return(0); } now i have several questions: why step 1 works why step 2 works too, looks like a "pointer to pointer" for p to me, very different from step 1 why step 3 is the only one that doesn't work and is the only one that makes some sort of sense to me, honestly how can i write an array of pointers or a pointer to pointers structure to store methods ( static or non-static from real objects ) what is the best syntax and coding style for generating a pointer to a method?

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  • Numerical Integration of a vector in matlab?

    - by clowny
    Hello all! Well i have an issue. I have a vector of 358 numbers. I'd like to make a numerical integration of this vector, but i don't know the function of this one. I found that we can use trapz or quad, but i don't really understand how to integrate without the function. Thanx for any help!

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  • Improper integral calculation using numerical intergration

    - by Andrei Taptunov
    I'm interested in calculation of Improper Integral for a function. Particularly it's a Gauss Integral. Using a numerical integration does make sense for a definite integrals but how should I deal with improper integrals ? Is there any was to extrapolate the function "around" negative infinity or should I just remove this part and start integration from some particular value because cumulative sum near "negative infinity" is almost non-existent for Gauss integral? Perhaps there are some algorithms that I'm not aware about.

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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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  • Numerical Pattern Matching

    - by Timothy Strimple
    A project I'm researching requires some numerical pattern matching. My searches haven't turned up many relevant hits since most results tend to be around text pattern matching. The idea is we'll have certain wave patterns we'll need to be watching for and trying to match incoming data vs the wave database we will be building. Here is and example of one of the wave patterns we'll need to be matching against. There is clearly a pattern there, but the peaks will not have the exact same values, but the overall shape of the wave iterations will be very similar. Does anyone have any advice on how to go about storing and later matching these patterns, and / or other search terms I can use to find more information on the subject of pattern matching? Thanks, Tim.

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  • Prefer extension methods for encapsulation and reusability?

    - by tzaman
    edit4: wikified, since this seems to have morphed more into a discussion than a specific question. In C++ programming, it's generally considered good practice to "prefer non-member non-friend functions" instead of instance methods. This has been recommended by Scott Meyers in this classic Dr. Dobbs article, and repeated by Herb Sutter and Andrei Alexandrescu in C++ Coding Standards (item 44); the general argument being that if a function can do its job solely by relying on the public interface exposed by the class, it actually increases encapsulation to have it be external. While this confuses the "packaging" of the class to some extent, the benefits are generally considered worth it. Now, ever since I've started programming in C#, I've had a feeling that here is the ultimate expression of the concept that they're trying to achieve with "non-member, non-friend functions that are part of a class interface". C# adds two crucial components to the mix - the first being interfaces, and the second extension methods: Interfaces allow a class to formally specify their public contract, the methods and properties that they're exposing to the world. Any other class can choose to implement the same interface and fulfill that same contract. Extension methods can be defined on an interface, providing any functionality that can be implemented via the interface to all implementers automatically. And best of all, because of the "instance syntax" sugar and IDE support, they can be called the same way as any other instance method, eliminating the cognitive overhead! So you get the encapsulation benefits of "non-member, non-friend" functions with the convenience of members. Seems like the best of both worlds to me; the .NET library itself providing a shining example in LINQ. However, everywhere I look I see people warning against extension method overuse; even the MSDN page itself states: In general, we recommend that you implement extension methods sparingly and only when you have to. (edit: Even in the current .NET library, I can see places where it would've been useful to have extensions instead of instance methods - for example, all of the utility functions of List<T> (Sort, BinarySearch, FindIndex, etc.) would be incredibly useful if they were lifted up to IList<T> - getting free bonus functionality like that adds a lot more benefit to implementing the interface.) So what's the verdict? Are extension methods the acme of encapsulation and code reuse, or am I just deluding myself? (edit2: In response to Tomas - while C# did start out with Java's (overly, imo) OO mentality, it seems to be embracing more multi-paradigm programming with every new release; the main thrust of this question is whether using extension methods to drive a style change (towards more generic / functional C#) is useful or worthwhile..) edit3: overridable extension methods The only real problem identified so far with this approach, is that you can't specialize extension methods if you need to. I've been thinking about the issue, and I think I've come up with a solution. Suppose I have an interface MyInterface, which I want to extend - I define my extension methods in a MyExtension static class, and pair it with another interface, call it MyExtensionOverrider. MyExtension methods are defined according to this pattern: public static int MyMethod(this MyInterface obj, int arg, bool attemptCast=true) { if (attemptCast && obj is MyExtensionOverrider) { return ((MyExtensionOverrider)obj).MyMethod(arg); } // regular implementation here } The override interface mirrors all of the methods defined in MyExtension, except without the this or attemptCast parameters: public interface MyExtensionOverrider { int MyMethod(int arg); string MyOtherMethod(); } Now, any class can implement the interface and get the default extension functionality: public class MyClass : MyInterface { ... } Anyone that wants to override it with specific implementations can additionally implement the override interface: public class MySpecializedClass : MyInterface, MyExtensionOverrider { public int MyMethod(int arg) { //specialized implementation for one method } public string MyOtherMethod() { // fallback to default for others MyExtension.MyOtherMethod(this, attemptCast: false); } } And there we go: extension methods provided on an interface, with the option of complete extensibility if needed. Fully general too, the interface itself doesn't need to know about the extension / override, and multiple extension / override pairs can be implemented without interfering with each other. I can see three problems with this approach - It's a little bit fragile - the extension methods and override interface have to be kept synchronized manually. It's a little bit ugly - implementing the override interface involves boilerplate for every function you don't want to specialize. It's a little bit slow - there's an extra bool comparison and cast attempt added to the mainline of every method. Still, all those notwithstanding, I think this is the best we can get until there's language support for interface functions. Thoughts?

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  • The speed of .NET in numerical computing

    - by Yin Zhu
    In my experience, .net is 2 to 3 times slower than native code. (I implemented L-BFGS for multivariate optimization). I have traced the ads on stackoverflow to http://www.centerspace.net/products/ the speed is really amazing, the speed is close to native code. How can they do that? They said that: Q. Is NMath "pure" .NET? A. The answer depends somewhat on your definition of "pure .NET". NMath is written in C#, plus a small Managed C++ layer. For better performance of basic linear algebra operations, however, NMath does rely on the native Intel Math Kernel Library (included with NMath). But there are no COM components, no DLLs--just .NET assemblies. Also, all memory allocated in the Managed C++ layer and used by native code is allocated from the managed heap. Can someone explain more to me? Thanks!

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  • curious ill conditioned numerical problem

    - by aaa
    hello. somebody today showed me this curious ill conditioned problem (apparently pretty famous), which looks relatively simple ƒ = (333.75 - a^2)b^6 + a^2 (11a^2 b^2 - 121b^4 - 2) + 5.5b^8 + a/(2^b) where a = 77617 and b = 33096 can you determine correct answer?

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  • C++ Numerical truncation error

    - by Andrew
    Hello everyone, sorry if dumb but could not find an answer. #include <iostream> using namespace std; int main() { double a(0); double b(0.001); cout << a - 0.0 << endl; for (;a<1.0;a+=b); cout << a - 1.0 << endl; for (;a<10.0;a+=b); cout << a - 10.0 << endl; cout << a - 10.0-b << endl; return 0; } Output: 0 6.66134e-16 0.001 -1.03583e-13 Tried compiling it with MSVC9, MSVC10, Borland C++ 2010. All of them arrive in the end to the error of about 1e-13. Is it normal to have such a significant error accumulation over only a 1000, 10000 increments?

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  • Thoughts on C# Extension Methods

    - by Damon
    I'm not a huge fan of extension methods.  When they first came out, I remember seeing a method on an object that was fairly useful, but when I went to use it another piece of code that method wasn't available.  Turns out it was an extension method and I hadn't included the appropriate assembly and imports statement in my code to use it.  I remember being a bit confused at first about how the heck that could happen (hey, extension methods were new, cut me some slack) and it took a bit of time to track down exactly what it was that I needed to include to get that method back.  I just imagined a new developer trying to figure out why a method was missing and fruitlessly searching on MSDN for a method that didn't exist and it just didn't sit well with me. I am of the opinion that if you have an object, then you shouldn't have to include additional assemblies to get additional instance level methods out of that object.  That opinion applies to namespaces as well - I do not like it when the contents of a namespace are split out into multiple assemblies.  I prefer to have static utility classes instead of extension methods to keep things nicely packaged into a cohesive unit.  It also makes it abundantly clear where utility methods are used in code.  I will concede, however, that it can make code a bit more verbose and lengthy.  There is always a trade-off. Some people harp on extension methods because it breaks the tenants of object oriented development and allows you to add methods to sealed classes.  Whatever.  Extension methods are just utility methods that you can tack onto an object after the fact.  Extension methods do not give you any more access to an object than the developer of that object allows, so I say that those who cry OO foul on extension methods really don't have much of an argument on which to stand.  In fact, I have to concede that my dislike of them is really more about style than anything of great substance. One interesting thing that I found regarding extension methods is that you can call them on null objects. Take a look at this extension method: namespace ExtensionMethods {   public static class StringUtility   {     public static int WordCount(this string str)     {       if(str == null) return 0;       return str.Split(new char[] { ' ', '.', '?' },         StringSplitOptions.RemoveEmptyEntries).Length;     }   }   } Notice that the extension method checks to see if the incoming string parameter is null.  I was worried that the runtime would perform a check on the object instance to make sure it was not null before calling an extension method, but that is apparently not the case.  So, if you call the following code it runs just fine. string s = null; int words = s.WordCount(); I am a big fan of things working, but this seems to go against everything I've come to know about instance level methods.  However, an extension method is really a static method masquerading as an instance-level method, so I suppose it would be far more frustrating if it failed since there is really no reason it shouldn't succeed. Although I'm not a fan of extension methods, I will say that if you ever find yourself at an impasse with a die-hard fan of either the utility class or extension method approach, then there is a common ground.  Extension methods are defined in static classes, and you call them from those static classes as well as directly from the objects they extend.  So if you build your utility classes using extension methods, then you can have it your way and they can have it theirs. 

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  • Numerical stability in continuous physics simulation

    - by Panda Pajama
    Pretty much all of the game development I have been involved with runs afoul of simulating a physical world in discrete time steps. This is of course very simple, but hardly elegant (not to mention mathematically inaccurate). It also has severe disadvantages when large values are involved (either very large speeds, or very large time intervals). I'm trying to make a continuous physics simulation, just for learning, which goes like this: time = get_time() while true do new_time = get_time() update_world(new_time - time) render() time = new_time end And update_world() is a continuous physical simulation. Meaning that for example, for an accelerated object, instead of doing object.x = object.x + object.vx * timestep object.vx = object.vx + object.ax * timestep -- timestep is fixed I'm doing something like object.x = object.x + object.vx * deltatime + object.ax * ((deltatime ^ 2) / 2) object.vx = object.vx + object.ax * deltatime However, I'm having a hard time with the numerical stability of my solutions, especially for very large time intervals (think of simulating a physical world for hundreds of thousands of virtual years). Depending on the framerate, I get wildly different solutions. How can I improve the numerical stability of my continuous physical simulations?

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  • Prefer class members or passing arguments between internal methods?

    - by geoffjentry
    Suppose within the private portion of a class there is a value which is utilized by multiple private methods. Do people prefer having this defined as a member variable for the class or passing it as an argument to each of the methods - and why? On one hand I could see an argument to be made that reducing state (ie member variables) in a class is generally a good thing, although if the same value is being repeatedly used throughout a class' methods it seems like that would be an ideal candidate for representation as state for the class to make the code visibly cleaner if nothing else. Edit: To clarify some of the comments/questions that were raised, I'm not talking about constants and this isn't relating to any particular case rather just a hypothetical that I was talking to some other people about. Ignoring the OOP angle for a moment, the particular use case that I had in mind was the following (assume pass by reference just to make the pseudocode cleaner) int x doSomething(x) doAnotherThing(x) doYetAnotherThing(x) doSomethingElse(x) So what I mean is that there's some variable that is common between multiple functions - in the case I had in mind it was due to chaining of smaller functions. In an OOP system, if these were all methods of a class (say due to refactoring via extracting methods from a large method), that variable could be passed around them all or it could be a class member.

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  • What modern alternatives to Numerical Recipes exist?

    - by Stewart
    In the past, the Numerical Recipes book was considered the gold standard reference for numerical algorithms. The earliest Fortran Edition was followed by editions in C and C++ and others, bringing it then more up-to-date. Through these, it provided reference code for the state-of-the-art algorithms of the day. Older editions are available online for free nowadays. Unfortunately, I think it is now mostly useful only as a historic tome. The "software engineering" practises seem to me to be outdated, and the actual content hasn't kept pace with the literature. What comprehensive yet approachable references should the modern programmer look at instead?

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