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  • Is it safe to run an operating system from an USB flash drive?

    - by Georg
    I've got a laptop that has a broken harddisk controller. Replacing the motherboard is quite expensive. I thought about buying a flash drive and installing & running the system from it. However, I'm concerned about some things. Speed: Are they fast enough for swap memory (I've got only 1GB RAM installed.) I'm considering buying 2 or 3 of them and making them into a RAID. What about limited write cycles? How long will it last for a system that has a filesystem with journaling enabled? I'd hate to abandon it. Are there significant differences between internal SSD which are used in modern laptops like MacBooks and USB flash drives? What should I expect in 10 years when the memory wear starts kicking in?

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  • System.Uri("") in default browser?

    - by Hallgaws
    I am using special program where it loads some url imagine it like window with automatically load www.google.com - program starts and it load the site - BUT when you click on some links in the program window it opens in default browser - how it can be opened in default browser - I am using this code: <Global.Microsoft.VisualBasic.CompilerServices.DesignerGenerated()> _ Partial Class Form1 Inherits System.Windows.Forms.Form <System.Diagnostics.DebuggerNonUserCode()> _ Protected Overrides Sub Dispose(ByVal disposing As Boolean) Try If disposing AndAlso components IsNot Nothing Then components.Dispose() End If Finally MyBase.Dispose(disposing) End Try End Sub Private components As System.ComponentModel.IContainer <System.Diagnostics.DebuggerStepThrough()> _ Private Sub InitializeComponent() Me.components = New System.ComponentModel.Container Dim resources As System.ComponentModel.ComponentResourceManager = New System.ComponentModel.ComponentResourceManager(GetType(Form1)) Me.WB = New System.Windows.Forms.WebBrowser ..... ..... Me.WB.Url = New System.Uri("http://www.google.com/", System.UriKind.Absolute) Using Visual Basic 2008

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  • The "System" process has my BCD open(windows 7)?

    - by Epic_orange
    I discovered this when I tried to use EasyBCD to well, edit by bcd, but it said "The current file is in use and cannot be opened by EasyBCD..." So I tried to use handle.exe to stop it but it said \Handle>handle -c 15C -p 4 Handle v3.46 Copyright (C) 1997-2011 Mark Russinovich Sysinternals - www.sysinternals.com 15C: File (---) C:\Boot\BCD Close handle 15C in System (PID 4)? (y/n) y Error closing handle: The handle is invalid. Why does system have my bcd open and how can i stop it? I have tried rebooting and googling.

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  • Get positions for NAs only in the "middle" of a matrix column

    - by Abiel
    I want to obtain an index that refers to the positions of NA values in a matrix where the index is true if a given cell is NA and there is at least one non-NA value before and after it in the column. For example, given the following matrix [,1] [,2] [,3] [,4] [1,] NA 1 NA 1 [2,] 1 NA NA 2 [3,] NA 2 NA 3 the only value of the index that comes back TRUE should be [2,2]. Is there a compact expression for what I want to do? If I had to I could loop through columns and use something like min(which(!is.na(x[,i]))) to find the first non-NA value in each column, and then set all values before that to FALSE (and the same for all values after the max). This way I would not select leading and trailing NA values. But this seems a bit messy, so I'm wondering if there is a cleaner expression that does this without loops.

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  • Tool to diagonalize large matrices

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

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  • What are best monitoring tool customizable for cluster / distributed system?

    - by Adil
    I am working on a system having multiple servers. I am interested in monitoring some server specific data like CPU/memory usage, disk/filesystem usage, network traffic, system load etc. and some other my process specific data. What are available open source that can serve my purpose? If it provides to customize the parameter to be monitored and monitor your own data by creating plugin / agent. Any suggestions? I heard of Nagios, Zabbix and Pandora but not sure if they provide such interface.

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  • extract data from an array without using loop in R

    - by Manolo
    I have a vector v with row positions: v<-c(10,3,100,50,...) with those positions I want to extract elements of a matrix, having a column fixed, for example lets suppose my column number is 2, so I am doing: data<-c() data<-c(matrix[[v]][[2]]) matrix has the data in the following format: [[34]] [1] "200_s_at" "4853" "1910" "3554" "2658" So for example, I want to extract from the row 342 the value 1910 only, column 2, and do the same with the next rows but I got an error when I want to do that, is it possible to do it directly? or should I have a loop that read one by one the positions in v and fill the data vector like: #algorithm for i<-1 to length(v) pos<-v[i] data[[i]]<-c(matriz[[pos]][[2]]) next i Thanks

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  • Compute rolling window covariance matrix

    - by user1665355
    I am trying to compute a rolling window (shifting by 1 day) covariance matrix for a number of assets. Say my df looks like this: df <- data.frame(x = 0:4, y = 5:9,z=1:5,u=4:8) How would a possible for loop look like if I want to calculate a covariance matrix on a rolling basis by shifting the rolling window by 1 day? Or should I use some apply family function? What time series class would be preferrable if I want to create a time series object for the loop above? I simply can't get it... Best Regards

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  • Generate (in R) a matrix of all possible outcomes for throwing n dice (ignoring order)

    - by Brani
    In cases where order does matter, it's rather easy to generate the matrix of all possible outcomes. One way for doing this is using expand.grid as shown here. What if it doesn't? If I'm right, the number of possible combinations is (S+N-1)!/S!(N-1)!, where S is the number of dice, each with N sides numbered 1 through N. (It is different from the well known combinations formula because it is possible for the same number to appear on more than one dice). For example, when throwing four six-sided dice, N=6 and S=4, so the number of possible combinations is (4+6-1)!/4!(6-1)! = 9!/4!x5! = 126. How can I generate a matrix of these 126 possible outcomes? Thank you.

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  • parallel to usb driver windows 7 dot matrix printer

    - by user975234
    I recently built a new Core i5 system with Windows 7 Professional. I had planned to hook up my TVS MSP 250 XL printer using a USB to parallel cable Once I plug the cable in, Windows 7 recognizes it as an IEEE-1284 controller and automatically installs the appropriate driver. However, in the status window it reports the following: "USB Printing Support -- Ready to use" "No Printer Attached -- Ready to use" When I then go ahead and manually add the printer using the "virtual printer port for USB" I can add the printer seemingly without problem. Once finished, it appears in the Devices and Printers panel. Yet, all attempts to print on this printer fail. It appears that simply no data is sent to the printer (either by programs like word or adobe or by attempting to print a test page. Does anybody know how to fix this?

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  • Generate a matrix of all possible outcomes for throwing n dice (ignoring order)

    - by Brani
    In cases where order does matter, it's rather easy to generate the matrix of all possible outcomes. One way for doing this is using expand.grid as shown here. What if it doesn't? If I'm right, the number of possible combinations is (S+N-1)!/S!(N-1)!, where S is the number of dice, each with N sides numbered 1 through N. (It is different from the well known combinations formula because it is possible for the same number to appear on more than one dice). For example, when throwing four six-sided dice, N=6 and S=4, so the number of possible combinations is (4+6-1)!/4!(6-1)! = 9!/4!x5! = 126. How can I generate a matrix of these 126 possible outcomes? Thank you.

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  • Calculating the null space of a matrix

    - by Ainsworth
    I'm attempting to solve a set of equations of the form Ax = 0. A is known 6x6 matrix and I've written the below code using SVD to get the vector x which works to a certain extent. The answer is approximately correct but not good enough to be useful to me, how can I improve the precision of the calculation? Lowering eps below 1.e-4 causes the function to fail. from numpy.linalg import * from numpy import * A = matrix([[0.624010149127497 ,0.020915658603923 ,0.838082638087629 ,62.0778180312547 ,-0.336 ,0], [0.669649399820597 ,0.344105317421833 ,0.0543868015800246 ,49.0194290212841 ,-0.267 ,0], [0.473153758252885 ,0.366893577716959 ,0.924972565581684 ,186.071352614705 ,-1 ,0], [0.0759305208803158 ,0.356365401030535 ,0.126682113674883 ,175.292109352674 ,0 ,-5.201], [0.91160934274653 ,0.32447818779582 ,0.741382053883291 ,0.11536775372698 ,0 ,-0.034], [0.480860406786873 ,0.903499596111067 ,0.542581424762866 ,32.782593418975 ,0 ,-1]]) def null(A, eps=1e-3): u,s,vh = svd(A,full_matrices=1,compute_uv=1) null_space = compress(s <= eps, vh, axis=0) return null_space.T NS = null(A) print "Null space equals ",NS,"\n" print dot(A,NS)

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  • Windows 7 System Image Restore to SSD - SSD not detected?

    - by steviey
    I recently bought a new SSD (OCZ Vertex 2 120GB) as a replacement for my laptop HDD . I created a System Image on an external drive using the Win 7 backup and restore tool and then restored this image to the SSD using the Win 7 disk. AHCI is enabled in Win 7 and the BIOS, but it seems that windows is not detecting the disk as an SSD because, in the defragger, the disk is still available for defragging. Prefetch and Superfetch were also still enabled. I have manually disabled scheduled defragging, prefetch and superfetch. I have also checked that TRIM is enabled using: fsutil behavior query DisableDeleteNotify and the result is '0' (it is). Performance seems ok, but not quite as fast as I expected. Was restoring from a win7 generated system image a mistake? Am I getting optimal SSD performance?

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  • Transform LINQ Dataset into a Matrix for export

    - by Mad Halfling
    Hi folks, I've got a data table with columns in which include Item, Category and Value (and others, but those are the only relevant ones for this problem) that I access via LINQ in a C# ASP.Net MVC app. I want to transform these into a matrix and output that as a CSV file to pull into Excel as matrix with the items down the side, the categories across the top and the values in the row cells. However, I don't know how many, or what, categories there will be in this table, nor will there always be a record for each item/category combination. I've written this by looping round, getting my "master category" list, then looking again for each item, filling in either blank or Value, depending on whether the item/category record exists, but as there are currently 27000 records in the table, this isn't as fast as I'd like. Is there a slicker and faster way I can do this, maybe via LINQ (firing into a quicker SQL statement so the DB server can do the leg-work), or will any method essentially come back to what I am doing? Thx MH

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  • Why is the System process listening on Port 443?

    - by Cornelius
    I am having problems starting my apache server, because port 443 is already in use. It turns out, the system process (PID 4) uses the port 443. I don't have IIS installed, the services.msc shows (predicatbly) no Exchange server running, nor WWW-Services, nor IIS. I have no idea how to find out what service uses that port short of just disabling each service one after the other, and I am not even sure that would help. I would be grateful if someone could point me towards how I can get my SSL port back, thank you :) P.S.: Of course "just switch apache to another port for SSL" would solve the problem of not being able to start apache. But I'd still like to know what is so insistent about hogging port 443. :) Edit: I by now took the 'hard route' and disabled services one after the other. It turned out that the "Routing and RAS" service was the culprit. Thank you all for the valuable input and the new tools in the combat against "WTF does my system do now".

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  • Is there an application that will move an active window to the system tray without minimizing it

    - by Robert
    I have an active application (lets say an instant messaging client) whose buddy list I have up (active) on one side of my screen at all times in Windows 7. I would like to remove that icon from the taskbar either by moving it to the system tray (as happens when the app is minimized) or by just removing it altogether. To be clear: I do not want to minimize the window to the system tray (as described in Windows app to hide application from taskbar) I want to keep the window its NORMAL size and location and just get rid of the taskbar icon for the app. I'm looking for any tool, third party, native, registry hack, to accomplish this.

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  • handling matrix data in python

    - by Ovisek
    I was trying to progressively subtract values of a 3D matrix. The matrix looks like: ATOM 1223 ZX SOD A 11 2.11 -1.33 12.33 ATOM 1224 ZY SOD A 11 -2.99 -2.92 20.22 ATOM 1225 XH HEL A 12 -3.67 9.55 21.54 ATOM 1226 SS ARG A 13 -6.55 -3.09 42.11 ... here the last three columns are representing values for axes x,y,z respectively. now I what I wanted to do is, take the values of x,y,z for 1st line and subtract with 2nd,3rd,4th line in a iterative way and print the values for each axes. I was using: for line in map(str.split,inp): x = line[-3] y = line[-2] z = line[-1] for separating the values, but how to do in iterative way. should I do it by using Counter.

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  • Where is this System.MissingMethodException occurring? How can I tell?

    - by Jeremy Holovacs
    I am a newbie to ASP.NET MVC (v2), and I am trying to use a strongly-typed view tied to a model object that contains two optional multi-select listbox objects. Upon clicking the submit button, these objects may have 0 or more values selected for them. My model class looks like this: using System; using System.Web.Mvc; using System.Collections.Generic; namespace ModelClasses.Messages { public class ComposeMessage { public bool is_html { get; set; } public bool is_urgent { get; set; } public string message_subject { get; set; } public string message_text { get; set; } public string action { get; set; } public MultiSelectList recipients { get; set; } public MultiSelectList recipient_roles { get; set; } public ComposeMessage() { this.is_html = false; this.is_urgent = false; this.recipients = new MultiSelectList(new Dictionary<int, string>(), "Key", "Value"); this.recipient_roles = new MultiSelectList(new Dictionary<int, string>(), "Key", "Value"); } } } My view looks like this: <%@ Page Title="" Language="C#" MasterPageFile="~/Views/Shared/Site.Master" Inherits="System.Web.Mvc.ViewPage<ModelClasses.Messages.ComposeMessage>" %> <asp:Content ID="Content1" ContentPlaceHolderID="TitleContent" runat="server">Compose A Message </asp:Content> <asp:Content ID="Content2" ContentPlaceHolderID="MainContent" runat="server"> <h2> Compose A New Message:</h2> <br /> <span id="navigation_top"> <%= Html.ActionLink("\\Home", "Index", "Home") %><%= Html.ActionLink("\\Messages", "Home") %></span> <% using (Html.BeginForm()) { %> <fieldset> <legend>Message Headers</legend> <label for="message_subject"> Subject:</label> <%= Html.TextBox("message_subject")%> <%= Html.ValidationMessage("message_subject")%> <label for="selected_recipients"> Recipient Users:</label> <%= Html.ListBox("recipients") %> <%= Html.ValidationMessage("selected_recipients")%> <label for="selected_recipient_roles"> Recipient Roles:</label> <%= Html.ListBox("recipient_roles") %> <%= Html.ValidationMessage("selected_recipient_roles")%> <label for="is_urgent"> Urgent?</label> <%= Html.CheckBox("is_urgent") %> <%= Html.ValidationMessage("is_urgent")%> </fieldset> <fieldset> <legend>Message Text</legend> <%= Html.TextArea("message_text") %> <%= Html.ValidationMessage("message_text")%> </fieldset> <input type="reset" name="reset" id="reset" value="Reset" /> <input type="submit" name="action" id="send_message" value="Send" /> <% } %> <span id="navigation_bottom"> <%= Html.ActionLink("\\Home", "Index", "Home") %><%= Html.ActionLink("\\Messages", "Home") %></span> </asp:Content> <asp:Content ID="Content3" ContentPlaceHolderID="Scripts" runat="server"> </asp:Content> I have a parameterless ActionResult in my MessagesController like this: [Authorize] public ActionResult ComposeMessage() { ModelClasses.Messages.ComposeMessage FormData = new ModelClasses.Messages.ComposeMessage(); Common C = (Common)Session["Common"]; FormData.recipients = new MultiSelectList(C.AvailableUsers, "Key", "Value"); FormData.recipient_roles = new MultiSelectList(C.AvailableRoles, "Key", "Value"); return View(FormData); } ...and my model-based controller looks like this: [Authorize, AcceptVerbs(HttpVerbs.Post)] public ActionResult ComposeMessage(DCASS3.Classes.Messages.ComposeMessage FormData) { DCASSUser CurrentUser = (DCASSUser)Session["CurrentUser"]; Common C = (Common)Session["Common"]; //... (business logic) return View(FormData); } Problem is, I can access the page fine before a submit. When I actually make selections and press the submit button, however, I get: Server Error in '/' Application. No parameterless constructor defined for this object. Description: An unhandled exception occurred during the execution of the current web request. Please review the stack trace for more information about the error and where it originated in the code. Exception Details: System.MissingMethodException: No parameterless constructor defined for this object. Source Error: An unhandled exception was generated during the execution of the current web request. Information regarding the origin and location of the exception can be identified using the exception stack trace below. Stack Trace: [MissingMethodException: No parameterless constructor defined for this object.] System.RuntimeTypeHandle.CreateInstance(RuntimeType type, Boolean publicOnly, Boolean noCheck, Boolean& canBeCached, RuntimeMethodHandle& ctor, Boolean& bNeedSecurityCheck) +0 System.RuntimeType.CreateInstanceSlow(Boolean publicOnly, Boolean fillCache) +86 System.RuntimeType.CreateInstanceImpl(Boolean publicOnly, Boolean skipVisibilityChecks, Boolean fillCache) +230 System.Activator.CreateInstance(Type type, Boolean nonPublic) +67 System.Activator.CreateInstance(Type type) +6 System.Web.Mvc.DefaultModelBinder.CreateModel(ControllerContext controllerContext, ModelBindingContext bindingContext, Type modelType) +307 System.Web.Mvc.DefaultModelBinder.BindSimpleModel(ControllerContext controllerContext, ModelBindingContext bindingContext, ValueProviderResult valueProviderResult) +495 System.Web.Mvc.DefaultModelBinder.BindModel(ControllerContext controllerContext, ModelBindingContext bindingContext) +473 System.Web.Mvc.DefaultModelBinder.GetPropertyValue(ControllerContext controllerContext, ModelBindingContext bindingContext, PropertyDescriptor propertyDescriptor, IModelBinder propertyBinder) +45 System.Web.Mvc.DefaultModelBinder.BindProperty(ControllerContext controllerContext, ModelBindingContext bindingContext, PropertyDescriptor propertyDescriptor) +642 System.Web.Mvc.DefaultModelBinder.BindProperties(ControllerContext controllerContext, ModelBindingContext bindingContext) +144 System.Web.Mvc.DefaultModelBinder.BindComplexElementalModel(ControllerContext controllerContext, ModelBindingContext bindingContext, Object model) +95 System.Web.Mvc.DefaultModelBinder.BindComplexModel(ControllerContext controllerContext, ModelBindingContext bindingContext) +2386 System.Web.Mvc.DefaultModelBinder.BindModel(ControllerContext controllerContext, ModelBindingContext bindingContext) +539 System.Web.Mvc.ControllerActionInvoker.GetParameterValue(ControllerContext controllerContext, ParameterDescriptor parameterDescriptor) +447 System.Web.Mvc.ControllerActionInvoker.GetParameterValues(ControllerContext controllerContext, ActionDescriptor actionDescriptor) +173 System.Web.Mvc.ControllerActionInvoker.InvokeAction(ControllerContext controllerContext, String actionName) +801 System.Web.Mvc.Controller.ExecuteCore() +151 System.Web.Mvc.ControllerBase.Execute(RequestContext requestContext) +105 System.Web.Mvc.ControllerBase.System.Web.Mvc.IController.Execute(RequestContext requestContext) +36 System.Web.Mvc.<c_DisplayClass8.b_4() +65 System.Web.Mvc.Async.<c_DisplayClass1.b_0() +44 System.Web.Mvc.Async.<c__DisplayClass81.<BeginSynchronous>b__7(IAsyncResult _) +42 System.Web.Mvc.Async.WrappedAsyncResult1.End() +140 System.Web.Mvc.Async.AsyncResultWrapper.End(IAsyncResult asyncResult, Object tag) +54 System.Web.Mvc.Async.AsyncResultWrapper.End(IAsyncResult asyncResult, Object tag) +40 System.Web.Mvc.MvcHandler.EndProcessRequest(IAsyncResult asyncResult) +52 System.Web.Mvc.MvcHandler.System.Web.IHttpAsyncHandler.EndProcessRequest(IAsyncResult result) +36 System.Web.CallHandlerExecutionStep.System.Web.HttpApplication.IExecutionStep.Execute() +8677678 System.Web.HttpApplication.ExecuteStep(IExecutionStep step, Boolean& completedSynchronously) +155 Version Information: Microsoft .NET Framework Version:2.0.50727.3603; ASP.NET Version:2.0.50727.3082 This error shows up before I can trap it. I have no idea where it's choking, or what it's choking on. I don't see any point of this model that cannot be created with a parameterless constructor, and I can't find out where it's dying... Help is appreciated, thanks. -Jeremy

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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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