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  • Multiplying Block Matrices in Numpy

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

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  • How can calculus and linear algebra be useful to a system programmer?

    - by Victor
    I found a website saying that calculus and linear algebra are necessary for System Programming. System Programming, as far as I know, is about osdev, drivers, utilities and so on. I just can't figure out how calculus and linear algebra can be helpful on that. I know that calculus has several applications in science, but in this particular field of programming I just can't imagine how calculus can be so important. The information was on this site: http://www.wikihow.com/Become-a-Programmer Edit: Some answers here are explaining about algorithm complexity and optimization. When I made this question I was trying to be more specific about the area of System's Programming. Algorithm complexity and optimization can be applied to any area of programming not just System's Programming. That may be why I wasn't able to came up with such thinking at the time of the question.

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  • In R draw two lines, with slopes double and half the value of the best fit line

    - by D W
    I have data with a best fit line draw. I need to draw two other lines. One needs to have double the slope and the other need to have half the slope. Later I will use the region to differentially color points outside it as per: http://stackoverflow.com/questions/2687212/conditionally-colour-data-points-outside-of-confidence-bands-in-r Example dataset: ## Dataset from http://www.apsnet.org/education/advancedplantpath/topics/RModules/doc1/04_Linear_regression.html ## Disease severity as a function of temperature # Response variable, disease severity diseasesev<-c(1.9,3.1,3.3,4.8,5.3,6.1,6.4,7.6,9.8,12.4) # Predictor variable, (Centigrade) temperature<-c(2,1,5,5,20,20,23,10,30,25) ## For convenience, the data may be formatted into a dataframe severity <- as.data.frame(cbind(diseasesev,temperature)) ## Fit a linear model for the data and summarize the output from function lm() severity.lm <- lm(diseasesev~temperature,data=severity) # Take a look at the data plot( diseasesev~temperature, data=severity, xlab="Temperature", ylab="% Disease Severity", pch=16, pty="s", xlim=c(0,30), ylim=c(0,30) ) title(main="Graph of % Disease Severity vs Temperature") par(new=TRUE) # don't start a new plot abline(severity.lm, col="blue")

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  • How can I provide users with the functionality of the DBUnit DatabaseOperation methods from a web in

    - by reckoner
    I am currently updating a java-based web application which allows database developers to create stored procedure regression test suites for database testing. Currently, for test setup, execution and clean-up stages, the user is provided with text boxes where they are able to enter SQL code which is executed by the isql command. I would like to extend the application to use DB Unit’s DatabaseOperation methods to provide more ways to setup the state of the database than just SQL statements. The main reason for using Db Unit rather than just SQL statements is to be able to create and store xml and xls DataSets on a server where they can be associated with their test cases and used for data setup. My question is: How can I provide users with the functionality of the DBUnit DatabaseOperation methods from a web interface? I have considered: Creating a simple programming language and a parser to read some simple syntax involving the DB Unit method names which accept a parameter being the file location to an xml or xls DataSet. I was thinking of allowing the user to register the files they need with the web app which would catalogue them and provide each file with an identifier which could passed as a parameter to the methods in this simple programming language. Creating an XML DTD which provides the user with the ability to specify operations and parameters. If I went this approach, how can I execute the methods and their parameters that I parse from the XML document? Creating a table in the database which stores the method and a FK relation to a catalogued DataSet file, however I don’t think this would be good solution due to the fact that data entry would be tedious. Thanks for your help.

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  • regressions with many nested categorical covariates

    - by eric
    I have a few hundred thousand measurements where the dependent variable is a probability, and would like to use logistic regression. However, the covariates I have are all categorical, and worse, are all nested. By this I mean that if a certain measurement has "city - Phoenix" then obviously it is certain to have "state - Arizona" and "country - U.S." I have four such factors - the most granular has some 20k levels, but if need be I could do without that one, I think. I also have a few non-nested categorical covariates (only four or so, with maybe three different levels each). What I am most interested in is prediction - given a new observation in some city, I would like to know the relevant probability/dependent variable. I am not interested as much in the related inferential machinery - standard deviations, etc - at least as of now. I am hoping I can afford to be sloppy. However, I would love to have that information unless it requires methods that are more computationally expensive. Does anyone have any advice on how to attack this? I have looked into mixed effects, but am not sure it is what I am looking for.

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  • Python fit polynomial, power law and exponential from data

    - by Nadir
    I have some data (x and y coordinates) coming from a study and I have to plot them and to find the best curve that fits data. My curves are: polynomial up to 6th degree; power law; and exponential. I am able to find the best fit for polynomial with while(i < 6): coefs, val = poly.polyfit(x, y, i, full=True) and I take the degree that minimizes val. When I have to fit a power law (the most probable in my study), I do not know how to do it correctly. This is what I have done. I have applied the log function to all x and y and I have tried to fit it with a linear polynomial. If the error (val) is lower than the others polynomial tried before, I have chosen the power law function. Am I correct? Now how can I reconstruct my power law starting from the line y = mx + q in order to draw it with the original points? I need also to display the function found. I have tried with: def power_law(x, m, q): return q * (x**m) using x_new = np.linspace(x[0], x[-1], num=len(x)*10) y1 = power_law(x_new, coefs[0], coefs[1]) popt, pcov = curve_fit(power_law, x_new, y1) but it seems not to work well.

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  • How to use Vendor Properties in Multiple Backgrounds?

    - by barraponto
    I want to use multiple backgrounds in css, which are currently supported by Firefox 3.61, Chrome/Safari, supposedly Opera10.5 (doesn't run on gnu/linux). It is working fine, however i would like to use linear-gradients as a background. it works ok for Firefox, doesn't work at all with Chrome, yet i can't figure out how to make it work for both at the same time. any clues? http://snook.ca/archives/html_and_css/multiple-bg-css-gradients came the closest to match what i need, but i couldn't get it to work with chrome yet.

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  • C Programming arrays, I dont understand how I would go about making this program, If anyone can just guide me through the basic outline please :) [on hold]

    - by Rashmi Kohli
    Problem The temperature of a car engine has been measured, from real-world experiments, as shown in the table and graph below: Time (min) Temperature (oC) 0 20 1 36 2 61 3 68 4 77 5 110 Use linear regression to find the engine’s temperature at 1.5 minutes, 4.3 minutes, and any other time specified by the user. Background In engineering, many times we measure several data points in an experiment, but then we need to predict a value that we have not measured which lies between two measured values, such as the problem statement above. If the relation between the measured parameters seems to be roughly linear, then we can use linear regression to find the relationship between those parameters. In the graph of the problem statement above, the relation seems to be roughly linear. Hence, we can apply linear regression to the above problem. Assuming y {y0, y1, …yn-1} has a linear relation with x {x0, x1, … xn-1}, we can say that: y = mx+b where m and b can be found with linear regression as follows: For the problem in this lab, using linear regression gives us the following line (in blue) compared to the measured curve (in red). As you can see, there is usually a difference between the measured values and the estimated (predicted) values. What linear regression does is to minimize those differences and still give us a straight line (blue). Other methods, such as non-linear regression, are also possible to achieve higher accuracy and better curve fitting. Requirements Your program should first print the table of the temperatures similar to the way it’s printed in the problem statement. It should then calculate the temperature at minute 1.5 and 4.3 and show the answers to the user. Next, it should prompt the user to enter a time in minutes (or -1 to quit), and after reading the user’s specified time it should give the value of the engine’s temperature at that time. It should then go back to the prompt. Hints •Use a one dimensional array to store the temperature values given in the problem statement. •Use functions to separate tasks such as calculating m, calculating b, calculating the temperature at a given time, printing the prompt, etc. You can then give your algorithm as well as you pseudo code per function, as opposed to one large algorithm diagram or one large sequence of pseudo code.

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  • Good way to capture/replay sessions from Apache Log?

    - by Mark Harrison
    For performance testing, I would like to capture some traffic from a production server and use that as a basis to replay the request to a test server in order to simulate a realistic load in our development environment. These are all stateless queries, so no issues regarding cookies, sessions, etc. The Apache log timestamps everything down to a 1 second resolution, but that's not fine enough granularity for our peak times. What's the best way to capture more fine-grained timestamps for replay? And is there some ab-like load generating program that can use this data to replicate load?

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  • Why doesn't this loop terminate?

    - by David
    Here's the sample code: public static void col (int n) { if (n % 2 == 0) n = n/2 ; if (n % 2 != 0) n = ((n*3)+1) ; System.out.println (n) ; if (n != 1) col (n) ; } this works just fine until it gets down to 2. then it outputs 2 4 2 4 2 4 2 4 2 4 infinitely. it seems to me that if 2 is entered as n then (n % 2 == 0) is true 2 will be divided by 2 to yeild 1. then 1 will be printed and since (n != 1) is false the loop will terminate. Why doesn't this happen?

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  • How to write automated tests for SQL queries?

    - by James
    The current system we are adopting at work is to write some extremely complex queries which perform multiple calculations and have multiple joins / sub-queries. I don't think I am experienced enough to say if this is correct or not so I am agreeing and attempting to function with this system as it has clear benefits. The problem we are having at the moment is that the person writing the queries makes a lot of mistakes and assumes everything is correct. We have now assigned a tester to analyse all of the queries but this still proves extremely time consuming and stressful. I would like to know how we could create an automated procedure (without specifically writing it with code if possible as I can work out how to do that the long way) to verify a set of 10+ different inputs, verify the output data and say if the calculations are correct. I know I could write a script using specific data in the database and create a script using c# (the db is SQL Server) and verify all the values coming out but I would like to know what the official "standard" is as my experience is lacking in this area and I would like to improve. I am happy to add more information if required, add a comment if necessary. Thank you. Edit: I am using c#

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  • Measuring the limit of a point on a smooth.spline in R

    - by Subtle Array
    I'm not sure if that's the right terminology. I've entered some data into R, and I've put a smoothingSpline through it using the following command. smoothingSpline = smooth.spline(year, rate, spar=0.35) plot(x,y) lines(smoothingSpline) Now I'd like to measure some limits (or where the curve is at a given y point), and maybe to some predictive analysis on points that extend beyond the graph. Are there commands in R for doing this?

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  • HTML5 CSS3 layout not working

    - by John.Weland
    I have been asked by a local MMA (Mixed Martial Arts) School to help them develop a website. For the life of me I CANNOT get the layout to work correctly. When I get one section set where it should be another moves out of place! here is a pic of the layout: here The header should be a set height as should the footer the entire site at its widest point should be 1250px with the header/content area/footer and the like being 1240px the black in the picture is a scaling background to expand wider as larger resolution systems are viewing them. The full site should be a minimum-height of 100% but scale virtually as content in the target area deems necessary. My biggest issue currently is that my "sticky" footer doesn't stick once the content has stretched the content target area virtually. the Code is not pretty but here it is: HTML5 <!doctype html> <html> <head> <link rel="stylesheet" href="menu.css" type="text/css" media="screen"> <link rel="stylesheet" href="master.css" type="text/css" media="screen"> <meta charset="utf-8"> <title>Untitled Document</title> </head> <body bottommargin="0" leftmargin="0" rightmargin="0" topmargin="0"> <div id="wrap" class="wrap"><div id="logo" class="logo"><img src="images/comalogo.png" width="100" height="150"></div> <div id="header" class="header">College of Martial Arts</div> <div id="nav" class="nav"> <ul id="menu"><b> <li><a href="#">News</a></li> <li>·</li> <li><a href="#">About Us</a> <ul> <li><a href="#">The Instructors</a></li> <li><a href="#">Our Arts</a></li> </li> </ul> <li>·</li> <li><a href="#">Location</a></li> <li>·</li> <li><a href="#">Gallery</a></li> <li>·</li> <li><a href="#">MMA.tv</a></li> <li>·</li> <li><a href="#">Schedule</a></li> <li>·</li> <li><a href="#">Fight Gear</a></li></b> </div> <div id="social" class="social"> <a href="http://www.facebook.com/pages/Canyon-Lake-College-of-Martial-Arts/189432551104674"><img src="images/soc/facebook.png"></a> <a href="https://twitter.com/#!/CanyonLakeMMA"><img src="images/soc/twitter.png"></a> <a href="https://plus.google.com/108252414577423199314/"><img src="images/soc/google+.png"></a> <a href="http://youtube.com/user/clmmatv"><img src="images/soc/youtube.png"></a></div> <div id="mid" class="mid">test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br>test <br></div> <div id="footer" class="footer"> <div id="contact" style="left:0px;">tel: (830) 214-4591<br /> e: [email protected]<br /> add: 1273 FM 2673, Sattler, TX 78133<br /> </div> <div id="affiliates" style="right:0px;">Hwa Rang World Tang soo Do</div> <div id="copyright">Copyright © College of Martial Arts</div> </div> </body> </html> CSS3 -Dropdown Menu- @charset "utf-8"; /* CSS Document */ /* Main */ #menu { width: 100%; margin: 0; padding: 10px 0 0 0; list-style: none; background: #444; background: -moz-linear-gradient(#000, #333); background: -webkit-gradient(linear,left bottom,left top,color-stop(0, #444),color-stop(1, #000)); background: -webkit-linear-gradient(#000, #333); background: -o-linear-gradient(#000, #333); background: -ms-linear-gradient(#000, #333); background: linear-gradient(#000, #333); -moz-border-radius: 5px; border-radius: 5px; -moz-box-shadow: 0 2px 1px #9c9c9c; -webkit-box-shadow: 0 2px 1px #9c9c9c; box-shadow: 0 8px 8px #9c9c9c; /* outline:#000 solid thin; */ } #menu li { left:150px; float: left; padding: 0 0 10px 0; position:relative; color: #FC0; font-size:15px; font-family:'freshman' cursive; line-height:15px; } #menu a { float: left; height: 15px; line-height:15px; padding: 0 10px; color: #FC0; font-size:15px; text-decoration: none; text-shadow: 1 1px 0 #000; text-align:center; } #menu li:hover > a { color: #fafafa; } *html #menu li a:hover /* IE6 */ { color: #fafafa; } #menu li:hover > ul { display: block; } /* Sub-menu */ #menu ul { list-style: none; margin: 0; padding: 0; display: none; position: absolute; top: 25px; left: 0; z-index: 99999; background: #444; background: -moz-linear-gradient(#000, #333); background: -webkit-gradient(linear,left bottom,left top,color-stop(0, #111),color-stop(1, #444)); background: -webkit-linear-gradient(#000, #333); background: -o-linear-gradient(#000, #333); background: -ms-linear-gradient(#000, #333); background: linear-gradient(#000, #333); -moz-border-radius: 5px; border-radius: 5px; /* outline:#000 solid thin; */ } #menu ul li { left:0; -moz-box-shadow: none; -webkit-box-shadow: none; box-shadow: none; } #menu ul a { padding: 10px; height: auto; line-height: 1; display: block; white-space: nowrap; float: none; text-transform: none; } *html #menu ul a /* IE6 */ { height: 10px; width: 200px; } *:first-child+html #menu ul a /* IE7 */ { height: 10px; width: 200px; } /*#menu ul a:hover { background: #000; background: -moz-linear-gradient(#000, #333); background: -webkit-gradient(linear, left top, left bottom, from(#04acec), to(#0186ba)); background: -webkit-linear-gradient(#000, #333); background: -o-linear-gradient(#000, #333); background: -ms-linear-gradient(#000, #333); background: linear-gradient(#000, #333); }*/ /* Clear floated elements */ #menu:after { visibility: hidden; display: block; font-size: 0; content: " "; clear: both; height: 0; } * html #menu { zoom: 1; } /* IE6 */ *:first-child+html #menu { zoom: 1; } /* IE7 */ CSS3 -Master Style Sheet- @charset "utf-8"; /* CSS Document */ a:link {color:#FC0; text-decoration:none;} /* unvisited link */ a:visited {color:#FC0; text-decoration:none;} /* visited link */ a:hover {color:#FFF; text-decoration:none;} /* mouse over link */ a:active {color:#FC0; text-decoration:none;} /* selected link */ ul.a {list-style-type:none;} ul.b {list-style-type:inherit} html { } body { /*background-image:url(images/cagebg.jpg);*/ background-repeat:repeat; background-position:top; } div.wrap { margin: 0 auto; min-height: 100%; position: relative; width: 1250px; } div.logo{ top:25px; left:20px; position:absolute; float:top; height:150px; } /*Freshman FONT is on my computer needs to be uploaded to the webhost and rendered host side like a webfont*/ div.header{ background-color:#999; color:#FC0; margin-left:5px; height:80px; width:1240px; line-height:70px; font-family:'freshman' cursive; font-size:50px; text-shadow:8px 8px #9c9c9c; text-outline:1px 1px #000; text-align:center; background-color:#999; clear: both; } div.social{ height:50px; margin-left:5px; width:1240px; font-family:'freshman' cursive; font-size:50px; text-align:right; color:#000; background-color:#999; line-height:30px; box-sizing: border-box; ms-box-sizing: border-box; webkit-box-sizing: border-box; moz-box-sizing: border-box; padding-right:5px; } div.mid{ position:absolute; min-height:100%; margin-left:5px; width:1240px; font-family:'freshman' cursive; font-size:50px; text-align:center; color:#000; background-color:#999; } /*SIDE left and right should be 40px wide and a minimum height (100% the area from nav-footer) to fill between the NAV and the footer yet stretch as displayed content streatches the page longer (scrollable)*/ div #side.sright{ top:96px; right:0; position:absolute; float:right; height:100%; min-height:100%; width:40px; background-image:url(images/border.png); } /*Container should vary in height in acordance to content displayed*/ div #content.container{ } /*Footer should stick at ABSOLUTE BOTTOM of the page*/ div #footer{ font-family:'freshman' cursive; position:fixed; bottom:0; background-color:#000000; margin-left:5px; width:1240px; color:#FC0; clear: both; /*this clear property forces the .container to understand where the columns end and contain them*/ } /*HTML 5 support - Sets new HTML 5 tags to display:block so browsers know how to render the tags properly.*/ header, section, footer, aside, nav, article, figure { display: block; } Eventually once the layout is correct I have to use PHP to make calls for where data should be displayed from what database. If anyone can help me to fix this layout and clean up the crap code, I'd be much appreciated.. I've spent weeks trying to figure this out.

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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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  • Examples of 2D side-scrollers that achieve open non-linear feel?

    - by Milosz Falinski
    I'm working on a 2.5D platformer prototype that aims for an open feel while maintaining familiar core mechanics. Now, there's some obvious challenges with creating a non constricted feel in a spatially constricted environment. What I'm interested in, is examples of how game designers deal with the "here's a level, beat the bad guys/puzzles to get to the next level" design that seems so natural to most platformers (eg. Mario/Braid/Pid/Meat Boy to name a few). Some ideas for achieving openness I've come across include: One obvious successful example is Terraria, which achieves openness simply through complexity and flexibility of the game-system Another example that comes to mind is Cave Story. Game is non-linear, offers multiple choices and side-stories Mario, Rayman and some other 'classics' with a top-down level selection. I actually really dislike this as it never did anything for me emotionally and just seems like a bit of a lazy way to do things. Note: I've not actually had much experience with most of the 'classical' console platformers, apart from the obvious Marios/Zeldas/Metroids, since I've grown up on adventure games. By that I mean, it's entirely possible that I simply missed some games that solve the problem really well and are by some considered obvious 'classics'.

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  • ILOG CPLEX: how to populate IloLPMatrix while using addGe to set up the model?

    - by downer
    I have a queatoin about IloLPMatrix and addGe. I was trying to follow the example of AdMIPex5.java to generate user defined cutting planes based on the solution to the LP relaxation. The difference is that eh initial MIP model is not read in from a mps file, but set up in the code using methods like addGe, addLe etc. I think this is why I ran into problems while copying the exampe to do the following. IloLPMatrix lp = (IloLPMatrix)cplex.LPMatrixIterator().next(); lp from the above line turns to be NULL. I am wondering 1. What is the relationship between IloLPMatrix and the addLe, addGe commands? I tried to addLPMatrix() to the model, and then used model.addGe methods. but the LPMatrix seems to be empty still. How do I populate the IloLPMatrix of the moel according to the value that I had set up using addGe and addLe. Is the a method to this easily, or do I have to set them up row by row myself? I was doing this to get the number of variables and their values by doing lp.getNumVars(). Is there other methods that I can use to get the number of variables and their values wihout doing these, since my system is set up by addLe, addGe etc? Thanks a lot for your help on this.

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  • Efficient 4x4 matrix inverse (affine transform)

    - by Budric
    Hi, I was hoping someone can point out an efficient formula for 4x4 affine matrix transform. Currently my code uses cofactor expansion and it allocates a temporary array for each cofactor. It's easy to read, but it's slower than it should be. Note, this isn't homework and I know how to work it out manually using 4x4 co-factor expansion, it's just a pain and not really an interesting problem for me. Also I've googled and came up with a few sites that give you the formula already (http://www.euclideanspace.com/maths/algebra/matrix/functions/inverse/fourD/index.htm). However this one could probably be optimized further by pre-computing some of the products. I'm sure someone came up with the "best" formula for this at one point or another? Thanks.

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  • Android VideoView LinearLayout.LayoutParams

    - by Chris
    I am playing a video using VideoView in my app. When I play it on Droid with linearlayout params FILL_PARENT, FILL_PARENT, it plays well. The same params do not work well for a myTouch. What params can I use that will work well with most devices? Thanks Chris

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  • This is more a matlab/math brain teaser than a question

    - by gd047
    Here is the setup. No assumptions for the values I am using. n=2; % dimension of vectors x and (square) matrix P r=2; % number of x vectors and P matrices x1 = [3;5] x2 = [9;6] x = cat(2,x1,x2) P1 = [6,11;15,-1] P2 = [2,21;-2,3] P(:,1)=P1(:) P(:,2)=P2(:) modePr = [-.4;16] TransPr=[5.9,0.1;20.2,-4.8] pred_modePr = TransPr'*modePr MixPr = TransPr.*(modePr*(pred_modePr.^(-1))') x0 = x*MixPr Then it was time to apply the following formula to get myP , where µij is MixPr. I used this code to get it: myP=zeros(n*n,r); Ptables(:,:,1)=P1; Ptables(:,:,2)=P2; for j=1:r for i = 1:r; temp = MixPr(i,j)*(Ptables(:,:,i) + ... (x(:,i)-x0(:,j))*(x(:,i)-x0(:,j))'); myP(:,j)= myP(:,j) + temp(:); end end Some brilliant guy proposed this formula as another way to produce myP for j=1:r xk1=x(:,j); PP=xk1*xk1'; PP0(:,j)=PP(:); xk1=x0(:,j); PP=xk1*xk1'; PP1(:,j)=PP(:); end myP = (P+PP0)*MixPr-PP1 I tried to formulate the equality between the two methods and seems to be this one. To make things easier, I ignored from both methods the summation of matrix P. where the first part denotes the formula that I used, while the second comes from his code snippet. Do you think this is an obvious equality? If yes, ignore all the above and just try to explain why. I could only start from the LHS, and after some algebra I think I proved it equals to the RHS. However I can't see how did he (or she) think of it in the first place.

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  • Metric 3d reconstruction

    - by srand
    I'm trying to reconstruct 3D points from 2D image correspondences. My camera is calibrated. The test images are of a checkered cube and correspondences are hand picked. Radial distortion is removed. After triangulation the construction seems to be wrong however. The X and Y values seem to be correct, but the Z values are about the same and do not differentiate along the cube. The 3D points look like as if the points were flattened along the Z-axis. What is going wrong in the Z values? Do the points need to be normalized or changed from image coordinates at any point, say before the fundamental matrix is computed? (If this is too vague I can explain my general process or elaborate on parts)

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  • Best open source Mixed Integer Optimization Solver

    - by Mark
    I am using CPLEX for solving huge optimization models (more than 100k variables) now I'd like to see if I can find an open source alternative, I solve mixed integer problems (MILP) and CPLEX works great but it is very expensive if we want to scale so I really need to find an alternative or start writing our own ad-hoc optimization library (which will be painful) Any suggestion/insight would be much appreciated

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  • Rotation Matrix calculates by column not by row

    - by pinnacler
    I have a class called forest and a property called fixedPositions that stores 100 points (x,y) and they are stored 250x2 (rows x columns) in MatLab. When I select 'fixedPositions', I can click scatter and it will plot the points. Now, I want to rotate the plotted points and I have a rotation matrix that will allow me to do that. The below code should work: theta = obj.heading * pi/180; apparent = [cos(theta) -sin(theta) ; sin(theta) cos(theta)] * obj.fixedPositions; But it wont. I get this error. ??? Error using == mtimes Inner matrix dimensions must agree. Error in == landmarkslandmarks.get.apparentPositions at 22 apparent = [cos(theta) -sin(theta) ; sin(theta) cos(theta)] * obj.fixedPositions; When I alter forest.fixedPositions to store the variables 2x250 instead of 250x2, the above code will work, but it wont plot. I'm going to be plotting fixedPositions constantly in a simulation, so I'd prefer to leave it as it, and make the rotation work instead. Any ideas? Also, fixed positions, is the position of the xy points as if you were looking straight ahead. i.e. heading = 0. heading is set to 45, meaning I want to rotate points clockwise 45 degrees. Here is my code: classdef landmarks properties fixedPositions %# positions in a fixed coordinate system. [x, y] heading = 45; %# direction in which the robot is facing end properties (Dependent) apparentPositions end methods function obj = landmarks(numberOfTrees) %# randomly generates numberOfTrees amount of x,y coordinates and set %the array or matrix (not sure which) to fixedPositions obj.fixedPositions = 100 * rand([numberOfTrees,2]) .* sign(rand([numberOfTrees,2]) - 0.5); end function obj = set.apparentPositions(obj,~) theta = obj.heading * pi/180; [cos(theta) -sin(theta) ; sin(theta) cos(theta)] * obj.fixedPositions; end function apparent = get.apparentPositions(obj) %# rotate obj.positions using obj.facing to generate the output theta = obj.heading * pi/180; apparent = [cos(theta) -sin(theta) ; sin(theta) cos(theta)] * obj.fixedPositions; end end end P.S. If you change one line to this: obj.fixedPositions = 100 * rand([2,numberOfTrees]) .* sign(rand([2,numberOfTrees]) - 0.5); Everything will work fine... it just wont plot.

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  • Static Typing and Writing a Simple Matrix Library

    - by duckworthd
    Aye it's been done a million times before, but damnit I want to do it again. I'm writing a simple Matrix Library for C++ with the intention of doing it right. I've come across something that's fairly obvious in mathematics, but not so obvious to a strongly typed system -- the fact that a 1x1 matrix is just a number. To avoid this, I started walking down the hairy path of matrices as a composition of vectors, but also stumbled upon the fact that two vectors multiplied together could either be a number or a dyad, depending on the orientation of the two. My question is, what is the right way to deal with this situation in a strongly typed language like C++ or Java?

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