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  • Less Mathematical Approaches to Machine Learning?

    - by Ed
    Out of curiosity, I've been reading up a bit on the field of Machine Learning, and I'm surprised at the amount of computation and mathematics involved. One book I'm reading through uses advanced concepts such as Ring Theory and PDEs (note: the only thing I know about PDEs is that they use that funny looking character). This strikes me as odd considering that mathematics itself is a hard thing to "learn." Are there any branches of Machine Learning that use different approaches? I would think that a approaches relying more on logic, memory, construction of unfounded assumptions, and over-generalizations would be a better way to go, since that seems more like the way animals think. Animals don't (explicitly) calculate probabilities and statistics; at least as far as I know.

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  • System Variables, Stored Procedures or Functions for Meta Data

    - by BuckWoody
    Whenever you want to know something about SQL Server’s configuration, whether that’s the Instance itself or a database, you have a few options. If you want to know “dynamic” data, such as how much memory or CPU is consumed or what a particular query is doing, you should be using the Dynamic Management Views (DMVs) that you can read about here: http://msdn.microsoft.com/en-us/library/ms188754.aspx  But if you’re looking for how much memory is installed on the server, the version of the Instance, the drive letters of the backups and so on, you have other choices. The first of these are system variables. You access these with a SELECT statement, and they are useful when you need a discrete value for use, say in another query or to put into a table. You can read more about those here: http://msdn.microsoft.com/en-us/library/ms173823.aspx You also have a few stored procedures you can use. These often bring back a lot more data, pre-formatted for the screen. You access these with the EXECUTE syntax. It is a bit more difficult to take the data they return and get a single value or place the results in another table, but it is possible. You can read more about those here: http://msdn.microsoft.com/en-us/library/ms187961.aspx Yet another option is to use a system function, which you access with a SELECT statement, which also brings back a discrete value that you can use in a test or to place in another table. You can read about those here: http://msdn.microsoft.com/en-us/library/ms187812.aspx  By the way, many of these constructs simply query from tables in the master or msdb databases for the Instance or the system tables in a user database. You can get much of the information there as well, and there are even system views in each database to show you the meta-data dealing with structure – more on that here: http://msdn.microsoft.com/en-us/library/ms186778.aspx  Some of these choices are the only way to get at a certain piece of data. But others overlap – you can use one or the other, they both come back with the same data. So, like many Microsoft products, you have multiple ways to do the same thing. And that’s OK – just research what each is used for and how it’s intended to be used, and you’ll be able to select (pun intended) the right choice. Share this post: email it! | bookmark it! | digg it! | reddit! | kick it! | live it!

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  • Multiple render targets and pixel shader outputs terminology

    - by Rei Miyasaka
    I'm a little confused on the jargon: does Multiple Render Targets (MRT) refer to outputting from a pixel shader to multiple elements in a struct? That is, when one says "MRT is to write to multiple textures", are multiple elements interleaved in a single output texture, or do you specify multiple discrete output textures? By the way, from what I understand, at least for DX9, all the elements of this struct need to be of the same size. Does this restriction still apply to DX11?

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  • Breaking down CS courses for freshmen

    - by Avinash
    I'm a student putting together a slide geared towards freshmen level students who are trying to understand what the importance of various classes in the CS curriculum are. Would it be safe to say that this list is fairly accurate? Data structures: how to store stuff in programs Discrete math: how to think logically Bits & bytes: how to ‘speak’ the machine’s language Advanced data structures: how to store stuff in more ways Algorithms: how to compute things efficiently Operating systems: how to do manage different processes/threads Thanks!

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  • The musical instrument software developer

    - by Peter Mortensen
    There is a correlation between playing a musical instrument and being a great software developer (the same for mathematics). But what is the causation (if any)? That is, should a software developer learn to play a musical instrument to become a better software developer? Or does proficiency in software development make it more likely that an interest in performing on a musical instrument will develop? Update: a very similar question was asked in podcast .NET Rocks, episode 614 (from Øredev 2010), 35 min 40 secs.

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  • Why no fortran standard library ?

    - by Stefano Borini
    To be a language focused on mathematics and scientific computing, I am always baffled by the total lack of useful mathematical routines in the Fortran standard library. One would expect it to be shipped at least with a routine to compute standard deviation and mean, but this is not the case. In particular with the introduction of Fortran 90 and the addition of modules (thus reducing namespace pollution), I don't see any reason why of this critical lack of services. I would like to hear your knowledge about why this is the case.

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  • Should certain math classes be required for a Computer Science degree?

    - by sunpech
    For a Computer Science degree at many colleges and universities, certain math courses are required: Calculus, Linear Algebra, and Discrete Mathematics are few examples. However, since I've started working in the real world as a software developer, I have yet to truly use the knowledge I had at once acquired from taking those classes. My question is: Should these math classes be required to obtain a computer science degree? Or would they better served as electives? A Slashdot post: CS Profs Debate Role of Math In CS Education

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  • Game programming course materials: What should it include?

    - by Esa
    I am tasked to create the course materials for a game programming class, and I’d like your opinion on what aspects and areas of game programming, such as game state management, game object storing or simple AI, should I include in it? The course is intented to be the first step into game programming for students with novice skills in programming. There will be mathematics as well, but I found that there are multiple questions, with good answers, on that subject already.

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  • spinning a 2d Cube

    - by Rahul Verma
    I know that a cube is actually a 3d shape , but i have some other problem over here. I have been doing 2D Game dev using libgdx but have never touched 3D rendering. Now what I want in my 2D game is that instead of coins I make my player collect magical cubes. But those cubes need to be spinning on one Diagonal, same can be seen in popular game Vector. Here is a screenshot. Can someone explaing the mathematics of such an animation

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  • Are there studies on what programming languages does to the brain? [closed]

    - by Eduard Florinescu
    Are there studies on what effects have programming languages on the brain or for that matter any other artificial languages in general, like mathematics ? Speaking from my personal experience I feel very different every time I speak Italian, I feel like a virtuoso on a quest but at the other end when I coded in machine code in debug.exe I felt like the main charcter inp(Movie). Why do I bring this up because I am suspecting that languages affect your mind and popular legends back this up too often: are full of mathematicians that crossed the Rubicon.

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  • Are proofs worth the effort?

    - by Shashank Jain
    I bought the de-facto book for learning about data structures and algorithms (CLRS). The book is though quite good but the singularity is in the proofs. The book is filled with Lemmas, theorems, peculiar symbols and unimaginable recurrence relations which are very hard to understand. I am able to somehow get the algorithms but the discrete mathematics just not for me. So should I leave them out and just concentrate on algorithims?

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  • Mutable Records in F#

    - by MarkPearl
    I’m loving my expert F# book – today I thought I would give a post on using mutable records as covered in Chapter 4 of Expert F#. So as they explain the simplest mutable data structures in F# are mutable records. The whole concept of things by default being immutable is a new one for me from my C# background. Anyhow… lets look at some C# code first. using System; using System.Collections.Generic; using System.Linq; using System.Text; namespace MutableRecords { public class DiscreteEventCounter { public int Total { get; set; } public int Positive { get; set; } public string Name { get; private set; } public DiscreteEventCounter(string name) { Name = name; } } class Program { private static void recordEvent(DiscreteEventCounter s, bool isPositive) { s.Total += 1; if (isPositive) s.Positive += 1; } private static void reportStatus (DiscreteEventCounter s) { Console.WriteLine("We have {0} {1} out of {2}", s.Positive, s.Name, s.Total); } static void Main(string[] args) { var longCounter = new DiscreteEventCounter("My Discrete Counter"); recordEvent(longCounter, true); recordEvent(longCounter, true); reportStatus(longCounter); Console.ReadLine(); } } } Quite simple, we have a class that has a few values. We instantiate an instance of the class and perform increments etc on the instance. Now lets look at an equivalent F# sample. namespace EncapsulationNS module Module1 = open System type DiscreteEventCounter = { mutable Total : int mutable Positive : int Name : string } let recordEvent (s: DiscreteEventCounter) isPositive = s.Total <- s.Total+1 if isPositive then s.Positive <- s.Positive+1 let reportStatus (s: DiscreteEventCounter) = printfn "We have %d %s out of %d" s.Positive s.Name s.Total let newCounter nm = { Total = 0; Positive = 0; Name = nm } // // Using it... // let longCounter = newCounter "My Discrete Counter" recordEvent longCounter (true) recordEvent longCounter (true) reportStatus longCounter System.Console.ReadLine() Notice in the type declaration of the DiscreteEventCounter we had to explicitly declare that the total and positive value holders were mutable. And that’s it – a very simple example of mutable types.

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  • Why Keep These 10 A Penny " How Long Will It Take To Program in Blah Blah Blah Language Questions" Open Yet Close This Question..? [migrated]

    - by user866190
    Why keep this question open and others like it open which basically asks how long does it take to learn a programming language? Yet I ask a valid question which basically asks how esteemed is a mathematics degree from the UK open university and would software employers hire whilst you are the course.. and it gets closed I love the advice I get from this site and I appreciate the fact that Software Technicians of all types use this site, but it's a bit shallow if you can't ask a question with a little bit of the real world involved

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  • Ubuntu 12.04 + AMD Radeon driver 12.8 problem

    - by wpinacz
    I have a Lenovo G570 laptop with AMD Radeon 6370M GPU. I wanted to install new 12.8 driver from AMD but with no success, after install and reboot, I got a screen with reconfigure graphics driver and it won't work. If I install 12.6 driver it works but I cannot switch to my integrated Intel GPU, only discrete (AMD) GPU is working. Please help with my problem (installing 12.8 driver or switching GPU under 12.6 driver).

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  • Where to after a year of java? [closed]

    - by avatarX
    I've just finished a my first year of programming Java at varsity and I have a three month break. In terms of my development would it be better to: Cover Java in more depth to acquire a more intermediate level of ability Learn a new programming language (if so which) to a similar level as my current Java ability Spend timing learning introductory discrete maths, algorithms and data structures I'm also open to any other possibilities that would be beneficial but that could be covered in about 3 months.

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  • Live cd and usb install failure blank screen when trying to install on an HP Pavilion dv6

    - by Ajian
    I recently bought a new computer, and have been trying to install linux on it, 11.10 x64. It is a HP pavilion dv6-6117dx. 2.4GHz/1.5GHz VISION A8 Technology from AMD with AMD Quad-Core A8-3500M Accelerated Processor AMD Radeon HD 6620G Discrete-Class Graphics I am pretty sure i picked a unsupported graphics card or something. I have tried booting from usb as well, but the screen becomes blank after rebooting.

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  • Google Page Rank - The Ultimate Popularity Contest

    Although you might initially think of it as simply the way that the Google search engine ranks pages, the term Google Page Rank is actually a trademarked term that actually belongs to Stanford University. The term is a tribute to its creator, Larry Page, and refers to a complex mathematics algorithm that allows today's advanced search engines, like Google, to index and rank the millions and millions of pages that exist on the internet today.

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  • Windows cannot open directory with too long name created by Linux

    - by Tim
    Hello! My laptop has two OSes: Windows 7 and Ubuntu 10.10. A partition of Windows 7 of format NTFS is mounted in Ubuntu. In Ubuntu, I created a directory under somehow deep path and with a long name for itself, specifically, the name for that directory is "a set of size-measurable subsets ie sigma algebra". Now in Windows, I cannot open the directory, which I guess is because of the name is too long, nor can I rename it. I was wondering if there is some way to access that directory under Windows? Better without changing the directory if possible, but will have to if necessary. Thanks and regards! Update: This is the output using "DIR /X" in cmd.exe, which does not shorten the directory name: F:\science\math\Foundations of mathematics\set theory\whether element of a set i s also a set\when element is set\when element sets are subsets of a universal se t\closed under some set operations\sigma algebra of sets>DIR /X Volume in drive F is Data Volume Serial Number is 0492-DD90 Directory of F:\science\math\Foundations of mathematics\set theory\whether elem ent of a set is also a set\when element is set\when element sets are subsets of a universal set\closed under some set operations\sigma algebra of sets 03/14/2011 10:43 AM <DIR> . 03/14/2011 10:43 AM <DIR> .. 03/08/2011 10:09 AM <DIR> a set of size-measurable sub sets ie sigma algebra 02/12/2011 04:08 AM <DIR> example 02/17/2011 12:30 PM <DIR> general 03/13/2011 02:28 PM <DIR> mapping from sigma algebra t o R or C i.e. measure 02/12/2011 04:10 AM <DIR> msbl mapping from general ms bl space to Borel msbl R or C 02/12/2011 04:10 AM 4,928 new file~ 03/14/2011 10:42 AM <DIR> temp 03/02/2011 10:58 AM <DIR> with Cartesian product of se ts 1 File(s) 4,928 bytes 9 Dir(s) 39,509,340,160 bytes free

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  • Ada and 'The Book'

    - by Phil Factor
    The long friendship between Charles Babbage and Ada Lovelace created one of the most exciting and mysterious of collaborations ever to have resulted in a technological breakthrough. The fireworks that created by the collision of two prodigious mathematical and creative talents resulted in an invention, the Analytical Engine, which went on to change society fundamentally. However, beyond that, we just don't know what the bulk of their collaborative work was about:;  it was done in strictest secrecy. Even the known outcome of their friendship, the first programmable computer, was shrouded in mystery. At the time, nobody, except close friends and family, had any idea of Ada Byron's contribution to the invention of the ‘Engine’, and how to program it. Her great insight was published in August 1843, under the initials AAL, standing for Ada Augusta Lovelace, her title then being the Countess of Lovelace. It was contained in a lengthy ‘note’ to her translation of a publication that remains the best description of Babbage's amazing Analytical Engine. The secret identity of the person behind those enigmatic initials was finally revealed by Prince de Polignac who, seventy years later, wrote to Ada's daughter to seek confirmation that her mother had, indeed, been the author of the brilliant sentences that described so accurately how Babbage's mechanical computer could be programmed with punch-cards. L.F. Menabrea's paper on the Analytical Engine first appeared in the 'Bibliotheque Universelle de Geneve' in October 1842, and Ada translated it anonymously for Taylor's 'Scientific Memoirs'. Charles Babbage was surprised that she had not written an original paper as she already knew a surprising amount about the way the machine worked. He persuaded her to at least write some explanatory notes. These notes ended up extending to four times the length of the original article and represented the first published account of how a machine could be programmed to perform any calculation. Her example of programming the Bernoulli sequence would have worked on the Analytical engine had the device’s construction been completed, and gave Ada an unassailable claim to have invented the art of programming. What was the reason for Ada's secrecy? She was the only legitimate child of Lord Byron, who was probably the best known celebrity of the age, so she was already famous. She was a senior aristocrat, with titles, a fortune in money and vast estates in the Midlands. She had political influence, and was the cousin of Lord Melbourne, who was the Prime Minister at that time. She was friendly with the young Queen Victoria. Her mathematical activities were a pastime, and not one that would be considered by others to be in keeping with her roles and responsibilities. You wouldn't dare to dream up a fictional heroine like Ada. She was dazzlingly beautiful and talented. She could speak several languages fluently, and play some musical instruments with professional skill. Contemporary accounts refer to her being 'accomplished in science, art and literature'. On top of that, she was a brilliant mathematician, a talent inherited from her mother, Annabella Milbanke. In her mother's circle of literary and scientific friends was Charles Babbage, and Ada's friendship with him dates from her teenage zest for Mathematics. She was one of the first people he'd ever met who understood what he had attempted to achieve with the 'Difference Engine', and with whom he could converse as intellectual equals. He arranged for her to have an education from the most talented academics in the country. Ada melted the heart of the cantankerous genius to the point that he became a faithful and loyal father-figure to her. She was one of the very few who could grasp the principles of the later, and very different, ‘Analytical Engine’ which was designed from the start to tackle a variety of tasks. Sadly, Ada Byron's life ended less than a decade after completing the work that assured her long-term fame, in November 1852. She was dying of cancer, her gambling habits had caused her to run up huge debts, she'd had more than one affairs, and she was being blackmailed. Her brilliant but unempathic mother was nursing her in her final illness, destroying her personal letters and records, and repaying her debts. Her husband was distraught but helpless. Charles Babbage, however, maintained his steadfast paternalistic friendship to the end. She appointed her loyal friend to be her executor. For years, she and Babbage had been working together on a secret project, known only as 'The Book'. We have a clue to what it was in a letter written by her nine years earlier, on 11th August 1843. It was a joint project by herself and Lord Lovelace, her husband, and was intended to involve Babbage's 'undivided energies'. It involved 'consulting your Engine' (it required Babbage’s computer). The letter gives no hint about the project except for the high-minded nature of its purpose, and its highly mathematical nature.  From then on, the surviving correspondence between the two gives only veiled references to 'The Book'. There isn't much, since Babbage later destroyed any letters that could have damaged her reputation within the Establishment. 'I cannot spare the book today, which I am very sorry for. At the moment I want it for constant reference, but I think you can have it tomorrow' (Oct 1844)  And 'I will send you the book directly, and you can say, when you receive it, how long you will want to keep it'. (Nov 1844)  The two of them were obviously intent on the work: She writes, four years later, 'I have an engagement for Wednesday which will prevent me from attending to your wishes about the book' (Dec 1848). This was something that they both needed to work on, but could not do in parallel: 'I will send the book on Tuesday, and it can be left with you till Friday' (11 Feb 1849). After six years work, it had been so well-handled that it was beginning to fall apart: 'Don't forget the new cover you promised for the book. The poor book is very shabby and wants one' (20 Sept 1849). So what was going on? The word 'book' was not a code-word: it was a real book, probably a 'printer's blank', plain paper, but properly bound so printers and publishers could show off how the published work might look. The hints from the correspondence are of advanced mathematics. It is obvious that the book was travelling between them, back and forth, each one working on it for less than a week before passing it back. Ada and her husband were certainly involved in gambling large sums of money on the horses, and so most biographers have concluded that the three of them were trying to calculate the mathematical odds on the horses. This theory has three large problems. Firstly, Ada's original letter proposing the project refers to its high-minded nature. Babbage was temperamentally opposed to gambling and would scarcely have given so much time to the project, even though he was devoted to Ada. Secondly, Babbage would have very soon have realized the hopelessness of trying to beat the bookies. This sort of betting never attracts his type of intellectual background. The third problem is that any work on calculating the odds on horses would not need a well-thumbed book to pass back and forth between them; they would have not had to work in series. The original project was instigated by Ada, along with her husband, William King-Noel, 1st Earl of Lovelace. Charles Babbage was invited to join the project after the couple had come up with the idea. What could William have contributed? One might assume that William was a Bertie Wooster character, addicted only to the joys of the turf, but this was far from the truth. He was a scientist, a Cambridge graduate who was later elected to be a Fellow of the Royal Society. After Eton, he went to Trinity College, Cambridge. On graduation, he entered the diplomatic service and acted as secretary under Lord Nugent, who was Lord Commissioner of the Ionian Islands. William was very friendly with Babbage too, able to discuss scientific matters on equal terms. He was a capable engineer who invented a process for bending large timbers by the application of steam heat. He delivered a paper to the Institution of Civil Engineers in 1849, and received praise from the great engineer, Isambard Kingdom Brunel. As well as being Lord Lieutenant of the County of Surrey for most of Victoria's reign, he had time for a string of scientific and engineering achievements. Whatever the project was, it is unlikely that William was a junior partner. After Ada's death, the project disappeared. Then, two years later, Babbage, through one of his occasional outbursts of temper, demonstrated that he was able to decrypt one of the most powerful of secret codes, Vigenère's autokey cipher.  All contemporary diplomatic and military messages used a variant of this cipher. Babbage had made three important discoveries, namely, the mathematical law of this cipher, the principle of the key periodicity, and the technique of the symmetry of position. The technique is now known as the Kasiski examination, also called the Kasiski test, but Babbage got there first. At one time, he listed amongst his future projects, the writing of a book 'The Philosophy of Decyphering', but it never came to anything. This discovery was going to change the course of history, since it was used to decipher the Russians’ military dispatches in the Crimean war. Babbage himself played a role during the Crimean War as a cryptographical adviser to his friend, Rear-Admiral Sir Francis Beaufort of the Admiralty. This is as much as we can be certain about in trying to make sense of the bulk of the time that Charles Babbage and Ada Lovelace worked together. Nine years of intensive work, involving the 'Engine' and a great deal of mathematics and research seems to have been lost: or has it? I've argued in the past http://www.simple-talk.com/community/blogs/philfactor/archive/2008/06/13/59614.aspx that the cracking of the Vigenère autokey cipher, was a fundamental motive behind the British Government's support and funding of the 'Difference Engine'. The Duke of Wellington, whose understanding of the military significance of being able to read enemy dispatches, was the most steadfast advocate of the project. If the three friends were actually doing the work of cracking codes by mathematical techniques that used the techniques of key periodicity, and symmetry of position (the use of a book being passed quickly to and fro is very suggestive), intending to then use the 'Engine' to do the routine cracking of each dispatch, then this is a rather different story. The project was Ada and William's idea. (William had served in the diplomatic service and would be familiar with the use of codes). This makes Ada Lovelace the initiator of a project which, by giving both Britain, and probably the USA, a diplomatic and military advantage in the second part of the Nineteenth century, changed world history. Ada would never have wanted any credit for cracking the cipher, and developing the method that rendered all contemporary military and diplomatic ciphering techniques nugatory; quite the reverse. And it is clear from the gaps in the record of the letters between the collaborators that the evidence was destroyed, probably on her request by her irascible but intensely honorable executor, Charles Babbage. Charles Babbage toyed with the idea of going public, but the Crimean war put an end to that. The British Government had a valuable secret, and intended to keep it that way. Ada and Charles had quite often discussed possible moneymaking projects that would fund the development of the Analytic Engine, the first programmable computer, but their secret work was never in the running as a potential cash cow. I suspect that the British Government was, even then, working on the concealment of a discovery whose value to the nation depended on it remaining so. The success of code-breaking in the Crimean war, and the American Civil war, led to the British and Americans  subsequently giving much more weight and funding to the science of decryption. Paradoxically, this makes Ada's contribution even closer to the creation of Colossus, the first digital computer, at Bletchley Park, specifically to crack the Nazi’s secret codes.

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  • Excel export displaying '#####...'

    - by Cypher
    I'm trying to export an Excel database into .txt (Tab Delimited), but some of my cells are quite large. When I export into a txt some of the cells are exported as '#######....' which is surprisingly useless. Has this happened to anyone else? Do you know an easy fix? Data from one cell of my column: Accounting, African Studies, Agricultural/Bioresource Engineering, Agricultural Economics, Agricultural Science, Anatomy/Cell Biology, Animal Biology, Animal Science, Anthropology, Applied Zoology, Architecture, Art History, Atmospheric/Oceanic Science, Biochemistry, Biology, Botanical Sciences, Canadian Studies, Chemical Engineering, Chemistry/Bio-Organic/Environmental/Materials,ChurchMusicPerformance, Civil Engineering/Applied Mechanics, Classics, Composition, Computer Engineering,ComputerScience, ContemporaryGerman Studies, Dietetics, Early Music Performance, Earth/Planetary Sciences, East Asian Studies, Economics, Electrical Engineering, English Literature/ Drama/Theatre/Cultural Studies, Entrepreneurship, Environment, Environmental Biology, Finance, Food Science, Foundations of Computing, French Language/Linguistics/Literature/Translation, Geography, Geography/ Urban Systems, German, German Language/Literature/Culture, Hispanic Languages/Literature/Culture,History,Humanistic Studies, Industrial Relations, Information Systems, International Business, International Development Studies, Italian Studies/Medieval/Renaissance, Jazz Performance, Jewish Studies, Keyboard Studies, Kindergarten/Elementary Education, Kindergarten/Elementary Education/Jewish Studies,Kinesiology, Labor/Management Relations, Latin American/Caribbean Studies, Linguistics, Literature/Translation, Management Science, Marketing, Materials Engineering,Mathematics,Mathematics/Statistics,Mechanical Engineering, Microbiology, Microbiology/Immunology, Middle Eastern Studies, Mining Engineering, Music, Music Education, MusicHistory,Music Technology,Music Theory,North American Studies, Nutrition,OperationsManagement,OrganizationalBehavior/Human Resources Management, Performing Arts, Philosophy, Physical Education, Physics, Physiology, Plant Sciences, Political Science, Psychology, Quebec Studies, Religious Studies/Scriptures/Interpretations/World Religions,ResourceConservation,Russian, Science for Teachers,Secondary Education, Secondary Education/Music, Secondary Education/Science, SocialWork, Sociology, Software Engineering, Soil Science, Strategic Management, Teaching of French/English as a Second Language, Theology, Wildlife Biology, Wildlife Resources, Women’s Studies.

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  • Scientific Algorithms that can produce imagery, pseudocode perhaps?

    - by Ross
    Hello, I have a client who are based in the field of mathematics. We are developing, amongst other things, a website. I like to create a mock-up of a drawing tool that an produce some imagery in the background based on some scientific algorithms. The intention being that the may client, later, may create there own. (They use emacs for everything, great client.) I'm look for an answer of where or what to go looking for. Not code specific, pseudocode even, as we can adapt and have not yet settled on a platform. I'm afraid my mathematic stops at the power of two and some trigonometry. Appreciated if they're are any mathematics related students/academics how could enlighten me? What to search for will be accepted? Edit: To summarise/clarify, I want to draw pretty pictures (the design perspective). I want them to have some context (i.e. not just for the sake of pretty images but have some explanation available). In essence I would to create a rendering engine which we can draw the images and we set the style parameters: line, colour, etc... But to pursue this option I want to experiment myself. Thanks Ross

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  • 3D and AI basics. The foundation before the coding.

    - by Allan
    Hi, everyone. (If you have the time and patience:) I've recently made the decision to study programming seriously and I'm about to order TAOCP and Concrete Mathematics to begin my studies (please don't get caught up on this). I'm very much interested in learning and understanding how 3D works but I'm aware that if I plan to do it right there's still a long walk before I get to actually play with 3D coding. Now to the question.. (tl;dr) Excluding programming itself, what disciplines do I have to be familiar with to code 3D? What kinds of mathematics? Physics? What else? What books do you recommend on such subjects? Now read it all again but replacing "3D" with "AI". Please don't recommend computer-specific books. The question is about the foundation to be learned before using the machine. Also, if possible, please keep the list brief; I plan to order one book on each subject but no more than that for now. Excuse me for any English mistakes, it's not my first language. Thank you.

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  • Parallelism in .NET – Part 13, Introducing the Task class

    - by Reed
    Once we’ve used a task-based decomposition to decompose a problem, we need a clean abstraction usable to implement the resulting decomposition.  Given that task decomposition is founded upon defining discrete tasks, .NET 4 has introduced a new API for dealing with task related issues, the aptly named Task class. The Task class is a wrapper for a delegate representing a single, discrete task within your decomposition.  We will go into various methods of construction for tasks later, but, when reduced to its fundamentals, an instance of a Task is nothing more than a wrapper around a delegate with some utility functionality added.  In order to fully understand the Task class within the new Task Parallel Library, it is important to realize that a task really is just a delegate – nothing more.  In particular, note that I never mentioned threading or parallelism in my description of a Task.  Although the Task class exists in the new System.Threading.Tasks namespace: Tasks are not directly related to threads or multithreading. Of course, Task instances will typically be used in our implementation of concurrency within an application, but the Task class itself does not provide the concurrency used.  The Task API supports using Tasks in an entirely single threaded, synchronous manner. Tasks are very much like standard delegates.  You can execute a task synchronously via Task.RunSynchronously(), or you can use Task.Start() to schedule a task to run, typically asynchronously.  This is very similar to using delegate.Invoke to execute a delegate synchronously, or using delegate.BeginInvoke to execute it asynchronously. The Task class adds some nice functionality on top of a standard delegate which improves usability in both synchronous and multithreaded environments. The first addition provided by Task is a means of handling cancellation via the new unified cancellation mechanism of .NET 4.  If the wrapped delegate within a Task raises an OperationCanceledException during it’s operation, which is typically generated via calling ThrowIfCancellationRequested on a CancellationToken, or if the CancellationToken used to construct a Task instance is flagged as canceled, the Task’s IsCanceled property will be set to true automatically.  This provides a clean way to determine whether a Task has been canceled, often without requiring specific exception handling. Tasks also provide a clean API which can be used for waiting on a task.  Although the Task class explicitly implements IAsyncResult, Tasks provide a nicer usage model than the traditional .NET Asynchronous Programming Model.  Instead of needing to track an IAsyncResult handle, you can just directly call Task.Wait() to block until a Task has completed.  Overloads exist for providing a timeout, a CancellationToken, or both to prevent waiting indefinitely.  In addition, the Task class provides static methods for waiting on multiple tasks – Task.WaitAll and Task.WaitAny, again with overloads providing time out options.  This provides a very simple, clean API for waiting on single or multiple tasks. Finally, Tasks provide a much nicer model for Exception handling.  If the delegate wrapped within a Task raises an exception, the exception will automatically get wrapped into an AggregateException and exposed via the Task.Exception property.  This exception is stored with the Task directly, and does not tear down the application.  Later, when Task.Wait() (or Task.WaitAll or Task.WaitAny) is called on this task, an AggregateException will be raised at that point if any of the tasks raised an exception.  For example, suppose we have the following code: Task taskOne = new Task( () => { throw new ApplicationException("Random Exception!"); }); Task taskTwo = new Task( () => { throw new ArgumentException("Different exception here"); }); // Start the tasks taskOne.Start(); taskTwo.Start(); try { Task.WaitAll(new[] { taskOne, taskTwo }); } catch (AggregateException e) { Console.WriteLine(e.InnerExceptions.Count); foreach (var inner in e.InnerExceptions) Console.WriteLine(inner.Message); } .csharpcode, .csharpcode pre { font-size: small; color: black; font-family: consolas, "Courier New", courier, monospace; background-color: #ffffff; /*white-space: pre;*/ } .csharpcode pre { margin: 0em; } .csharpcode .rem { color: #008000; } .csharpcode .kwrd { color: #0000ff; } .csharpcode .str { color: #006080; } .csharpcode .op { color: #0000c0; } .csharpcode .preproc { color: #cc6633; } .csharpcode .asp { background-color: #ffff00; } .csharpcode .html { color: #800000; } .csharpcode .attr { color: #ff0000; } .csharpcode .alt { background-color: #f4f4f4; width: 100%; margin: 0em; } .csharpcode .lnum { color: #606060; } Here, our routine will print: 2 Different exception here Random Exception! Note that we had two separate tasks, each of which raised two distinctly different types of exceptions.  We can handle this cleanly, with very little code, in a much nicer manner than the Asynchronous Programming API.  We no longer need to handle TargetInvocationException or worry about implementing the Event-based Asynchronous Pattern properly by setting the AsyncCompletedEventArgs.Error property.  Instead, we just raise our exception as normal, and handle AggregateException in a single location in our calling code.

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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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  • How To Make NVIDIA’s Optimus Work on Linux

    - by Chris Hoffman
    Many new laptops come with NVIDIA’s Optimus technology – the laptop includes both a discrete NVIDIA GPU for gaming power and an onboard Intel GPU for power savings. The notebook switches between the two when necessary. However, this isn’t yet well-supported on Linux. Linus Torvalds had some choice words for NVIDIA regarding Optimus not working on Linux, and NVIDIA is now currently working on official support. However, if you have a laptop with Optimus support, you don’t have to wait for NVIDIA — you can use the Bumblebee project’s solution to enable Optimus on Linux today. Image Credit: Jemimus on Flickr How To Create a Customized Windows 7 Installation Disc With Integrated Updates How to Get Pro Features in Windows Home Versions with Third Party Tools HTG Explains: Is ReadyBoost Worth Using?

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