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  • SQL Server 2012 : Changes to system objects in RC0

    - by AaronBertrand
    As with every new major milestone, one of the first things I do is check out what has changed under the covers. Since RC0 was released yesterday, I've been poking around at some of the DMV and other system changes. Here is what I have noticed: New objects in RC0 that weren't in CTP3 Quick summary: We see a bunch of new aggregates for use with geography and geometry. I've stayed away from that area of programming so I'm not going to dig into them. There is a new extended procedure called sp_showmemo_xml....(read more)

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  • The Breakpoint Ep. 4 —The Tour De Timeline

    The Breakpoint Ep. 4 —The Tour De Timeline Ask and vote for questions at: goo.gl The DevTools' Timeline shows the heartbeat and health of your application's performance. In this episode we'll do a deep deep dive into how to uncover the cost of internal browser operations like parsing HTML, decoding images, invalidating layout geometry and painting to screen. Paul and Addy will show you how best to approach improving the performance of your CSS and JS. From: GoogleDevelopers Views: 0 0 ratings Time: 01:00:00 More in Science & Technology

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  • how to start LXDE session automatically after tightvncserver starts to make me able see desktop when connecting to the host via vncclient?

    - by Oleksandr Dudchenko
    I have system which is equipped with Intel Celeron processor 1.1 GHz s370 with 384 Mb of RAM on Intel d815egew motherboard which supports wake-on-lan function. I want to use such a PC for Internet sharing to the local network. Also this PC is a DHCP+DNS server as well as router/gateway. Based on above I decided to install Lubuntu as it is lightweight system. I installed Lubuntu 10.04.4 LTS from alternate ISO. System has no auto login. System boots and has acceptable performance. Host PC has onboard 4 network adapters: eth0 – ethernet controller which is used for Local Network connections. Has static address 10.0.0.1 eth1 – ethernet controller which is not used and not configured so far, I plan to connect printer here later on. eth2 - ethernet controller which is used to connect to Internet, which we plan to share for the local network wlan0 – wireless controller, it is used in role of access poit for local Network and has address 10.0.0.2 We want to control our gateway remotely. So, we need to be able to power it on remotely. To allow this I’ve done the following things: $ cd /etc/init.d/ made a new file with command $ sudo vim wakeonlanconfig Wrote the following lines to the newly created file, saved and closed it #!/bin/bash ethtool -s eth0 wol g ethtool -s eth2 wol g exit Made the abovementioned file executable $ sudo chmod a+x wakeonlanconfig Then included it into autostart sequence during boot. $ sudo update-rc.d -f wakeonlanconfig defaults after system reboot we will be able to poweron system remotely. Than we need to have a possibility to connect remotely to the host via SSH and VNC. So, I installed following packets with the following commands: $ sudo apt-get update $ sudo apt-get install openssh-server tightvncserver Add ssh daemon into autostart sequence during boot. $ sudo update-rc.d -f ssh defaults Power off the host PC $ sudo halt Then I went to remote place, send magic paket and powered the Host up. System started... And I connected to the host via Putty from remote system under Windows. Than logged in and run the command to start vnc server. $ tightvncserver -geometry 800x600 -depth 16 :2 VNC server successfully started and I got message like follows. New 'X' desktop is gateway:2 Starting applications specified in /home/dolv/.vnc/xstartup Log file is /home/dolv/.vnc/gateway:2.log Using UltraVNC Viewer programm under windows I connected to the host's vnc server, enterd the password and.... sow only mouse cursor in form of cross on a grey background of 800x600 dots, no desktop. Here is my .vnc/xstartup file #!/bin/sh xrdb $HOME/.Xresources xsetroot -solid grey #x-terminal-emulator -geometry 80x24+10+10 -ls -title "$VNCDESKTOP Desktop" & #x-window-manager & # Fix to make GNOME work export XKL_XMODMAP_DISABLE=1 /etc/X11/Xsession The Question: What I have to change and where to make LXDE session start automatically after tightvncserver starts?

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  • The need for user-defined index types

    - by Greg Low
    Since the removal of the 8KB limit on serialization, the ability to define new data types using SQL CLR integration is now almost at a usable level, apart from one key omission: indexes. We have no ability to create our own types of index to support our data types. As a good example of this, consider that when Microsoft introduced the geometry and geography (spatial) data types, they did so as system CLR data types but also needed to introduce a spatial index as a new type of index. Those of us that...(read more)

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  • Engine Rendering pipeline : Making shaders generic

    - by fakhir
    I am trying to make a 2D game engine using OpenGL ES 2.0 (iOS for now). I've written Application layer in Objective C and a separate self contained RendererGLES20 in C++. No GL specific call is made outside the renderer. It is working perfectly. But I have some design issues when using shaders. Each shader has its own unique attributes and uniforms that need to be set just before the main draw call (glDrawArrays in this case). For instance, in order to draw some geometry I would do: void RendererGLES20::render(Model * model) { // Set a bunch of uniforms glUniformMatrix4fv(.......); // Enable specific attributes, can be many glEnableVertexAttribArray(......); // Set a bunch of vertex attribute pointers: glVertexAttribPointer(positionSlot, 2, GL_FLOAT, GL_FALSE, stride, m->pCoords); // Now actually Draw the geometry glDrawArrays(GL_TRIANGLES, 0, m->vertexCount); // After drawing, disable any vertex attributes: glDisableVertexAttribArray(.......); } As you can see this code is extremely rigid. If I were to use another shader, say ripple effect, i would be needing to pass extra uniforms, vertex attribs etc. In other words I would have to change the RendererGLES20 render source code just to incorporate the new shader. Is there any way to make the shader object totally generic? Like What if I just want to change the shader object and not worry about game source re-compiling? Any way to make the renderer agnostic of uniforms and attributes etc?. Even though we need to pass data to uniforms, what is the best place to do that? Model class? Is the model class aware of shader specific uniforms and attributes? Following shows Actor class: class Actor : public ISceneNode { ModelController * model; AIController * AI; }; Model controller class: class ModelController { class IShader * shader; int textureId; vec4 tint; float alpha; struct Vertex * vertexArray; }; Shader class just contains the shader object, compiling and linking sub-routines etc. In Game Logic class I am actually rendering the object: void GameLogic::update(float dt) { IRenderer * renderer = g_application->GetRenderer(); Actor * a = GetActor(id); renderer->render(a->model); } Please note that even though Actor extends ISceneNode, I haven't started implementing SceneGraph yet. I will do that as soon as I resolve this issue. Any ideas how to improve this? Related design patterns etc? Thank you for reading the question.

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  • T-SQL Tuesday #006 Round-up!

    - by Mike C
    T-SQL Tuesday this month was all about LOB (large object) data. Thanks to all the great bloggers out there who participated! The participants this month posted some very impressive articles with information running the gamut from Reporting Services to SQL Server spatial data types to BLOB-handling in SSIS. One thing I noticed immediately was a trend toward articles about spatial data (SQL Server 2008 Geography and Geometry data types, a very fun topic to explore if you haven’t played around with...(read more)

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  • Strategy to prevent players from seeing through walls in an online FPS?

    - by geneotech
    Why do we still moan on wallhackers in multiplayer first-person shooters ? Isn't it possible to perform occlusion culling for all players server-side ? For example, send player xyz information to client only when the player is visible in client's frustum and not occluded by any object ? Even if the collision-geometry is very simplified, most of the time cheater won't receive tactical information. Why not do this ?

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  • Uniform not being applied to proper mesh

    - by HaMMeReD
    Ok, I got some code, and you select blocks on a grid. The selection works. I can modify the blocks to be raised when selected and the correct one shows. I set a color which I use in the shader. However, I am trying to change the color before rendering the geometry, and the last rendered geometry (in the sequence) is rendered light. However, to debug logic I decided to move the block up and make it white, in which case one block moves up and another block becomes white. I checked all my logic and it knows the correct one is selected and it is showing in, in the correct place and rendering it correctly. When there is only 1 it works properly. Video Of the bug in action, note how the highlighted and elevated blocks are not the same block, however the code for color and My Renderer is here (For the items being drawn) public void render(Renderer renderer) { mGrid.render(renderer, mGameState); for (Entity e:mGameEntities) { UnitTypes ut = UnitTypes.valueOf((String)e.getObject(D.UNIT_TYPE.ordinal())); if (ut == UnitTypes.Soldier) { renderer.testShader.begin(); renderer.testShader.setUniformMatrix("u_mvpMatrix",mEntityMatrix); renderer.texture_soldier.bind(0); Vector2 pos = (Vector2) e.getObject(D.COORDS.ordinal()); mEntityMatrix.set(renderer.mCamera.combined); if (mSelectedEntities.contains(e)) { mEntityMatrix.translate(pos.x, 1f, pos.y); renderer.testShader.setUniformf("v_color", 0.5f,0.5f,0.5f,1f); } else { mEntityMatrix.translate(pos.x, 0f, pos.y); renderer.testShader.setUniformf("v_color", 1f,1f,1f,1f); } mEntityMatrix.scale(0.2f, 0.2f, 0.2f); renderer.model_soldier.render(renderer.testShader,GL20.GL_TRIANGLES); renderer.testShader.end(); } else if (ut == UnitTypes.Enemy_Infiltrator) { renderer.testShader.begin(); renderer.testShader.setUniformMatrix("u_mvpMatrix",mEntityMatrix); renderer.testShader.setUniformf("v_color", 1.0f,1,1,1.0f); renderer.texture_enemy_infiltrator.bind(0); Vector2 pos = (Vector2) e.getObject(D.COORDS.ordinal()); mEntityMatrix.set(renderer.mCamera.combined); mEntityMatrix.translate(pos.x, 0f, pos.y); mEntityMatrix.scale(0.2f, 0.2f, 0.2f); renderer.model_enemy_infiltrator.render(renderer.testShader,GL20.GL_TRIANGLES); renderer.testShader.end(); } } }

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  • Achieve Spatial Data Support in SSIS

    Overview SQL Server 2008 introduced a new category of datatypes known as spatial datatypes which stores spatial information. The new spatial datatypes are geography and geometry. SQL Server Management Studio comes with good good support for these spatial data ... [Read Full Article]

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  • Really weird GL Behaviour, uniform not "hitting" proper mesh? LibGdx

    - by HaMMeReD
    Ok, I got some code, and you select blocks on a grid. The selection works. I can modify the blocks to be raised when selected and the correct one shows. I set a color which I use in the shader. However, I am trying to change the color before rendering the geometry, and the last rendered geometry (in the sequence) is rendered light. However, to debug logic I decided to move the block up and make it white, in which case one block moves up and another block becomes white. I checked all my logic and it knows the correct one is selected and it is showing in, in the correct place and rendering it correctly. When there is only 1 it works properly. Video Of the bug in action, note how the highlighted and elevated blocks are not the same block, however the code for color and My Renderer is here (For the items being drawn) public void render(Renderer renderer) { mGrid.render(renderer, mGameState); for (Entity e:mGameEntities) { UnitTypes ut = UnitTypes.valueOf((String)e.getObject(D.UNIT_TYPE.ordinal())); if (ut == UnitTypes.Soldier) { renderer.testShader.begin(); renderer.testShader.setUniformMatrix("u_mvpMatrix",mEntityMatrix); renderer.texture_soldier.bind(0); Vector2 pos = (Vector2) e.getObject(D.COORDS.ordinal()); mEntityMatrix.set(renderer.mCamera.combined); if (mSelectedEntities.contains(e)) { mEntityMatrix.translate(pos.x, 1f, pos.y); renderer.testShader.setUniformf("v_color", 0.5f,0.5f,0.5f,1f); } else { mEntityMatrix.translate(pos.x, 0f, pos.y); renderer.testShader.setUniformf("v_color", 1f,1f,1f,1f); } mEntityMatrix.scale(0.2f, 0.2f, 0.2f); renderer.model_soldier.render(renderer.testShader,GL20.GL_TRIANGLES); renderer.testShader.end(); } else if (ut == UnitTypes.Enemy_Infiltrator) { renderer.testShader.begin(); renderer.testShader.setUniformMatrix("u_mvpMatrix",mEntityMatrix); renderer.testShader.setUniformf("v_color", 1.0f,1,1,1.0f); renderer.texture_enemy_infiltrator.bind(0); Vector2 pos = (Vector2) e.getObject(D.COORDS.ordinal()); mEntityMatrix.set(renderer.mCamera.combined); mEntityMatrix.translate(pos.x, 0f, pos.y); mEntityMatrix.scale(0.2f, 0.2f, 0.2f); renderer.model_enemy_infiltrator.render(renderer.testShader,GL20.GL_TRIANGLES); renderer.testShader.end(); } } }

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  • How do I get a collision event from a KActor subclass?

    - by Almo
    I have a subclass of KActor, and I want an event when it collides with things. event RigidBodyCollision seems to be what I want according to this http://wiki.beyondunreal.com/UE3:Actor_events_%28UDK%29#RigidBodyCollision Called when a PrimitiveComponent this Actor owns has: -bNotifyRigidBodyCollision set to true -ScriptRigidBodyCollisionThreshold greater than 0 -it is involved in a physics collision where the relative velocity exceeds ScriptRigidBodyCollisionThreshold As far as I can tell, I have these set up, and the event is not called for any collisions (KActor-KActor, KActor-world geometry, etc). Is there something else I need to do?

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  • Why does this loopback device creation malfunction?

    - by user50118
    The stackoverflow people thought this was more appropriate here, I put it there as it is part of a program but I can see their POV, so here it is: At the bottom of the code you can see it failing. In fact, I'll put it here at the start too because it is the problem I need to solve: [350591.924819] EXT4-fs (loop0): bad geometry: block count 9750806 exceeds size of device (9750168 blocks) I don't understand why the device is supposedly too small. I made this partition two days ago with normal fdisk, it was created and formatted with ext4 supplying no options other than the partition (/dev/sdb2) to format. The only explaination I can think of is that ext4 has the size of the partition wrong somehow but that seems very unlikely. What is wrong with my math? The offset is correct, you can see that with the file command, and the size should be correct too because End - Start comes to the same number of sectors minus 1, just like it should (A disk starting on sector 1 and ending on sector 2 would be 2 - 1 = 1 and have two sectors). # sfdisk -luS /dev/sdb Disk /dev/sdb: 9729 cylinders, 255 heads, 63 sectors/track Units = sectors of 512 bytes, counting from 0 Device Boot Start End #sectors Id System /dev/sdb2 78295040 156296384 78001345 83 Linux # losetup -r -f --show -o $((78295040 * 512)) --sizelimit $((78001345 * 512)) /dev/sdb /dev/loop0 # file -s /dev/loop0 /dev/loop0: Linux rev 1.0 ext4 filesystem data (needs journal recovery) (extents) (large files) (huge files) # mount -o ro -t ext4 /dev/loop0 /mnt mount: wrong fs type, bad option, bad superblock on /dev/loop0, missing codepage or helper program, or other error In some cases useful info is found in syslog - try dmesg | tail or so # dmesg | tail -n 1 [350591.924819] EXT4-fs (loop0): bad geometry: block count 9750806 exceeds size of device (9750168 blocks)

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  • flash core engine by Dinesh [closed]

    - by hdinesh
    This post was a dump of the following code (without the highlights). No question, just a dump. Please update this q. with a real question to have it reopened. You (the asker) risk to be flagged as spammer (if not already) and a bad reputation. This is a q/a site, not a site to promote your own code libraries. package facers { import flash.display.*; import flash.events.*; import flash.geom.ColorTransform; import flash.utils.Dictionary; import org.papervision3d.cameras.*; import org.papervision3d.scenes.*; import org.papervision3d.objects.*; import org.papervision3d.objects.special.*; import org.papervision3d.objects.primitives.*; import org.papervision3d.materials.*; import org.papervision3d.events.FileLoadEvent; import org.papervision3d.materials.special.*; import org.papervision3d.materials.shaders.*; import org.papervision3d.materials.utils.*; import org.papervision3d.lights.*; import org.papervision3d.render.*; import org.papervision3d.view.*; import org.papervision3d.events.InteractiveScene3DEvent; import org.papervision3d.events.*; import org.papervision3d.core.utils.*; import org.papervision3d.core.geom.renderables.Vertex3D; import caurina.transitions.*; public class Main extends Sprite { public var viewport :BasicView; public var displayObject :DisplayObject3D; private var light :PointLight3D; private var shadowPlane :Plane; private var dataArray :Array; private var material :BitmapFileMaterial; private var planeByContainer :Dictionary = new Dictionary(); private var paperSize :Number = 0.5; private var cloudSize :Number = 1500; private var rotSize :Number = 360; private var maxAlbums :Number = 50; private var num :Number = 0; public function Main():void { trace("START APPLICATION"); viewport = new BasicView(1024, 690, true, true, CameraType.FREE); viewport.camera.zoom = 50; viewport.camera.extra = { goPosition: new DisplayObject3D(),goTarget: new DisplayObject3D() }; addChild(viewport); displayObject = new DisplayObject3D(); viewport.scene.addChild(displayObject); createAlbum(); addEventListener(Event.ENTER_FRAME, onRenderEvent); } private function createAlbum() { dataArray = new Array("images/thums/pic1.jpg", "images/thums/pic2.jpg", "images/thums/pic3.jpg", "images/thums/pic4.jpg", "images/thums/pic5.jpg", "images/thums/pic6.jpg", "images/thums/pic7.jpg", "images/thums/pic8.jpg", "images/thums/pic9.jpg", "images/thums/pic10.jpg", "images/thums/pic1.jpg", "images/thums/pic2.jpg", "images/thums/pic3.jpg", "images/thums/pic4.jpg", "images/thums/pic5.jpg", "images/thums/pic6.jpg", "images/thums/pic7.jpg", "images/thums/pic8.jpg", "images/thums/pic9.jpg", "images/thums/pic10.jpg"); for (var i:int = 0; i < dataArray.length; i++) { material = new BitmapFileMaterial(dataArray[i]); material.doubleSided = true; material.addEventListener(FileLoadEvent.LOAD_COMPLETE, loadMaterial); } } public function loadMaterial(event:Event) { var plane:Plane = new Plane(material, 300, 180); displayObject.addChild(plane); var _x:int = Math.random() * cloudSize - cloudSize/2; var _y:int = Math.random() * cloudSize - cloudSize/2; var _z:int = Math.random() * cloudSize - cloudSize/2; var _rotationX:int = Math.random() * rotSize; var _rotationY:int = Math.random() * rotSize; var _rotationZ:int = Math.random() * rotSize; Tweener.addTween(plane, { x:_x, y:_y, z:_z, rotationX:_rotationX, rotationY:_rotationY, rotationZ:_rotationZ, time:5, transition:"easeIn" } ); } protected function onRenderEvent(event:Event):void { var rotY: Number = (mouseY-(stage.stageHeight/2))/(900/2)*(1200); var rotX: Number = (mouseX-(stage.stageWidth/2))/(600/2)*(-1200); displayObject.rotationY = viewport.camera.x + (rotX - viewport.camera.x) / 50; displayObject.rotationX = viewport.camera.y + (rotY - viewport.camera.y) / 30; viewport.singleRender(); } } } package designLab.events { import flash.display.BlendMode; import flash.display.Sprite; import flash.events.Event; import flash.filters.BlurFilter; // Import designLab import designLab.layer.IntroLayer; import designLab.shadow.ShadowCaster; import designLab.utils.LayerConstant; // Import Papervision3D import org.papervision3d.cameras.*; import org.papervision3d.scenes.*; import org.papervision3d.objects.*; import org.papervision3d.objects.special.*; import org.papervision3d.objects.primitives.*; import org.papervision3d.materials.*; import org.papervision3d.materials.special.*; import org.papervision3d.materials.shaders.*; import org.papervision3d.materials.utils.*; import org.papervision3d.lights.*; import org.papervision3d.render.*; import org.papervision3d.view.*; import org.papervision3d.events.InteractiveScene3DEvent; import org.papervision3d.events.*; import org.papervision3d.core.utils.*; import org.papervision3d.core.geom.renderables.Vertex3D; public class CoreEnging extends Sprite { public var viewport :BasicView; // Create BasicView public var displayObject :DisplayObject3D; // Create DisplayObject public var shadowCaster :ShadowCaster; // Create ShadowCaster private var light :PointLight3D; // Create PointLight private var shadowPlane :Plane; // Create Plane private var layer :LayerConstant; // Create constant resource layer private static var instance :CoreEnging; // Create CoreEnging class static instance // CoreEnging class static instance mathod function public static function getinstance() { if (instance != null) return instance; else { instance = new CoreEnging(); return instance; } } // CoreEnging constrictor public function CoreEnging () { trace("INFO: Design Lab Application : Core Enging v0.1"); layer = new LayerConstant(); viewport = new BasicView(900, 600, true, true, CameraType.FREE); // pass the width, height, scaleToStage, interactive, cameraType to BasicView viewport.camera.zoom = 100; // Define the zoom level of camera addChild(viewport); createFloor(); // Create the floor displayObject = new DisplayObject3D(); // Create new instance of DisplayObject viewport.scene.addChild(displayObject); // Add the DisplayObject to the BasicView light = new PointLight3D(); // Create new instance of PointLight light.z = -50; // Position the Z of create instance light.x = 0; //Position the X of create instance light.rotationZ = 45; //Position the rotation angel of the Z of create instance light.y = 500; //Position the Y of create instance shadowCaster = new ShadowCaster("shadow", 0x000000, BlendMode.MULTIPLY, .1, [new BlurFilter(20, 20, 1)]); // pass shadowcaster name, color, blend mode, alpha and filters shadowCaster.setType(ShadowCaster.SPOTLIGHT); // Define the shadow type addEventListener(Event.ENTER_FRAME, onRenderEvent); // Add frame render event } // Start create floor public function createFloor() { var spr:Sprite = new Sprite(); // Create Sprite spr.graphics.beginFill(0xFFFFFF); // Define the fill color for sprite spr.graphics.drawRect(0, 0, 600, 600); // Define the X, Y, width, height of the sprite var sprMaterial:MovieMaterial = new MovieMaterial(spr, true, true, true); //Create a texture from an existing sprite instance shadowPlane = new Plane(sprMaterial, 2000, 2000, 1, 1); // create new instance of the Plane and pass the texture material, width, height, segmentsW and segmentsH shadowPlane.rotationX = 80; //Position the rotation angel of the X of Plane shadowPlane.y = -200; //Position the Y of Plane viewport.scene.addChild(shadowPlane); // Add the Plane to the BasicView } // switch method function of the page layer control public function addLayer(type:String) { switch (type) { case layer.INTRO: var intro:IntroLayer = new IntroLayer(); break; } } // Create get mathod function for DisplayObject public function getDisplayObject():DisplayObject3D { return displayObject; } // Create get mathod function for BasicView public function getViewport():BasicView { return viewport; } // Rendering function protected function onRenderEvent(event:Event):void { var rotY: Number = (mouseY-(stage.stageHeight/2))/(900/2)*(1200); var rotX: Number = (mouseX-(stage.stageWidth/2))/(600/2)*(-1200); displayObject.rotationY = viewport.camera.x + (rotX - viewport.camera.x) / 50; displayObject.rotationX = viewport.camera.y + (rotY - viewport.camera.y) / 30; // Remove the shadow shadowCaster.invalidate(); // create new shadow on DisplayObject move shadowCaster.castModel(displayObject, light, shadowPlane); viewport.singleRender(); } } } package designLab.layer { import flash.display.Sprite; import flash.events.Event; // Import designLab import designLab.materials.iBusinessCard; import designLab.events.CoreEnging; // Import Papervision3D import org.papervision3d.objects.primitives.Cube; import org.papervision3d.materials.ColorMaterial; import org.papervision3d.materials.MovieMaterial; public class IntroLayer { // IntroLayer constrictor public function IntroLayer() { trace("INFO: Load Intro layer"); var indexDP:DP_index = new DP_index(); //Create the library MovieClip var blackMaterial:MovieMaterial = new MovieMaterial(indexDP, true); //Create a texture from an existing library MovieClip instance blackMaterial.smooth = true; blackMaterial.doubleSided = false; var mycolor:ColorMaterial = new ColorMaterial(0x000000); //Create solid color material var mycard:iBusinessCard = new iBusinessCard(blackMaterial, blackMaterial, mycolor, 372, 10, 207); // Create custom 3D cube object to pass the Front, Back, All, CubeWidth, CubeDepth and CubeHeight CoreEnging.getinstance().getDisplayObject().addChild(mycard.create3DCube()); // Add the custom 3D cube to the DisplayObject } } } package designLab.materials { import flash.display.*; import flash.events.*; // Import Papervision3D import org.papervision3d.materials.*; import org.papervision3d.materials.utils.MaterialsList; import org.papervision3d.objects.primitives.Cube; public class iBusinessCard extends Sprite { private var materialsList :MaterialsList; private var cube :Cube; private var Front :MovieMaterial = new MovieMaterial(); private var Back :MovieMaterial = new MovieMaterial(); private var All :ColorMaterial = new ColorMaterial(); private var CubeWidth :Number; private var CubeDepth :Number; private var CubeHeight :Number; public function iBusinessCard(Front:MovieMaterial, Back:MovieMaterial, All:ColorMaterial, CubeWidth:Number, CubeDepth:Number, CubeHeight:Number) { setFront(Front); setBack(Back); setAll(All); setCubeWidth(CubeWidth); setCubeDepth(CubeDepth); setCubeHeight(CubeHeight); } public function create3DCube():Cube { materialsList = new MaterialsList(); materialsList.addMaterial(Front, "front"); materialsList.addMaterial(Back, "back"); materialsList.addMaterial(All, "left"); materialsList.addMaterial(All, "right"); materialsList.addMaterial(All, "top"); materialsList.addMaterial(All, "bottom"); cube = new Cube(materialsList, CubeWidth, CubeDepth, CubeHeight); cube.x = 0; cube.y = 0; cube.z = 0; cube.rotationY = 180; return cube; } public function setFront(Front:MovieMaterial) { this.Front = Front; } public function getFront():MovieMaterial { return Front; } public function setBack(Back:MovieMaterial) { this.Back = Back; } public function getBack():MovieMaterial { return Back; } public function setAll(All:ColorMaterial) { this.All = All; } public function getAll():ColorMaterial { return All; } public function setCubeWidth(CubeWidth:Number) { this.CubeWidth = CubeWidth; } public function getCubeWidth():Number { return CubeWidth; } public function setCubeDepth(CubeDepth:Number) { this.CubeDepth = CubeDepth; } public function getCubeDepth():Number { return CubeDepth; } public function setCubeHeight(CubeHeight:Number) { this.CubeHeight = CubeHeight; } public function getCubeHeight():Number { return CubeHeight; } } } package designLab.shadow { import flash.display.Sprite; import flash.filters.BlurFilter; import flash.geom.Point; import flash.geom.Rectangle; import flash.utils.Dictionary; import org.papervision3d.core.geom.TriangleMesh3D; import org.papervision3d.core.geom.renderables.Triangle3D; import org.papervision3d.core.geom.renderables.Vertex3D; import org.papervision3d.core.math.BoundingSphere; import org.papervision3d.core.math.Matrix3D; import org.papervision3d.core.math.Number3D; import org.papervision3d.core.math.Plane3D; import org.papervision3d.lights.PointLight3D; import org.papervision3d.materials.MovieMaterial; import org.papervision3d.objects.DisplayObject3D; import org.papervision3d.objects.primitives.Plane; public class ShadowCaster { private var vertexRefs:Dictionary; private var numberRefs:Dictionary; private var lightRay:Number3D = new Number3D() private var p3d:Plane3D = new Plane3D(); public var color:uint = 0; public var alpha:Number = 0; public var blend:String = ""; public var filters:Array; public var uid:String; private var _type:String = "point"; private var dir:Number3D; private var planeBounds:Dictionary; private var targetBounds:Dictionary; private var models:Dictionary; public static var DIRECTIONAL:String = "dir"; public static var SPOTLIGHT:String = "spot"; public function ShadowCaster(uid:String, color:uint = 0, blend:String = "multiply", alpha:Number = 1, filters:Array=null) { this.uid = uid; this.color = color; this.alpha = alpha; this.blend = blend; this.filters = filters ? filters : [new BlurFilter()]; numberRefs = new Dictionary(true); targetBounds = new Dictionary(true); planeBounds = new Dictionary(true); models = new Dictionary(true); } public function castModel(model:DisplayObject3D, light:PointLight3D, plane:Plane, faces:Boolean = true, cull:Boolean = false):void{ var ar:Array; if(models[model]) { ar = models[model]; }else{ ar = new Array(); getChildMesh(model, ar); models[model] = ar; } var reset:Boolean = true; for each(var t:TriangleMesh3D in ar){ if(faces) castFaces(light, t, plane, cull, reset); else castBoundingSphere(light, t, plane, 0.75, reset); reset = false; } } private function getChildMesh(do3d:DisplayObject3D, ar):void{ if(do3d is TriangleMesh3D) ar.push(do3d); for each(var d:DisplayObject3D in do3d.children) getChildMesh(d, ar); } public function setType(type:String="point"):void{ _type = type; } public function getType():String{ return _type; } public function castBoundingSphere(light:PointLight3D, target:TriangleMesh3D, plane:Plane, scaleRadius:Number=0.8, clear:Boolean = true):void{ var planeVertices:Array = plane.geometry.vertices; //convert to target space? var world:Matrix3D = plane.world; var inv:Matrix3D = Matrix3D.inverse(plane.transform); var lp:Number3D = new Number3D(light.x, light.y, light.z); Matrix3D.multiplyVector(inv, lp); p3d.setNormalAndPoint(plane.geometry.faces[0].faceNormal, new Number3D()); var b:BoundingSphere = target.geometry.boundingSphere; var bounds:Object = planeBounds[plane]; if(!bounds){ bounds = plane.boundingBox(); planeBounds[plane] = bounds; } var tbounds:Object = targetBounds[target]; if(!tbounds){ tbounds = target.boundingBox(); targetBounds[target] = tbounds; } var planeMovie:Sprite = Sprite(MovieMaterial(plane.material).movie); var movieSize:Point = new Point(planeMovie.width, planeMovie.height); var castClip:Sprite = getCastClip(plane); castClip.blendMode = this.blend; castClip.filters = this.filters; castClip.alpha = this.alpha; if(clear) castClip.graphics.clear(); vertexRefs = new Dictionary(true); var tlp:Number3D = new Number3D(light.x, light.y, light.z); Matrix3D.multiplyVector(Matrix3D.inverse(target.world), tlp); var center:Number3D = new Number3D(tbounds.min.x+tbounds.size.x*0.5, tbounds.min.y+tbounds.size.y*0.5, tbounds.min.z+tbounds.size.z*0.5); var dif:Number3D = Number3D.sub(lp, center); dif.normalize(); var other:Number3D = new Number3D(); other.x = -dif.y; other.y = dif.x; other.z = 0; other.normalize(); var cross:Number3D = Number3D.cross(new Number3D(plane.transform.n12, plane.transform.n22, plane.transform.n32), p3d.normal); cross.normalize(); //cross = new Number3D(-dif.y, dif.x, 0); //cross.normalize(); cross.multiplyEq(b.radius*scaleRadius); if(_type == DIRECTIONAL){ var oPos:Number3D = new Number3D(target.x, target.y, target.z); Matrix3D.multiplyVector(target.world, oPos); Matrix3D.multiplyVector(inv, oPos); dir = new Number3D(oPos.x-lp.x, oPos.y-lp.y, oPos.z-lp.z); } //numberRefs = new Dictionary(true); var pos:Number3D; var c2d:Point; var r2d:Point; //_type = SPOTLIGHT; pos = projectVertex(new Vertex3D(center.x, center.y, center.z), lp, inv, target.world); c2d = get2dPoint(pos, bounds.min, bounds.size, movieSize); pos = projectVertex(new Vertex3D(center.x+cross.x, center.y+cross.y, center.z+cross.z), lp, inv, target.world); r2d = get2dPoint(pos, bounds.min, bounds.size, movieSize); var dx:Number = r2d.x-c2d.x; var dy:Number = r2d.y-c2d.y; var rad:Number = Math.sqrt(dx*dx+dy*dy); castClip.graphics.beginFill(color); castClip.graphics.moveTo(c2d.x, c2d.y); castClip.graphics.drawCircle(c2d.x, c2d.y, rad); castClip.graphics.endFill(); } public function getCastClip(plane:Plane):Sprite{ var planeMovie:Sprite = Sprite(MovieMaterial(plane.material).movie); var movieSize:Point = new Point(planeMovie.width, planeMovie.height); var castClip:Sprite;// = new Sprite(); if(planeMovie.getChildByName("castClip"+uid)) return Sprite(planeMovie.getChildByName("castClip"+uid)); else{ castClip = new Sprite(); castClip.name = "castClip"+uid; castClip.scrollRect = new Rectangle(0, 0, movieSize.x, movieSize.y); //castClip.alpha = 0.4; planeMovie.addChild(castClip); return castClip; } } public function castFaces(light:PointLight3D, target:TriangleMesh3D, plane:Plane, cull:Boolean=false, clear:Boolean = true):void{ var planeVertices:Array = plane.geometry.vertices; //convert to target space? var world:Matrix3D = plane.world; var inv:Matrix3D = Matrix3D.inverse(plane.transform); var lp:Number3D = new Number3D(light.x, light.y, light.z); Matrix3D.multiplyVector(inv, lp); var tlp:Number3D; if(cull){ tlp = new Number3D(light.x, light.y, light.z); Matrix3D.multiplyVector(Matrix3D.inverse(target.world), tlp); } //Matrix3D.multiplyVector(Matrix3D.inverse(target.transform), tlp); //p3d.setThreePoints(planeVertices[0].getPosition(), planeVertices[1].getPosition(), planeVertices[2].getPosition()); p3d.setNormalAndPoint(plane.geometry.faces[0].faceNormal, new Number3D()); if(_type == DIRECTIONAL){ var oPos:Number3D = new Number3D(target.x, target.y, target.z); Matrix3D.multiplyVector(target.world, oPos); Matrix3D.multiplyVector(inv, oPos); dir = new Number3D(oPos.x-lp.x, oPos.y-lp.y, oPos.z-lp.z); } var bounds:Object = planeBounds[plane]; if(!bounds){ bounds = plane.boundingBox(); planeBounds[plane] = bounds; } var castClip:Sprite = getCastClip(plane); castClip.blendMode = this.blend; castClip.filters = this.filters; castClip.alpha = this.alpha; var planeMovie:Sprite = Sprite(MovieMaterial(plane.material).movie); var movieSize:Point = new Point(planeMovie.width, planeMovie.height); if(clear) castClip.graphics.clear(); vertexRefs = new Dictionary(true); //numberRefs = new Dictionary(true); var pos:Number3D; var p2d:Point; var s2d:Point; var hitVert:Number3D = new Number3D(); for each(var t:Triangle3D in target.geometry.faces){ if( cull){ hitVert.x = t.v0.x; hitVert.y = t.v0.y; hitVert.z = t.v0.z; if(Number3D.dot(t.faceNormal, Number3D.sub(tlp, hitVert)) <= 0) continue; } castClip.graphics.beginFill(color); pos = projectVertex(t.v0, lp, inv, target.world); s2d = get2dPoint(pos, bounds.min, bounds.size, movieSize); castClip.graphics.moveTo(s2d.x, s2d.y); pos = projectVertex(t.v1, lp, inv, target.world); p2d = get2dPoint(pos, bounds.min, bounds.size, movieSize); castClip.graphics.lineTo(p2d.x, p2d.y); pos = projectVertex(t.v2, lp, inv, target.world); p2d = get2dPoint(pos, bounds.min, bounds.size, movieSize); castClip.graphics.lineTo(p2d.x, p2d.y); castClip.graphics.lineTo(s2d.x, s2d.y); castClip.graphics.endFill(); } } public function invalidate():void{ invalidateModels(); invalidatePlanes(); } public function invalidatePlanes():void{ planeBounds = new Dictionary(true); } public function invalidateTargets():void{ numberRefs = new Dictionary(true); targetBounds = new Dictionary(true); } public function invalidateModels():void{ models = new Dictionary(true); invalidateTargets(); } private function get2dPoint(pos3D:Number3D, min3D:Number3D, size3D:Number3D, movieSize:Point):Point{ return new Point((pos3D.x-min3D.x)/size3D.x*movieSize.x, ((-pos3D.y-min3D.y)/size3D.y*movieSize.y)); } private function projectVertex(v:Vertex3D, light:Number3D, invMat:Matrix3D, world:Matrix3D):Number3D{ var pos:Number3D = vertexRefs[v]; if(pos) return pos; var n:Number3D = numberRefs[v]; if(!n){ n = new Number3D(v.x, v.y, v.z); Matrix3D.multiplyVector(world, n); Matrix3D.multiplyVector(invMat, n); numberRefs[v] = n; } if(_type == SPOTLIGHT){ lightRay.x = light.x; lightRay.y = light.y; lightRay.z = light.z; }else{ lightRay.x = n.x-dir.x; lightRay.y = n.y-dir.y; lightRay.z = n.z-dir.z; } pos = p3d.getIntersectionLineNumbers(lightRay, n); vertexRefs[v] = pos; return pos; } } } package designLab.utils { public class LayerConstant { public const INTRO:String = "INTRO"; // Intro layer string constant } }*emphasized text*

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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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  • NET Math Libraries

    - by JoshReuben
    NET Mathematical Libraries   .NET Builder for Matlab The MathWorks Inc. - http://www.mathworks.com/products/netbuilder/ MATLAB Builder NE generates MATLAB based .NET and COM components royalty-free deployment creates the components by encrypting MATLAB functions and generating either a .NET or COM wrapper around them. .NET/Link for Mathematica www.wolfram.com a product that 2-way integrates Mathematica and Microsoft's .NET platform call .NET from Mathematica - use arbitrary .NET types directly from the Mathematica language. use and control the Mathematica kernel from a .NET program. turns Mathematica into a scripting shell to leverage the computational services of Mathematica. write custom front ends for Mathematica or use Mathematica as a computational engine for another program comes with full source code. Leverages MathLink - a Wolfram Research's protocol for sending data and commands back and forth between Mathematica and other programs. .NET/Link abstracts the low-level details of the MathLink C API. Extreme Optimization http://www.extremeoptimization.com/ a collection of general-purpose mathematical and statistical classes built for the.NET framework. It combines a math library, a vector and matrix library, and a statistics library in one package. download the trial of version 4.0 to try it out. Multi-core ready - Full support for Task Parallel Library features including cancellation. Broad base of algorithms covering a wide range of numerical techniques, including: linear algebra (BLAS and LAPACK routines), numerical analysis (integration and differentiation), equation solvers. Mathematics leverages parallelism using .NET 4.0's Task Parallel Library. Basic math: Complex numbers, 'special functions' like Gamma and Bessel functions, numerical differentiation. Solving equations: Solve equations in one variable, or solve systems of linear or nonlinear equations. Curve fitting: Linear and nonlinear curve fitting, cubic splines, polynomials, orthogonal polynomials. Optimization: find the minimum or maximum of a function in one or more variables, linear programming and mixed integer programming. Numerical integration: Compute integrals over finite or infinite intervals, over 2D and higher dimensional regions. Integrate systems of ordinary differential equations (ODE's). Fast Fourier Transforms: 1D and 2D FFT's using managed or fast native code (32 and 64 bit) BigInteger, BigRational, and BigFloat: Perform operations with arbitrary precision. Vector and Matrix Library Real and complex vectors and matrices. Single and double precision for elements. Structured matrix types: including triangular, symmetrical and band matrices. Sparse matrices. Matrix factorizations: LU decomposition, QR decomposition, singular value decomposition, Cholesky decomposition, eigenvalue decomposition. Portability and performance: Calculations can be done in 100% managed code, or in hand-optimized processor-specific native code (32 and 64 bit). Statistics Data manipulation: Sort and filter data, process missing values, remove outliers, etc. Supports .NET data binding. Statistical Models: Simple, multiple, nonlinear, logistic, Poisson regression. Generalized Linear Models. One and two-way ANOVA. Hypothesis Tests: 12 14 hypothesis tests, including the z-test, t-test, F-test, runs test, and more advanced tests, such as the Anderson-Darling test for normality, one and two-sample Kolmogorov-Smirnov test, and Levene's test for homogeneity of variances. Multivariate Statistics: K-means cluster analysis, hierarchical cluster analysis, principal component analysis (PCA), multivariate probability distributions. Statistical Distributions: 25 29 continuous and discrete statistical distributions, including uniform, Poisson, normal, lognormal, Weibull and Gumbel (extreme value) distributions. Random numbers: Random variates from any distribution, 4 high-quality random number generators, low discrepancy sequences, shufflers. New in version 4.0 (November, 2010) Support for .NET Framework Version 4.0 and Visual Studio 2010 TPL Parallellized – multicore ready sparse linear program solver - can solve problems with more than 1 million variables. Mixed integer linear programming using a branch and bound algorithm. special functions: hypergeometric, Riemann zeta, elliptic integrals, Frensel functions, Dawson's integral. Full set of window functions for FFT's. Product  Price Update subscription Single Developer License $999  $399  Team License (3 developers) $1999  $799  Department License (8 developers) $3999  $1599  Site License (Unlimited developers in one physical location) $7999  $3199    NMath http://www.centerspace.net .NET math and statistics libraries matrix and vector classes random number generators Fast Fourier Transforms (FFTs) numerical integration linear programming linear regression curve and surface fitting optimization hypothesis tests analysis of variance (ANOVA) probability distributions principal component analysis cluster analysis built on the Intel Math Kernel Library (MKL), which contains highly-optimized, extensively-threaded versions of BLAS (Basic Linear Algebra Subroutines) and LAPACK (Linear Algebra PACKage). Product  Price Update subscription Single Developer License $1295 $388 Team License (5 developers) $5180 $1554   DotNumerics http://www.dotnumerics.com/NumericalLibraries/Default.aspx free DotNumerics is a website dedicated to numerical computing for .NET that includes a C# Numerical Library for .NET containing algorithms for Linear Algebra, Differential Equations and Optimization problems. The Linear Algebra library includes CSLapack, CSBlas and CSEispack, ports from Fortran to C# of LAPACK, BLAS and EISPACK, respectively. Linear Algebra (CSLapack, CSBlas and CSEispack). Systems of linear equations, eigenvalue problems, least-squares solutions of linear systems and singular value problems. Differential Equations. Initial-value problem for nonstiff and stiff ordinary differential equations ODEs (explicit Runge-Kutta, implicit Runge-Kutta, Gear's BDF and Adams-Moulton). Optimization. Unconstrained and bounded constrained optimization of multivariate functions (L-BFGS-B, Truncated Newton and Simplex methods).   Math.NET Numerics http://numerics.mathdotnet.com/ free an open source numerical library - includes special functions, linear algebra, probability models, random numbers, interpolation, integral transforms. A merger of dnAnalytics with Math.NET Iridium in addition to a purely managed implementation will also support native hardware optimization. constants & special functions complex type support real and complex, dense and sparse linear algebra (with LU, QR, eigenvalues, ... decompositions) non-uniform probability distributions, multivariate distributions, sample generation alternative uniform random number generators descriptive statistics, including order statistics various interpolation methods, including barycentric approaches and splines numerical function integration (quadrature) routines integral transforms, like fourier transform (FFT) with arbitrary lengths support, and hartley spectral-space aware sequence manipulation (signal processing) combinatorics, polynomials, quaternions, basic number theory. parallelized where appropriate, to leverage multi-core and multi-processor systems fully managed or (if available) using native libraries (Intel MKL, ACMS, CUDA, FFTW) provides a native facade for F# developers

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  • Wubi without downloading ubuntu-12.04.1-wubi-i386.tar.xz

    - by kaveh
    I have a class of computational physics. I want to install Ubuntu on 24 systems. unfortunately our access to Internet is limited. On the other hand I do not like to make new partitions for Linux. So I have to use Wubi but Wubi needs a large file .i.e. "ubuntu-12.04.1-wubi-i386.tar.xz". Unfortunately I could not make a trick to Wubi because when I put "ubuntu-12.04.1-wubi-i386.tar.xz" manually in the ubuntu/disks directory, wubi starts to complain about the existence of already installed ubuntu and all thing should be done from scratch. Does anybody know a solution for this problem? Thanks

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  • Podcast interview with Michael Kane

    - by mhornick
    In this podcast interview with Michael Kane, Data Scientist and Associate Researcher at Yale University, Michael discusses the R statistical programming language, computational challenges associated with big data, and two projects involving data analysis he conducted on the stock market "flash crash" of May 6, 2010, and the tracking of transportation routes bird flu H5N1. Michael also worked with Oracle on Oracle R Enterprise, a component of the Advanced Analytics option to Oracle Database Enterprise Edition. In the closing segment of the interview, Michael comments on the relationship between the data analyst and the database administrator and how Oracle R Enterprise provides secure data management, transparent access to data, and improved performance to facilitate this relationship. Listen now...

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  • Computer Science or Computer Engineering for Data Science and Machine Learning

    - by ATMathew
    I'm a 25 year old data consultant who is considering returning to school to get a second bachelors degree in computer science or engineering. My interest is data science and machine learning. I use programming as a means to an end, and use languages like Python, R, C, Java, and Hadoop to find meaning in large data sets. Would a computer science or computer engineering degree be better for this? I realize that a statistics degree may be even more beneficial, but I'll be at a school which dosn't have a stats department or a computational math department.

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  • Linked List is now Patented?

    - by John Isaiah Carmona
    Linked list Ming-Jen Wang Patent number: 7028023 Filing date: Sep 26, 2002 Issue date: Apr 11, 2006 Application number: 10/260,471 A computerized list is provided with auxiliary pointers for traversing the list in different sequences. One or more auxiliary pointers enable a fast, sequential traversal of the list with a minimum of computational time. Such lists may be used in any application where lists may be reordered for various purposes. Does this mean that I need to acquire permission before using a linked list in my codes? What about the codes I write from my previous apps that uses a linked list? What about the framework that implements the linked list?

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  • Buzzword for "performance-aware" software development

    - by errantlinguist
    There seems to be an overabundance of buzzwords for software development styles and methodologies: Agile development, extreme programming, test-driven development, etc... well, is there any sort of buzzword for "performance-aware" development? By "performance awareness", I don't necessarily mean low-latency or low-level programming, although the former would logically fall under the blanket term I'm looking for. I mean development in which resources are recognised to be finite and so there is a general emphasis on low computational complexity, good resource management, etc. If I was to be snarky, I would say "good programming", but that doesn't seem to get the message across so well...

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  • Parallel Computing in .Net 4.0

    - by kaleidoscope
    Technorati Tags: Ram,Parallel Computing in .Net 4.0 Parallel computing is the simultaneous use of multiple compute resources to solve a computational problem: To be run using multiple CPUs A problem is broken into discrete parts that can be solved concurrently Each part is further broken down to a series of instructions Instructions from each part execute simultaneously on different CPUs Parallel Extensions in .NET 4.0 provides a set of libraries and tools to achieve the above mentioned objectives. This supports two paradigms of parallel computing Data Parallelism – This refers to dividing the data across multiple processors for parallel execution.e.g we are processing an array of 1000 elements we can distribute the data between two processors say 500 each. This is supported by the Parallel LINQ (PLINQ) in .NET 4.0 Task Parallelism – This breaks down the program into multiple tasks which can be parallelized and are executed on different processors. This is supported by Task Parallel Library (TPL) in .NET 4.0 A high level view is shown below:

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  • GTK applications do not start

    - by Greg
    Hello, I have a fresh install of Ubuntu 10.04 Server on nodes of a computational cluster, and I access the nodes via ssh. I configured a X server, which I start with the command startx -- -ac. The server is running fine on port :0. Then, I set the environment variable DISPLAY to :0. Now, when I run a GTK application on the node, it fails with the following error: Error: Unable to initialize gtk, is DISPLAY set properly? Now, my question is, is there any runtime library that I need for running GTK applications on top of a X server? I'm probably missing something obvious here, but I can't tell what :P

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  • When do you use float and when do you use double

    - by Jakub Zaverka
    Frequently in my programming experience I need to make a decision whether I should use float or double for my real numbers. Sometimes I go for float, sometimes I go for double, but really this feels more subjective. If I would be confronted to defend my decision, I would probably not give sound reasons. When do you use float and when do you use double? Do you always use double, only when memory constraints are present you go for float? Or you use always float unless the precision requirement requires you to use double? Are there some substantial differences regarding computational complexity of basic arithemtics between float and double? What are the pros and cons of using float or double? And have you even used long double?

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  • Solving a probabilistic problem

    - by ????????????
    So I am interested in Computational Investing and came across this problem on a wiki page: Write a program to discover the answer to this puzzle:"Let's say men and women are paid equally (from the same uniform distribution). If women date randomly and marry the first man with a higher salary, what fraction of the population will get married?" I don't have much knowledge in probability theory, so I'm not really sure how to implement this in code. My thinking: Populate two arrays(female,male) with random salary values from a uniform distribution. Randomly pair one female and one male array element and see if condition of higher salary is met. If it is, increment a counter. Divide counter by population and get percentage. Is this the correct logic? Do woman continually date until there is no males left with higher salaries than women?

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  • Given two sets of DNA, what does it take to computationally "grow" that person from a fertilised egg and see what they become? [closed]

    - by Nicholas Hill
    My question is essentially entirely in the title, but let me add some points to prevent some "why on earth would you want to do that" sort of answers: This is more of a mind experiment than an attempt to implement real software. For fun. Don't worry about computational speed or the number of available memory bytes. Computers get faster and better all of the time. Imagine we have two data files: Mother.dna and Father.dna. What else would be required? (Bonus point for someone who tells me approx how many GB each file will be, and if the size of the files are exactly the same number of bytes for everyone alive on Earth!) There would ideally need to be a way to see what the egg becomes as it becomes a human adult. If you fancy, feel free to outline the design. I am initially thinking that there'd need to be some sort of volumetric voxel-based 3D environment for simulation purposes.

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