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• Objective-C 2D Interpolation

  - by Brett

  Has anyone come across an Objective-C equivalent to the MATLAB function interp2()? ...a 2D interpolation. Cheers, Brett

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• Interpolate air drag for my game?

  - by Valentin Krummenacher

  So I have a little game which works with small steps, however those steps vary in time, so for example I sometimes have 10 Steps/second and then I have 20 Steps/second. This changes automatically depending on how many steps the user's computer can take. To avoid inaccurate positioning of the game's player object I use y=v0*dt+g*dt^2/2 to determine my objects y-position, where dt is the time since the last step, v0 is the velocity of my object in the beginning of my step and g is the gravity. To calculate the velocity in the end of a step I use v=v0+g*dt what also gives me correct results, independent of whether I use 2 steps with a dt of for example 20ms or one step with a dt of 40ms. Now I would like to introduce air drag. For simplicity's sake I use a=k*v^2 where a is the air drag's acceleration (I am aware that it would usually result in a force, but since I assume 1kg for my object's mass the force is the same as the resulting acceleration), k is a constant (in this case I'm using 0.001) and v is the speed. Now in an infinitely small time interval a is k multiplied by the velocity in this small time interval powered by 2. The problem is that v in the next time interval would depend on the drag of the last which again depends on the v of the last interval and so on... In other words: If I use a=k*v^2 I get different results for my position/velocity when I use 2 steps of 20ms than when I use one step of 40ms. I used to have this problem for my position too, but adding +g*dt^2/2 to the formula for my position fixed the problem since it takes into account that the position depends on the velocity which changes slightly in every infinitely small time interval. Does something like that exist for air drag too? And no, I dont mean anything like Adding air drag to a golf ball trajectory equation or similar, for that kind of method only gives correct results when all my steps are the same. (I hope you can understand my intermediate english, it's not my main language so I would like to say sorry for all the silly mistakes I might have made in my question)

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• Computing pixel's screen position in a vertex shader: right or wrong?

  - by cubrman

  I am building a deferred rendering engine and I have a question. The article I took the sample code from suggested computing screen position of the pixel as follows: VertexShaderFunction() { ... output.Position = mul(worldViewProj, input.Position); output.ScreenPosition = output.Position; } PixelShaderFunction() { input.ScreenPosition.xy /= input.ScreenPosition.w; float2 TexCoord = 0.5f * (float2(input.ScreenPosition.x,-input.ScreenPosition.y) + 1); ... } The question is what if I compute the position in the vertex shader (which should optimize the performance as VSF is launched significantly less number of times than PSF) would I get the per-vertex lighting insted. Here is how I want to do this: VertexShaderFunction() { ... output.Position = mul(worldViewProj, input.Position); output.ScreenPosition.xy = output.Position / output.Position.w; } PixelShaderFunction() { float2 TexCoord = 0.5f * (float2(input.ScreenPosition.x,-input.ScreenPosition.y) + 1); ... } What exactly happens with the data I pass from VS to PS? How exactly is it interpolated? Will it give me the right per-pixel result in this case? I tried launching the game both ways and saw no visual difference. Is my assumption right? Thanks. P.S. I am optimizing the point light shader, so I actually pass a sphere geometry into the VS.

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• Interpolating Matrices

  - by sebf

  Hello, Apologies if I am missing something very obvious (likely!) but is there anything wrong with interpolating between two matrices by: float d = (float)(targetTime.Ticks - keyframe_start.ticks) / (float)(keyframe_end.ticks - keyframe_start.ticks); return ((keyframe_start.Transform * (1 - d)) + (keyframe_end.Transform * d)); As in my app, when I try an use this to interpolate between two keyframes, the model begins to 'shrink' - the severity based on how far between the two keyframes the target time is; its worst when the transform split is ~50/50.

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• bitmap interpolation c#

  - by Raghav

  Grid size : 160*160 No of row* columns = 16*16 I have created a bitmap for this. Each cell of the grid is filled with different colors. I need to perform color interpolation.

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• Server-side Input

  - by Thomas

  Currently in my game, the client is nothing but a renderer. When input state is changed, the client sends a packet to the server and moves the player as if it were processing the input, but the server has the final say on the position. This generally works really well, except for one big problem: falling off edges. Basically, if a player is walking towards an edge, say a cliff, and stops right before going off the edge, sometimes a second later, he'll be teleported off of the edge. This is because the "I stopped pressing W" packet is sent after the server processes the information. Here's a lag diagram to help you understand what I mean: http://i.imgur.com/Prr8K.png I could just send a "W Pressed" packet each frame for the server to process, but that would seem to be a bandwidth-costly solution. Any help is appreciated!

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• Extrapolation breaks collision detection

  - by user22241

  Before applying extrapolation to my sprite's movement, my collision worked perfectly. However, after applying extrapolation to my sprite's movement (to smooth things out), the collision no longer works. This is how things worked before extrapolation: However, after I implement my extrapolation, the collision routine breaks. I am assuming this is because it is acting upon the new coordinate that has been produced by the extrapolation routine (which is situated in my render call ). After I apply my extrapolation How to correct this behaviour? I've tried puting an extra collision check just after extrapolation - this does seem to clear up a lot of the problems but I've ruled this out because putting logic into my rendering is out of the question. I've also tried making a copy of the spritesX position, extrapolating that and drawing using that rather than the original, thus leaving the original intact for the logic to pick up on - this seems a better option, but it still produces some weird effects when colliding with walls. I'm pretty sure this also isn't the correct way to deal with this. I've found a couple of similar questions on here but the answers haven't helped me. This is my extrapolation code: public void onDrawFrame(GL10 gl) { //Set/Re-set loop back to 0 to start counting again loops=0; while(System.currentTimeMillis() > nextGameTick && loops < maxFrameskip){ SceneManager.getInstance().getCurrentScene().updateLogic(); nextGameTick+=skipTicks; timeCorrection += (1000d/ticksPerSecond) % 1; nextGameTick+=timeCorrection; timeCorrection %=1; loops++; tics++; } extrapolation = (float)(System.currentTimeMillis() + skipTicks - nextGameTick) / (float)skipTicks; render(extrapolation); } Applying extrapolation render(float extrapolation){ //This example shows extrapolation for X axis only. Y position (spriteScreenY is assumed to be valid) extrapolatedPosX = spriteGridX+(SpriteXVelocity*dt)*extrapolation; spriteScreenPosX = extrapolationPosX * screenWidth; drawSprite(spriteScreenX, spriteScreenY); } Edit As I mentioned above, I have tried making a copy of the sprite's coordinates specifically to draw with.... this has it's own problems. Firstly, regardless of the copying, when the sprite is moving, it's super-smooth, when it stops, it's wobbling slightly left/right - as it's still extrapolating it's position based on the time. Is this normal behavior and can we 'turn it off' when the sprite stops? I've tried having flags for left / right and only extrapolating if either of these is enabled. I've also tried copying the last and current positions to see if there is any difference. However, as far as collision goes, these don't help. If the user is pressing say, the right button and the sprite is moving right, when it hits a wall, if the user continues to hold the right button down, the sprite will keep animating to the right, while being stopped by the wall (therefore not actually moving), however because the right flag is still set and also because the collision routine is constantly moving the sprite out of the wall, it still appear to the code (not the player) that the sprite is still moving, and therefore extrapolation continues. So what the player would see, is the sprite 'static' (yes, it's animating, but it's not actually moving across the screen), and every now and then it shakes violently as the extrapolation attempts to do it's thing....... Hope this help

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• Why is there no string interpolation in Scala?

  - by Alex R

  This is not just an idle quip... I wonder if anybody knows if there's an actual design reason why Scala does not support interpolation similar to Groovy and other "syntactically better Javas"? e.g. var str1 = "World"; var str2 = "Hello, ${str1}";

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• Interpolating between two networked states?

  - by Vaughan Hilts

  I have many entities on the client side that are simulated (their velocities are added to their positions on a per frame basis) and I let them dead reckon themselves. They send updates about where they were last seen and their velocity changes. This works great and other players see this work find. However, after a while these players begin to desync after some time. This is because of latency. I'd like to know how I can interpolate between states so they appear to be in the correct position. I know where the player was LAST seen and their current velocity but interpolating to the last seen state causes the player to actually move -backwards-. I could not use velocity at all for other clients and simply 'lerp' them towards the appropriate direction but I feel this would cause jaggy movement. What are the alternatives?

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• How do I interpolate air drag with a variable time step?

  - by Valentin Krummenacher

  So I have a little game which works with small steps, however those steps vary in time, so for example I sometimes have 10 Steps/second and then I have 20 Steps/second. This changes automatically depending on how many steps the user's computer can take. To avoid inaccurate positioning of the game's player object I use y=v0*dt+g*dt^2/2 to determine my objects y-position, where dt is the time since the last step, v0 is the velocity of my object in the beginning of my step and g is the gravity. To calculate the velocity in the end of a step I use v=v0+g*dt what also gives me correct results, independent of whether I use 2 steps with a dt of for example 20ms or one step with a dt of 40ms. Now I would like to introduce air drag. For simplicity's sake I use a=k*v^2 where a is the air drag's acceleration (I am aware that it would usually result in a force, but since I assume 1kg for my object's mass the force is the same as the resulting acceleration), k is a constant (in this case I'm using 0.001) and v is the speed. Now in an infinitely small time interval a is k multiplied by the velocity in this small time interval powered by 2. The problem is that v in the next time interval would depend on the drag of the last which again depends on the v of the last interval and so on... In other words: If I use a=k*v^2 I get different results for my position/velocity when I use 2 steps of 20ms than when I use one step of 40ms. I used to have this problem for my position too, but adding +g*dt^2/2 to the formula for my position fixed the problem since it takes into account that the position depends on the velocity which changes slightly in every infinitely small time interval. Does something like that exist for air drag too? And no, I dont mean anything like Adding air drag to a golf ball trajectory equation or similar, for that kind of method only gives correct results when all my steps are the same. (I hope you can understand my intermediate english, it's not my main language so I would like to say sorry for all the silly mistakes I might have made in my question)

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• How to make other semantics behave like SV_Position?

  - by object

  I'm having a lot of trouble with shadow mapping, and I believe I've found the problem. When passing vectors from the vertex shader to the pixel shader, does the hardware automatically change any of the values based on the semantic? I've compiled a barebones pair of shaders which should illustrate the problem. Vertex shader : struct Vertex { float3 position : POSITION; }; struct Pixel { float4 position : SV_Position; float4 light_position : POSITION; }; cbuffer Matrices { matrix projection; }; Pixel RenderVertexShader(Vertex input) { Pixel output; output.position = mul(float4(input.position, 1.0f), projection); output.light_position = output.position; // We simply pass the same vector in screenspace through different semantics. return output; } And a simple pixel shader to go along with it: struct Pixel { float4 position : SV_Position; float4 light_position : POSITION; }; float4 RenderPixelShader(Pixel input) : SV_Target { // At this point, (input.position.z / input.position.w) is a normal depth value. // However, (input.light_position.z / input.light_position.w) is 0.999f or similar. // If the primitive is touching the near plane, it very quickly goes to 0. return (0.0f).rrrr; } How is it possible to make the hardware treat light_position in the same way which position is being treated between the vertex and pixel shaders? EDIT: Aha! (input.position.z) without dividing by W is the same as (input.light_position.z / input.light_position.w). Not sure why this is.

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• linear interpolation on 8bit microcontroller

  - by JB

  I need to do a linear interpolation over time between two values on an 8 bit PIC microcontroller (Specifically 16F627A but that shouldn't matter) using PIC assembly language. Although I'm looking for an algorithm here as much as actual code. I need to take an 8 bit starting value, an 8 bit ending value and a position between the two (Currently represented as an 8 bit number 0-255 where 0 means the output should be the starting value and 255 means it should be the final value but that can change if there is a better way to represent this) and calculate the interpolated value. Now PIC doesn't have a divide instruction so I could code up a general purpose divide routine and effectivly calculate (B-A)/(x/255)+A at each step but I feel there is probably a much better way to do this on a microcontroller than the way I'd do it on a PC in c++ Has anyone got any suggestions for implementing this efficiently on this hardware?

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• OpenGL billboard interpolation issue

  - by PeanutPower

  I have a billboard quad with a texture mapped onto it. This is basically some text with transparency. The billboard floats forwards and backwards from the camera's perspective. As the billboard moves away (and appears smaller) there is an flickering effect around the edges of the text where there is a stroke border on the actual texture. I think this is because interpolation is needed as the image which is normally X pixels wide is now shown as only a % of X and some pixels need to be merged together. I guess it's doing nearest neighbour or something? Can anyone point me in the right direction for opengl settings to control this, I'm guessing there is some way of preventing this effect from happening by adjusting the method for how the texture is handled ?

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• WPF Animate a Matrix using interpolation

  - by Mark

  I'm having a issue with my application that is using touch gestures to scale, translate and rotate my scene. I was using a TransformGroup which contained TranslateTransform, ScaleTransform and a RotateTransform but I could not get the movement correct, it always jumps and skips, so I moved to a MaxtrixTransform which I was able to use much easier to get my scene to be zoomable, rotatable and panable nicely. However, what I later found out was that you cannot animate smoothly (using interpolation) the values of a Matrix, for what reason I have no idea, but its part of the MSDN doco and the properties of the Matrix are not dependency properties anyways... Has anyone had any luck animating a matrix to make it smooth? The only idea(s) I have had is to animate a few different, custom DP which all have callbacks that I update the matrix from OR To convert the matrix to a set of Transform objects that I then animate and then afterwords convert back. Is there a smarter way to do this?

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• WPF Animate a Maxtrix using interpolation

  - by Mark

  Im having a issue with my application that is using touch gestures to scale, translate and rotate my scene. I was using a TransformGroup which contained TranslateTransform, ScaleTransform and a RotateTransform but I could not get the movement correct, it always jumps and skips, so I moved to a MaxtrixTransform which I was able to use much easier to get my scene to be zoomable, rotatable and panable nicely. However, what I later found out was that you cannot animate smoothly (using interpolation) the values of a Matrix, for what reason I have no idea, but its part of the MSDN doco and the properties of the Matrix are not dependency properties anyways... Has anyone had any luck animating a maxtrix to make it smooth? The only idea(s) I have had is to animate a few different, custom DP which all have callbacks that I update the Matrix from OR To convert the Maxtix to a set of Transform object that I then animate and then afterwords convert back. Is there a smarter way to do this?

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• Java error on bilinear interpolation of 16 bit data

  - by Jon

  I'm having an issue using bilinear interpolation for 16 bit data. I have two images, origImage and displayImage. I want to use AffineTransformOp to filter origImage through an AffineTransform into displayImage which is the size of the display area. origImage is of type BufferedImage.TYPE_USHORT_GRAY and has a raster of type sun.awt.image.ShortInterleavedRaster. Here is the code I have right now displayImage = new BufferedImage(getWidth(), getHeight(), origImage.getType()); try { op = new AffineTransformOp(atx, AffineTransformOp.TYPE_BILINEAR); op.filter(origImage, displayImage); } catch (Exception e) { e.printStackTrace(); } In order to show the error I have created 2 gradient images. One has values in the 15 bit range (max of 32767) and one in the 16 bit range (max of 65535). Below are the two images 15 bit image 16 bit image These two images were created in identical fashions and should look identical, but notice the line across the middle of the 16 bit image. At first I thought that this was an overflow problem however, it is weird that it's manifesting itself in the center of the gradient instead of at the end where the pixel values are higher. Also, if it was an overflow issue than I would suspect that the 15 bit image would have been affected as well. Any help on this would be greatly appreciated.

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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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• Flipping issue when interpolating Rotations using Quaternions

  - by uhuu

  I use slerp to interpolate between two quaternions representing rotations. The resulting rotation is then extracted as Euler angles to be fed into a graphics lib. This kind of works, but I have the following problem; when rotating around two (one works just fine) axes in the direction of the green arrow as shown in the left frame here the rotation soon jumps around to rotate from the opposite site to the opposite visual direction, as indicated by the red arrow in the right frame. This may be logical from a mathematical perspective (although not to me), but it is undesired. How could I achieve an interpolation with no visual flipping and changing of directions when rotating around more than one axis, following the green arrow at all times until the interpolation is complete? Thanks in advance.

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• Perl Search and Replace Avoid Variable Interpolation

  - by Justin

  I'm really getting my butt kicked here. I can not figure out how to write a search and replace that will properly find this string. String: $QData{"OrigFrom"} $Text{"wrote"}: Note: That is the actual STRING. Those are NOT variables. I didn't write it. I need to replace that string with nothing. I've tried escaping the $, {, and }. I've tried all kinds of combinations but it just can't get it right. Someone out there feel like taking a stab at it? Thanks!

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• Scipy interpolation on a numpy array

  - by dassouki

  I have a lookup table that is defined the following way: TR_ua1 = np.array([ [3.6, 6.5, 9.1, 11.5, 13.8], [3.9, 7.3, 10.0, 13.1, 15.9], [4.5, 9.2, 12.2, 14.8, 18.2] ]) The header row elements are (hh) <1,2,3,4,5+ The header column (inc) elements are <10000, 20000, 20001+ The user will input a value ex (1.3, 25,000) or (0.2, 50,000). Scipy.interpolate() should interpolate to determine the correct value. Currently, the only way i can do this is with a bunch of if/elifs as exemplified below. I'm pretty sure there is a better, more efficient way of doing this Here's what i've got so far import numpy as np from scipy import interplate if (ua == 1): if (inc <= low_inc): #low_inc = 10,000 if (hh <= 1): return TR_ua1[0][0] elif (hh >= 1 & hh < 2): return interpolate( (1,2), (TR_ua1[0][1], TR_ua1[0][2]) )

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• Non-Linear color interpolation?

  - by user146780

  If I have a straight line that mesures from 0 to 1, then I have colorA(255,0,0) at 0 on the line, then at 0.3 I have colorB(20,160,0) then at 1 on the line I have colorC(0,0,0). How could I find the color at 0.7? Thanks

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• Interpolation not working on Rails generator

  - by Tom

  For some reason the code I have included below does not interpolate the variables into the template. It simply copies the file over verbatim. I cannot figure out why. https://gist.github.com/60484f7b57b06b6eb3e3 The Rails version is 2.3.4. Thanks in advance!

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• How to use string interpolation when rendering templates?

  - by Senthil

  I found this code in a Rails cookbook. class BlogController < ApplicationController def display_by_date year = params[:year] month = params[:month] day = params[:day] day ='0'+day if day && day.size == 1 @day = day if ( year && month && day ) render(:template => "blog/#{year}/#{month}/#{day}") elsif ( year ) render(:template => "blog/#{year}/list") end end end I'm not sure what to name the templates so the router can find them. Thanks for your help.

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• Downsampling the number of entries in a list (without interpolation)

  - by Dave

  I have a Python list with a number of entries, which I need to downsample using either: A maximum number of rows. For example, limiting a list of 1234 entries to 1000. A proportion of the original rows. For example, making the list 1/3 its original length. (I need to be able to do both ways, but only one is used at a time). I believe that for the maximum number of rows I can just calculate the proportion needed and pass that to the proportional downsizer: def downsample_to_max(self, rows, max_rows): return downsample_to_proportion(rows, max_rows / float(len(rows))) ...so I really only need one downsampling function. Any hints, please? EDIT The list contains objects, not numeric values so I do not need to interpolate. Dropping objects is fine.

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• GSL interpolation error, values must be x values must be monotonically increasing

  - by pyCthon

  Hi my problem is that my data set is monotonically increasing but towards the end the of the data it looks like it does below ,where some of the x[i-1] = x[i] as shown below. This causes an error to be raised in GSL because it thinks that the values are not monotonically increasing. Is there a solution, fix or work around for this problem? the values are already double precision ,this particular data set starts at 9.86553e-06 and ends at .999999 would the only solution be to offset every value in a for loop? 0.999981 0.999981 0.999981 0.999982 0.999982 0.999983 0.999983 0.999983 0.999984 0.999984 0.999985 0.999985 0.999985

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