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• How to implement Horner's scheme for multivariate polynomials?

  - by gsreynolds

  Background I need to solve polynomials in multiple variables using Horner's scheme in Fortran90/95. The main reason for doing this is the increased efficiency and accuracy that occurs when using Horner's scheme to evaluate polynomials. I currently have an implementation of Horner's scheme for univariate/single variable polynomials. However, developing a function to evaluate multivariate polynomials using Horner's scheme is proving to be beyond me. An example bivariate polynomial would be: 12x^2y^2+8x^2y+6xy^2+4xy+2x+2y which would factorised to x(x(y(12y+8))+y(6y+4)+2)+2y and then evaluated for particular values of x & y. Research I've done my research and found a number of papers such as: staff.ustc.edu.cn/~xinmao/ISSAC05/pages/bulletins/articles/147/hornercorrected.pdf citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.40.8637&rep=rep1&type=pdf www.is.titech.ac.jp/~kojima/articles/B-433.pdf Problem However, I'm not a mathematician or computer scientist, so I'm having trouble with the mathematics used to convey the algorithms and ideas. As far as I can tell the basic strategy is to turn a multivariate polynomial into separate univariate polynomials and compute it that way. Can anyone help me? If anyone could help me turn the algorithms into pseudo-code that I can implement into Fortran myself, I would be very grateful.

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• Finding the Formula for a Curve

  - by Mystagogue

  Is there a program that will take "response curve" values from me, and provide a formula that approximates the response curve? It would be cool if such a program would take a numeric "percent correct" (perhaps with a standard deviation) so that it returns simplified formulas when laxity is permissable, and more precise (viz. complex) formulas when the curve needs to be approximated closely. My interest is to play with the response curve values and "laxity" factor, until such a tool spits out a curve-fit formula simple enough that I know it will be high performance during machine computations.

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• Unique keys for Sphinx along three vectors instead of two

  - by Brendon Muir

  I'm trying to implement thinking-sphinx across multiple 'sites' hosted under a single rails application. I'm working with the developer of thinking-sphinx to sort through the finer details and am making good progress, but I need help with a maths problem: Usually the formula for making a unique ID in a thinking-sphinx search index is to take the id, multiply it by the total number of models that are searchable, and add the number of the currently indexed model: id * total_models + current_model This works well, but now I also through an entity_id into the mix, so there are three vextors for making this ID unique. Could someone help me figure out the equation to gaurantee that the id's will never collide using these three variables: id, total_models, total_entities The entity ID is an integer. I thought of: id * (total_models + total_entities) + (current_model + current_entity) but that results in collisions. Any help would be greatly appreciated :)

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• Does Python/Scipy have a firls( ) replacement (i.e. a weighted, least squares, FIR filter design)?

  - by delicasso

  I am porting code from Matlab to Python and am having trouble finding a replacement for the firls( ) routine. It is used for, least-squares linear-phase Finite Impulse Response (FIR) filter design. I looked at scipy.signal and nothing there looked like it would do the trick. Of course I was able to replace my remez and freqz algorithsm, so that's good. On one blog I found an algorithm that implemented this filter without weighting, but I need one with weights. Thanks, David

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• How to compute the cross-product?

  - by WizardOfOdds

  I have the following piece of pseudo-C/Java/C# code: int a[]= { 30, 20 }; int b[] = { 40, 50 }; int c[] = {12, 12}; How do I compute the cross-product ABxAC?

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• Divide and conquer method to compute roots [SOLVED]

  - by hellsoul153

  Hello, Knowing that we can use Divide-and-Conquer algorithm to compute large exponents, for exemple 2 exp 100 = 2 exp(50) * 2 exp(50), which is quite more efficient, is this method efficient using roots ? For exemple 2 exp (1/100) = (2 exp(1/50)) exp(1/50) ? In other words, I'm wondering if (n exp(1/x)) is more efficient to (n exp(1/y)) for x < y and where x and y are integers.

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• posmax: like argmax but gives the position(s) of the element x for which f[x] is maximal

  - by dreeves

  Mathematica has a built-in function ArgMax for functions over infinite domains, based on the standard mathematical definition. The analog for finite domains is a handy utility function. Given a function and a list (call it the domain of the function), return the element(s) of the list that maximize the function. Here's an example of finite argmax in action: http://stackoverflow.com/questions/471029/canonicalize-nfl-team-names/472213#472213 And here's my implementation of it (along with argmin for good measure): (* argmax[f, domain] returns the element of domain for which f of that element is maximal -- breaks ties in favor of first occurrence. *) SetAttributes[{argmax, argmin}, HoldFirst]; argmax[f_, dom_List] := Fold[If[f[#1]>=f[#2], #1, #2]&, First[dom], Rest[dom]] argmin[f_, dom_List] := argmax[-f[#]&, dom] First, is that the most efficient way to implement argmax? What if you want the list of all maximal elements instead of just the first one? Second, how about the related function posmax that, instead of returning the maximal element(s), returns the position(s) of the maximal elements?

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• finding ALL cycles in a huge sparse matrix

  - by Andy

  Hi there, First of all I'm quite a Java beginner, so I'm not sure if this is even possible! Basically I have a huge (3+million) data source of relational data (i.e. A is friends with B+C+D, B is friends with D+G+Z (but not A - i.e. unmutual) etc.) and I want to find every cycle within this (not necessarily connected) directed graph. I've found this thread (http://stackoverflow.com/questions/546655/finding-all-cycles-in-graph/549402#549402) which has pointed me to Donald Johnson's (elementary) cycle-finding algorithm which, superficially at least, looks like it'll do what I'm after (I'm going to try when I'm back at work on Tuesday - thought it wouldn't hurt to ask in the meanwhile!). I had a quick scan through the code of the Java implementation of Johnson's algorithm (in that thread) and it looks like a matrix of relations is the first step, so I guess my questions are: a) Is Java capable of handling a 3+million*3+million matrix? (was planning on representing A-friends-with-B by a binary sparse matrix) b) Do I need to find every connected subgraph as my first problem, or will cycle-finding algorithms handle disjoint data? c) Is this actually an appropriate solution for the problem? My understanding of "elementary" cycles is that in the graph below, rather than picking out A-B-C-D-E-F it'll pick out A-B-F, B-C-D etc. but that's not the end of the world given the task. E / \ D---F / \ / \ C---B---A d) If necessary, I can simplify the problem by enforcing mutuality in relations - i.e. A-friends-with-B <== B-friends-with-A, and if really necessary I can maybe cut down the data size, but realistically it is always going to be around the 1mil mark. z) Is this a P or NP task?! Am I biting off more than I can chew? Thanks all, any help appreciated! Andy

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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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• multidimensional vector rotation and angle computation -- how?

  - by macias

  Input: two multidimensional (for example dim=8) vectors a and b. I need to find out the "directed" angle (0-2*Pi, not 0-Pi) between those vectors a and b. And if they are not parallel I need to rotate vector b in plane a,b by "directed" angle L. If they are parallel, plane does not matter, but angle of rotation is still the same L. For 2d and 3d this is quite easy, but for more dimensions I am lost, I didn't find anything on google, and I prefer using some already proved&tested equations (avoiding errors introduced by my calculations :-D). Thank you in advance for tips, links, etc. Edit: dimension of the space is the same as dimension of the vectors.

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• OPTICS Clustering algorithm. How to get the best epsilon

  - by Marco Galassi

  I am implementing a project which needs to cluster geographical points. OPTICS algorithm seems to be a very nice solution. It needs just 2 parameters as input(MinPts and Epsilon), which are, respectively, the minimum number of points needed to consider them as a cluster, and the distance value used to compare if two points are in can be placed in same cluster. My problem is that, due to the extreme variety of the points, I can't set a fixed epsilon. Just look at the image below. The same points structure but in a different scale would result very different. Suppose to set MinPts=2 and epsilon = 1Km. On the left, the algorithm would create 2 clusters(red and blue), but on the right it would create one single cluster containing all of the points(red), but I would like to obtain 2 clusters even on the right. So my question is: is there any kind of way to calculate dynamically the epsilon value to get this result? Thank you very much and excuse my for my poor english. Marco

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• How can I write a power function myself?

  - by Koning WWWWWWWWWWWWWWWWWWWWWWW

  Since I was 12, I was always wondering how I can make a function which calculates the power (e.g. 23) myself. In most languages these are included in the standard library, mostly as pow(double x, double y), but how can I write it myself? I was thinking about for loops, but it think my brain got in a loop (when I wanted to do a power with a non-integer exponent, like 54.5 or negatives 2-21) and I went crazy ;) So, how can I write a function which calculates the power of a real number? Thanks

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• How to get bit rotation function to accept any bit size?

  - by calccrypto

  i have these 2 functions i got from some other code def ROR(x, n): mask = (2L**n) - 1 mask_bits = x & mask return (x >> n) | (mask_bits << (32 - n)) def ROL(x, n): return ROR(x, 32 - n) and i wanted to use them in a program, where 16 bit rotations are required. however, there are also other functions that require 32 bit rotations, so i wanted to leave the 32 in the equation, so i got: def ROR(x, n, bits = 32): mask = (2L**n) - 1 mask_bits = x & mask return (x >> n) | (mask_bits << (bits - n)) def ROL(x, n, bits = 32): return ROR(x, bits - n) however, the answers came out wrong when i tested this set out. yet, the values came out correctly when the code is def ROR(x, n): mask = (2L**n) - 1 mask_bits = x & mask return (x >> n) | (mask_bits << (16 - n)) def ROL(x, n,bits): return ROR(x, 16 - n) what is going on and how do i fix this?

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• Python - How to find a correlation between two vectors ?

  - by psihodelia

  Given two vectors X and Y I have to find their correlation, i.e. their linear dependence/independence. Both vectors have equal dimension. A resulted answer should be a floating point number from [-1.0 .. 1.0]. Example: X=[-1, 2, 0] Y=[ 4, 2, -0.3] Find y=cor(X,Y) such that y belongs to [-1.0 .. 1.0]. It should be a simple construction involving a list-comprehension. No external library is allowed. UPDATE: ok, if dot product is enough, then here is my solution: nX = 1/(sum([x*x for x in X]) ** 0.5) nY = 1/(sum([y*y for y in Y]) ** 0.5) cor = sum([(x*nX)*(y*nY) for x,y in zip(X,Y) ]) right?

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• Express any number as the sum of 4 prime numbers [Doubts]

  - by WarDoGG

  I was give a problem to express any number as sum of 4 prime numbers. Conditions: Not allowed to use any kind of database. Maximum execution time : 3 seconds Numbers only till 100,000 If the splitting is NOT possible, then return -1 What i did : using the sieve of eratosthenes, i calculated all prime numbers till the specified number. looked up a concept called goldbach conjecture which expresses an even number as the summation of 2 primes. However, i am stuck beyond that. Can anyone help me on this one as to what approach u might take ? The sieve of eratosthenes is taking 2 seconds to count primes till 100,000 :(((

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• st-ordering library function?

  - by chang

  I'm in the search for an implementation of an ear-decomposition algorithm (http://www.ics.uci.edu/~eppstein/junkyard/euler/ear.html). I examined networkx and didn't find one. Although the algorithm layout is vaguely in my mind, I'd like to see some reference implementation, too. Side problem: First step could be an st-ordering of a graph. Are there any implementations for st-ordering algorithms you know? Thanks for your input. I'd really like to contribute e.g. to networkx by implementing the ear-decomposition algorithm in python.

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• Calculating distance from latitude, longitude and height using a geocentric co-ordinate system

  - by Sarge

  I've implemented this method in Javascript and I'm roughly 2.5% out and I'd like to understand why. My input data is an array of points represented as latitude, longitude and the height above the WGS84 ellipsoid. These points are taken from data collected from a wrist-mounted GPS device during a marathon race. My algorithm was to convert each point to cartesian geocentric co-ordinates and then compute the Euclidean distance (c.f Pythagoras). Cartesian geocentric is also known as Earth Centred Earth Fixed. i.e. it's an X, Y, Z co-ordinate system which rotates with the earth. My test data was the data from a marathon and so the distance should be very close to 42.26km. However, the distance comes to about 43.4km. I've tried various approaches and nothing changes the result by more than a metre. e.g. I replaced the height data with data from the NASA SRTM mission, I've set the height to zero, etc. Using Google, I found two points in the literature where lat, lon, height had been transformed and my transformation algorithm is matching. What could explain this? Am I expecting too much from Javascript's double representation? (The X, Y, Z numbers are very big but the differences between two points is very small). My alternative is to move to computing the geodesic across the WGS84 ellipsoid using Vincenty's algorithm (or similar) and then calculating the Euclidean distance with the two heights but this seems inaccurate. Thanks in advance for your help!

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• Is the binormal of a vertex the cross between its normal and tangent?

  - by meds

  I'm trying to find out what a binormal is in the context of graphics programming but coming up short, I saw on a site that the binormal was being calculated as the corss between the normal and tangent (i.e. cross(normal, tangent)), is this the correct way to calculate a binormal?

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• Can't get my object to point at the mouse.

  - by melignus

  I'm using a combination of SDL and OpenGL in a sort of crash course project to teach myself how this all works. I'm really only interested in OpenGL as a way to use acceleration in 2D games so I just need this to work in a 2D plane. I have been having a lot of problems today with my current issue, I would like an object to point towards the mouse while the mouse button is clicked and then of course stay pointing in that direction after the mouse is lifted. void Square::handle_input() { //If a key was pressed if( event.type == SDL_KEYDOWN ) { //Adjust the velocity switch( event.key.keysym.sym ) { case SDLK_UP: upUp = false; yVel = -1; break; case SDLK_DOWN: downUp = false; yVel = 1; break; case SDLK_LEFT: leftUp = false; xVel = -1; break; case SDLK_RIGHT: rightUp = false; xVel = 1; break; case SDLK_w: wUp = false; sAng = 1; break; case SDLK_s: sUp = false; sAng = -1; break; } } //If a key was released else if( event.type == SDL_KEYUP ) { //Adjust the velocity switch( event.key.keysym.sym ) { case SDLK_UP: upUp = true; yVel = 0; break; case SDLK_DOWN: downUp = true; yVel = 0; break; case SDLK_LEFT: leftUp = true; xVel = 0; break; case SDLK_RIGHT: rightUp = true; xVel = 0; break; case SDLK_w: wUp = true; sAng = 0; break; case SDLK_s: sUp = true; sAng = 0; break; } } //If a mouse button was pressed if( event.type == SDL_MOUSEBUTTONDOWN ) { switch ( event.type ) { case SDL_MOUSEBUTTONDOWN: mouseUp = false; mousex == event.button.x; mousey == event.button.y; break; case SDL_MOUSEBUTTONUP: mouseUp = true; break; } } } And then this is called at the end of my Object::Move call which also handles x and y translation if (!mouseUp) { xVect = mousex - x; yVect = mousey - y; radAng = atan2 ( mousey - y, mousex - x ); sAng = radAng * 180 / 3.1415926l; } Right now when I click the object turns and faces down to the bottom left but then no longer changes direction. I'd really appreciate any help I could get here. I'm guessing there might be an issue here with state versus polled events but from all the tutorials that I've been through I was pretty sure I had fixed that. I've just hit a wall and I need some advice!

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• How to make scipy.interpolate give a an extrapolated result beyond the input range?

  - by Salim Fadhley

  I'm trying to port a program which uses a hand-rolled interpolator (developed by a mathematitian colleage) over to use the interpolators provided by scipy. I'd like to use or wrap the scipy interpolator so that it has as close as possible behavior to the old interpolator. A key difference between the two functions is that in our original interpolator - if the input value is above or below the input range, our original interpolator will extrapolate the result. If you try this with the scipy interpolator it raises a ValueError. Consider this program as an example: import numpy as np from scipy import interpolate x = np.arange(0,10) y = np.exp(-x/3.0) f = interpolate.interp1d(x, y) print f(9) print f(11) # Causes ValueError, because it's greater than max(x) Is there a sensible way to make it so that instead of crashing, the final line will simply do a linear extrapolate, continuing the gradients defined by the first and last two pouints to infinity. Note, that in the real software I'm not actually using the exp function - that's here for illustration only!

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• floor of double(time_t)

  - by plok

  I cannot understand why this throws undefined reference to `floor'": double curr_time = (double)time(NULL); return floor(curr_time); Hasn't it been casted to double, which is what floor receives?

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• Generate RTT values

  - by Jean Gauthier

  Hi all, I'm writing a Java applet where I should be able to simulate a connection between two hosts. Hence I have to generate packet round-trip times at random. These RTTs can go from ~0 to infinity, but are typically oscillating around some average value (i.e. an extremely large or small value is very improbable but possible). I was wondering if anyone had an idea of how I could do this? Thanks in advance

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• Mapping Hilbert values to 3D points

  - by Alexander Gladysh

  I have a set of Hilbert values (length from the start of the Hilbert curve to the given point). What is the best way to convert these values to 3D points? Original Hilbert curve was not in 3D, so I guess I have to pick by myself the Hilbert curve rank I need. I do have total curve length though (that is, the maximum value in the set). Perhaps there is an existing implementation? Some library that would allow me to work with Hilbert curve / values? Language does not matter much.

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• Count bits used in int

  - by sigvardsen

  If you have the binary number 10110 how can I get it to return 5? e.g a number that tells how many bits are used? There are some likewise examples listed below: 101 should return 3 000000011 should return 2 11100 should return 5 101010101 should return 9 How can this be obtained the easiest way in Java? I have come up with the following method but can i be done faster: public static int getBitLength(int value) { int l = 1; if (value >> 16 > 0) { value = value >> 16; l += 16; } if (value >> 8 > 0) { value = value >> 8; l += 8; } if (value >> 4 > 0) { value = value >> 4; l += 4; } if (value >> 2 > 0) { value = value >> 2; l += 2; } if (value >> 1 > 0) { value = value >> 1; l += 1; } return l; }

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• Base 62 conversion in Python

  - by mikl

  How would you convert an integer to base 62 (like hexadecimal, but with these digits: '0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ'). I have been trying to find a good Python library for it, but they all seems to be occupied with converting strings. The Python base64 module only accepts strings and turns a single digit into four characters. I was looking for something akin to what URL shorteners use.

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