I've written a simple graphing implementation in C#, and I can graph things by comparing each pixel to the position on the graph it represents and plugging that position into the function I have to see if it is on the curve. That's all well and good.
The problem I'm having is USING a generated taylor polynomial. For example, I am able to create the nth taylor polynomial of a transcendent function f centered at c by doing
summation of this from 0 to to n with the counter variable being k = ((kth derivative of f(c)) * (x-c)^k)/k!
I am not sure how to do math markup on stackoverflow nor am I too competent with doing that on the web, but I hope that is understandable. The left side could be written as sigma _k=0 ^n or something like that with _ representing the section under sigma and the ^ representing the part above...
So I end up generating a Taylor polynomial of the 6th degree for cos(x) centered at 0(maclaurin, I know) that looks something like
"1 - x^2/2! + x^4/4! - x^6/6!"
This can be done through simple string manipulation in C#. I can just loop through and add the next term to the string.
I really can't comprehend how I would actually be able to use the string as a function to compare to graph positions to see if that graph position is actually on this graph to therefore graph it. So essentially: How would I use a string as an actual mathematical function in C#, or is there a better way of doing this.
Really sorry if it's confusing...really trying my best to explain it in a way that people can help.
You need a parser of string -> function. See MathParser for an example, that probably does everything you mentioned you need.
From a general perspective, anytime you want to convert a string into something that does work, you have to implement a parser, which will interpret the string and perform the actions dictated by it. For mathematical formulas, an expression tree may be of use to maintain order of operations and grouping. There are probably some math expression libraries available that will do this, or you can roll your own. This is not a trivial task, but it's certainly possible.
Once you have the expression tree, to figure out if a value f(x) for a given x is graphable, just evaluate it. For an f(x) graph, you can test x first to see if it falls in the visible domain of the graphing area. If it does, evaluate f(x) and if the point (x,f(x)) is graphable then draw the point.
Related
How do I get powers of a trig function in Math.net?
Expr x = Expr.Variable("x");
Expr g = (2 * x).Sinh().Pow(2);
g.ToString() gives the output: (sinh(2*x))^2
What I want is sinh^2(2*x)
How do I do that?
Edit:
As per Christoph's comment below this can be done in v0.21.0 which implements Expr.ToCustomString(true)
It does not currently support this notation. However, I see three options:
We define powers of trigonometric functions as new functions, or make them parametric.
This is something I do not wish to do.
We introduce un-applied functions as first class concept, which can be manipulated in such ways.
This is something we likely want to explore at some point, but this is a larger topic, and likely overkill for what you need.
Extend our visual expressions to support positive integer powers of functions.
This is something which we could implement.
Option 3 would save one set of parentheses in such expressions, which would lead to a more compact rendering, especially also for the LaTeX formatter where the result would be more readable. When building a visual expression from an expression, it would automatically pull positive integer powers to the applied function.
To my understanding your concern is only about how it is printed, so it seems to me this would solve your problem as well?
I'm looking for a simple function in C# to interpolate my 3D data.
Given is already a list with around 100-150 data sets and 3 double values.
-25.000000 -0.770568 2.444945
-20.000000 -0.726583 2.467809
-15.000000 -0.723274 2.484167
-10.000000 -0.723114 2.506445
and so on...
The chart created by these values looks usually like this, I'm not sure if this counts as scattered or rather still gridded data ...
In the end I want to hand over two double values and get the third then from the interpolation function. It shouldn't flatten the surface, it should still go through all the given data points.
Since I'm not given the time to look into all possible algorithms and lack the mathematical background I'm a bit overwhelmed by all the possibilities that I get thrown at: Kriging, Delauney triangulation, NURBs and many more ...
In addition to that most solutions I found in the net were either for a different language, outdated or are charged by the time (e.g ilnumerics, still not sure if they have the solution)
In matlab there exists a griddata function that does exactly this (and is based on a kriging algorithm as far as I know) but in this case C# is mandatory for me.
Thank you for your help and criticism and suggestions are welcome.
In order to find the FWHM I need to find a Gaussian f corresponding to a set of (x,f(x)) values. The available fitting methods (I'm restricted to C#) assume the solution to be a polynomium of n'th degree (or that's what I've been able to find so far). I'm wondering if there exists a specialized fitting method/scheme aimed at finding Gaussians. Or is there a generalized method out there that converges fast?
I can provide a good guess for the middle of the bell curve and its height but no more than that.
Solved this by observing that
ln(y) = ln(a) - ½(x-b)^2/c^2
which gives
c = sqrt(½(x-b)^2/ln(a-y))
So with a and b known it was a no-brainer :)
I apologize if this has already been asked - I'm not certain of the right terms to use here, so if it has, hopefully it will help others like me find whatever this gets marked as a dupe of.
I'm looking to create a formula for a curve in code (C# or Javascript ideally) from 3 points - the formula should be of the form y = a/(t+b) + c where t is time - the horizontal axis - and y is the vertical axis. Obviously a, b, and c are just there for graph fit.
How would I go about this? Is there an existing library I should be using?
The source data has a lot more than 3 data points available, I'm just looking for the simplest way to fit a 1/x curve to the data - so if for example 4 points are required for accuracy that's easy to provide as input.
If you are looking to fit a function of the form
y(t) = a/(t + b) + c
to a set of data points you are faced with a nonlinear least-squares problem for which you can use Gauss-Newton or Levenberg-Marquardt methods. However, there is an old algorithm that goes by the name of Loeb's algorithm that can be used to generate good (but not best - it can be shown that it will not converge to the best approximation) approximations when your approximation is a ratio of polynomials. It works by linearising the least-squares problem and results in an iterative least-squares solution (although in practice you will get good results with a single iteration). I studied this algorithm for my doctorate and I would strongly recommend it for any practical problem in which you want to approximate data points using a polynomial ratio (of which your case is very simple example).
The downside is that this algorithm is very old and you may struggle to find decent documentation of it. If you can it is no more complicated to implement than a standard linear least-squares approximation. If you get no better answer here to your problem, consider googling for it. If you cant find and decent information, let me know and I will upload my thesis to my website (contains implementation details of the method) and you can download it.
As I say you may get a far simpler answer here but if not it will certainly be an option open to you.
I have 2D world maps that are basically Mercator-Like projections, (If you walk west long enough you end up east of where you started)
Question I have: Can you use A* for computing paths on these types of maps as well?
I can't think of any reason why you couldn't (I'm thinking that you would simply represent the edge map nodes such that the North, South, East, Wed, "border" nodes simply connected to the opposite side).
Thanks in advance, if anyone has seen something like this before or can give me a few hints I would appreciate it.
Pathfinding algorithms don't really care about global topology of the map. The only tricky part is to get a good estimator for A* but using the 3D distance should be ok if your map is indeed a surface in a 3d space and step cost is step length.
Your map can have all sort of strange "connections" (including for example knotted bridges) and this will not be a problem if you implement A* correctly.
I can't imagine why a Mercator-Like projections would cause a problem for A*, as long as your heuristic function approximates distances correctly. I think something along the below function should work fine
float heuristic(point from, point to, size mapsize) {
float x = from.x - to.x;
if (abs(x) > mapsize.x/2)
x = mapsize.x - x;
float y = from.y - to.y;
if (abs(y) > mapsize.y/2)
y = mapsize.y - y;
return sqrt(x*x+y*y);
}
Edited: I realize know I was misled by the non-graph theoretical) use of the word edge (where the question title strongly suggested a graph algorithm question :))
Why do you suppose there are no edges? There are many logical discrete locations that you could model, and limited connections between (i.e. not where a wall is :)). There you go: you have your edges.
What you probably mean is that you don't want to represent your edges in data (which you don't have to, but still there are the logical edges that connect locations/points.)
That said:
you ask whether someone has seen things like this before. I vaguely recall seeing something relevant to this in Knuths Dancing Links article (DLX) which is an implementation technique for A* algorithms.
http://en.wikipedia.org/wiki/Dancing_Links
original publication [PDF]
The article specifically treats states as 'cells' (in a grid) with east/west/north/south links. It's been a long time so I don't quite recall how you would map (no pun intended) your problem on that algorithm.
The dance steps. One good way to implement algorithm X is to represent each 1 in the
matrix A as a data object x with five fields L[x]; R[x]; U [x]; D[x]; C[x]. Rows of the matrix
are doubly linked as circular lists via the L and R fields ("left" and "right"); columns are
doubly linked as circular lists via the U and D fields ("up" and "down"). Each column
list also includes a special data object called its list header.