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For a school project I have to determine a function u(t) of time. I have derived an expression of the following form:
(https://i.stack.imgur.com/vNrYb.png)
with a,b,c,d constants (not necessarily integers). I have figured out that this problem is only solvable with numerical integration with initial condition u(0)=u_0, yet I don't know how to do this particular problem.
I have looked at all the numerical integration methods I have learnt so far, but they all seem to apply for polynomials or for functions where you know the function evaluations at specific points.
There are lots of ways to calculate an approximate value for u(t), some simple but requiring a lot of iterations, and more complex requiring fewer iterations. Assuming a,b,c,d are real numbers, and u_0 = u(0) then for t > 0, one could just split the interval between 0 and t into N sub-intervals and calculate
u_(i+1) = u_i + (du/dt)(t_i)*t/N
where t_i = i*t/N
then,
u_N = u(t).
If N is not sufficiently large, the result will be inaccurate. To obtain a satisfactory N is more art than science. Just printing the results for increasing N should give you an idea of how large N needs to be to obtain the level of accuracy you need. Adding higher order terms (d^2u/dt^2 etc.) can sometimes improve speed and accuracy.
You can't numerically integrate anything unless you have values for all those constants.
I don't know what numerical integration schemes you looked at, but I think Euler's method or Runga-Kutta would both be worth trying.
You don't say which language you want to use. Python would be a fine choice. So would Java. Lots of libraries to help.
Wolfram Alpha has a closed-form solution here. It's a separable, non-linear ODE. You'll need to know hypergeometric functions to evaluate.
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I'm preparing for competitive programming, I keep running into different symbols in problems that I'm expected to know.
(example image of such symbols)
Can someone make a summary of all of these different symbols and explain what they mean? Thanks in advance.
The big Sigma stands for "sum". The part you've circled in red means:
a_l + a_(l+1) + ... + a_(r-1)
It's a sum of r-l elements. And then you're dividing this sum by r-l, so you end up with the average of those r-l elements.
A sum like this can usually be implemented with a for-loop:
s = 0;
for (i = l; i <= r-1; i++)
{
s += a[i];
}
But note that in many cases the sum can be computed directly without a slow loop. For instance if the a_i follow an arithmetic progression or geometric progression then you can calculate the sum directly without actually summing all the terms one by one.
The arrow presumably means "tends to", giving information about a limit of this quantity. However, normally we never use this arrow alone; there should be some context to tell which limit we're talking about. For instance, "3x+4 tends to 13", alone, makes no sense. What would make sense would be "3x+4 tends to 13 when x tends to 3".
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I am by no means a math person, but I am really trying to figure out how create a graphable function from some data plots I measure from a chemical titration. I have been trying to learn R and I would like to know if anyone can explain to me or point me to a guide to create a mathmatic function of the titration graph below.
Thanks in advance.
What you are looking for is a Interpolation. I'm not a R programmer, but I'll try to answer anyway.
Some of the more common ways to achieve this function you want is by Polynomial Interpolation which usually gives back a Nth degree polynomial function, where N is the number of data points minus one (1 point gives a constant, 2 points make a line, 3 makes a*x^2 + b*x + c and so on).
Other common alternatives I've learn are used in Computer Graphics are Splines, B-spline, Bézier curve and Hermite interpolation. Those make the curve smoother and good looking (I've told they originated in the car industry so they are less true to the data points).
TL;DR: I've found evidence that there is a implementation of spline in R from the question Interpolation in R which may lead you to your solution.
Hope you get to know better your tool and do a great work.
When doing this kind of work in Computer Science we call it Numerical Methods (at least here in my university), I've done some class and homework in this area while attending to the Numerical Methods Course (it can be found at github) but it's nothing worth noting.
I would add a lot of links to Wikipedia but StackOverflow didn't allow it.
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I have a species abundance dataset with quite a few zeros in it and even when I set trymax = 1000 for metaMDS() the program is unable to find a stable solution for the stress. I have already tried combining data (collapsing multiple years together to reduce the number of zeros) and I can't do any more. I was just wondering if anyone knows - is it scientifically valid to pick what R gives me at the end (the lowest of the 1000 solutions) or should I not be using NMDS because it cannot find a stable spot? There seems to be very little information about this on the internet.
One explanation for this is that you are trying to use too few dimensions for the mapping. I presume you are using the default k = 2? If so, try k = 3 and compare the stress from the best solution you got from the 1000 tries for the k = 2 solution.
I would be a little concerned to take one solution out of 1000 just because it had the best/lowest stress.
You could also try 1000 more random starts and see if it converges if you run more iterations. When you saved the output from metaMDS(), you can supply that object to another call to metaMDS() via the previous.best argument. It will then do trymax further random starts but compare any lower-stress solutions with the previous best and converge if it finds one similar to it, rather than have to find two similar low-stress solutions in the 1000 starts.
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I'm programming some program which calculates the limit of markov chain.
if the markov matrix diverges, I should transform it into the form
dA + (1-d)E, where both A and E are n * n matrix, and all of the elements of E are 1/n.
But if I apply that transformation when the input converges, the wrong value comes out.
Is there any easy way to check if the markov matrix converges?
I'm not going to go into detail, because it's an entire field unto itself. Although the general convergence theorem states that any finite Markov chain that is aperiodic and irreducible converges (to its stationary distribution). Irreducibility is simple to check (it's equivalent to connectedness in graphs), and periodicity is also easy to check (the definition of both is found in the first chapter of the book below, and the convergence theorem is proved in chapter 4 of the book).
It's worth noting that if there isn't irreducibility that can be easily solved in the symmetrical case by splitting the state space into "connected components", and considering each one separately. While periodicity can be patched by doing something similar to what you're suggesting. It's called creating the lazy Markov chain. If you want to understand the whole topic a little better (Mixing times for example will be very helpful in your convergence algorithm), this is an excellent book (available for free):
http://pages.uoregon.edu/dlevin/MARKOV/markovmixing.pdf
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i'd like to know how to solve a definite integral in Mathematica.
I do know all variables except b, and need to solve for F(b)=0.
How can i solve it in Mathematica?
Here is my try:
NSolve[Integrate[1/(8*(1 - ff) (2 Pi)^0.5) E^(-0.5*((x - 1.1)/(1 - ff)/8)^2), {x, 0, 9999}] == -0.44531779637243296, ff]
These integrals can be trivially expressed in terms of an error function: Wiki, Mathworld. Hence what you need here is a library to (i) calculate error functions, (ii) numerically solve non-linear equations. Virtually any language has this, so pick anything you're familiar with. In Mathematica, look up Erf and NSolve.
I'd start by plugging it into Wolfram Alpha and see what it gives you.
Mathematica should be able to do it. I think of statistics first when R comes up; I don't know about its calculus capabilities. Excel is not the first choice.
If I were you, I'd be less worried about the software and more worried about the solution itself. A function of this form might be well known. Plot each one and visually check to see what the functions look like and how easy they might be to integrate.
Like this:
http://www.wolframalpha.com/input/?i=graph+exp%28-%28%28x%2B5%29%2F1.5%29%5E2%29
You should be wondering why it's three similar looking integrals. Those singularities in the plot tell you why.
If there's no closed form solutions, you'll have to go with a numerical one. You'll have to choose an algorithm (simple Euler or Runga Kutta or something else), interval sizes, etc. You'll want to know about singular points and how best to tackle them.
Choosing a package is just the start.
You might find http://r.789695.n4.nabble.com/calculus-using-R-td1676727.html helpful.