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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.
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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 currently teach myself programming and I came to a point where I was introduced to the recursive functions. I understand the basic principle behind them but whenever I try to read a code which contains recursive functions I have tough times tracing them. Honestly, If I don't write down the whole function on a paper and manually follow it I can't really understand what it does, I can't follow it mentally is what I am saying. Can you give some tips on how to do it mentally? I am pretty much average intelligent ( IQ 117 ), maybe that's the problem? Thanks in advance.
I think this is an excellent question. Many people make the mistake of trying to follow the call-stack of a recursive function, which is the wrong approach.
I find the only way is to look at what is written and ask yourself whether it is logically correct and whether it will end. If those things are true, it is going to work and you don't really need to understand how.
In more detail:
Are all possible input values considered?
Are the statements true in each case?
Is there a (are there) value(s) at which point the function will not recurse?
Do all remaining input values ultimately lead to that value?
For example (pseudo code):
factorial(n):
if (n < 0) throw InvalidArgument
if (n == 0) return 1
else return factorial(n - 1) * n
All input values are covered. The statements are true. There is a value at which it does not recurse (<=0) and the all remaining values reduce by 1 until the exit value is reached. Therefore it works.
You may of course make mistakes in your thinking. A good IQ does help here, and you may need at some point to resort to tracing or debugging to check your assumptions and find the error in your logic, but this should not be the first approach.
That is the beauty of recursive solutions: they tend to be statements of fact rather than procedural instructions!
I think IQ does not matter. Anyway, i think it's all about practice. Try to write some recursive functions. And after alot of practice and experience you will find it more easy to understand.
There is some good ideas.
Good luck.
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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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My employer asked me to do a project for our local team. Actually, it will be a way to help our work to finished faster.
We have a local database where we add exercises divided in two fields. The question and the solution. My employer wants since we are a team and we work at the same time, to create a system like stackoverflow's similar questions. When one of the team tries to submit a new data in the database, then it will check if there are other fields which may be duplicates.
The reason he asked me is because I have done something similar in the past but only for text using techniques like TF-IDF and Latent Semantic Analysis. But now, since the math symbols are all in Latex, I cannot find a way to check for duplicates.
I have tried to apply TF-IDF to the text only, but it doesn't work.
Any suggestion?
Edit:
Sorry for the broad topic. I will try to give more examples about my problem.
All the texts are exercises of primary and secondary schools. It is a mix of text and numbers-equations-symbols. If there were only text, I could use TF-IDF to find possible duplicates. Now, several exercises have a little or are without text.
Examples:
1) a. Solve the following equation: (x+1)*(x-1) = 5
b. Find the x: x^2 - 1 = 5
They are the same equation but with a different expression. So, I don't want to mark them as duplicates.
2) a. Solve the following equation: 3x + 7 = 12
b. Find the solution: 7 + 3x = 12
c. Find the x: 3x = 12 - 7
a and b should be duplicate whereas the c will not be.
You could try using MathJax to convert the LaTeX equation into MathML an XML format. You could then use tools to examine that structure. There are probably a few other tools which can convert your equation into some kind of tree structure.
Equality of mathematical expressions is a complex problem. There are question that should you treat (x+1)*(x-1) as being equal to x^2-1, algebraically they are the same.
You might want to investigate computer algebra systems which have a lot of sophisticated features for manipulating expressions.
One technique is to evaluate the expression at a number of points. If the values agree then its a good indication that the expressions are the same.
It might be easier to give a better answer if there was some idea of the type of problems you are working with, polynomials, integrals etc.?
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How would I go about creating the below transfer function?
It would take two parameters:
position of bulge in range (-1.0, +1.0)
sharpness of bump
That picture only demonstrates movement in the first parameter.
I can think of a few possible approaches:
figure out a formula
Bezier curves?
start with a few points and do some kind of chain-links type physics dynamical simulation, where each link exerts a force of its neighbours, and the end links are held low, a particular link is held high
something like the above but starting out with a crude shape and filtering out high frequencies
However I can't see any simple way to set out on any of the above approaches.
Can anyone see a clean way to crack it?
Looks like a normal distribution with variable skew to me. I would look for something like that before I'd go for Bezier curves.