Write a recursive function that, given a positive integer k, computes the product (1-1/2)(1-1/3)...(1-1/k).
I have tried to make a loop, but don't know where to start.
Any ideas?
Sure! This example fits the pattern of section 9.3 (Natural Numbers) from How To Design Programs, v2e. You should find a lot of helpful guidance there.
Related
Beloew Hyperlink shows Orthogonal Functions.
I used different commands in maple but i can't apply these Integral expressions in Maple.
How can i integrate such conditional Integrals ??? (For Example the Integral with red box around it)
Orthogonal Functions
(This is more of a math Question than a programming Question, so it probably should've gone to math.stackexchange.com.)
You need to use an assuming clause to tell Maple that m and n are integer, and you need to use option AllSolutions to int to tell it to do a case-by-case analysis of the parameters. For example,
int(sin(n*Pi*x/L)*sin(m*Pi*x/L), x= 0..L, AllSolutions)
assuming n::posint, m::posint, L>0;
I've assumed positivity of all parameters simply to reduce the number of cases presented in Maple's answer.
I need to write my own function which has the form f(x,y)=Integrate(g(x,y,z),z from 0 to inf). so the code I used was:
function y=f(x,y)
g=#(z)exp(-z.^2)./(z.^x).*(z.^2+y.^2).^(x/2);% as a function of x,y and z
y=quadgk(g,0,inf)
and if I call it for a single value like f(x0,y0), it works but if I try to calculate something like f([1:10],y0), then the error message says that there is something wrong with the times and dimension. In principle I can use for loops but then my code slows down and takes forever. Is there any help I can get from you guys? or references?
I'm trying to avoid the for loop since in matlab it's much faster to use matrix computation than to use for loop. I wonder if there is any trick that I can take advantage of this feature.
Thanks for any help in advance,
Lynn
Perhaps you can try to transpose the interval, creating row based values instead of column based f([1:10]',y0). Otherwise something in your function might be wrong, for example to get x^y to work with lists as input, you have to prefix with a dot x.^y. The same for mulitply and division I think..
If loop is no problem for you, you should do something like:
function y2=f(x,y)
y2=zeros(size(x));
for n=1:numel(x)
g=#(z)exp(-z.^2)./(z.^x(n)).*(z.^2+y.^2).^(x(n)/2);% as a function of x,y and z
y2(n)=quadgk(g,0,inf)
end
The problem here is that quadk itself uses vectors as argument for g. Then you have in g somethink like z.^x, which is the power of two vectors that is only defined if z and x have the same dimension. But this is not what you want.
I assume that you want to evaluate the function for all arguments in x and that the output vector has the same dimension as x. But this does not seem to be possible since even this simple example
g=#(x)[x;x.^2]
quad(g,0,1)
does not work:
Error using quad (line 79)
The integrand function must return an output vector of the same length as the
input vector.
A similar error shows when using quadgk. The documentation also says that this routine works only for scalar functions and this is not surprising since an adaptive quadrature rule would in general use different points for each function to evaluate the integral.
You have to use quadvinstead, which can integrate vector valued functions. But this gives wrong results since your function is integrated in the interval [0,\infty).
How to find the following Maximum or supremum by computer software such as Mathematica and Matlab: $\sup\frac{(1+s)^{4}+(s+t)^{4}+t^{4}}{1+s^{4}+t^{4}}$?
Instead of numerical approximation, what is the accurate maximum?
Thanks.
Since the question seems a bit like homework, here's an answer that starts a bit like a lecture:
ask yourself what happens to the function as s and t go to small and to large positive and negative values; this will help you to identify the range of values you should be examining; both Mathematica and Matlab can help your figure this out;
draw the graph of your function over the range of values of interest, develop a feel for its shape and try to figure out where it has maxima; for this the Mathematic Plot3D[] function and the Matlab plot() function will both be useful;
since this is a function of 2 variables, you should think about plotting some of its sections, ie hold s (or t) constant, and make a 2D plot of the section function; again, develop some understanding of how the function behaves;
now you should be able to do some kind of search of the s,t values around the maxima of the function and get an acceptably accurate result.
If this is too difficult then you could use the Mathematica function NMaximize[]. I don't think that Matlab has the same functionality for symbolic functions built-in and you'll have to do the computations numerically but the function findmax will help.
In Matlab, one would create a vector/matrix with s and t values, and a corresponding vector with the function values. Then you can pinpoint the maximum using the function max
In Mathematica, use FindMaximum like this:
f[s_,t_]:= ((1+s)^4 + (s+t)^4 + t^4)/(1+s^4+t^4)
FindMaximum[ f[s,t],{s,0},{t,0} ]
This searches for a maximum starting from (s,t)=(0,0).
For more info, see http://reference.wolfram.com/mathematica/ref/FindMaximum.html
I am working on a project to create a generic equation solver... envision this to take the form of 25-30 equations that will be saved in a table- variable names along with the operators.
I would then call this table for solving any equation with a missing variable and it would move operators/ other pieces to the other side of the missing variable
e.g. 2x+ 3y=z and if x were missing variable. I would call equation with values for y and z and it would convert to solve for x=(z-3y)/2
equations could be linear, polynomial, binary(yes/no result)...
i am not sure if i can get any light-weight library available or whether this needs to built from scratch... any pointers or guidance will be appreciated
See Maxima.
I rather like it for my symbolic computation needs.
If such a general black-box algorithm could be made accurate, robust and stable, pigs could fly. Solutions can be nonexistent, multiple, parametrized, etc.
Even for linear equations it gets tricky to do it right.
Your best bet is some form of Newton algorithm, but generally you tailor it to your problem at hand.
EDIT: I didn't see you wanted something symbolic, rather than numerical. It's another bag of worms.
Given maths is not my strongest point I'm implementing a bezier curve for 3D animation.
The formula is shown here, and as you can see it is quite nasty. In my programming I use descriptive names, and like to break complex lines down to smaller manageable ones.
How is the best way to handle a scenario like this?
Is it to ignore programming best practices and stick with variable names such as x, y, and t?
In my opinion when you have a predefined mathematical equation it is perfectly acceptable to use short variable names: x, y, t, P_0 etc. which correspond to the equation. Make sure to reference the formula clearly though.
if the formulas is extrated to its own function i'd certainly use the canonical maths representation, and maybe add the wiki page url in a comment
if its imbedded in code with a specific usage of the function then keeping the domain names from your code might be better
it depends
Seeing as only the mathematician in you is actually going to understand the formula, my advice would be to go with a style that a mathematician would be most comfortable with (so letters as variables etc...)
I would also definitely put a comment in there somewhere that clearly states what the formula is, and what it does, for example "This method returns a series of points along a quadratic Bezier curve". That way whenever the programmer in you revisits the code you can safely ignore the mathematical complexity with the assumption that your inner mathematician has already checked to make sure its all ok.
I'd encourage you to use mathematic's best practices and denote variables with letters. Just provide explanation for the variables above the formula. And if you can split the formula to smaller subformulas, even better.
Don't bother. Just reference the documentation (the wikipedia page in this case or even better your own documentation) and make sure the variable names match your documentation. Code comments are just not well suited (nor need them to) describe mathematical formulation.
Sometimes a reference is better than 40 lines of comments or even suggestive variable names.
Make the formula in C# (or other language of preference) resemble the mathematical formula as closely as possible, and include a reference to the formula, including a description of the variables. The idea in coding is to be readable, and if you're dealing with mathematical formulae the most readable representation is the one that looks most like mathematics.
You could key the formula into wolfram alpha ... it will try to simplify for you.
It'll also output in a mathematica friendly style ... funnily enough ;)
Kindness,
Dan
I tend to break an equation down into its root parts.
def sum(array)
array.inject(0) { |result, item| result + item }
end
def average(array)
sum(array) / array.length
end
def sum_squared_error(array)
avg = average(array)
array.inject(0) { |result, item| result + (item - avg) ** 2 }
end
def variance(array)
sum_squared_error(array) / (array.length - 1)
end
def standard_deviation(array)
Math.sqrt(variance(array))
end
You might consider using a domain-specific language to handle this. Mathematica would allow you to write out the equation just as it appears in mathematical notion.
The more your final form resembles the original equation, the more maintainable it will be in the long run (otherwise you have to interpret the code every time you see it).