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
Related
I have plotted a simple velocity against time graph about vibration on GNU Octave and I would like to know the exact values when the time is 1 and 10s. I'm not very sure on this, please help me on this.
As long as your time array t contains those exact values, and assuming your velocity array is v, you could use
v(find(sum(t==[1 10].')))
and put as many values into that bracketed vector as you need.
If t doesn't have those exact values, like t = [0 0.9 10.1], check out the intelligent use of the unique function at this page
https://www.mathworks.com/matlabcentral/answers/375710-find-nearest-value-to-specific-number
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.
I'm a total novice in Maple (I usually use R) but I'm trying to do some analytic transformations on a user-defined function, hence my choice of Maple.
The function is in t and I have a vector of known values (basically, known constants) (t_1,..., t_n) which is passed to the function and I also know the values for alpha, beta and miu.
I cannot figure out how to code the summation of exponentials in Maple: in R I would simply use sum().
I have looked at the examples in http://www.maplesoft.com/support/help/maple/view.aspx?path=sum but none seems to apply to my case. I don't want to evaluate the function but I want to be able to find, analytically, the integral or the derivative of it (or of functions of it.)
Links to manuals or web-pages that deal with similar problems/cases are very much welcome!
Make the vector of constants global, not passed in. I'll call it T. Then the function is
lambda:= t-> mu+alpha*add(exp(-beta*(t-tau)), tau= T);
I am attempting to fit a circle to some data. This requires numerically solving a set of three non-linear simultaneous equations (see the Full Least Squares Method of this document).
To me it seems that the NEWTON function provided by IDL is fit for solving this problem. NEWTON requires the name of a function that will compute the values of the equation system for particular values of the independent variables:
FUNCTION newtfunction,X
RETURN, [Some function of X, Some other function of X]
END
While this works fine, it requires that all parameters of the equation system (in this case the set of data points) is hard coded in the newtfunction. This is fine if there is only one data set to solve for, however I have many thousands of data sets, and defining a new function for each by hand is not an option.
Is there a way around this? Is it possible to define functions programmatically in IDL, or even just pass in the data set in some other manner?
I am not an expert on this matter, but if I were to solve this problem I would do the following. Instead of solving a system of 3 non-linear equations to find the three unknowns (i.e. xc, yc and r), I would use an optimization routine to converge to a solution by starting with an initial guess. For this steepest descent, conjugate gradient, or any other multivariate optimization method can be used.
I just quickly derived the least square equation for your problem as (please check before use):
F = (sum_{i=1}^{N} (xc^2 - 2 xi xc + xi^2 + yc^2 - 2 yi yc + yi^2 - r^2)^2)
Calculating the gradient for this function is fairly easy, since it is just a summation, and therefore writing a steepest descent code would be trivial, to calculate xc, yc and r.
I hope it helps.
It's usual to use a COMMON block in these types of functions to pass in other parameters, cached values, etc. that are not part of the calling signature of the numeric routine.
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).