I need to find the roots from the equations as follows (Mathematica):
Sqrt[3]/2*x-(I-x*Sqrt[3]/2*c^2)*I/Sqrt[2*Pi]/d^3*Integrate[t*Exp[-t^2/2/d^2]/(Sqrt[3]/2*x+I*(t+b0)),{t,-Inf,Inf}]=0
i.e. as the picture shows:
where c, d, and b0 is given parameters, x is a complex root needs to find.
I have tried several methods, including scanning the real and imagine part of x and the iteration approach, but non of them could resolve all the cases.
Are there any general approaches that could solve such kind of equation efficiently, or by MATLAB/Mathematica?
did you try Matlab's mupad? it is a powerful symbolic tool, very similar to Maple wich gives really good results in non-numerical mathematics. Try there. declare the equation, give assumptions to the software ,i.e assume c real positive (don't copy this, I dont remember the proper syntax) and then solve! It will very likely find a solution if it exits (sometimes in some mathematical cases that you even don't know!)
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I am studying Stochastic calculus, and occasionally we need to compute an integral (from -infinity to +infinity) for some complex distribution. In this case, it was
with the answer on the right. This is the code I put into Matlab (and I have the symbolic math toolbox), which Matlab simply cannot process:
>> syms x t
>> f = exp(1+2*x)*(1/((2*pi*t)^0.5))*exp(-(x^2)/(2*t))
>> int(f,-inf,inf)
ans =
-((2^(1/2)*pi^(1/2)*exp(2*t + 1)*limit(erf((2^(1/2)*((x*1i)/t - 2i))/(2*(-1/t)^(1/2))), x, -Inf)*1i)/(2*(-1/t)^(1/2)) - (2^(1/2)*pi^(1/2)*exp(2*t + 1)*limit(erf((2^(1/2)*((x*1i)/t - 2i))/(2*(-1/t)^(1/2))), x, Inf)*1i)/(2*(-1/t)^(1/2)))/(2*pi*t)^(1/2)
This answer at the end looks like nonsense, while Wolfram (via their free tool), gives me the answer that the picture above has. Am I missing something fundamental about doing such integrations in Matlab that the basic Mathworks pages don't cover? Where am I proceeding incorrectly?
In order to explain what is happening, we need some theory:
Symbolic systems such as Matlab or Mathematica calculate integrals symbolically by the Risch algorithm (yes, there is a method to mechanically calculate integrals, just like derivatives).
However, the Risch algorithms works differently than using derivation rules. Strictly spoken, it is not an algorithm but a semi-algorithm. This is, it is not deterministic one (as algorithms are).
This (semi) algorithm makes a series of transformations on the input expression (the one to be integrated), and in a specific point, it requires to ask if the transformed expression is equals to zero, because if it were zero, it cannot continue (the input is not integrable using a finite set of terms).
The problem (and the reason of the "semi-algoritmicity") is that, the (apparently simple) equation:
E = 0
Is indecidable (it is also called the constant problem). It means that there cannot exist a formal method to solve the constant problem, for any expression E. Of course, we know to solve the constant problem for specific forms of the expression E (i.e. polynomials), but it is impossible to solve the problem for the general case.
It also means that the Risch algorithm cannot be perfect (being able to solve any integral -integrable in finite terms-). In other words, the Risch algorithm will be as powerful as our ability to solve the constant problem for as many forms of the expression E as we can, but losing any hope of solving for the general case.
Different symbolic systems have similar, but different methods to try to solve equations (and therefore the constant problem), it explains why some of them can "solve" different sets of integrals than others.
And generalizing, because no symbolic system will never be able to solve the constant problem for the general case, it will neither be able to solve any integral (integrable in finite terms).
The second parameter of int() needs to be the variable you're integrating over (which looks like t in this case):
syms x t
f = exp(1+2*x)*(1/((2*pi*t)^0.5))*exp(-(x^2)/(2*t))
int(f,'t',-inf,inf) % <- integrate over t
I have been looking for a Matlab function that can do a nonlinear total least square fit, basically fit a custom function to data which has errors in all dimensions. The easiest case being x and y data-points with different given standard deviations in x and y, for every single point. This is a very common scenario in all natural sciences and just because most people only know how to do a least square fit with errors in y does not mean it wouldn't be extremely useful. I know the problem is far more complicated than a simple y-error, this is probably why most (not even physicists like myself) learned how to properly do this with multidimensional errors.
I would expect that a software like matlab could do it but unless I'm bad at reading the otherwise mostly useful help pages I think even a 'full' Matlab license doesn't provide such fitting functionality. Other tools like Origin, Igor, Scipy use the freely available fortran package "ODRPACK95", for instance. There are few contributions about total least square or deming fits on the file exchange, but they're for linear fits only, which is of little use to me.
I'd be happy for any hint that can help me out
kind regards
First I should point out that I haven't practiced MATLAB much since I graduated last year (also as a Physicist). That being said, I remember using
lsqcurvefit()
in MATLAB to perform non-linear curve fits. Now, this may, or may not work depending on what you mean by custom function? I'm assuming you want to fit some known expression similar to one of these,
y = A*sin(x)+B
y = A*e^(B*x) + C
It is extremely difficult to perform a fit without knowning the form, e.g. as above. Ultimately, all mathematical functions can be approximated by polynomials for small enough intervals. This is something you might want to consider, as MATLAB does have lots of tools for doing polynomial regression.
In the end, I would acutally reccomend you to write your own fit-function. There are tons of examples for this online. The idea is to know the true solution's form as above, and guess on the parameters, A,B,C.... Create an error- (or cost-) function, which produces an quantitative error (deviation) between your data and the guessed solution. The problem is then reduced to minimizing the error, for which MATLAB has lots of built-in functionality.
Matlab has functionalities that allow you to work with known functions that you must define.
But, sometimes I want to do a complex symbolic calculation using a general function, Say A(x), without specifying A(x).
In other words, is it possible for me to make a statement like
diff(A(x^2+1),x), where the answer should involve a symbolic derivative of A???
diff(A(x^2+1),x) = A' diff(x^2+1,x)
That is, if A' is the derivative of A.
Yes. The functionality you describe is part of the symbolic algebra toolkit -- note that it comes with some fairly significant limitations, but, in short, all you would require would be
syms x A(x)
diff(A(x), x)
Note that ' is reserved for transpose, even with symbolic functions. (Although, personally, I'd frankly suggest Mathematica for any serious symbolic algebra any day over matlab -- it's really the intended purpose of the whole product, whereas the symbolic algebra toolkit is exactly that: a toolkit add-on to the core features of Matlab, namely fast linear algebra).
I've got a problem with my equation that I try to solve numerically using both MATLAB and Symbolic Toolbox. I'm after several source pages of MATLAB help, picked up a few tricks and tried most of them, still without satisfying result.
My goal is to solve set of three non-polynomial equations with q1, q2 and q3 angles. Those variables represent joint angles in my industrial manipulator and what I'm trying to achieve is to solve inverse kinematics of this model. My set of equations looks like this: http://imgur.com/bU6XjNP
I'm solving it with
numeric::solve([z1,z2,z3], [q1=x1..x2,q2=x3..x4,q3=x5..x6], MultiSolutions)
Changing the xn constant according to my needs. Yet I still get some odd results, the q1 var is off by approximately 0.1 rad, q2 and q3 being off by ~0.01 rad. I don't have much experience with numeric solve, so I just need information, should it supposed to look like that?
And, if not, what valid option do you suggest I should take next? Maybe transforming this equation to polynomial, maybe using a different toolbox?
Or, if trying to do this in Matlab, how can you limit your solutions when using solve()? I'm thinking of an equivalent to Symbolic Toolbox's assume() and assumeAlso.
I would be grateful for your help.
The numerical solution of a system of nonlinear equations is generally taken as an iterative minimization process involving the minimization (i.e., finding the global minimum) of the norm of the difference of left and right hand sides of the equations. For example fsolve essentially uses Newton iterations. Those methods perform a "deterministic" optimization: they start from an initial guess and then move in the unknowns space essentially according to the opposite of the gradient until the solution is not found.
You then have two kinds of issues:
Local minima: the stopping rule of the iteration is related to the gradient of the functional. When the gradient becomes small, the iterations are stopped. But the gradient can become small in correspondence to local minima, besides the desired global one. When the initial guess is far from the actual solution, then you are stucked in a false solution.
Ill-conditioning: large variations of the unknowns can be reflected into large variations of the data. So, small numerical errors on data (for example, machine rounding) can lead to large variations of the unknowns.
Due to the above problems, the solution found by your numerical algorithm will be likely to differ (even relevantly) from the actual one.
I recommend that you make a consistency test by choosing a starting guess, for example when using fsolve, very close to the actual solution and verify that your final result is accurate. Then you will discover that, by making the initial guess more far away from the actual solution, your result will be likely to show some (even large) errors. Of course, the entity of the errors depend on the nature of the system of equations. In some lucky cases, those errors could keep also very small.
I am willing to compute a Frechet/Gateaux derivative of a function which is not entirely explicit and my question is : What would be the most efficient way to do it ? Which language would you recommend me to use ?
Precisely, my problem is that I have a function, say F, which is the square of the euclidean norm of the sum of products of pairs of multidimensional functions (i.e. from R^n to R^k).
AFAIK, If I use Maple or Maxima, they will ask me to explicit the functions involved in the formula whereas I would like to keep them abstract. Then, I necessarly need to compute a Frechet/Gateaux derivative so as to keep the expressions simple. Indeed, when I proceed the standard way, I start to develop the square of the euclidean norm as a sum of squares and there is a lot of indexes. My goal being to make a Taylor developpement with integral remainder to the third order, the expression becomes, according to me, humanly infeasible (the formula is more than one A4 page long).
So I would prefer to use a Frechet/Gateaux derivative, which would allow me, among other, to keep scalar products instead of sums.
As the functions invloved have some similarities with their derivatives (due to the presence of exponentials) there is just a small amount of rules to know. So I thought that I might make such a one-purposed computer algebra system by myself.
And I started to learn LISP, as I read that it would be efficient for my problem, but I am a little bit lost now, since this language is very different and I am still used to think in terms of C/Python/Perl...
Here is another question : would you have some links to courses or articles about how an algebra system for symbolic computations is made (preferably in LISP) ? Any suggestions are welcome.
Thank you very much for your answers.
My advice is to use Maxima. Maxima is inspired by Lisp, and implemented in Lisp, so using Maxima will save you a tremendous amount of time and trouble. If Lisp is suitable for your problem, Maxima is even more so.
Maxima will allow you to use undefined terms in an expression; it is not necessary to define all terms.
Post a message to the Maxima mailing list (maxima#math.utexas.edu) to ask for specific advice. Please explain in detail about what you are trying to accomplish.