Can I find ode solvers (ode23, ode45, and ode113) in Scilab?
I use these solvers in MATLAB, but I have no idea if there is the same option in Scilab or not.
Thanks in advance.
Did you try the search function? The answer in Convert ode45() to scilab should give an idea, even if RKF is not DoPri5.
Read the documentation on the other available steppers.
The default stepper without type parameter uses lsoda, which can be seen as comparable to ode113
With "stiff" you get lsode, which is approximately equivalent to ode15s.
"adams" could substitute for ode23, there are no explicit low-order methods available, so adaptive-step and order Adams-Bashford is the best you get for a fast integration. And, as mentioned,
"rkf" is an embedded explicit 4(5) method that can substitute for the embedded explicit (4)5 Dormand-Prince method of ode45.
There exist more modern solvers and step-size heuristics, using dense output, an advanced event "root->action" mechanism etc. Scilab is not alone in having a stalled development in this regard. The default is good enough for small projects and prototyping, for massive number-crunching use a compiled language.
Related
Here is a good linear solver named GotoBLAS. It is available for download and runs on most computing platforms. My question is, is there an easy way to link this solver with the Mathematica kernel, so that we can call it like LinearSolve? One thing most of you may agree on for sure is that if we have a very large Linear system then we better get it solved by some industry standard Linear solver. The inbuilt solver is not meant for really large problems.
Now that Mathematica 8 has come up with better compilation and library link capabilities we can expect to use some of those solvers from within Mathematica. The question is does that require little tuning of the source code, or you need to be an advanced wizard to do it. Here in this forum we may start linking some excellent open source programs like GotoBLAS with Mathematica and exchange our views. Less experienced people can get some insight from the pro users and at the end we get a much stronger Mathematica. It will be an open project for the ever increasing Mathematica community and a platform where these newly introduced capabilities of Mathematica 8 could be transparently documented for future users.
I hope some of you here will give solid ideas on how we can get GotoBLAS running from within Mathematica. As the newer compilation and library link capabilities are usually not very well documented, they are not used by the common users very often. This question can act as a toy example to document these new capabilities of Mathematica. Help in this direction by the experienced forum members will really lift the motivation of new users like me as well as it will teach us a very useful thing to extend Mathematica's number crunching arsenal.
The short answer, I think, is that this is not something you really want to do.
GotoBLAS, as I understand it, is a specific implementation of BLAS, which stands for Basic Linear Algebra Subroutines. "Basic" really means quite basic here - multiply a matrix times a vector, for example. Thus, BLAS is not a solver that a function like LinearSolve would call. LinearSolve would (depending on the exact form of the arguments) call a LAPACK command, which is a higher level package built on top of BLAS. Thus, to really link GotoBLAS (or any BLAS) into Mathematica, one would really need to recompile the whole kernel.
Of course, one could write a C/Fortran program that was compiled against GotoBLAS and then link that into Mathematica. The resulting program would only use GotoBLAS when running whatever specific commands you've linked into Mathematica, however, which rather misses the whole point of BLAS.
The Wolfram Kernel (Mathematica) is already linked to the highly-optimized Intel Math Kernel Library, and is distributed with Mathematica. The MKL is multithreaded and vectorized, so I'm not sure what GotoBLAS would improve upon.
since Excel Solver is quite slow to run on thousands of optimizations (the reason being that it uses the spreadsheet as interface), I'm trying to implement a similar (problem-specific) solver in C++ (with Visual Studio 2010, on a Win 7 64-bit platform). I would include the DLL via a Declare statement in VBA and already have experience in doing this, so this is not the problem.
My problem would be minimizing the sum of squared errors between empirical data and a target function which is non-linear but smooth, and the problem would include non-negativity (X>=0) or even positivity constraints (e.g. X>=0.00000001), with X denoting the decision variable.
I'm looking for a robust, proven implementation. It may be part of an established library.
For example, I've already looked into what ALGLIB has in store (see http://www.alglib.net/optimization/) and it seems only one of their algorithms accepts bounded constraints. But I don't know what it's worth, though, that's why I'm trying to gather some opinions.
Or, on another note, would it be advisable to augment ALGLIB's Levenberg-Marquardt algorithm with such basic constraints, for example by rejecting every intermediate solution that does not satisfy my constraints? (guess that won't do it, but it's still worth asking)
There are modifications of the Levenberg-Marquardt method that add support for inequality constraints. I know about one library that implements such an algorithm:
levmar (GPL).
If you would like to modify an existing algorithm, rejecting bad solutions won't do, the optimization will likely get stuck. But you can make a variable substitution, e.g. to ensure that X > 0.1 you can use t^2+0.1 instead of X.
I use this method as a workaround for the lack of built-in box constraints in my program. Here is a quote from Data fitting in the chemical sciences by Peter Gans that describes it better:
https://github.com/wojdyr/fityk/wiki/InequalityConstraints
We find OPTIF9 and UNCMIN to be the standard methods of choice.
You should be able to link them in a library, and call them from C++,
if you don't want to bother compiling Fortran.
A way to put limits on the search space is to transform the parameters, such as by a logit function.
Have you looked into the Microsoft Solver Foundation? The express edition is free, and comes with a .NET 4.0 dll. I found it fairly easy to use. On the other hand, I don't know how large of a problem you are talking: there are some limitations in the number of variables in the express edition.
I am doing some numerical analysis work in VB6, and the question arises of which of
sqr(x)
or
x^0.5
I should use.
Is there any difference in the method used to evaluate these two expressions, and if so, which of them should I prefer?
VB6 does not document the method is uses to evaluate sqr() or x^0.5. Empirically, sqr() is much faster, which could mean that they are using a dedicated root finding algorithm here. The use of a specialized algorithm could mean that sqr() also has better numerical stability, but I have no information regarding this.
I remember solving a lot of indefinite integration problems. There are certain standard methods of solving them, but nevertheless there are problems which take a combination of approaches to arrive at a solution.
But how can we achieve the solution programatically.
For instance look at the online integrator app of Mathematica. So how do we approach to write such a program which accepts a function as an argument and returns the indefinite integral of the function.
PS. The input function can be assumed to be continuous(i.e. is not for instance sin(x)/x).
You have Risch's algorithm which is subtly undecidable (since you must decide whether two expressions are equal, akin to the ubiquitous halting problem), and really long to implement.
If you're into complicated stuff, solving an ordinary differential equation is actually not harder (and computing an indefinite integral is equivalent to solving y' = f(x)). There exists a Galois differential theory which mimics Galois theory for polynomial equations (but with Lie groups of symmetries of solutions instead of finite groups of permutations of roots). Risch's algorithm is based on it.
The algorithm you are looking for is Risch' Algorithm:
http://en.wikipedia.org/wiki/Risch_algorithm
I believe it is a bit tricky to use. This book:
http://www.amazon.com/Algorithms-Computer-Algebra-Keith-Geddes/dp/0792392590
has description of it. A 100 page description.
You keep a set of basic forms you know the integrals of (polynomials, elementary trigonometric functions, etc.) and you use them on the form of the input. This is doable if you don't need much generality: it's very easy to write a program that integrates polynomials, for example.
If you want to do it in the most general case possible, you'll have to do much of the work that computer algebra systems do. It is a lifetime's work for some people, e.g. if you look at Risch's "algorithm" posted in other answers, or symbolic integration, you can see that there are entire multi-volume books ("Manuel Bronstein, Symbolic Integration Volume I: Springer") that have been written on the topic, and very few existing computer algebra systems implement it in maximum generality.
If you really want to code it yourself, you can look at the source code of Sage or the several projects listed among its components. Of course, it's easier to use one of these programs, or, if you're writing something bigger, use one of these as libraries.
These expert systems usually have a huge collection of techniques and simply try one after another.
I'm not sure about WolframMath, but in Maple there's a command that enables displaying all intermediate steps. If you do so, you get as output all the tried techniques.
Edit:
Transforming the input should not be the really tricky part - you need to write a parser and a lexer, that transforms the textual input into an internal representation.
Good luck. Mathematica is very complex piece of software, and symbolic manipulation is something that it does the best. If you are interested in the topic take a look at these books:
http://www.amazon.com/Computer-Algebra-Symbolic-Computation-Elementary/dp/1568811586/ref=sr_1_3?ie=UTF8&s=books&qid=1279039619&sr=8-3-spell
Also, going to the source wouldn't hurt either. These book actually explains the inner workings of mathematica
http://www.amazon.com/Mathematica-Book-Fourth-Stephen-Wolfram/dp/0521643147/ref=sr_1_7?ie=UTF8&s=books&qid=1279039687&sr=1-7
I'd like to do some work with the nitty-gritties of computer imaging. I'm looking for a way to read single pixels of data, analyze them programatically, and change them. What is the best language to use for this (Python, c++, Java...)? What is the best fileformat?
I don't want any super fancy software/APIs... I'm looking for the bare basics.
If you need speed (you'll probably always want speed with image processing) you definitely have to work with raw pixel data.
Java has some real disadvantages as you cannot access memory directly which makes pixel access quite slow compared to accessing the memory directly.
C++ is definitely the language of choice for production use image processing. But you can, for example, also use C# as it allows for unsafe code in specific areas. (Take a look at the scan0 pointer property of the bitmapdata class.)
I've used C# successfully for image processing applications and they are definitely much faster than their java counterparts.
I would not use any scripting language or java for such a purpose.
It's very east to manipulate the large multi-dimensional or complex arrays of pixel information that are pictures using high-level languages such as Python. There's a library called PIL (the Python Imaging Library) that is quite useful and will let you do general filters and transformations (change the brightness, soften, desaturate, crop, etc) as well as manipulate the raw pixel data.
It is the easiest and simplest image library I've used to date and can be extended to do whatever it is you're interested in (edge detection in very little code, for example).
I studied Artificial Intelligence and Computer Vision, thus I know pretty well the kind of tools that are used in this field.
Basically: you can use whatever you want as long as you know how it works behind the scene.
Now depending on what you want to achieve, you can either use:
C language, but you will lose a lot of time in bugs checking and memory management when implementing your algorithms. So theoretically, this is the fastest language to do that kind of job, but if your algorithms are not computationnally efficient (in terms of complexity) or if you lose too much time in bugs checking, this is clearly not worth it. So I would advise to first implement your application in another language, and then later you can always optimize small parts of your code with C bindings.
Octave/MatLab: very efficient language, almost as much as C, and you can make very elegant and succinct algorithms. If you are into vectorization, matrix and linear operations, you should go with that. However, you won't be able to develop a whole application with this language, it's more focused on algorithms, but then you can always develop an interface using another language later.
Python: all-in-one elegant and accessible language, used in gigantically large scale applications such as Google and Facebook. You can do pretty much everything you want with Python, any kind of application. It will be perfectly adapted if you want to make a full application (with client interaction and all, not only algorithms), or if you want to quickly draft a prototype using existent libraries since Python has a very large set of high quality libraries, like OpenCV. However if you only want to make algorithms, you should better use Octave/MatLab.
The answer that was selected as a solution is very biaised, and you should be careful about this kind of archaic comment.
Nowadays, hardware is cheaper than wetware (humans), and thus, you should use languages where you will be able to produce results faster, even if it's at the cost of a few CPU cycles or memory space.
Also, a lot of people tends to think that as long as you implement your software in C/C++, you are making the Saint Graal of speedness: this is just not true. First, because algorithms complexity matters a lot more than the language you are using (a bad algorithm will never beat a better algorithm, even if implemented in the slowest language in the universe), and secondly because high-level languages are nowadays doing a lot of caching and speed optimization for you, and this can make your program run even faster than in C/C++.
Of course, you can always do everything of the above in C/C++, but how much of your time are you willing to waste to reinvent the wheel?
Not only will C/C++ be faster, but most of the image processing sample code you find out there will be in C as well, so it will be easier to incorporate things you find.
if you are looking to numerical work on your images (think matrix) and you into Python check out http://www.scipy.org/PyLab - this is basically the ability to do matlab in python, buddy of mine swears by it.
(This might not apply for the OP who only wanted the bare basics -- but now that the speed issue was brought up, I do need to write this, just for the record.)
If you really need speed, it's better to forget about working on the pixel-by-pixel level, and rather see whether the operations that you need to perform could be vectorized. For example, for your C/C++ code you could use the excellent Intel IPP library (no, I don't work for Intel).
It depends a little on what you're trying to do.
If runtime speed is your issue then c++ is the best way to go.
If speed of development is an issue, though, I would suggest looking at java. You said that you wanted low level manipulation of pixels, which java will do for you. But the other thing that might be an issue is the handling of the various file formats. Java does have some very nice APIs to deal with the reading and writing of various image formats to file (in particular the java2d library. You choose to ignore the higher levels of the API)
If you do go for the c++ option (or python come to think of it) I would again suggest the use of a library to get you over the startup issues of reading and writing files. I've previously had success with libgd
What language do you know the best? To me, this is the real question.
If you're going to spend months and months learning one particular language, then there's no real advantage in using Python or Java just for their (to be proven) development speed.
I'm particularly proficient in C++ and I think that for this particular task I can be as speedy as a Java programmer, for example. With the aid of some good library (OpenCV comes to mind) you can create anything you need in a matter of a couple of lines of C++ code, really.
Short answer: C++ and OpenCV
Short answer? I'd say C++, you have far more flexibility in manipulating raw chunks of memory than Python or Java.