WOLFRAM MATHEMATICA 7 MATHID HOW TO SPEED UP

Companion journals for Mathematics include: Foundations and AppliedMath. Physics leads by a fair margin, fol-Sections: published in thirteen topical sections. Close behind computer science Business/Social Sciences 6 Life Sciences 6 Mathematical Sciences 12 come the physical sciences. Mathematica users by field Other 7 Engineering 28 Computer Sciences 21 who are responsi- ble for compu- ters on which Mathematica is used but who do not themselves use Mathematica.

Use floating-point numbers if you can, and use them early.I know that in Wolfram Mathematica I can specify solving method in NDSolve function, but I cant find a full list of available methods in Stack Exchange Network Stack Exchange network consists of 178 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build. For the latest updates and corrections to this manual: visit reference.wolfram.com For information on additional copies of this documentation: visit the Customer Service website at or email Customer Service at infowolfram.com, 2014) and has You should take these derivatives using pencil and paper (or some computer algebra program such as Maple, Mathematica or the symbolic toolbox in.1. I thought I would share the list of things that I look for first when trying to optimize Mathematica code.For use with Wolfram Mathematica® 7.0 and later.

Wolfram Mathematica 7 Mathid How To Speed Up

Here is a simple example where working with floating-point numbers is nearly 50.6 times faster than doing the computation exactly and then converting the result to a decimal afterward. In Mathematica any number with a decimal point and less than 16 digits of input is automatically treated as a machine float, so always use the decimal point if you want speed ahead of accuracy (e.g. That difference can be pretty important when you hit nasty, numerically unstable problems, but in the majority of tasks, floating-point numbers are good enough and, importantly, much faster. 1/3 is the same thing as 0.33333333333333. Unnecessary use of exact arithmetic is the most common case.In most numerical software, there is no such thing as exact arithmetic. But compare to other languages in case of Mathematica is much harder to figure out how to speed up things.Of the most common issues that I see when I review slow code is that the programmer has inadvertently asked Mathematica to do things more carefully than needed.

More than the average person would care to sit down and learn in one go. There is more overhead in the compilation stage, but the DLL runs directly on your CPU, not on the Mathematica virtual machine, so the results can be even faster.Mathematica has a lot of functions. …and use Compile to generate C code.Furthermore, if your code is compilable, then you can also use the option CompilationTarget->“ C” to generate C code, call your C compiler to compile it to a DLL, and link the DLL back into Mathematica, all automatically.

But the key feature in this context is the profiler that lets you see which lines of code used up the time, and how many times they were called.Take this example, a truly horrible way (computationally speaking) to implement Fibonacci numbers. First it lets you debug and organize large code projects better, and having clean, organized code should make it easier to write good code. That’s great when you just need to get a quick answer, but it will also let you get away with less than optimal solutions without realizing it.Workbench helps in several ways. Specifying such details might be comparatively fiddly, but staying within Flatten to do the whole flattening job turns out to be nearly 4 times faster than re-implementing that sub-feature yourself.So remember—do a search in the Help menu before you implement anything.Mathematica can be quite forgiving of some kinds of programming mistakes—it will proceed happily in symbolic mode if you forget to initialize a variable at the right point and doesn’t care about recursion or unexpected data types. If I have a list of a million 2×2 matrices that I want to turn into a list of a million flat lists of 4 elements, the conceptually easiest way might be to Map the basic Flatten operation down the list of them.But Flatten knows how to do this whole task on its own when you specify that levels 2 and 3 of the data structure should be merged and level 1 be left alone. Not only is it a waste of time re-implementing work that is already done, but our guys are paid to worry about what the best algorithms are for different kinds of input and how to implement them efficiently, so most built-in functions are really fast.If you find something close-but-not-quite-right, then check the options and optional arguments often they generalize functions to cover many specialized uses or abstracted applications.Here is such an example.

You are trading speed for memory here, so it isn’t appropriate if your function is likely to be called for a huge number of values, but rarely the same ones twice. The Mathematica construct that you will want to know is this:It saves the result of calling f on any value, so that if it is called again on the same value, Mathematica will not need to work it out again. Remember values that you will need in the future.This is good programming advice in any language. The main rule is invoked 9,227,464 times, and the fib value is requested 18,454,929 times.Being told what your code really does, rather than what you thought it would do, can be a real eye-opener.5.

Looking up previous results prevents the need to repeatedly recurse down to fib.An increasing number of Mathematica operations will automatically parallelize over local cores (most linear algebra, image processing, and statistics), and, as we have seen, so does Compile if manually requested. Change the first rule to this:And it becomes immeasurably fast, since fib now only requires the main rule to be evaluated 33 times. Here it is rescuing the program that I used to illustrate tip 3.

Your Mathematica comes with four compute kernels, and you can scale up with grid Mathematica if you have access to additional CPU power. There is some overhead for sending the task and retrieving the result, so there is a trade-off of time gained versus time lost. Each of these automatically takes care of communication, worker management, and collection of results.

(And the construct data=Append is the same as AppendTo.)Instead use Sow and Reap. This makes it progressively slower as the data accumulates. As a result, AppendTo must create a fresh copy of all of the data, restructured to accommodate the appended information. Use Sow and Reap to accumulate large amounts of data (not AppendTo).Because of the flexibility of Mathematica data structures, AppendTo can’t assume that you will be appending a number, because you might equally append a document or a sound or an image.