Performance tuning in Wolfram Language

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NOTE: Please see the original version of this post HERE. Cross-posted here per suggestion of @Vitaliy Kaurov.

Since Mathematica is a symbolic system, with symbolic evaluator much more
general than in numerically-based languages, it is not surprising that performance-tuning can be
more tricky here. There are many techniques, but they can all be understood
from a single main principle. It is:

Avoid full Mathematica symbolic evaluation process as much as possible.

All techniques seem to reflect some facet of it. The main idea here is that most of the
time, a slow Mathematica program is such because many Mathematica functions are very general. This
generality is a great strength, since it enables the language to support better and more powerful abstractions, but in many places in the program such generality, used without care, can be a (huge) overkill.

I won’t be able to give many illustrative examples in the limited space, but they can be found in
several places, including some WRI technical reports (Daniel Lichtblau’s
one on efficient data structures in Mathematica comes to mind), a very good
book of David Wagner on Mathematica programming, and most notably, many Mathgroup
posts. I also discuss a limited subset of them in my book. I will supply more references soon.

Here are a few most common ones (I only list those available within Mathematica
language itself, not mentioning CUDA \ OpenCL, or links to other languages, which are
of course also the possibilities):

  1. Push as much work into the kernel at once as possible, work with as large
    chunks of data at a time as possible, without breaking them into pieces

    1.1. Use built-in functions whenever possible. Since they are implemented
    in the kernel, in a lower-level language (C), they are typically (but not always!)
    much faster than user-defined ones solving the same problem. The more specialized
    version of a built-in function you are able to use, the more chances you have for a speed-up.

    1.2. Use functional programming (Map, Apply, and friends). Also, use pure functions
    in #-& notation when you can, they tend to be faster than Function-s with named
    arguments or those based on patterns (especially for not computationally-intensive
    functions mapped on large lists).

    1.3. Use structural and vectorized operations (Transpose, Flatten, Partition, Part and friends), they are even faster than functional.

    1.4. Avoid using procedural programming (loops etc), because this programming
    style tends to break large structures into pieces (array indexing etc).
    This pushes larger part of the computation outside of the kernel and makes it slower.

  2. Use machine-precision whenever possible

    2.1. Be aware and use Listability of built-in numerical functions, applying them to
    large lists of data rather than using Map or loops.

    2.2. Use Compile, when you can. Use the new capabilities of Compile, such as CompilationTarget->"C",
    and making our compile functions parallel and Listable.

    2.3. Whenever possible, use vectorized operations (UnitStep, Clip, Sign, Abs, etc)
    inside Compile, to realize “vectorized control flow” constructs such as If, so that
    you can avoid explicit loops (at least as innermost loops) also inside Compile. This
    can move you in speed from Mathematica byte-code to almost native C speed, in some cases.

    2.4. When using `Compile`, make sure that the compiled function doesn't bail out to non-compiled evaluation.  See examples [in this MathGroup thread](https://groups.google.com/d/topic/comp.soft-sys.math.mathematica/XOXapJm_Q1Q/discussion).

  3. Be aware that Lists are implemented as arrays in Mathematica

    3.1. Pre-allocate large lists

    3.2. Avoid Append, Prepend, AppendTo and PrependTo in loops, for building
    lists etc (because they copy entire list to add a single element, which leads
    to quadratic rather than linear complexity for list-building)

    3.3. Use linked lists (structures like {1,{2,{3,{}}}} ) instead of plain lists
    for list accumulation in a program. The typical idiom is a = {new element, a}.
    Because a is a reference, a single assignment is constant-time.

    3.4. Be aware that pattern-matching for sequence patterns (BlankSequence,
    BlankNullSequence) is also based on Sequences being arrays. Therefore, a rule
    {fst_,rest___}:>{f[fst],g[rest]} will copy the entire list when applied. In particular, don’t
    use recursion in a way which may look natural in other languages. If you want to use recursion on lists, first convert your lists to linked lists.

  4. Avoid inefficient patterns, construct efficient patterns

    4.1. Rule-based programming can be both very fast and very slow, depending on how
    you build your structures and rules, but in practice it is easier to inadvertently
    make it slow. It will be slow for rules which force the pattern-matcher to make many
    a priory doomed matching attempts, for example by under-utilizing each run of the
    pattern-matcher through a long list (expression). Sorting elements is a good example:
    list//.{left___,x_,y_,right___}/;x>y:>{left,y,x,right} - has a cubic complexity in the
    size of the list (explanation is e.g. here).

    4.2. Build efficient patterns, and corresponding structures to store your data, making
    pattern-matcher to waste as little time on false matching attempts as possible.

    4.3. Avoid using patterns with computationally intensive conditions or tests. The
    pattern-matcher will give you the most speed when patterns are mostly syntactic in
    nature (test structure, heads, etc). Every time when condition (/;) or pattern test (?)
    is used, for every potential match, the evaluator is invoked by the pattern-matcher,
    and this slows it down.

  5. Be aware of immutable nature of most Mathematica built-in functions

    Most Mathematica built-in functions which process lists create a copy of an original list and
    operate on that copy. Therefore, they may have a linear time (and space) complexity in the
    size of the original list, even if they modify a list in only a few places. One universal
    built-in function that does not create a copy, modifies the original expression and does not
    have this issue, is Part.

    5.1. Avoid using most list-modifying built-in functions for a large number of
    small independent list modifications, which can not be formulated as a single step
    (for example, NestWhile[Drop[#,1]&,Range[1000],#<500&] )

    5.2. Use extended functionality of Part to extract and modify a large number of
    list (or more general expression) elements at the same time. This is very fast,
    and not just for packed numerical arrays (Part modifies the original list).

    5.3. Use Extract to extract many elements at different levels at once, passing
    to it a possibly large list of element positions.

  6. Use efficient built-in data structures

    The following internal data structures are very efficient and can be used in
    many more situations than it may appear from their stated main purpose. Lots of such examples can be found by searching the Mathgroup archive, particularly contributions of Carl Woll.

    6.1. Packed arrays

    6.2. Sparse arrays

  7. Use hash - tables.

    Starting with version 10, immutable associative arrays are available in Mathematica (Associations)

7.1 Associations

the fact that they are immutable does not prevent them to have efficient insertion and deletion of key-value pairs (cheap copies different from the original association by the presence, or absence, of a given key-value pair). They represent the idiomatic associative arrays in Mathematica, and have very good performance characteristics.

For earlier versions,the following alternatives work pretty well, being based on internal Mathematica’s hash-tables:

7.2. Hash-tables based on DownValues or SubValues

7.3. Dispatch

  1. Use element - position duality

    Often you can write faster functions to work with positions of elements rather than
    elements themselves, since positions are integers (for flat lists). This can give you
    up to an order of magnitude speed-up, even compared to generic built-in functions
    (Position comes to mind as an example).

  2. Use Reap - Sow

    Reap and Sow provide an efficient way of collecting intermediate results, and generally
    “tagging” parts you want to collect, during the computation. These commands also go well
    with functional programming.

  3. Use caching, dynamic programming, lazy evaluation

10.1. Memoization is very easily implemented in Mathematica, and can save a lot of execution 
time for certain problems.

10.2. In Mathematica, you can implement more complex versions of memoization, where you can 
define functions (closures) at run-time, which will use some pre-computed parts in their
definitions and therefore will be faster.

10.3. Some problems can benefit from lazy evaluation. This seems more relevant to memory - 
efficiency, but can also affect run-time efficiency. Mathematica's symbolic constructs make 
it easy to 	implement.

A successful performance - tuning process usually employs a combination of these techniques,
and you will need some practice to identify cases where each of them will be beneficial.

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Thanks for sharing!

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Glad that you found this useful. Just keep in mind that this is a direct repost of a post that is 6 years old, and as such, it can be a little dated.

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you said avoid patterns and symbolism then also to use efficient patterns?

did you know mathematica uses a form of hashing to search for symbols ? (associative arrays, like awk does)

the Mathematica books discuss efficiency as well. for example: Dispatch is an old feature which was built for efficiency

but you are right. besides “testing every equation”, it would be useful to know what parts of Mathematica are doing and what they’re Big O is.

In[2119]:= Unprotect[In, Out];

In[2120]:= Clear[In, Out];

In[2121]:= Protect[In, Out];

In[2122]:= $HistoryLength = 5;

In[2123]:= MemoryInUse[]

Out[2123]= foo

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You can use patterns efficiently or not. Just like anything else. The only difference is that it is easier to construct inefficient patterns for users with less experience, which is why I generally suggest to avoid pattern-matching in performance-sensitive code in favor of other programming styles. A good reference on this subject is an old book by David Wagner “Power Programming with Mathematica: the kernel” from 1996.

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