Mathematica终极参考资源列表
Where can I find examples of good Mathematica programming practice?
哪里能找到优质的Mathematica学习实践的示例？
（下面的每一个链接都指向一个学习的资源）
I consider myself a pretty good Mathematica programmer, but I'm always looking out for ways to either improve my way of doing things in Mathematica, or to see if there's something nifty that I haven't encountered yet. Where (books, websites, etc.) do I look for examples of good (best?) practices of Mathematica programming?
reference-requestprogramming
shareimprove this question asked Jan 17 '12 at 21:51
community wiki
Ｊ. Ｍ.
We're looking for long answers that provide some explanation and context. Don't just give a one-line answer; explain why your answer is right, ideally with citations. Answers that don't include explanations may be removed.
add a comment
Answers 答案
Here's a collection of resources that I started on Mathgroup (a collection of Mathematica learning resources) and updated here at Stack Overflow. As this site is dedicated to Mathematica it makes more sense to maintain it here. This represents a huge amount of information; of course it's not exhaustive so feel free to improve it! Also, don't hesitate to share it and suggest other interesting links! Remember, you can always search the online Documentation Center of Mathematica, that is identical to the built-in help of the latest software version.
Links to more advanced aspects of the program that you can start to appreciate once you understand the basics are provided in separate
answers (below) as this post became too large.
?Tips and Tricks
?Advanced evaluation, patterns and neat algorithms
Introduction 入门级教程
?If you're just beginning try to have a look at these videos.
Mathematica Basics, Elementary Programming in Mathematica
Hands-on Start to Mathematica
Several introductory videos by Jon McLoone
and many other video introductions and tutorials from the official Wolfram website
?Is it necessary to have a prior computational background or is it possible to learn Mathematica as a first programming language?
?What are the most common pitfalls awaiting new users?
?How To-s: full solutions for particular tasks from the online documentation
?Easy-to-understand animations explaining common Mathematica functions
?Sal Mangano's videos for using pure functions, Part and patterns ?Introductory videos of various applications of Mathematica
?What is the best Mathematica tutorial for young people?
Basic advices for people new to Mathematica Functional style函数式编程风格
Avoid iterative programming using loops like For or Do, use instead functional programming functions Map, Scan, MapThread, Fold, FoldList, ... and pure functions. This makes the code cleaner and faster.
?Functional Programming, Functional Programming: Quick Start
?Pure functions
What does # mean in Mathematica?
?Alternatives to procedural loops and iterating over lists in Mathematica
?An example: Programming a numerical method in the functional style ?How to understand the usage of Inner and Outer figuratively? Transpose and dimensions
?Something not easy to guess alone at the beginning: if you have
x={1,2} and y={3,4}, doing Transpose[{x,y}] or {x,y}ESC tr ESC in the front end will produce {{1,3},{2,4}} (format compatible with ListPlot). This animation helps understand why.
?You can also use the second argument of Transpose to reorder the indices of a multidimensional list.
?Don't forget to regularly control the output of the lists you generate using Dimensions.
Get familiar with shorthand syntax (@, &, ##, /@, /., etc.)
快捷语法糖
?Operator Input Forms
?when is f@g not the same as f[g]?
Programming easily 获取帮助和自动补全
?Getting help: Execute ?Map for example for a short description of
a function, or press F1 on a function name for more details and
examples about it. You can solve many problems by adapting examples to your needs.
?Auto-completion: Start typing the name of a function and (in Mathematica 9+) select from the pop-up auto-completion menu, or press Ctrl+k to get a list of functions which names start with what has already been entered. Once the name of the function is written completely press Ctrl+Shift+k (on Mac, Cmd+k) to get a list of its arguments.
?Function templates: In Mathematica 9, after typing a function name, press Ctrl+Shift+k (on Mac, Cmd+Shift+k) and click on the desired form from the pop-up menu to insert a template with named
placeholders for the arguments.
Other useful shortcuts are described in the post Using the
Mathematica front-end efficiently for editing notebooks.
?Use palettes in the Palettes menu especially when you're beginning.
?In Mathematica 8, use the natural input capability of Wolfram Alpha, for example type "= graph 2 x + 1 between 0 and 3" without the quotes and see the command associated with the result. Tutorials
?Fundamentals of Mathematica Programming(by Richard Gaylord, great tutorial for an overview of the logic behind Mathematica: patterns)
?Introduction to Mathematica (by Thomas Hahn, another succinct overview of Mathematica)
?Tutorial Collection by WRI (lots of extra documentation and examples, available as free PDFs, also available and up-to-date in Help>Virtual Book in Mathematica).
?Programming Paradigms via Mathematica (A First Course)
?Mathematica Tutorial: A New Resource for Developers
?Wolfram's Mathematica 101
?http://bmia.bmt.tue.nl/Software/Downloads/Campus/TrainingMathe maticaEnglish.zip
http://bmia.bmt.tue.nl/Software/Mathematica/Tutorials/index.ht
ml
? A problem centered approach
? A beginner's guide to Mathematica
?https://www.sodocs.net/doc/0710993168.html,/MathematiClub/tutorials.htm
?https://www.sodocs.net/doc/0710993168.html,/mmcguff/mathematica/
?https://www.sodocs.net/doc/0710993168.html,/courses/hnichols/phys303/
?https://www.sodocs.net/doc/0710993168.html,/courses/ap1601y/ (Introduction to Computational Mathematics and Physics)
?https://www.sodocs.net/doc/0710993168.html,.au/pub/MATH2200/2012/Lectures/ (Applied Mathematics)
https://www.sodocs.net/doc/0710993168.html,.au/pub/MATH2200/2009/Lectures (path for some lectures in pdf)
?https://www.sodocs.net/doc/0710993168.html,/wiki/Mathematica
?https://www.sodocs.net/doc/0710993168.html,/homes/ayg/CS590C/www/mathematica/math .html (Basic tutorial)
?https://www.sodocs.net/doc/0710993168.html,/questions/4430998/mathematica-what-is -symbolic-programming (What is symbolic programming) ?http://www.cer.ethz.ch/resec/people/tsteger/Econ_Model_Math_1.
?https://www.sodocs.net/doc/0710993168.html,/enp/jjkelly (An introduction to Mathematica as well as some physics courses)
?Do you know of any web-based university course that is entirely Mathematica based?
?http://homepage.cem.itesm.mx/jose.luis.gomez/data/mathematica (Tutorials in Spanish)
?Mathematica programming(some examples of the various programming paradigms that can be used in Mathematica)
FAQ 常见问答
?https://www.sodocs.net/doc/0710993168.html,/my_notes/faq/mma_notes/MMA.htm (FAQ)
?https://www.sodocs.net/doc/0710993168.html,/questions/tagged/mathematica?sort=faq
&pagesize=15 (FAQ on Stack Overflow)
?https://www.sodocs.net/doc/0710993168.html,/questions?sort=faq (FAQ on this site)
?https://www.sodocs.net/doc/0710993168.html,/webMathematica/wiki/wiki.jsp?
pageName=FAQs
?https://www.sodocs.net/doc/0710993168.html,/conferences/conference98/Lichtblau/ SymbolicFAQ.nb (Symbolic FAQ)
Books 图书
?Stephen Wolfram's The Mathematica Book (online, version 5.2), available for free
?Mathematica programming: an advanced introduction (online) by Leonid Shifrin, available for free
?Tutorial Collection by WRI (lots of extra documentation and examples, available as free pdfs, also available and up-to-date in Help > Virtual Book in Mathematica).
?Mathematica Cookbook by Sal Mangano (O'Reilly, 2010)
?Mathematica in Action by Stan Wagon (Springer, 2010)
?Mathematica: A Problem-Centered Approach by RoozbehHazrat (Springer, 2010)
?Mathematica Navigator by HeikkiRuskeepaa (Academic Press, 2009) ?The MathematicaGuideBooks (for Programming, Numerics, Graphics, Symbolics) by Michael Trott (Springer, 2004-2005) ?An introduction to programming with Mathematica by Paul R. Wellin, Richard J. Gaylord and Samuel N. Kamin (Cambridge University Press, 2005); contains an example of Domain Specific Language (DSL)
creation.
?Mastering Mathematica by John W. Gray (Academic Press, 1997) ?Programming in Mathematica by Roman Maeder (Addison-Wesley Professional, 1997)
?Programming with Mathematica?: An Introduction by Paul Wellin (Cambridge University Press, 2013)
?Power Programming With Mathematica: The Kernel, by David B. Wagner (Mcgraw-Hill, 1997), out of print but scanned copy available here.
?https://www.sodocs.net/doc/0710993168.html,/2014/01/10/read-up-on-mathematica-in-m any-subjects
Wolfram Websites 官方网站
Learn
?https://www.sodocs.net/doc/0710993168.html,/broadcast/
?https://www.sodocs.net/doc/0710993168.html,/training/courses (Online video courses, most are free)
https://www.sodocs.net/doc/0710993168.html,/training/special-event/ (Links to videos of past conferences)
?Slides of seminars
?https://www.sodocs.net/doc/0710993168.html,/user/WolframResearch
Examples丰富的示例代码
?https://www.sodocs.net/doc/0710993168.html,
?How To-s
?https://www.sodocs.net/doc/0710993168.html,/mathematica/new-in-8
https://www.sodocs.net/doc/0710993168.html,/mathematica/new-in-9
https://www.sodocs.net/doc/0710993168.html,/mathematica/new-in-10/
https://www.sodocs.net/doc/0710993168.html,/training/special-event/new-in-mathemati
ca-10/
? A plot gallery for Mathematica 9
?https://www.sodocs.net/doc/0710993168.html,/language/
https://www.sodocs.net/doc/0710993168.html,/language/fast-introduction-for-programm
ers
https://https://www.sodocs.net/doc/0710993168.html,/datadrop/quick-reference/home/ Resources档案资料库
?https://www.sodocs.net/doc/0710993168.html,/mathematica/resources
?https://www.sodocs.net/doc/0710993168.html,/ (Great amount of resources here) ?https://www.sodocs.net/doc/0710993168.html,/kb/topic/mathematica (Knowledge base)
?https://www.sodocs.net/doc/0710993168.html,
?Help
Help > Virtual Book
?https://www.sodocs.net/doc/0710993168.html,/support/learn/
?https://www.sodocs.net/doc/0710993168.html,/books/
?https://www.sodocs.net/doc/0710993168.html,
?++Mathematica https://www.sodocs.net/doc/0710993168.html,/language
Blogs官方博客和微博
?https://www.sodocs.net/doc/0710993168.html,
?https://www.sodocs.net/doc/0710993168.html,
?https://www.sodocs.net/doc/0710993168.html,
?https://www.sodocs.net/doc/0710993168.html,
?https://www.sodocs.net/doc/0710993168.html,/#!/mathematicatip
Other related sites其他相关站点
?https://www.sodocs.net/doc/0710993168.html,
SMP
https://www.sodocs.net/doc/0710993168.html,/2013/06/there-was-a-time-before
-mathematicahttps://www.sodocs.net/doc/0710993168.html,/data/uploads/2013/0
6/SMPHandbook.pdf
?https://www.sodocs.net/doc/0710993168.html,
?Wolfram Science: the official site of Stephen Wolfram's New Kind of Science
NKS forum
Lecture notes from NKS summer schools
Programs from the notes
Demonstrations
?https://www.sodocs.net/doc/0710993168.html,/
?https://www.sodocs.net/doc/0710993168.html, (Some interactive basic math courses, useful for curious young people)
?https://www.sodocs.net/doc/0710993168.html,/webresources.html (other Mathematica related sites)
Virtual conferences虚拟会议
?https://www.sodocs.net/doc/0710993168.html,/events/virtual-conference/spring-2013?https://www.sodocs.net/doc/0710993168.html,/events/virtual-conference/2012
?https://www.sodocs.net/doc/0710993168.html,/events/virtual-conference/2011 Mathematica one-liner competition 在线竞赛
?https://www.sodocs.net/doc/0710993168.html,/events/techconf2010/competition.html
?https://www.sodocs.net/doc/0710993168.html,/events/technology-conference/2011/one-l iners.html
?https://www.sodocs.net/doc/0710993168.html,/training/special-event/mathematica-expe rts-live-one-liner-competition-2012
Wolfram technology conferences 会议
?2014, https://www.sodocs.net/doc/0710993168.html,/events/technology-conference/2014?2013, https://www.sodocs.net/doc/0710993168.html,/events/technology-conference/2013?2012, https://www.sodocs.net/doc/0710993168.html,/events/technology-conference/2012?2011, https://www.sodocs.net/doc/0710993168.html,/events/technology-conference/2011?2010, https://www.sodocs.net/doc/0710993168.html,/events/techconf2010
?2009, 2007, 2006, 2005, 2004, 2003, 2001, 1999, 1998, 1997, 1994, 1992
?https://www.sodocs.net/doc/0710993168.html,/infocenter/Conferences/
Blogs 博客
?https://www.sodocs.net/doc/0710993168.html,/
?https://www.sodocs.net/doc/0710993168.html,/ (Blog of YaroslavBulatov dedicated to graph theory)
?https://www.sodocs.net/doc/0710993168.html, (Blog of Nasser M. Abbasi)
?https://www.sodocs.net/doc/0710993168.html,/ (Various interesting demos by Matt Henderson)
?https://www.sodocs.net/doc/0710993168.html, (Blog of Mike Honeychurch) ?https://www.sodocs.net/doc/0710993168.html,/(Blog about data mining in texts)
?https://www.sodocs.net/doc/0710993168.html, (Blog partly about Mathematica)
?https://www.sodocs.net/doc/0710993168.html,/ (Lunchtime playground)
?https://www.sodocs.net/doc/0710993168.html, (Blog of Nilo de Roock) ?https://www.sodocs.net/doc/0710993168.html, (Blog about scientific programming languages)
?https://www.sodocs.net/doc/0710993168.html,/(Rip’s Applied Mathematics Blog, Mathematica is used to demonstrate various concepts of graduate level)
?https://www.sodocs.net/doc/0710993168.html,/ (Blog of Kris Carlson)
I liked this post: Functional-Procedural Fusion, this function is
useful for this style of programming:
MapEach[function_]:=(function/@#)& (for example:
{1,2}//MapEach[2 #&])
?https://www.sodocs.net/doc/0710993168.html,/
Personal websites 个人爱好者站点
?https://www.sodocs.net/doc/0710993168.html,/MathematiClub/ (Games, various interesting notebooks)
?https://www.sodocs.net/doc/0710993168.html,/BrainRot/ (Conversation with Theodore Gray one of the historical developer of Mathematica) ?https://www.sodocs.net/doc/0710993168.html,/ (Super Widget Package, can be interesting, but using Dynamic with built-in Mathematica GUI components is now easier)
?https://www.sodocs.net/doc/0710993168.html,/mathematica/code.html (Various interesting links to other packages)
?https://www.sodocs.net/doc/0710993168.html, (Mathematica users wiki)
?https://www.sodocs.net/doc/0710993168.html,/wiki/Main_Page(Knot Atlas, package available) ?https://www.sodocs.net/doc/0710993168.html,/resources(Some applications to mathematical finance, package working as an interface to Quantlib) ?https://www.sodocs.net/doc/0710993168.html,/~djmpark/Mathematica.html(Site of David
Park, Expression manipulation and some interesting packages) ?https://www.sodocs.net/doc/0710993168.html,(Site about finance, with some links to demonstrations)
?https://www.sodocs.net/doc/0710993168.html,/infocenter/Conferences/8073/ (A formal approach for modeling and simulation)
?https://www.sodocs.net/doc/0710993168.html,/ (Paul Nylander)
?https://www.sodocs.net/doc/0710993168.html,/groups/mathematica/videos (Some cool videos in 3D made with MM)
?https://www.sodocs.net/doc/0710993168.html,/noeckel/MathematicaGraphics.html (Creating an Post-Processing Mathematica Graphics (on Mac OS X)) ?https://www.sodocs.net/doc/0710993168.html, (Some interesting (non free) packages, like JavaTools which provides Mathematica links to
Scala, C# and F#)
?https://www.sodocs.net/doc/0710993168.html,/sierpinski.htm(Very cool pictures and math using Mathematica, code available)
?https://www.sodocs.net/doc/0710993168.html,/ (Useful Mathematica scripts among others. Connection with iTunes,Twitter, YouTube, Font
outlines,Mapping GPS Data are some of them)
?https://www.sodocs.net/doc/0710993168.html, (Package on natural language processing)
?http://www.familydahl.se/mathematica/ (Ingolf Dahl, package implementing several interpolation methods)
Calculus 微积分
?https://www.sodocs.net/doc/0710993168.html,/~richard.mercer/Calculus/Lab/Download/in dex.html (Various notebooks about mathematics)
?https://https://www.sodocs.net/doc/0710993168.html,/site/calcuplus/ (Introductory calculus course with CDF demos)
?https://www.sodocs.net/doc/0710993168.html,/infocenter/TechNotes/6111/
(Numerical analysis, code of a lot of common numerical methods) Resources on other languages其他语言
?Having used Mathematica as a "gateway" language, where to from here?
?Online interpreters can be found here Online-REPs-and-REPLs in order to test ideas in a lot of different languages.
https://c9.io (Cloud 9)
?This resource is very useful for common languages like Java, C++ or VBA
https://www.sodocs.net/doc/0710993168.html,
?C++ reference books
?Tutorials on a lot of current technologies (but Mathematica is
missing there ...) https://www.sodocs.net/doc/0710993168.html,
?https://www.sodocs.net/doc/0710993168.html,/wiki/Mathematica (Several algorithms in Mathematica together with other languages)
?https://www.sodocs.net/doc/0710993168.html,(Cheat sheets for a lot of languages) ?Wikibooks, can be handy for a quick reference on many languages.
?The Archive of Interesting Code
?dGSD, a great tool for organizing your projects and knowledge (Based on TiddlyWiki, a wiki stored in a single HTML file).
?Alternatives to Mathematica
?google-styleguide
Forums 论坛
MathGroup
?https://https://www.sodocs.net/doc/0710993168.html,/group/comp.soft-sys.math.mathematica https://www.sodocs.net/doc/0710993168.html,/Uwe/Forums.aspx/mathematica/201107/1
https://www.sodocs.net/doc/0710993168.html,/mathgroup/
Stack Exchange sites
?Mathematica questions on Stack Overflow
?Mathematica.StackExchange(look at "favorites" in the profiles of active users for reading past interesting answers) ?Mathematica questions on Stack Exchange: all questions tagged mathematica on the Stack Exchange Network
Links to some packages 各种程序包
?What third-party packages do you use?
?http://www.xact.es/index.html(Open source tensor package suite) also see this post Differential geometry add-ons for Mathematica
and Tensor analysis
?FeynRules by Neil D. Christensen, Claude Duhr& Benjamin Fuks (latest version: 1.6.0, package to calculate Feynman rules) ?Automatic physical units, by Jon McLoone (2010, available from the Wolfram Library Archive)
https://www.sodocs.net/doc/0710993168.html,/infocenter/MathSource/7655/?NCAlgebra (Non Commutative Algebra Package)
?Sym (Symmetry analysis of differential equations)
?Some packages written by Leonid Shifrin
https://www.sodocs.net/doc/0710993168.html,/additional_resources.html
https://https://www.sodocs.net/doc/0710993168.html,/lshifr
?Various links to common websites (Google, Amazon,...), financial data, and some useful functions for manipulating time series.
https://https://www.sodocs.net/doc/0710993168.html,/fmeinberg?tab=repositories
?RUBI (symbolic integration rules)
Packages for preparing publication-quality scientific figures ?LevelScheme by Mark Caprio (latest version: 3.52, Sep 2011, for Mathematica 6 and higher)
?Presentations and other packages by David Park (latest version:
25 Aug 2011)
? A WorkLife Framework by Scientific Arts LLC (extendable and customizable toolset that broadens Mathematica's scope across many aspects of daily work)
?FeynArts by Thomas Hahn (latest version: 3.7, 27 Mar 2012, package for generation and visualization of Feynman diagrams and
amplitudes)
?Writing and Publishing a Book with Mathematica by Paul R. Wellin (2005, available from the Wolfram Library Archive)
Useful non-free tools for development, deployment, distribution, linking, etc.需要付费的资源
?Workbench by WRI, current version: 2 (official Mathematica IDE based on Eclipse, great for projects involving several packages and generating integrated documentation)
?webMathematica by WRI, current version: 3 ("Deploy high-powered applications as interactive websites")
?gridMathematica by WRI, current version: 8 ("Easily control CPUs and GPUs to solve large problems fast")
?SystemModeler by WRI, current version: 3. Integrated symbolic modellingplatform.Some related links: MathModelicaPaper,
ObjectMath, Modelica, OpenModelica
How to do System dynamics simulations / diagrams in Mathematica?
?Finance Platform by WRI, current version: 1. Includes option pricing, risk analysis, enterprise system development, and
interactive reporting
?Mathematica Applications and Add-Ons (all add-ons marketed via WRI, related to Mathematica)
?Mathematica Link for Excel by Episoft, Inc., current version: 3.5
(links Microsoft Excel with Mathematica)
shareimprove this answer edited Mar 15 at 10:46
community wiki
================================华丽的分割线================================
My original post above became too big so I'm splitting it in two posts. The tips and tricks section that I used to keep track of interesting new posts mostly on this site was big so here it is.
Note the tags at the end of each question on this site in order to read related questions.
I'm further dividing this post. See my third answer on Advanced evaluation, patterns and neat algorithms below.
Tips and tricks 各种奇技淫巧
Top posts
?What are the most common pitfalls awaiting new users?
Programming paradigm change
?Can one identify the design patterns of Mathematica?
Mathematica style guide?
?What is in your Mathematica Toolbag
General tricks
?https://www.sodocs.net/doc/0710993168.html,/mathematica/tips/Tricks.html(Ted Ersek's tricks)
https://www.sodocs.net/doc/0710993168.html,/infocenter/MathSource/4557/ (Ted
Ersek's tricks as notebooks)
?https://www.sodocs.net/doc/0710993168.html,/pelican/pages/mathematica.html (Szabolcs' tricks)
?https://www.sodocs.net/doc/0710993168.html,/mathematica/tips/tips.html (Verbeia's tricks)
?https://www.sodocs.net/doc/0710993168.html,/~morrison/Mathematica/ (Tip sheets)
?As pointed out by Mr. Wizard, have a look at the available packages in your install directory for getting new ideas!
?Tricks of the trade in the Mathematica Journal
?How do I designate arguments in a nested map?
?Levels: how do they work?
?Elegant operations on matrix rows and columns
https://www.sodocs.net/doc/0710993168.html,/mathematica/tutorial/LinearAlgebra
MatrixAndTensorOperations.html
Tips for writing faster code 写出性能更佳的代码
?Performance tuning in Mathematica?
?https://www.sodocs.net/doc/0710993168.html,/2011/12/07/10-tips-for-writing-fast-mat hematica-code
?Programming for speed
?Benchmarking expressions (timeAvg function)
Profiling from Mathematica
?https://www.sodocs.net/doc/0710993168.html,/m_performance/m_performance .html
?https://www.sodocs.net/doc/0710993168.html,/infocenter/Conferences/7005/ (Principles of efficient MM programs, tips for writing fast code) ?https://www.sodocs.net/doc/0710993168.html,/infocenter/Conferences/320/ (Generally Efficient Numerics Programming in Mathematica) ?Writing Efficient Mathematica Code
?Dispatch tables
Using Mathematica's Dispatch
Efficiently extracting an array subset given a separate array
How to speed up calculations with large number of replacement rules?
(Experimental`OptimizeExpression)
?Alternatives to procedural loops and iterating over lists in Mathematica
Are there any cases when For[] loops are reasonable?
Alternatives to While Loops?
Is Table the only functional way to construct nested loops in
Mathematica?
How to transform Do loop to more efficient codes?
Replace For-loop with functional code
Equivalent Nested Loop Structure
"do ... while" loop equivalent in Mathematica
?Evaluate while external command is being run
?Use Internal`Bag as a linked list (see links in Data Structures paragraph below)
Reap and Sow 表达式中间状态跟踪
?Collecting Expressions during Evaluation
Reap and Sow can be used with StepMonitor
?SelectEquivalents (my personal favorite Mathematica function, basically it allows to define classes of equivalence and apply a
function on the elements of each class, using Reap and Sow)
How would you do a PivotTable in Mathematica?
?Reap and Sow can also be used for debugging, see for example how it's used in ShowIt
?How to collect result continuously (interruptible calculation) when running parallel calculations?
Compile编译相关
?How to compile effectively?
How well does Mathematica code exported to C compare to code
directly written for C? (Note the function
Experimental`OptimizeExpression)
Has this implementation of FDM touched the speed limit of
Mathematica?
?Note the instruction (< StringFreeQ[CompilePrint[cf], "MainEvaluate"]) to quickly verify that a compiled function doesn't need calls to external functions, available as from v8. ?https://www.sodocs.net/doc/0710993168.html,/mathematica/Compile/tutorial/Overv iew.html ?https://www.sodocs.net/doc/0710993168.html,/infocenter/Conferences/7968/ (Effective Use of the Mathematica Compiler and Code Generation) What kinds of uses of SymbolicC (introduced in Mathematica 8) do you foresee? Are there any neat examples already? ?List of compilable functions Why is there a huge performance gap using Map with more than 100 List entries ?Shaving the last 50 ms off NMinimize (Advanced tricks for a very fast Nelder-Mead optimizer) Minimization by Nelder-Mead Using Apply inside Compile Removing calls to MainEvalute when using inlined compiled closures ?JIT compilation Implementing a function which generalizes the merging step in merge sort A fast, robust DropWhile ?Retaining and reusing a one-to-one mapping from a sort ?Is it possible to use Compile on a function with optional arguments? ?Compiling more functions How can I compile this function Compiling more functions that don't call MainEvaluate ?Does Mathematica have a "MATLAB Builder" equivalent? https://www.sodocs.net/doc/0710993168.html,/mathematica/CCodeGenerator/tutoria l/CodeGeneration.html https://www.sodocs.net/doc/0710993168.html,/mathematica/CCodeGenerator/ref/Lib raryGenerate.html ?Third argument of Compile Using links to other languages 和其他语言通信 ?System interfaces and deployment ?Faster alternatives for DayOfWeek (Java) Working with Java hashmaps from Mathematica ?WSTP Convert C program using mathlink to WSTP Handling error in WSTP program C/C++ A MathLink tutorial Mathlink Mode article ?Minimal effort method for integrating C++ functions into Mathematica Converting other C++ classes to MTensor in LibraryLink How can I use shared libraries in LibraryLink code and ensure Mathematica will find them? How to understand shared passing of a MTensor in LibraryLink? Using Boost with CreateLibrary Write C/C++ programs calling Mathematica functions Example program for calling Mathematica from C Returning multiple results from a LibraryLink function Calling Mathematica from Visual Studio - example Which Distributions can be Compiled using RandomVariate Librarylink made easier to use with C++11 Calling Mathematica from Visual Studio - example ?How to use C language functions in Mathematica? ?Mathematica Functions from VBA ?Linking FORTRAN with Librarylink ?MATLink, link between Mathematica and Matlab Is it possible to export the equations from Mathematica to MATLAB? ?Calling IronPython code from Mathematica ?JLink JLink classes' reference Packed arrays ?https://www.sodocs.net/doc/0710993168.html,/infocenter/TechNotes/391/ (Packed Arrays) ?What is a Mathematica packed array? (instruction for being warned when a packed array is unpacked) Mathematica memory management for large arrays ?Avoid mixing integers with reals in a list, which would unpack the list and slow computations. For example use 1. instead of 1 if you have to enter manually an entry in a list of reals. ?Isn't it possible to Pack a SparseArray? Debugging 程序调试技术 ? A debug utility to print or extract intermediate data from a program (ShowIt, a useful variant of Print) ?The clearest way to represent Mathematica's evaluation sequence How do I evaluate only one step of an expression? ?Debugging memory leaks Profiling memory usage in Mathematica Preventing Mathematica from eating up all your memory (not a bug but the info can be useful) How to un-eat memory? ?Using the Mathematica debugger, and other interesting possibilities Debugging Mathematica Code ?DebugTrace ?How to program a F::argx message? How to abort on any message generated? Best practices in error reporting How to leave function unevaluated programmatically? Setting up diagnostic error messages in large Mathematica projects Dealing with errors and resuming the code after one Interrupting package evaluation, handling error Prepend Information to Warning Messages Need help with exception handling Error checking and trapping techniques with Throw and Catch Exception/Error handling using returns codes How do I get Mathematica to return a function call unevaluated? ?Wolfram Workbench has an integrated debugger and profiler. See here for posts about it on this site: https://www.sodocs.net/doc/0710993168.html,/questions/tagged/workbench Inspecting non-variable state from a breakpoint in Workbench workbench: how to see values of global variables during debugging ?Mathematica Debuggability ?Automatically generating a dependency graph of an arbitrary Mathematica function? Is there an analogue of the Variables command for general expressions? Data Structures实现各种数据结构 ?Data structures and efficient algorithms, from Daniel Lichtblau Kd-list, Kd-tree (applied to Gr?bner bases) ?Expr Bag Internal`Bag inside Compile Efficiently collecting results inside a compiled function Most efficient way of splitting a file into length-prefixed blocks ?Linked lists Mathematica “linked lists” and performance Merge-sort Variant of the cutting-stock problem in Mathematica The answer of Leonid is a reference on the topic: Can one identify the design patterns of Mathematica? Searching linked lists that contain lists? ?Efficient circular buffer? ?Efficient priority queues? ?BTree Using Memoization with a Mutable Object How to speed up Minimum Spanning Tree algorithm? How to make Huffman coding by using tree representation What is the fastest way to maintain a large set of expressions? Generating an ordered list of pairs of elements from ordered lists Objects 对象模型 ?How to create strong types? How to implement FittedModel like objects How can one define a custom data object? ?Struct equivalent in Mathematica? Implementing a dictionary data structure Is there HashTable structure in Wolfram Mathematica? Is there a method to enumerate the keys/values of System`Utilities`HashTable A usage example in the answer of Oleksandr R. or in the answer to the post How can you give a Module a context and have its local variables and Modules belong to that context? What's fastest way of defining 10^5 down values? Return a Part by Reference? (For accessing hierarchical / structured data) ?Once more on object orientation in Mathematica: does it have to be so hard? Builder pattern for complex objects construction ?OOP applied to manipulate ?Question about designing a particular data structure ?Code Readability and Object-Oriented Code ?Polymorphism and the pattern-matcher ?Inheritance in Mathematica using pattern matching on UpValues ?How to implement the Observer design pattern? ?https://www.sodocs.net/doc/0710993168.html,/Documentation6/guide/ReferenceGuide.ht ml (non free, but still can give some inspiration) Traversal of an expression 表达式遍历 ?How to perform a depth-first preorder traversal of an expression? ?How to perform a breadth-first traversal of an expression? ?Determining all possible traversals of a tree ?Uses for MapAll (//@) myHold examples with MapAll (1,2) ?Can a Trie be implemented efficiently? ?Concatenate strings from different levels of list Graphs 图论 ?GraphUtilities ?How to play with Facebook data inside Mathematica? Optional Values 关于系统函数的各种Options ?Functions with Options ?What are the benefits of switching from Rule and /. to OptionsPattern[] and OptionValue in a large application? ?How can I create a function with optional arguments and options? ?Can a function be made to accept a variable amount of inputs? ?Dealing with nested options ?Evaluation of OptionValue[] ?Custom functions by delegating options in a specific way and using core functions (withOptionConfiguration) ?Specifying optional arguments with variables ?Is there any way to define pure functions with optional arguments? Is it possible to separately call the default value of a variable which isn't at the end of the sequence? ?Why doesn't Mathematica use uniform criteria for validating Options? ?How to inherit options by scoping? ?How to make a function with its own options as well as passing options to other functions ?Using a held option Accessing data in different ways不同方式访问数据 ?Cases[data,Colon[key,_]] vs. Cases[data,key:_] toward XPath, XQuery ?Emulating R data frame getters with UpValues Creating a R DataFrame like construct in Mathematica How to achieve Set+Part like behaviour in custom Set function? ?The gap between MapAt and ReplacePart for 2D data tables with headers ?Data Table Manipulation in Mathematica Data Table Manipulation in Mathematica: Step 2 How can a big table be treated as a database? ?Path queries for tree-structured data ?Splitting up delimited data in lists Partitioning with varying partition size (dynamicPartition) Splitting a list by specifying section headers ?Accessing list elements by name Association/Dataset 关联数组，哈希表 ?How to make use of Associations? Normal // Association // Dataset workaround for some user query operators Can't get Append to append a record (association) to a dataset How can I save a Manipulate's control settings to a dataset? V10's Operator Forms - what are they good for? What is the natural way of using Dataset to get a FittedModel? Dataset: get number of levels and elements Is there a comprehensive list of functions with V10 operator forms? Converting structured data to a dataset How can I add a column into a existing Dataset? GroupBy several keys while keeping the Dataset as a Table (List of Associations) Learning Resources for Data Science in the Wolfram Language (many usage examples of List/Association/Dataset) SQL-like computation on Dataset: Aggregate function Is there a faster way to Map an Association? Difference between Association and Dispatch Changing Values in an Association using Map (Answer on Immutability) Rules and replacement (the backbone of Mathematica, more advanced) 规则与替换，这是mathematica语言的特性 ?Where definitions are stored ?What is the distinction between DownValues, UpValues, SubValues, and OwnValues? Getting ideas from Lisp 借鉴Lisp的思想 ?Structure and interpretation of computer programs, Common Lisp The Language, On Lisp, Practical Common Lisp ... ?repl.it has an online interpreter for Scheme with some examples. ?Is there a Mathematica/Lisp link? ?Other functional languages : Scala, Haskell, Erlang, Clojure, Caml, F#, Maude ... ?FAQ on functional programming Memoization/Caching内存与缓存 ?The answer of Leonid is a reference on the topic: Can one identify the design patterns of Mathematica? ?What does the construct f[x_] := f[x] = ... mean? ?Avoid repeated calls to Interpolation (Interesting answers and links about caching values and expressions) ?Function that caches when it returns unevaluated ?Why is there no PositionFunction in Mathematica? ?Built-in Mathematica data: are they cached? how to speed up the loading? Undocumented features 非官方的特性 ?What are some useful, undocumented Mathematica functions? ?How can one find undocumented options or option values in Mathematica? ?What is the complete list of valid Front End Tokens? ?How do you convert a string containing a number in C scientific notation to a Mathematica number? ?What is the complete list of valid FrontEnd Packet types? ?How can I get the unchanged Box form of an arbitrary expression? ?How can I manipulate TemporalData? ?How to work with Experimental`NumericalFunction? ?Full documentation for AppearanceElements ?How to create internally optimized expression for computing with high WorkingPrecision? ?What are all the named Mathematica styles? Mathematica functions and environment 运行时环境 ?How to specify Mathematica as a programming language? https://www.sodocs.net/doc/0710993168.html,/mathematica/quick-revision-history.html https://www.sodocs.net/doc/0710993168.html,/my_notes/compare_mathematica/index.htm How to get complete Documentation Center graph of guide pages? Mathematica函数及使用方法 (来源：北峰数模) --------------------------------------------------------------------- 注：为了对Mathematica有一定了解的同学系统掌握Mathematica的强大功能，我们把它的一些资料性的东西整理了一下，希望能对大家有所帮助。 --------------------------------------------------------------------- 一、运算符及特殊符号 Line1; 执行Line，不显示结果 Line1,line2 顺次执行Line1，2，并显示结果 ?name 关于系统变量name的信息 ??name 关于系统变量name的全部信息 !command 执行Dos命令 n! N的阶乘 !!filename 显示文件内容 < Expr>> filename 打开文件写 Expr>>>filename 打开文件从文件末写 () 结合率 [] 函数 {} 一个表 <*Math Fun*> 在c语言中使用math的函数 (*Note*) 程序的注释 #n 第n个参数 ## 所有参数 rule& 把rule作用于后面的式子 % 前一次的输出 %% 倒数第二次的输出 %n 第n个输出 var::note 变量var的注释"Astring " 字符串 Context ` 上下文 a+b 加 a-b 减 a*b或a b 乘 a/b 除 a^b 乘方 base^^num 以base为进位的数 lhs&&rhs 且 lhs||rhs 或 !lha 非 ++,-- 自加1，自减1 +=,-=,*=,/= 同C语言 >,<,>=,<=,==,!= 逻辑判断（同c） Mathematica函数大全--运算符及特殊符号一、运算符及特殊符号 Line1;执行Line，不显示结果 Line1,line2顺次执行Line1，2，并显示结果 ?name关于系统变量name的信息 ??name关于系统变量name的全部信息 !command执行Dos命令 n! N的阶乘 !!filename显示文件内容 a-b减 a*b或a b 乘 a/b除 a^b 乘方 base^^num以base为进位的数 lhs&&rhs且 lhs||rhs或 !lha非 ++,-- 自加1，自减1 +=,-=,*=,/= 同C语言 >,<,>=,<=,==,!=逻辑判断（同c） lhs=rhs立即赋值 lhs:=rhs建立动态赋值 lhs:>rhs建立替换规则 expr//funname相当于filename[expr] expr/.rule将规则rule应用于expr expr//.rule 将规则rule不断应用于expr知道不变为止param_ 名为param的一个任意表达式（形式变量）param__名为param的任意多个任意表达式（形式变量） 二、系统常数 Pi 3.1415....的无限精度数值 E 2.17828...的无限精度数值 Catalan 0.915966..卡塔兰常数 EulerGamma 0.5772....高斯常数 GoldenRatio 1.61803...黄金分割数 Degree Pi/180角度弧度换算 I复数单位 Infinity无穷大 50Mathematica线性代数运算命令与例题 第五章 线性代数运算命令与例题 线性代数中常用的工具是矩阵(向量)和行列式。用这些工具可以表示工程技术，经济工作中一些需要用若干个数量从整体上反映其数量关系的问题。用这些工具可以简明凝练而准确地把所要研究的问题描述出来，以提高研究的效率。在线性代数课程中我们看到了用这些工具研究齐次和非齐次线性方程组解的理论和解的结构，矩阵的对角化，二次型化标准形等问题的有力，便捷. 5.1向量与矩阵的定义 数学上矩阵是这样定义的: 由n m ?个数排成m 行n 列的数表 mn m m n n a a a a a a a a a Λ M M M Λ Λ21 2222111211 称为m 行n 列矩阵，特别，当m=1时就是线性代数中的向量。 记作: ????? ?? ?????=mn m m n n a a a a a a a a a A ΛM M M ΛΛ2122221 11211 两个n m ?矩阵称为同型矩阵。 线性代数中的运算对象是向量和矩阵，因此首先介绍向量和矩阵的输入。 5.1.1输入一个矩阵 命令形式1:Table[f[i,j]，{i ，m}，{j ，n}] 功能: 输入n m ?矩阵，其中f 是关于i 和j 的函数，给出[i ， j]项的值. 命令形式2:直接用表的形式来输入 功能:用于矩阵元素表达式规律不易找到的矩阵的输入。 注意: 1.Mathematica 是采用一个二重表的形式来表示矩阵的，即用 {{…},{…},…,{…}} 其中表中的每个表元素都是等长的一维表，第一 个表元素是矩阵的第一行，第二个表元素是矩阵的第二行，一般，第n 个表元素是矩阵的第n 行。要看通常的矩阵形式可以用命令: MatrixForm[%] 2. 对应上述命令形式，输入一个向量的命令为 Table[f[j]，{j,n}]或直接输入一个一维表{a1,a2,…,an}，这里a1,a2,…,an 是数或字母。 例题 例 1.输入矩阵A=???? ??????---41381639121458561203 12、向量 b={1,4,7,-3}。 解:Mathematica 命令 In[1]:= a={{12，-3，0，2，1}，{56，-8，-45，21，91}，{3，6，81，13，4}} Out[1]:= {{12，-3，0，2，1}，{56，-8，-45，21，91}，{3，6，81，13，4}} In[2]:=b={1, 4, 7, -3} Out[2]:= {1, 4, 7, -3} Mathematica的内部常数 Pi , 或π(从基本输入工具栏输入, 或“Esc”+“p”+“Esc”）圆周率π E (从基本输入工具栏输入, 或“Esc”+“ee”+“Esc”）自然对数的底数e I (从基本输入工具栏输入, 或“Esc”+“ii”+“Esc”）虚数单位i Infinity, 或∞(从基本输入工具栏输入, 或“Esc”+“inf”+“Esc”）无穷大∞ Degree 或°(从基本输入工具栏输入,或“Esc”+“deg”+“Esc”）度 Mathematica的常用内部数学函数 指数函数Exp[x]以e为底数 对数函数Log[x]自然对数，即以e为底数的对数 Log[a，x]以a为底数的x的对数 开方函数Sqrt[x]表示x的算术平方根 绝对值函数Abs[x]表示x的绝对值 三角函数 （自变量的单位为弧度）Sin[x]正弦函数 Cos[x]余弦函数 Tan[x]正切函数 Cot[x]余切函数 Sec[x]正割函数 Csc[x]余割函数 反三角函数ArcSin[x]反正弦函数 ArcCos[x]反余弦函数 ArcTan[x]反正切函数 ArcCot[x]反余切函数 ArcSec[x]反正割函数 ArcCsc[x]反余割函数 双曲函数Sinh[x]双曲正弦函数 Cosh[x]双曲余弦函数 Tanh[x]双曲正切函数 Coth[x]双曲余切函数 Sech[x]双曲正割函数 Csch[x]双曲余割函数 反双曲函数ArcSinh[x]反双曲正弦函数 ArcCosh[x]反双曲余弦函数 ArcTanh[x]反双曲正切函数 ArcCoth[x]反双曲余切函数 ArcSech[x]反双曲正割函数 ArcCsch[x]反双曲余割函数 求角度函数ArcTan[x，y]以坐标原点为顶点，x轴正半轴为始边，从原点到点（x，y）的射线为终边的角，其单位为弧度 数论函数GCD[a,b,c,．．．]最大公约数函数 LCM[a,b,c,．．．]最小公倍数函数 Mathematica函数大全一、运算符及特殊符号 Line1; 执行Line，不显示结果 Line1,line2 顺次执行Line1，2，并显示结果 ?name 关于系统变量name的信息 ??name 关于系统变量name的全部信息 !command 执行Dos命令 n! N的阶乘 !!filename 显示文件内容 < >,<,>=,<=,==,!= 逻辑判断（同c） lhs=rhs 立即赋值 lhs:=rhs 建立动态赋值 lhs:>rhs 建立替换规则 lhs->rhs 建立替换规则 expr//funname 相当于filename[expr] expr/.rule 将规则rule应用于expr expr//.rule 将规则rule不断应用于expr知道不变为止 param_ 名为param的一个任意表达式（形式变量） param__ 名为param的任意多个任意表达式（形式变量） 二、系统常数 Pi 3.1415....的无限精度数值 E 2.17828...的无限精度数值 Catalan 0.915966..卡塔兰常数 EulerGamma 0.5772....高斯常数 GoldenRatio 1.61803...黄金分割数 Degree Pi/180角度弧度换算 I 复数单位 Infinity 无穷大 -Infinity 负无穷大 ComplexInfinity 复无穷大 Indeterminate 不定式 三、代数计算 Expand[expr] 展开表达式 Factor[expr] 展开表达式 Simplify[expr] 化简表达式 FullSimplify[expr] 将特殊函数等也进行化简 PowerExpand[expr] 展开所有的幂次形式 ComplexExpand[expr,{x1,x2...}] 按复数实部虚部展开 FunctionExpand[expr] 化简expr中的特殊函数 Collect[expr, x] 合并同次项 Collect[expr, {x1,x2,...}] 合并x1,x2,...的同次项 Together[expr] 通分 Apart[expr] 部分分式展开 Apart[expr, var] 对var的部分分式展开 Cancel[expr] 约分 ExpandAll[expr] 展开表达式 ExpandAll[expr, patt] 展开表达式 FactorTerms[poly] 提出共有的数字因子 FactorTerms[poly, x] 提出与x无关的数字因子 FactorTerms[poly, {x1,x2...}] 提出与xi无关的数字因子 Coefficient[expr, form] 多项式expr中form的系数 图3.3里的Mathematica程序(版本8.0) 在Mathematica8.0的命令窗口输入如下命令：Cal[n_,i_]=2i*Binomial[n,i]; Print[ScientificForm[Cal[500,8]*1.0,4]]; Print[ScientificForm[Cal[500,12]*1.0,4]]; Print[ScientificForm[Cal[500,16]*1.0,4]]; Print[ScientificForm[Cal[500,24]*1.0,4]]; Print[ScientificForm[Cal[720,8]*1.0,4]]; Print[ScientificForm[Cal[720,12]*1.0,4]]; Print[ScientificForm[Cal[720,16]*1.0,4]]; Print[ScientificForm[Cal[720,24]*1.0,4]]; 按Shift+Enter组合键,运行可得： 2.344×1019 1.828×1027 3.75×1034 9.199×1047 4.41×1020 1.514×1029 1.381×1037 6.913×1051 图4.3.2.4.1里的Mathematica程序(版本8.0) 在Mathematica8.0的命令窗口输入如下命令：Cal[m_,n_]=Binomial[m,n]; Fun1[B_,m_,n_,ρ_]=((Cal[m,ρ]*Cal[B-m,n-ρ])/Cal[B,n]); X1=i 826Fun1104,26,26,i Cal i,80.267810.267i 8 ; X2=i 1226Fun1104,26,26,i Cal i,120.2671210.267i 12; X3=i 1626Fun1104,26,26,i Cal i,160.2671610.267i 16; X4=i 2426Fun1104,26,26,i Cal i,240.2672410.267i 24; Y1=i 846Fun1248,46,46,i Cal i,80.267810.267i 8; Y2=i 1246Fun1248,46,46,i Cal i,120.2671210.267i 12; Y3=i 1646Fun1248,46,46,i Cal i,160.2671610.267i 16; Y4=i 2446Fun1248,46,46,i Cal i,240.2672410.267i 24; Print[X1 "\n",X2 "\n",X3 "\n",X4 "\n",Y1 "\n",Y2 "\n",Y3 "\n",Y4 "\n"]; 按Shift+Enter 组合键,运行可得： 0.0000951929 3.92256×10-9 5.19076×10-15 7.7419×10-33 0.00104287 5.9339×10-7 3.76588×10-11 3.96973×10-22 附录B ：Mathematica 的基本应用 1. 什么是Mathematica Mathematica 是美国Wolfram Research 公司开发的通用科学计算软件，主要用途是科学研究与工程技术中的计算,这里介绍的是第6版（2008年更新为第7版）。由于它的功能十分强大，使用非常简便，现在已成为大学师生进行教学和科研的有力工具。它的主要特点有： 1)既可以进行程序运行，又可以进行交互式运行。一句简单的Mathematic 命令常常可以完成普通的c 语言几十甚至几百个语句的工作。例如解方程：x 4 + x 3 + 3x -5 = 0只要运行下面的命令： Solve[x^4+x^3+3 x-5 0,x] 。 2) 既可以进行任意高精度的数值计算，又可以进行各种复杂的符号演算，如函数的微分、积分、幂级数展开、矩阵求逆等等。它使许多以前只能靠纸和笔解决的推理工作可以用计算机处理。例如求不定积分：? x 4 e -2x dx 只要运行下面的命令： Integrate[x^4*Exp[2 x],x]。 3) 既可以进行抽象计算，又可以用图形、动画和声音等形式来具体表现，使人能够直观地把握住研究对象的特性。例如绘制函数图形：y = e -x /2 cos x , x ∈ [0, π]，只要运行下面的命令： Plot[Exp[x/2]*Cos[x],{x,0,Pi}]。 4) Mathematica 把各种功能有机地结合在一个集成环境里，可以根据需要做不同的操作，给使用者带来极大的方便。 2. Mathematica 的基本功能 2.1 基本运算及其对象 Mathematica 的基本数值运算有加法、减法、乘法、除法和乘(开)方，分别用运算符“+”、“-”、“*”、“/”和“^”来表示（在不引起误解的情况下，乘号可以省略或用空格代替），例 如2.4*3^2 -(5/(6+3))^(1/3)表示3236534.2）（+÷-?。小括号“（”和“）”作为表示运算优先顺 序的符号，用于组合运算；中括号用于命令和函数，大括号用于集合和列表。 Mathematica 的关系运算符有：>、<、>=、<=、!=、== 等，它们的意义与通常的数学语言相同，要注意“!=”表示不等于，双等号“==”表示等于。而单等号“=”和冒号等号“:=”表示定义或赋值，不表示相等。逻辑运算符主要有：！、&&、||，它们的意义与c 语言中相同，分别是“非”、“与”、“或”。 Mathematica 的基本数值运算对象有常数、变数和函数，包含整数，有理数、实数和复数等数值类型。为了方便，Mathematica 预先用符号表示了一些重要常数，如Pi 表示圆周率π，E 表示自然对数的底e = 2.17828…，I 表示虚单位i ，Infinity 表示无穷大∞等。比如说，E^(2*Pi*I)表示i e π2。 Mathematica 还预先定义了大量数学函数以供调用，调用格式为“函数名[自变量]”，预定义的函数名用大写字母开始的标识符表示，常用的有 第五章 数值分析和数值计算 1． 如何求插值多项式 给定n 个点( x i ,y i )，(i=1,2,…,n)，构造一个次数不超过n-1的多项式函数f(x)，使得f(x i )=y i ，则称f(x)为拉格朗日插值多项式。 可以证明该多项式函数由公式 ))...()(())...()((...) )...()(())...()(())...()(())...()((1211212321231113121321--------++------+------=n n n n n n n n n n x x x x x x x x x x x x y x x x x x x x x x x x x y x x x x x x x x x x x x y y 唯一给定。 Mathematica 提供了根据插值点数据计算拉格朗日插值多项式的函数InterpolatingPolynomial ，下面是其调用格式： InterpolatingPolynomial[data,var] 作出以data 为插值点数据，以var 为变量名的插值多项式。 例： 在多数情况下，我们构造插值函数的目的在于计算函数f(x)的值，而并不在意插值多项式的具体表示形式。对于拉格朗日插值多项式，当n 较大时，得到的高次插值多项式由于截断误差和舍入误差的影响，往往误差较大。此时在实际应用中，一般采用分段插值。 Mathematica 提供了分段插值函数Interpolation ，其使用格式为： Interpolation[data,InterpolationOrder->n] 这里InterpolationOrder->n 指定插值多项式的次数，默认值为3。此外数据data 中还可以包括插值点处的导数，格式为：{{x1,{y1,dy1}},{x2,{y2,dy2}},…} 例：已知f(0)=0,f(1)=2,f’(0)=1,f’(1)=1,求3次插值多项式f(x)，并计算f(0.72)和画出函数f(x)在[0,1]区间上的图形。 表达式： Plot[4 x - 9, {x, 0, 9}] f[x_] = x^3 Plot[f[x], {x, 0, 9}] a = Plot[4 x - 9, {x, 0, 9}] b = Plot[x^3, {x, 0, 3}] 两图画在一个坐标系 Show[a, b] a = Plot[4 x - 9, {x, 0, 9}] b = Plot[x^3, {x, 0, 3}] 两图画在一起（一排） c = GraphicsArray[{a, b}] Show[c] a = Plot[4 x - 9, {x, 0, 9}] b = Plot[x^3, {x, 0, 3}] c = GraphicsArray[{a}, {b}] 两图画在一起（两排） Show[c] 二维画图： Automatic 默认值 DisplayFunction -> Identity 不出现图 DisplayFunction -> $DisplayFunction 出现图 PlotRange -> All 画出所有点，指定区域点 PlotStyle -> {RGBColor[1, 0, 0]} 图像颜色 PlotStyle -> {Dashing[{0.01}]} 图像成虚线 PlotStyle -> {Thickness[0.01]} 图像粗细 AxesLabel -> {"x/t", "y/cm"} 坐标标签 PlotLabel -> {"s-t"} 图像标签 Frame -> True 图像边框 Axes -> {True, True} 坐标轴的显示 AxesOrigin -> {0, -5} 设置坐标原点 GridLines -> {{-π, -π/2, 0, π/2, π}, {-1，-0.5，0, 0.5, 1}} 给坐标轴分网格 TextStyle -> {FontSize -> 30} 坐标字体大小AspectRatio -> Automatic 坐标比例一致 Ticks -> {{0, 1, 2, 3}, {0,10,20}} 在坐标轴上显示特定点ParametricPlot[x(t),y(t)},{t,0,6,}] 画参数方程 实验九数据的曲线拟合 一、实验目的与要求 学会利用Mathematica软件对已知数据进行拟合处理，并针对拟合结果的图形显示分析拟合函数的优劣 二、实验的基本知识 熟知一些曲线及其方程 三、实验的具体内容 例1现有一组实测数据 解输入数据表 L={{0,0.3},{0.2,0.45},{0.3,0.47},{0.52,0.50},{0.64,0.38},{0.7,0.33},{1.0,0.24}} 由于假设用一元二次函数拟合，因而经验函数表为{1 , x , x^2} 键入f=Fit[L,{1, x , x^2},x] 为观察拟合情况，我们在一个图上画出数据点和拟合函数，键入 ListPlot[L,PlotStyle→{RGBColor[0,1,0],PointSize[0.04]}] Plot[f,{x , -0.2 , 1.2}] Show[%,%%] 或键入fp= ListPlot[L,PlotStyle→{RGBColor[0,1,0],PointSize[0.04]}] gp= Plot[f,{x , -0.2 , 1.2}] Show[fp,gp] 运行可得拟合函数为0.33129+0.596026x-0.71812x2，并且从图形中可以观察拟合的结果，若散点图与曲线拟合不够理想，可以考虑用更高次的多项式或其它函数进行拟合。 例 2 在某化学反应里，由实验得到生物的浓度与时间的关系如下，求浓度与时间关系的拟合曲线 t（分） 1 2 3 4 5 6 7 8 y 4 6.4 8.0 8.4 9.28 9.5 9.7 9.86 t（分）9 10 11 12 13 14 15 16 y 10.0 10.2 10.32 10.42 10.5 10.55 10.58 10.6 解为确定拟合函数的类型，可先在直角坐标系中作出散点图，键入 t1={{1,4},{2,6.4,{3,8.0}},{4.8.4},{5,9.28},{6,9.5},{7,9.7},{8,9.86},{9,10.0},{10,10.2}, {11,10.32},{12,10.42},{13,10.5},{14,10.55},{15,10.58},{16,10.6}} t2=ListPlot[t1,PlotStyle→{RGBColor[0,1,0],PointSize[0.04]}] 若用四次多项式进行拟合，则键入 t3=Fit[t1,Table[x^I,{I,0,4}],x] t4=Plot[t3,{x,0,17},PlotStyle→{RGBColor[1,0,0]}] Show[t2,t4] 运行后，可得拟合函数的表达式以及散点图与拟合函数图，从图中可见二者的吻合情况是否满意。此例中，亦可用对数函数进行拟合，为此键入 t5=Fit[t1,{log[x],1},x] 【Mathematica 简介】 Mathematica 软件是由沃尔夫勒姆研究公司（Wolfram Research Inc.）研发的。Mathematica 版发布于1988年6月23日。发布之后，在科学、技术、媒体等领域引起了一片轰动，被认为是一个革命性的进步。几个月后，Mathematica 就在世界各地拥有了成千上万的用户。今天，Mathematica 已经在世界各地拥有了数以百万计的忠实用户。 Mathematica 已经被工业和教育领域被广泛地采用。实际上，Mathematica 负责将高级的数学和计算引入了传统上非技术的领域，极大的增加了科技软件的市场。一个包含应用、咨询、书籍、和课程软件的行业支持着国际化的 Mathematica 用户群，这个行业还在不断地膨胀。随着沃尔夫勒姆研究公司不断地扩大和 Mathematica 的使用被不断地扩展到不同的领域，将会看到 Mathematica 在全世界范围内对未来产品、重要研究发现、和教学的巨大影响。 数学软件是现在科研工作者的必备的工具，个人比较喜欢用Mathematica，因为它是最接近数学语言的。Mathematica 在15日发布，其最显著的变化是允许自由形式的英文输入，而不再需要严格按照Mathematica语法，这类似于Wolfram|Alpha搜索引擎。Mathematica 8允许用户按照自己习惯的思考过程输入方程式或问题，最令人激动的部分是软件不是逐行执行命令，而是能理解上下文背景。 1. Enter your queries in plain English using new free-form linguistic input 2. Access more than 10 trillion sets of curated, up-to-date, and ready-to-use data 3. Import all your data using a wider array of import/export formats 4. Use the broadest statistics and data visualization capabilities on the market 5. Choose from a full suite of engineering tools, such as wavelets and control systems 6. Use more powerful image processing and analysis capabilities 7. Create interactive tools for rapid exploration of your ideas 8. Develop faster and more powerful applications Wolfram Research 的 CEO 和创立者斯蒂芬·沃尔夫勒姆表示：“传统上，让计算机执行任务必须使用计算机语言或者使用点击式界面：前者要求用户掌握它的语法；而后者则限制了可访问函数的范围。”“自由格式语言学能够理解人类的语言，并将其转化为具有特定语法结构的语言。这是产品适用性上的一个突破。 Mathematica 8 是这种创新思想下的第一个产品，但是它已经能够大幅度提高用户的工作效率。” Mathematica简明教程 第1章Mathematica概述 运行和启动：介绍如何启动Mathematica软件，如何输入并运行命令 第8章Mathematica中的常用函数8.1 运算符及特殊符号 Linel 执行Line，不显示结果 Linel,line2 顺次执行Line1，Line2，并显示结果 ?name 关于系统变量name的信息 ??name 关于系统变量name的全部信息 !command 执行Dos命令 N! N的阶乘 !!filename 显示文件内容 < -Infinity 负无穷大 Complexlnfinity 复无穷大 Indeterminate 不定式 8.3 代数计算 Expand[expr] 展开表达式 Factor[expr] 展开表达式 Simplify[expr] 化简表达式 FullSimplify[expr] 将特殊函数也进行化简PowerExpand[expr] 展开所有的幂次形式ComplexExpand[expr,{x1，x2…}] 按复数实部虚部展开FunctionExpand[expr] 化简表达式中的特殊函数 Collect[expr,x] 合并同次项 Collect[expr,{x1，x2，…}] 合并x1，x2,...的同次项 Together[expr] 通分 Apart[expr] 部分分式展开 Apart[expr,var] 对var的部分分式展开 Cancel[expr] 约分 ExpandAll[expr] 展开表达式 ExpandAll[expr,patt] 展开表达式 FactorTermsrpoly] 提出共有的数字因子 FactorTerms[poly，x] 提出与x无关的数字因子 FactorTerms[poly,(x1,x2…)] 提出与xi无关的数字因子 Coefficient[expr,form] 多项式expr中form的系数 Coefficient[expr,form，n] 多项式expr中form^n的系数 Exponent[expr,form] 表达式expr中form的最高指数 Numerator[expr] 表达式expr的分子 Denominator[expr] 表达式expr的分母 ExpandNumerator[expr] 展开expr的分子部分 8.4 解方程 Solve[eqns,vats] 从方程组eqns中解出Vats Solve[eqns,vats,elims] 从方程组eqns中削去变量elims，解出vats DSolve[eqn,y,x] 解微分方程，其中、y是x的函数 DSolve[{eqnl,eqn2,…}，{y1，y2…},] 解微分方程组，其中yi是x的函数DSolve[eqn,y,{x1,x2…}]解偏微分方程 Eliminate[eqns，Vats] 把方程组eqns中变量vars约去SolveAlways[eqns，vars] 给出等式成立的所有参数满足的条件Reduce[eqns，Vats] 化简并给出所有可能解的条件LogicalExpand[expr] 用&&和，，将逻辑表达式展开InverseFunction[f] 求函数f的反函数 Root[f,k] 求多项式函数的第k个根 教师指导实验6 实验名称：简单数理统计 一、问题：求样本数据的特征数字值；绘制样本的频率分布条形图和直方图。 二、实验目的： 学会使用Mathematica求求样本数据的极差、中位数、均值、方差及标准差；绘制样本的频率分布直方图并作简单的修饰。 三、预备知识：本实验所用的Mathematica命令提示 1、SampleRange[data] 求样本数据data的极差（最大数减最小数）； Median[data] 求样本数据data的中位数； Mean[data] 求样本数据data的均值； 2、VarianceMLE[data] 求样本数据data的方差； StandardVarianceMLE[data] 求样本数据data的标准差； 3、BarChart[data1, data2,…] 分别绘制样本数据data1,data2,…的条形图 图形修饰选项： BarSpacing 设置两条形的总宽度，设置值是实际宽度相对于区间宽度的比值； BarGroupSpacing 设置相邻条形的宽度，设置值是条形的实际宽度相对于条形的总宽度的比值； BarStyle 条形风格设置；BarEdgeStyle 条形边界风格设置； BarLabels 条形标签设置，PlotLabel 图形名称设置， 4、Histogram[data] 绘制样本数据data的频率分布直方图 图形修饰选项： Ticks设置标记相对于条形的位置； HistogramScale 设置条形高度为频率密度，使条形的面积和为所设置的值。 四、实验的内容和要求： 1、取50个在1到20的随机整数，求这组数的极差、中位数、均值、方差及标准差； 2、对以上数据绘制样本频率分布直方图； 3、data1={1, 3, 4, 5, 3.5, 3}, data2={3, 2, 5, 3}，在同一图表中绘制data1和data2的条形图，并作一定的修饰。 五、操作提示 1、取50个在1到20的随机整数，求这组数的极差、中位数、均值、方差及标准差； In[1]:=< Mathematica for Windows 常用用法 一、Mathematica 的主要功能 Mathematica 是美国Wolfram 公司开发的一个功能强大的计算机数学系统，提供了范围广泛的数学计算功能，主要包括三个方面：符号演算、数值计算、图形。例如：多项式的四则运算、展开、因式分解，有理式的各种计算，有理方程、超越方程的解，向量和矩阵的各种计算，求极限、导数、极值、不定积分、定积分、幂级数展开式，求解微分方程，作一元、二元函数的图形等等。 二、Mathematica 的基本知识 1．输入表达式：直接输入一个表达式（包括算式和命令，长表达式用“Enter ”换行）后，按“Shift+Enter ”执行，执行后以“Out[命令序号]= ……”形式输出执行结果，输出的结果可在后续的表达式中使用。 若命令后有分号，则不输出执行结果（图形输出与Print 命令除外）。 “%”表示上一个输出，“%%”表示倒数第2个输出，“%ｉ”表示第ｉ个 命令的输出。 2．运算符：+、-、*、/、^ ，“*”可用空格代替，“^”表示乘方。 如：In[1]:=2^10，输出为“Out[1]= 1024”，其中“In[1]:=”不需要输入。 In[2]:=3+5，Out[2]= 8；In[3]:=%-2，Out[3]= 6； In[4]:=%2+4，Out[4]= 12； In[5]:=1/3-1/4，Out[5]=12 1 ；In[6]:=N[%]，Out[6]= 0.0833333； In[7]:=N[%5+12,10]，Out[7]= 12.08333333（注意字母的大小写） 3．变量赋值：变量=表达式，“x=.”或Clear[x] 表示清除对x 的赋值。 表达式/.t ->c ，将表达式中的t 全替换成c 。?x ，查x 信息。 4．常用的数学常数：Pi (π)、E(e)、Infinity (∞)、I (1-) 5．常用的数学函数：Abs, Sin, Cos, Tan, Cot, ArcSin, Log (自然对数), Sqrt, Exp 如：In[1]:=Sqrt[2]+1；In[2]:=Sin[2]+ArcSin[1]；In[3]:=Exp[2]+% （自变量用［ ］括，区分大小写，首字母大写） 三、常用运算 1．多项式运算：In[1]:= (2+4*x^2)*(1-x)^3 或 In[1]:= t = (2+4*x^2)*(1-x)^3 （将右端表达式赋值给t ）； In[2]:=a=t/.x->4 (计算表达式t 当x=4时的值，并赋值给变量a ) In[3]:=a=. (清除变量a ) In[3]:=Expand[t]（展开）；In[4]:=Factor[%]（把上一个结果因式分解） 2．解方程：In[1]:=Solve[x^2+3*x = = 2]；In[2]:=N[%]； In[3]:=Solve[a*x-b= = 0, x]； In[4]:=NSolve[{x-2*y= =0,x^2-y= =1},{x,y}]（解方程组并得到数值解） 3．自定义函数：In[1]:= f [x_ ]:=x^2+2*x ； In[2]:=f[5]+7； In[3]:=f[a+b] 4．求极限：In[1]:=Limit[Sin[x]/x, x ->0]； In[2]:=Limit[(1+1/n)^n, n->Infinity]，Out[2]=E 5．求（偏）导数：In[1]:=D[a*x^2+3, x]；In[2]:=D[x^2+y^3-Sin[2*y], y]（对y 的偏导数）； In[3]:=D[Log[x], {x,2}] (求对x 的二阶导数)； In[4]:=D[Sin[x+y]*Exp[z*y^2],x,y] （求对x 、y 的二阶混合偏导数）； In[5]:=Simplify[%] (对前一结果化简)； In[6]:=D[Sin[x+y]*Exp[z*y^2],{x,2},{y,3}] 6．求不定积分：In[1]:=Integrate[x^2,x]；In[2]:=Integrate[1/(x^2+a^2),x] 7．定积分：In[1]:=Integrate[x^2, {x,0,1}]；In[2]:=Integrate[x^2,{x,a,b}]； In[3]:=Integrate[x^2+y^2, {x,0,a},{y,0,b}]；（求矩形域上的二重积分） In[4]:=Integrate[1, {x,-1,1},{y,-Sqrt[1-x^2],Sqrt[1-x^2]}]；Out[4]=Pi （圆面积） 8．幂级数展开：In[1]:=Series[Exp[x],{x,0,4}]（在x=0处展开到x 的四次幂） 9．矩阵的输入和输出：In[1]:= a ={{1,2},{3,4}}（定义一个2x2的矩阵a ，按 行写）； In[2]:=MatrixForm[a]（输出为矩阵形式）；In[3]:=Transpose[a]（a 的转置）； In[4]:=a[[2]]（a 的第2行）；In[5]:=Tanspose[a][[2]]（a 的第2列）； In[6]:=Inverse[a]（求a 的逆矩阵）；In[7]:=Det[a]（矩阵的行列式）； In[8]:=Eigenvalues[a]（求特征值）；In[9]:=Eigenvectors[a]（求特征向量）； In[10]:=RowReduce[a]（把a 化为阶梯形，可用于求矩阵的秩、判断线性相关性）； In[11]:= b ={{5,6,7},{8,9,10}}；In[12]:= a .b （矩阵a 与b 的乘积） 10．解线性方程组： In[1]:= a ={{3,4,5,6},{6,8,10,12},{4,5,6,7},{5,6,7,8}}；（a 的秩为2） In[2]:= b ={1,2,3,5}（列向量）；（增广矩阵的秩也为2） In[3]:=LinearSolve[a,b]（求线性方程组ax=b 的一个特解）； In[4]:=NullSpace[a]（求线性方程组ax=0的一个基础解系）； In[5]:= x =k1%4[[1]]+k2%4[[2]]+%3（ax=b 的全部解，k1、k2为任意常数） 11．求和：In[1]:=NSum[Sin[n]/n^3,{n,1,Infinity}]（求级数∑ ∞=13sin n n n 的和） 12．求极小值：In[1]:=FindMinimum[Sin[x]*Cos[x],{x,0.5}]（求函数在0.5附 近的极小值）； In[2]:=FindMinimum[Sin[x*y]*Exp[x^2],{x,0.2}, {y,0.3}]（求多元函数极小值） 13．求解线性规划问题：Min cx ，mx ≥b ，x ≥0，求向量x 。 In[1]:= c ={2,-3}（列向量）；In[2]:= m ={{-1,-1},{1,-1},{1,0}}； In[3]:= b ={-10,2,1}； In[4]:=LinearProgramming[c,m,b] 14．数据拟合：In[1]:= d ={{1,2.18},{1.2,2.56},{1.6,3.0},{1.8,2.66}}； In[2]:= f =Fit[d,{1, x, x^2}, x]（求和上面4个点吻合最好的二次多项式f ）； 检验效果：In[3]:=ListPlot[d]（画d 中4个点的图）； In[4]:=Plot[f,{x,0.8,2.0}]（画多项式f 在x 从0.8到2.0之间的图）； In[5]:=Show[%3, %4]（把上面两个图画在一起） 注：函数集{1, x, x^2}可以是更高次的或其它函数集，如三角函数集等。 15．一元函数作图：In[1]:=Plot[Exp[-x^2]*Sin[6*x],{x,-2,2}]（如图1） 参数方程作图：In[2]:=ParametricPlot[{Sin[t]^3,Cos[t]^3},{t,0,2*Pi}] 16．二元函数作图：In[1]:=Plot3D[Sin[x*y],{x,-Pi, Pi},{y,-Pi, Pi}]；（如图2） In[2]:=Plot3D[Sin[x*y],{x,-Pi, Pi},{y,-Pi, Pi},PlotPoints->40, ViewPoint->{2,-3,2}] In[3]:=ParametricPlot3D[{Cos[u]*Cos[v],Sin[u]*Cos[v],Sin[v]},{u,0,2*P i},{v,-Pi/2,Pi/2}] 17．数据画图：In[1]:= d ={{1,2},{3,4},{7,6}}；In[2]:=ListPlot[d]； In[3]:=ListPlot[d, PlotStyle->{RGBColor[1,0,0], PointSize[0.02]}]（红色 的大点）； 或直接用 In[4]:=ListPlot[{1,2},{3,4},{7,6}] 代替“In[2]:=”。 18．作图范围：In[1]:=Plot[x-x^3/6,{x,-4,4}]； In[2]:=Plot[x-x^3/6,{x,-4,4},PlotRange->{-5,2}]（限定纵坐标（函数值）范围） 19．图形组合：In[1]:=Plot[{Sin[x],Cos[x]},{x,0,2*Pi}]；或 In[2]:= g1=Plot[Sin[x],{x,0,2*Pi}, PlotStyle->{RGBColor[1,0,0]}]； In[3]:= g2=Plot[Cos[x],{x,0,2*Pi}, PlotStyle->{RGBColor[0,0,1]}]； In[4]:=Show[g1,g2]（把g1、g2画在一起） 20．文件的使用：In[1]:= y =25；In[2]:= a ={{1,4},{2,6}}；In[3]:= f [x_ ]:=x^2 ； In[4]:= g =Plot[Sin[x],{x,0,2*Pi}, PlotStyle->{RGBColor[1,0,0]}]； In[5]:=Save[“abc .m”,a,y,f,g]（将a, y, f, g 保存在文件“abc .m ”中，扩 展名为m ）； In[6]:=!!abc .m （显示文件内容）； In[1]:=<Mathematica函数及使用方法
Mathematica函数大全(内置)
50Mathematica线性代数运算命令与例题
Mathematica的常用函数
Mathematica函数大全
Mathematica程序举例1
附录B：Mathematica的基本应用b
(完整版)Mathematica数值分析和数值计算
Mathematica常用指令
Mathematica软件进行拟合
Mathematica使用教程
Mathematica中的常用函数命令
Mathematica数学实验——简单数理统计
Mathematical常用功能大全-精简版
相关文档
- mathematica常用命令大全
- mathematica函数大全
- mathematica最常用的方法一览
- Mathematica命令符集
- Mathematica函数大全(内置)
- mathematica_函数表
- Mathematica完美教程-从入门到精通
- Mathematica使用教程
- 手册：mathematica命令大全
- 个人整理Mathematica 画图命令集
- 汇总mathematica数学常用命令大全
- mathematica命令大全
- Mathematic命令大全
- Mathematica画图函数命令大全
- mathematica命令大全
- Mathematica常用指令
- Mathematica命令快速参考
- Mathematica教程3
- mathematica命令大全
- Mathematica函数大全