Mathematica 10.0.0新特性抽查 - stereohomology的专栏
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A random walk through Mathematica 10
Introduction
Mathematica 10 was released yesterday amid the usual marketing storm we’ve come to expect for a new product release from Wolfram Research. Some of the highlights of this marketing information include
Without a doubt, there is a lot of great new functionality in this release and I’ve been fortunate enough to have access to some pre-releases for a while now. &There is no chance that I can compete with the in-depth treatments given by Wolfram Research of the&&in Mathematica 10 so I won’t try. &Instead, I’ll hop around some of the new features, settling on those that take my fancy.
Multiple Undo
I’ve been a Mathematica user since version 4 of the product which was released way back in 1999. &For the intervening 15 years, one aspect of Mathematica that always frustrated me was the fact that the undo feature could only undo the most recent action. I,
along with many other users, repeatedly asked for multiple undo to be implemented but were bitterly disappointed for release after release.
Few things have united Mathematica users more than the need for multiple undo:
Finally, with version 10, our hopes and dreams have been answered and multiple undo is finally here. To be perfectly honest, THIS is the biggest reason to upgrade to version 10. Everything else is just tasty gravy.
Key new areas of functionality
Most of the things considered in this article are whimsical and only scratch the surface of what’s new in Mathematica 10. &I support Mathematica (and several other products) at the University of Manchester in the UK and so I tend to be interested in things
that my users are interested in. Here’s a brief list of new functionality areas that I’ll be shouting about
Entities – a new data type for things that Mathematica knows stuff about
One of the new functions I’m excited about is&&that pulls down map data from Wolfram’s
servers and displays them in the notebook. &Obviously, I did not read the manual and so my first attempt at getting a map of the United Kingdom was
GeoGraphics[&United Kingdom&]
What I got was a map of my home town, Sheffield, surrounded by a red cell border indicating an error message
The error message is “United Kingdom is not a Graphics primitive or directive.”&&The practical upshot of this is that GeoGraphics is not built to take strings as arguments. &Fair enough, but why the map of Sheffield?
&Well, if you call GeoGraphics[] on its own, the default action is to return a map centred on the GeoLocation found by considering your current IP address and it seems that it also does this if you send something bizarre to GeoGraphics. &In all honesty, I’d
have preferred no map and a simple error message.
In order to get what I want, I have to pass an&&that represents the UK to the GeoGraphics
function. &Entities are a new data type in Mathematica 10 and, as far as I can tell, they formally represent ‘things’ that Mathematica knows about. There are several ways to create entities but here I use the new&&function
From the above, you can see that Entities have a special new&&but their&&looks
straightforward enough. One thing to bear in mind here is&that all of the above functions require a live internet connection in order to work. &For example, on thinking that I had gotten the hang of the Entity syntax,
I switched off my internet connection and did
mycity = Entity[&City&, &Manchester&]
During evaluation of In[92]:= URLFetch::invhttp: Couldn't resolve host name. &&
During evaluation of In[92]:= WolframAlpha::conopen: Using WolframAlpha requires internet connectivity. Please check your network connection. You may need to configure your firewall program or set a proxy in the Internet Connectivity tab of the Preferences dialog. &&
Out[92]= Entity[&City&, &Manchester&]
So, you need an internet connection even to create entities at this most fundamental level. &Perhaps it’s for validation? &Turning the internet connection back on and re-running the above command removes the error message but the thing that’s returned isn’t
in the new StandardForm:
If I attempt to display a map using the mycity variable, I get the map of Sheffield that I currently associate with something having gone wrong (If I’d tried this out at work,in Manchester, on the other hand, I would think it had worked perfectly!). &So, there
is something very wrong with the entity I am using here – It doesn’t look right and it doesn’t work right – clearly that connection to WolframAlpha during its creation was not to do with validation (or if it was, it hasn’t helped). &I turn back to the Interpreter
In[102]:= mycity2 = Interpreter[&City&][&Manchester&] // InputForm
Out[102]//InputForm:= Entity[&City&, {&Manchester&, &Manchester&, &UnitedKingdom&}]
So, clearly my guess at how a City entity should look was completely incorrect. &For now, I think I’m going to avoid creating Entities directly and rely on the various helper functions such as Interpreter.
What are the limitations of knowledge based computation in Mathematica?
All of the computable data resides in the&&which is a new name for the data store used by Wolfram
Alpha, Mathematica and many other Wolfram products. In his recent blog post, Stephen Wolfram says that they’ll soon release the&& which will allow large scale access to the Knowledgebase and indicated that&‘basic versions of&Mathematica&10 are just set up for small-scale data access.’&&I
have no idea what this means and what limitations are in place and can’t find anything in the documentation.
Until I understand what limitations there might be, I find myself unwilling to use these data-centric functions for anything important.
IntervalSlider – a new control for Manipulate
I’ll never forget the first time I saw a Mathematica&&– it was at the&&when version 6 was still in beta. A Wolfram employee created a fully functional, interactive graphical user interface with just a few lines of code in about 2 minutes –I’d never seen anything like it and was
seriously excited about the possibilities it represented.
8 years and 4 Mathematica versions later and we can see just how amazing this interactive functionality turned out to be. It forms the basis of the&&which currently has 9677 interactive demonstrations covering dozens of fields in engineering, mathematics and science.
Not long after Mathematica introduced Manipulate, the&&introduced a similar function called interact. The interact function
had something that Manipulate did not – an interval slider (see the interaction called&‘Numerical integrals with various rules’&at&&for
an example of it in use). This control allows the user to specify intervals on a single slider and is very useful in certain circumstances.
As of version 10, Mathematica has a similar control called an IntervalSlider. &Here’s some example code of it in use
Manipulate[
pl1 = Plot[Sin[x], {x, -Pi, Pi}];
pl2 = Plot[Sin[x], {x, range[[1]], range[[2]]}, Filling -& Axis,
PlotRange -& {-Pi, Pi}];
inset = Inset[&The Integral is approx & && ToString[
NumberForm[
Integrate[Sin[x], {x, range[[1]], range[[2]]}]
, {3, 3}, ExponentFunction -& (Null &)]], {2, -0.5}];
Show[{pl1, pl2}, Epilog -& inset], {{range, {-1.57, 1.57}}, -3.14,
3.14, ControlType -& IntervalSlider, Appearance -& &Labeled&}]
and here’s the result:
Associations – A new kind of brackets
Mathematica 10 brings a new fundamental data type to the language, associations. As far as I can tell, these are analogous to&&or&&since they consist of key,value pairs. &Since Mathematica has already used
every bracket type there is, Wolfram Research have had to invent a new one for associations.
Let’s create an association called scores that links 3 people to their test results
In[1]:= scores = &|&Mike& -& 50.2, &Chris& -& 100.00, &Johnathan& -& 62.3|&
Out[1]= &|&Mike& -& 50.2, &Chris& -& 100., &Johnathan& -& 62.3|&
We can see that the Head of the scores object is Association
In[2]:= Head[scores]
Out[2]= Association
We can pull out a value by supplying a key. For example, let’s see what value is associated with “Mike”
In[3]:= scores[&Mike&]
Out[3]= 50.2
All of the usual functions you expect to see for dictionary-type objects are there:-
In[4]:= Keys[scores]
Out[4]= {&Mike&, &Chris&, &Johnathan&}
In[5]:= Values[scores]
Out[5]= {50.2, 100., 62.3}
In[6]:= (*Show that the key &Bob& does not exist in scores*)
KeyExistsQ[scores, &Bob&]
Out[6]= False
If you ask for a key that does not exist this happens:
In[7]:= scores[&Bob&]
Out[7]= Missing[&KeyAbsent&, &Bob&]
There’s a new function called Counts that takes a list and returns an association which counts the unique elements in the list:
In[8]:= Counts[{q, w, e, r, t, y, u, q, w, e}]
Out[8]= &|q -& 2, w -& 2, e -& 2, r -& 1, t -& 1, y -& 1, u -& 1|&
Let’s use this to find something out something interesting, such as the most used words in the classic text, Moby Dick
In[1]:= (*Import Moby Dick from Project gutenberg*)
MobyDick =
Import[&http://www.gutenberg.org/cache/epub/2701/pg2701.txt&];
(*Split into words*)
words = StringSplit[MobyDick];
(*Convert all words to lowercase*)
words = Map[ToLowerCase[#] &, words];
(*Create an association of words and corresponding word count*)
wordcounts = Counts[words];
(*Sort the association by key value*)
wordcounts = Sort[wordcounts, Greater];
(*Show the top 10*)
wordcounts[[1 ;; 10]]
Out[6]= &|&the& -& 14413, &of& -& 6668, &and& -& 6309, &a& -& 4658,
&to& -& 4595, &in& -& 4116, &that& -& 2759, &his& -& 2485,
&it& -& 1776, &with& -& 1750|&
All told, associations are a useful addition to the Mathematica language and I’m happy to see them included. &Many existing functions have been updated to handle Associations making them a fundamental part of the language.
s/Mathematica/Wolfram Language/g
I’ve been programming in Mathematica for well over a decade but the language is no longer called ‘Mathematica’, it’s now called ‘The Wolfram Language.’ &I’ll not lie to you, this grates a little but I guess I’ll just have to get used to it. &Flicking through
the documentation, it seems that a global search and replace has happened and almost every occurrence of ‘Mathematica’ has been changed to ‘The Wolfram Language’
This is part of a huge marketing exercise for Wolfram Research and I guess that part of the reason for doing it is to shift the emphasis away from mathematics to general purpose programming. &I wonder if this marketing push will increase the popularity of The
Wolfram Language as measured by the&?&Neither ‘Mathematica’ or ‘The Wolfram
Language’ is listed in the top 100 and last time I saw more detailed results&.
Fractal exploration
One of Mathematica’s competitors,&, had a new release recently which saw the inclusion of a set of&. Although I found this a fun and interesting addition to the product, I did think it rather an odd thing to do. &After all, if any software vendor is stuck for functionality to implement, there is a whole host of things to do that
rank higher in most user’s list of priorities than a function that plots a standard fractal.
It seems, however, that both Maplesoft and Wolfram Research have seen a market for such functionality. &Mathematica 10 comes with a set of functions for exploring the Mandelbrot and Julia sets. The Mandelbrot set alone accounts for at least 5 of Mathematica
10′s 700 new functions:-&,&,&,&and&.
MandelbrotSetPlot[]
I found this more fun than is reasonable! &Mathematica can generate and recognize bar codes and QR codes in various formats. &For example
BarcodeImage[&&, &QR&]
Scanning the result using my mobile phone brings me right back home :)
Unit Testing
A decent unit testing framework is essential to anyone who’s planning to do serious software development. Python has had one for years, MATLAB got one in 2013a and now Mathematica has one. &This is good news! &I’ve not had chance to look at it in any detail,
however. For now, I’ll simply nod in approval and&.
Opinions welcomed.
Disappointments in Mathematica 10
There’s a lot to like in Mathematica 10 but there’s also several aspects that disappointed me
No update to RLink
Version 9 of Mathematica included integration with R which excited quite a few people I work with. Sadly, it seems that there has been no work on RLink at all between version 9 and 10. &Issues include:
This lack of work on RLink really is a shame since the original release was a very nice piece of work.
If the combination of R and notebook environment is something that interests you, I think that the current best solution is to use the&.
No update to CUDA/OpenCL functions
Mathematica introduced&but very little
appears to have been done in this area since. In contrast, MATLAB has improved on its CUDA functionality (it has never supported OpenCL) every release since its introduction in 2010b and is now superb!
Accelerating computations using GPUs is a big deal at the University of Manchester (my employer) which has a&made
up of around 250 researchers. Sadly, I’ll have nothing to report at the next meeting as far as Mathematica is concerned.
FinancialData is broken (and this worries me more than you might expect)
I wrote&&that used the FinancialData function and it suddenly stopped working because of some
issue with the underlying data source. In short, this happens:
In[12]:= FinancialData[&^FTAS&, &Members&]
Out[12]= Missing[&NotAvailable&]
This wouldn’t be so bad if it were not for the fact that an example given in Mathematica’s own documentation fails in exactly the same way! The documentation in both version 9 and 10 give this example:
In[1]:= FinancialData[&^DJI&, &Members&]
Out[1]= {&AA&, &AXP&, &BA&, &BAC&, &CAT&, &CSCO&, &CVX&, &DD&, &DIS&, \
&GE&, &HD&, &HPQ&, &IBM&, &INTC&, &JNJ&, &JPM&, &KFT&, &KO&, &MCD&, \
&MMM&, &MRK&, &MSFT&, &PFE&, &PG&, &T&, &TRV&, &UTX&, &VZ&, &WMT&, \
but what you actually get is
In[1]:= FinancialData[&^DJI&, &Members&]
Out[1]= Missing[&NotAvailable&]
For me, the implications of this bug are far more reaching than a few broken examples. &Wolfram Research are making a big deal of the fact that Mathematica gives you access to computable data sets, data sets that you can just use in your code and not worry
about the details.
Well, I did just as they suggest, and it broke!
I’ve had a lot of fun playing with Mathematica 10 but that’s all I’ve really done so far – play – something that’s probably obvious from my choice of topics in this article. Even through play, however, I can tell you that this is a very solid new release with
some exciting new functionality. Old-time Mathematica users will want to upgrade for multiple-undo alone and people new to the system have an awful lot of toys to play with.
Looking to the future of the system, I feel excited and concerned in equal measure. There is so much new functionality on offer that it’s almost overwhelming and I love the fact that its all integrated into the core system. I’ve always been grateful of the
fact that Mathematica hasn’t gone down the route of hiving functionality off into add-on products like MATLAB does with its numerous toolboxes.
My concerns center around the data and Stephen Wolfram’s comment&‘basic versions of&Mathematica&10 are just set up for small-scale data access.’& What does this mean?
What are the limitations and will this lead to serious users having to purchase add-ons that would effectively be data-toolboxes?
Have you used Mathematica 10 yet? If so, what do you think of it? Any problems? What’s your favorite function?
Mathematica 10 links
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Mathematica 10.0.1 超强大的科学计算软件 最新破解版
Mathematica是一款非常强大的具有划时代意义的科学计算软件。
它很好地结合了数值和符号计算引擎、图形系统、编程语言、文本系统和与其他应用程序的高级连接，很多功能在相应领域内处于世界领先地位，截至2009年，它也是为止使用最广泛的数学软件之一。人们常说，Mathematica的发布标志着现代科技计算的开始。
Mathematica是世界上通用计算系统中最强大的系统。在1988由美国科学家斯蒂芬&沃尔夫勒姆领导的沃尔夫勒姆研究公司（位于美国伊利诺伊州香 槟市）开发，软件名字&Mathematica&是由苹果创办人乔布斯向沃尔夫勒姆公司创立者斯蒂芬&沃尔夫勒姆提议的，并且沿用至今。发布以来，它已经 对如何在科技和其它领域运用计算机产生了深刻的影响。
Mathematica的基本概念是用一个连贯的和统一的方法创造一个能适用于科技计算各个方面的软件系统，拥有强大的数值计算和符号运算能力。实现这一 点的关键之处是发明了一种新的计算机符号语言。这种语言能仅仅用很少量的基本元素制造出广泛的物体，满足科技计算的广泛性。这在人类历史上还是第一次。
Mathematica适用于从事理论研究的数学工作者和其它科学工作者、以及从事实际工作的工程技术人员。可以用于解决各种领域的涉及复杂的符号计算和 数值计算的问题。它代替了许多以前仅仅只能靠纸和笔解决的工作，这种思维和解题工具的革新可能对各种研究领域和工程领域产生深远的影响。
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Mathematica教程进阶篇；一．Mathematica中的表（★好好看，很重；（1.以下用小麦表示mathematica；◆1.1什么是表，以及表的结构；在小麦里，形如“{{}，{}}”的结构表示“表”；成的一个整体；一共有三个元素；前面这个表的第二个元素是x；a={{1,2,3}，{4,5,6}，{7,8,；构可以很复杂，感兴趣的可以自己尝试构造一些
Mathematica教程进阶篇一．Mathematica中的表（★好好看，很重要★）。（1.以下用小麦表示mathematica。2.下面提到的任何一个函数都需要你实际操作一次。）◆1.1什么是表，以及表的结构。在小麦里，形如“{{
}”的结构表示“表”（Table）。表是一些表达式聚集成的一个整体。这些表达式称为这个表的元素。例如2是表{2，x，y}的一个元素，这个表一共有三个元素。表的元素是有序的，依次分别被叫做表的第一个元素，第二个元素……。前面这个表的第二个元素是x。由于表的元素还可以是表（表的内部可以嵌套子表，比如：a={{1,2,3}，{4,5,6}，{7,8,9}}，其中{1,2,3}是表a的元素，但其本身仍然是一个表，表的结构可以很复杂，感兴趣的可以自己尝试构造一些复杂的表），人们也把上面所说的表元素称为表的第一层元素。作为一种特殊情况，一个表可以没有元素，这样的元素成为“空表”，空表只有一个，就是{}。（空表有其特殊的作用，比如建立一个空表来盛放符合条件的数据）。◆1.2理解表的作用。表可以作为表示结构，小麦系统本身就这样使用了表，比如它表示平面或者空间点的坐标。另外表在系统内部还被用于表示向量、矩阵和集合。比如：
biao1 = {{1, 2}, {3, 4}, {;4 ,☆表示四个坐标：（1,2）、（3,4）、（4,5）、（6,7）。尝试用ListPlot[
]函数绘制这四个点坐标：输入：ListPlot[biao1]就可以得到散点图。☆表示矩阵：biao1看做矩阵的话，就是一个4行2列的矩阵。尝试用MatrixForm[
]函数将biao1用矩阵的形式输出。（可以等价使用biao1//MatrixForm）。☆表示向量：一层的表用于表示向量，这个是非常直截了当的。例如{1,2,3}表示空间向量（1,2,3）。◆1.3如何生成表。诚然通过键盘一个一个输入可以构造出需要的表，但是在数据量很大的时候工作量就很大你懂得，挨个输入就显得鸡肋了。小麦同样提供了生成表的众多函数，比如Range，Table等。我们先来学习下这两个函数的功能和用法（另外感兴趣的可以通过帮助系统学习Array,DiagonalMatrix,IdentityMatrix等函数的功能和用法）。☆Range函数。Range[最小值，最大值，步长]尝试用Range函数生成{1,2,3.，…,99,100}；☆Table函数。Table函数相对与Range函数功能更强，用法有两种。Table[表达式,{变量,最小值,最大值，步长}]；Table[表达式,{变量1,最小值,最大值},{变量2,最小值,最大值}]下面对Table函数给出两个实例：（1）生成1至12的所有偶数。 （2）生成形如（1…8,11…18,21…28,31…38）的表。 ☆常用函数Import和Export 在实际问题中（尤其是建模竞赛），附件经常会提供Excel文件，小麦调入的Excel或者 txt文档同样也是一个表。你可以尝试在F盘新建一个名为“123”的Excel表格，打开表格在里面的单元格随机输入几个数字并保存。调用Import函数导入这个表。（Excel文档后缀为xls） 同样可以用Export函数导出处理好的表。 这时候在F盘就可以找到名为“biao1”的Excel文档了。◆1.4如何使用表和如何获取表的信息。对于已经生成的表，在使用中常常有取出表中的某几个元素，去除表中的某几个元素，添加元素等操作。我们定义biao2={{1,2,3}，{4,5,6}，{7,8,9}}，这是一个三行三列的表。▲利用：biao2[[a,b]]的结构提取biao2第a行第b个元素另外形如biao2[[a]]的结构则取出biao2中第a行的所有元素。☆AppendTo和PrependToAppendTo函数用于将值添加到表的后面，我们定义biao3={1,2,3，4}；将5添加到biao2中。 PrependTo函数则恰恰相反，它的作用是将值添加到表的前面，你可以尝试将0添加到biao3的前面。AppendTo和PrependTo函数都有很广的应用，在处理数据时，常常建立一个空表，将数据中符合要求的值添加到空表里面进行进一步的处理。☆Take和Drop函数。Take函数用于取出表中的元素，Drop函数则用于删除表中的元素。这两个函数需要你运用小麦的帮助系统，U know也就是自学。对于一个表（尤其是Excel导入的表），常常需要知道这些表的一些信息，最基本的就是表的“长度”（行数）。使用Length函数可以很方便的做到这点，我们继续使用F\盘刚刚刚刚创建的123.xls文档。 可以看到biao4的长度为1，但是你或许觉得biao4是三行三列的，它的长度应该为3,实际上我们处理的时候也希望得到3。小麦没犯错误，你也没有，在导入Excel表的时候，Excel中的数据往往被放置到一个“大表”里面来输出，我们期望得到的是“大表”的一个元素，即“大表”的第一行。所以以后在导入Excel表的时候要有提出表第一个元素的习惯这时候输出的表则去除了最外层的{}；我们得到的Length变成了3。（如果不理解，可以把它记住）。★表的应用举例。上面提到小麦中的表可以看做是坐标，向量，矩阵等，问题中的很多信息和规律则蕴藏在表中需要我们去挖掘。我们简单看下常用的线性拟合函数Fit（另外还有FindFit函数）在处理表时候的用法。建立：biao5={{1,0.5}，{2,2}，{3,4.5}，{4,8}，{5,12.5}，{6,18}}；☆Fit函数线性拟合使用方法为：Fit[表达式,函数,变量] （Fit函数中data则为biao5）我们尝试拟合biao5为一元二次函数，则有： Fit函数输出了拟合的函数方程，另外通过绘制这个方程的图形和数据的散点图可以比较拟合的优劣，如图可看到拟合的还是不错的。实际上在给出biao5中的数据时不是随机给的（拟合的那么好也就不足为怪了），而是根据y=(x^2)/2给出的，拟合出的函数前面的两个量由于数值很小，可以忽略近似为c=0.5^2; 此处仅为配合介绍表的作用来扼要提出Fit函数的应用举例，在后面的学习中还会详细的学习功能更强大的函数。二．小麦程序设计。★2.1模块（Module，Block，感兴趣的可以自学）。★2.2条件结构。☆If，While，Which函数。条件结构用于执行编程流程中，执行一个判断的过程，并决定下一步走向。比如：简单的程序 If语句执行规则为：If[条件,对,错,其他情况]解释：首先判断条件的对和错，正确则执行“对”对应得语句，“错”则执行“错”对应的语句。其他情况则执行“其他情况”，多数时候省略最后一项。在执行上述程序时候，首先判断a&b的对错情况，显然错误，则执行“错”对应的语句“Print[a+2]”最后输出三。我们看到If[]语句中用2-3个逗号来分隔几个块，每个块里执行一个命令。其实程序可以执行多条语句，每条语句中需要“;”间隔开。 执行“条件”判断时候，小麦同样有与（&&）或（||）等判断规则。具体使用原则可以自行查找资料学习。例1：输出小于4的正整数解释：首先将n的初值置为1，然后进入while循环，首先判断n=1是否小于4，判断的结果为真，所以执行打印语句Print，显示1。接着将n的值进行累加一个1（n++等效于n=n+1）。又重新回到While的开头，判断n是否小于4，结果为真，继续打印当前的n的值。如此循环，直到n的值不满足小于4这个约束条件为止。循环结束★Which语句的使用请查看帮助文档自学。2.3循环结构。★For循环。比如： 同样在执行“过程”那块时候，可以用“；”执行多个命令，这是很常用的。我们不妨将“步长表达式”做空，将其放置到最后： 来计算1+2+3+…+99+100；For循环的“过程”同样可以执行If[]语句用于判断，这样的嵌套使用可以达到筛选需要数据的效果。以上内容，讲的比较笼统，需要大家具体多实践才会真正理解和掌握。我们下面通过一个具体的程序来总结以上内容：biao8={1,2，-8,10,22,77，-86，-95，-1,2,4,5,6,7,8,12，-8，-6，-7，-5,1，-8，-2,0,1}；问题是找出biao8中的所有负数，置于biao9中；作业：1.生成如下表：a= {1,4,7,10，…，100}b={{1,2,3,4,5,6,7,8,9},{11,12,13,14,15,16,17,18,19}..{}…{91,92,93,94,95,96,97,98,99}}c={{10,20,30…90},{11,21,31…91},{12,22,32,42,,,92},….{}..{19,29,39,…,99}}2.计算1到100中所有偶数的和。（可以用多种方法）3.Flatten函数可以将多行多列的表抹平为只有一行的表，Sort函数可以将表内的元素按某种规则排序，查看Flatten和Sort的帮助，学习其用法。并将下表中的元素按 从
大到小顺序排列。(下句复制到小麦中运行可以得到一个随机的10X10的表)biao=Table[RandomInteger[10],{10},{10}]4．PrimeQ[]；函数用于判断函数是否为质数（素数），试编程找出1000内的所有质数，置与表“biao2”中； 包含各类专业文献、中学教育、幼儿教育、小学教育、外语学习资料、应用写作文书、10mathematica教程1等内容。
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