IS: Useful Mathematica Functions (75.html)

In the examples below:
- the following sequences of letters each refer to one keyboard key not multiple keys
Esc, F1, F3, F7, F12, Alt, shift, ctrl, Enter
- iff means "if and only if"

After purchasing a $35 Raspberry computer in Feb 2015, David Cole, the author of this article, found it very easy to use it to surf the web. But he also discovered the free working copy of the Mathematica programming language on the Raspberry. David has been a computer programmer (and electrical engineer) since 1966. This article is a notebook recording his early interactions with this programming language.

The basic Mathematica is similar to other programming languages, but it differs in many important ways. The primary difference is that it is based on Functions. Such functions can minimally be used as mathematical functions. Each function creates an object after accepting an object as input. Anyone familiar with UNIX or Linux, notices that Mathematica is a major extension of the UNIX command line. In fact, the basic concept of "piping results from one function to the next" carries through into Mathematica. But the differences between Mathematica and its peers are immense. Back in the 60's when Iverson created his innovative APL programming language, goal was to make programs more concise. But they were also difficult to read (or understand) and they required a special ball on Selectric typewriter terminals. Some say that a trip is usually more about the voyage than the destination. When a program is being written, creation of a working algorithm is all-important. A program's comments and documentation are also important, but even more important is the mental process that was in the programmer's mind. Mathematica is less concise than APL, but the opportunity to follow the evolution or development of a program is fostered by Mathematica's concept of the "notebook".

Mathematica is more than a mere programming language. The "code" is a notebook in which one can type notes describing ongoing research, as in working thesis notes. Many typsetting features enable sentences and paragraphes to include mathematical expressions, equations etc. Mathematica can be used to list data, to evaluate the data on-the-spot by using the "actual data" (not data similar to it) through program segments. The user can write programs (within the notebook), solve equations, cite (and verify) references and much more. Mathematica can be used to draw figures and to create objects such as graphs, images and figures programatically. Resulting objects can be static or dynamic. Expressions are the main construct in Mathematica. Such expressions can be procedural, can contain logic and can contain formulae, numbers (integers, decimal fractions and complex numbers). In Mathematica, 1/0 produces Infinity, and 1/Infinity produces 0. Each expression is a sequence of functions that can even be recursive.

Interaction with Mathematica is primarily in the form of questions and answers. Each question is posed in the form of an expression eg. 23 * 17^2 followed by the Shift-Return key. Mathematica then evaulates the expression. If the expression ends with a ";" the result is not output to the screen for the user to see. Mathematica then evaluates the subsequent expression in the same way. After processing all the expressions in the notebook, Mathematica waits for the user to tell it what expression(s) to evaluate next. At this point, the user composes more expressions. A program's statements, per se, do not exist. Instead, we have a series of expressions written in the notebook. The results of such a Mathematica session are usually stored in one or more named notebooks on the user's storage device such as a hard drive or flash drive.

The author of Mathematica, Stephan Wolfram, has created a hierarchy of three engines:
-Wolfram - an interactive programming language which manipulates numbers and strings (c1980)
-Mathematica - an interactive programming language containing Wolfram and adding image functionality
-Wolfram|Alpha - a WebSite that answers questions. It uses the following:

Mathematica

An engine to understand questions that are posed in English

Data located elsewhere on the Internet.

This document restricts itself to notes about Mathematica, specifically Mathematica running under Raspian on the Raspberry Pi 2 B computer platform. More articles cited or written by David Cole can be found at ePC Articles by Cole .

The first group is a list of Built-In Mathematica Functions (not just Mathematica-9 functions) that I think might be useful to me. Special System Variables (Functions not needing arguments) are also listed. After this, some useful Mathematica Commands are listed, then an explanation of some Special Operator Coding Syntax for Mathematica. Finally a number of one-liners that a new coder must know. This is followed by a list of Related References.

Mathmatica uses a large collection of symbols in its lexicon. Rather than forcing its users to learn how to use a new keyboard, many constructs are included. These constructs permit the user to choose which method is best to write Mathematica code. But this does lead to some confusion. Hopefully this confusion is short-lived for every Mathematica novice. For example, the operator named "Rule" can be produced in any of the following ways:

x->4.

x\[Rule]4.

xEsc->Esc4.

Rule[x,4.]

Copy from another expression

Copy from another notebook

Not all operators can be input in all these ways. The user must use trial & error to discover which to use. Note that x==4. no longer works. It has been deprecated (removed) from having this meaning. The "==" now means something else. References to old Mathematica programs may still use this old construct. Such programs need to be identified before they are used. Special precautions must be taken to avoid such temporal problems. Fortunately, this rarely happens after a programming language has been in use for a long time.

Useful Mathematica Functions

( * indicates na [not available . . . yet] )

create a list of numbers from 1 to 5EdgeDetect[object]

Discipline...	Name	Description
All	Cases[n,n..._p ]	Returns cases that match the pattern
All	Now	Returns current time e.g. Wed 2 Apr 2014 13:34:45
All	Timing [e]	Returns elapsed time, value of e
All	SortBy[list,f ]	Sorts the elements by applying f to each element
Asm.	ByteArray[s ]	Converts a Base64 string to a byte array
Crypt.	Encrypt [ ]	Encrypts something using a key/code
Dict.	DictionaryLookup[s]	Looks up words using patterns
Editting	TextWords[ ]	Break into a list of words
Info.	WikipediaSearch*[s ]	Structured search for content in Wikipedia
Linux	UnixTime[date]	Converts Date/Time to seconds since 1970 A Jan 01
Math.	MinMax[l,d ]	Returns the min & max values padded by d
Math.	NestWhileList[f,e,?]	repeats f starting with e till test:? fails
Math.	Solve[{e,e},x]	Solves a list of equationsfor x
Math.	With[{x=x0..},e]	evaluates e with x=x0 etc
Math.	Range[n ]	Range[5]
Stats.	TTest[{d,d...}]	Probability that the mean (ie Average) is zero
Strings	StringJoin[s,s,s ]	Joins a list of strings
Strings	StringPartition[s,n ]	Partitions a string into a list of ..
Tables	Table[x^2,{x,3}]	create a Table with 1,4,9 in it.
Vision(IS)	EdgeDetect[j]	Creates an object showing edges
Vision(IS)	ImageIdentify[ ]	Tries to identify what the image is
	[ ]

Useful Mathematica Commands

Discipline...	Name	Description
All	Clear[]	clears previous definition of variables
All	Clear[f]	clears previous definition of function f
All	Clear[Global`*]	clears all previous Global definitions
Coding	ctrl-6	superscript
Coding	?Name	Look Up "Name" in Documentation
Coding	Print[s]	Output s to window pane just prior to next [Out-nn]
Coding	Print[s,s]	Output s,s to window pane
Coding	%	Previous Output Result
Coding	%n	Output Result number n
Coding	(* *)	These enclose comments
Coding	Word`	Context Names must end with `
Coding	SetAttributes[f,Listable]	function is defined by a list
Coding	FullForm["a"<>"b"]	Prints a string enclosed in double quotes
Coding	FullForm[2Pi]	Prints Times[2,Pi] (* showing the implied multiplication *)
Math.	InputArray(in menu)	accepts array elements from the user
Math.	Solve[{x^2-1==0},x]	Solves x^2-1 for x [iff x is undefined]
Math.	Solve*[{Equal[x^2-1,0]},x]	Solves x^2-1 for x
NoteBook	Print[$OperatingSystem]	Prints "Unix" for example

Voiced Equivalent of Mathematica Syntax (under Raspbian)

** indicates deprecated syntax (either "no longer valid" or "now has a different meaning")
!! indicates Warning: this similar construct does not have the same meaning
GT is voiced as Greater Than
LT is voiced as Less Than
EQ is Voiced as Equal To

Name....	Syntax	Voiced Equivalent	Example	Voiced As
Alias	Esc Esc	escaped	Esc->Esc	Escaped "->" is the alias of the Rule Operator
All	FullForm	print the full form of	FullForm[2.Pi]	print the full form of 2.Pi prints 6.283185307179586`
Anonymous Function	f[&]	see Pure Function		Can be piped (see Piping)
Apply	@@	apply to	Plus @@ {a,b,c}	Plus apply to list ie a+b+c
Arguments	_	1 argument	f[x_]	function f with only 1 argument
Arguments	__	1 or more arguments	f[x__]	function f with 1 or more arguments
Arguments	___	0, 1 or more arguments	f[x___]	function f with 0, 1 or more arguments
Assign.	:=	set equal to (deferred)	x:=b	x is set equal to b (deferred)
Assign.	=	set equal to	x=b	x is set equal to b
Assign.	=	set equal to	x=5.	x is set equal to 5.
Assign.	=	is given the value of	x=5.	x is given the value of 5.
Assign.	!	not	x!<y	x not less than y
Code	Alt-/	Uncomment	Alt-/	Uncomment a statement-removes (* & *)
Concat.	<>	concatenated to	a$<>b$	a$ concatenated to b$
Condition	>=	is GT or EQ to	y>=x^2-3	y is GT or EQ to x^2-3
Condition	<=	is LT or EQ to	y<=x^2-3	y is LT or EQ to x^2-3
Cond.	/;	such that	f[r_ /; r <a]:=n*r	function of r such that r<a is equal to deferred n times r
Full Name	\[ ]	full name of	\[Rule]	\[Rule] is the full name of the Rule operator.
Function	f[x_]=	f is the result of x being substituted in	f[x_]=Sin[x]	f is the result of x being substituted in Sin[x]
Iteration	+=	increment by then test after	i += 1	begin with i=first, increment by 1 then test for completion
Help	F1	Find Selected Function	Select Sin then F1	Wolfram Doc center explains Sin
Help	shift-F1	any symbol	shift-F1 then (# in search documentation box)	shift F1 explains any single symbol (#-first arg of a pure function)
Help	F3	Find Next	F3	Finds next occurance of current search string
Help	shift-F3	Find Previous	shift-F3	Finds previous occurance of cureent previous search string
Help	F7	Evaluate in Subsession	F7	?
Help	F12	Toggle Full Screen	F12	Toggles Full Screen Mode on/off
Iter.	{x,a,z,d}	iterated	{x,1,7,2}	x iterated from 1 to 7 by 2
Math.	==	is equal to	y==x^2-3	y is equal to x^2-3
Math.	Equal[ ]	is equal to	Equal[x^2-1,0]	x^2-1 is equal to 0
Math.	Equal[]	is equal to	Equal[y,x^2-3]	y is equal to x^2-3
Math.	Esc==Esc	is equal to	yEsc==Escx^2-3	y is equal to x^2-3
Math.	I	imaginary I	3+I*2	3 plus imaginary I times 2
Patterns	"d"	defined string	DictionaryLookup["sphinx"]	Searches for "sphinx"
Patterns	~~	joined to	DictionaryLookup["so"~~"x"]	Searches for "sox"
Patterns	___	any string	DictionaryLookup["x"~~___~~"x"]	Finds "xerox"
Patterns	y	variable name	y="x";DictionaryLookup[y~~___~~y]	Finds "xerox"
Patterns	_pattern	match a pattern	_Integer	match an Integer
Piping	//	piped to	Sin[.01] // ArcSin	Sin[.01] piped to ArcSin ie ArcSin[Sin[.01]]
Pure Function	f[&]	is a pure function	f[&]=Sin[#]	f is a pure function of Sin
Rule	->	goes to	y->x^2-3	y goes to x^2-3
Rule	Esc->Esc	goes to	yEsc->Escx^2-3	y goes to x^2-3
Rule	Rule[]	goes to	Rule[y,x^2-3]	y goes to x^2-3
Rule	:>	goes to (delayed)	y:>x^2-3	y goes to delayed x^2-3
Rule!!	=>	Invalid	y=>x^2-3	y=>x^2-3 (* bad syntax *)
Rule!!	Esc=>Esc	Implies	yEsc=>Escx^2-3	y Implies x^2-3
Rule!!	\[Implies]	Implies	y\[Implies]x^2-3	y Implies x^2-3
Rule!!	Implies[]	Implies	Implies[y,x^2-3]	y Implies x^2-3
Replacement	/.	given that	2.Sin[x]/.x->4.	2.Sin[x] given that x goes to 4. (* 2Sin[4] *)
Replacement	/.	given that	2.Sin[x]/.xEsc->Esc4.	2.Sin[x] given that x goes to 4. (* 2Sin[4] *)
R.Replacement	//..	repeated replacement

Mathematica Concepts (One-Liners)

Elements of lists are numbered beginning with 1.
Example variables in function definitions end with _.
Numbers for aritmetic should contain a decimal point.
Imaginary I must be capitalized.
Clear[] variable parameters before using them in functions or commands such as Solve[]
Clear[Global`*] is wise at the beginning of a NoteBook (if n GT 1 in line In[n])
The command "c=d" assigns the expression on the right (d) to the variable on the left (c)
The command "j:=k" does the same as "=" but it is deferred until the command is executed.
The operator "l==m" asserts that (l EQ m) is True. This is how an equation is defined.
The rule "p->q" asserts that the rule is True iff p goes to the expression on the right (q).It does not do p=q.
The operator "=>" is bad syntax. It means neither "EQ or GT" nor "Esc=>Esc" which would mean \[Implies]/
The expression 3.*x/.x->4. is equivalent to 3.*4. The expression 3.*x/.x->n is equivalent to 3.*n
Use StringJoin[] instead of the undefined function: Concatenate[]
The character "`" printed after a decimal number means "approximated i.e. truncated"
The integer 2 in "2 Pi" vs 2. in "2. Pi" is an unevaluated 2 (* quite different; try FullForm[] of each *)
A $ as the last letter of a string name is not necessary, but improves code readability
Plot IBM since 2008 using DateListPlot[TimeSeries[FinancialData["IBM","Jan.1,2008"]]];

References

Related Articles (to Mathematica)

Click on Title to view the Mathematica article

Discipline...	Title	Description
Math.	Mathematical Operators	List of such operators eg +, *, <>
Input Assistant	Wolfram Predictive Interface	Provides suggestions while typing into a notebook.
Intro.	Fast Introduction for Programmers.	First slides for coders.

Bibliography

The www link is on the web, the main link is "identical" but may also be archived on the Webmaster's server, just in case the www link disappears or is altered. The small icons after the name of the main link indicate the type or location of the reference.

External Sources

Book Sources
Book 1: Stephen Wolfram - www It's Been 10 Years: What's Happened with A New Kind of Science

, Date: 2012 E May 07

Web Sources

Web Source S075:01:www Mathematica Tips, Tricks, and Techniques

by Michael A. Morrison Date: 2000 D Apr 02

Web Source S075:02:www Mathematica Stack Exchange

by many authors Date: 2015 D Apr 02

Web Source S075:03:www Mathematica Stack Exchange: Tips and Tricks (very thorough)

by faysou Date: 2012 A Jan 09

Web Source S075:04:www Basic Advice for beginners (short excerpt from Source 3)

by faysou Date: 2015 C Mar 15

Web Source S075:05:www Code Refactoring

Wikipedia Date: 2015 C Mar 15

Web Source S075:06:www Refactoring

by Martin Fowler Date: 2015 C Mar 25

Web Source S075:07:www Mathematica 10 announcement

by Amoud Buzing of Wolfram Date: 2014 8 Aug 14

WebMaster: Thou Old King Cole

Click here to return to ePC Articles by Old King Cole

Updated: 2015 D Apr 11

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