计算机代数系统Computer Algebra Software: Mathematica, Maxima, Pari/GP

Computer Algebra Software: Mathematica, Maxima, Pari/GP

a side-by-side reference sheet

	mathematica	maxima	pari/gp
version used	8.0	5.21	2.3
show version	select About Mathematica in Mathematica menu	$ maxima --version	$ gp --version
grammar and invocation
	mathematica	maxima	pari/gp
interpreter		$ maxima -b foo.mac	$ gp -q foo.gp
repl	$ math	$ maxima	$ gp
block delimiters	( stmt; …)	block ( … )	{ … }
statement separator	; or sometimes newline ; before a newline suppresses output	; suppresses output: $	newline or ; a trailing semicolon suppresses output
end-of-line comment	none	none	1 + 1 \\ addition
multiple line comment	1 + (* addition *) 1	1 + /* addition */ 1	1 + /* addition */ 1
variables and expressions
	mathematica	maxima	pari/gp
assignment	a = 3 Set[a, 3]	a: 3	a = 3
parallel assignment	{a, b} = {3, 4} Set[{a, b}, {3, 4}]	[a, b]: [3, 4]	none
compound assignment	+= -= *= /= corresponding functions: AddTo SubtractFrom TimeBy DivideBy	none	+= -= *= /= %=
increment and decrement	++x --x PreIncrement[x] PreDecrement[x] x++ x-- Increment[x] Decrement[x]	none	return value after increment or decrement: x++ x--
null	Null
null test
undefined variable access	treated as an unknown number	treated as an unknown symbol	treated as an unknown number
remove variable binding	Clear[x] Remove[x]	kill(x)	kill(x)
conditional expression	If[x > 0, x, -x]	if is(x > 0) then x else -x	if(x > 0, x, -x)
arithmetic and logic
	mathematica	maxima	pari/gp
true and false	True False	true false	1 0
falsehoods	False	false	0
logical operators	! True \|\| (True && False) Or[Not[True], And[True, False]]	is(not true or (true and false))	! 1 \|\| (1 && 0)
relational operators	== != > < >= <= corresponding functions: Equal Unequal Greater Less GreaterEqual LessEqual	= # > < >= <=	== != > < >= <=
arithmetic operators	+ - * / Quotient Mod adjacent terms are multiplied, so * is not necessary. Quotient and Mod are functions, not binary infix operators. These functions are also available: Plus Subtract Times Divide	+ - * / quotient mod quotient and mod are functions, not binary infix operators	+ - * / none %
integer division	Quotient[a, b]	quotient(a, b)	divrem(a, b)[1]
integer division by zero	dividend is zero: Indeterminate otherwise: ComplexInfinity	error	error
float division	exact division: a / b	float(a) / b	exact division: a / b
float division by zero	dividend is zero: Indeterminate otherwise: ComplexInfinity	error	error
power	2 ^ 16 Power[2, 16]	2 ^ 16	2 ^ 16
sqrt	returns symbolic expression: Sqrt[2]	returns symbolic expression: sqrt(2)	returns float: sqrt(2)
sqrt -1	I	%i	1.000 * I
transcendental functions	Exp Log Sin Cos Tan ArcSin ArcCos ArcTan ArcTan ArcTan accepts 1 or 2 arguments	exp log sin cos tan asin acos atan atan2	exp log sin cos tan asin acos atan none
transcendental constants pi and the euler constant	Pi E	%pi %e	Pi exp(1)
float truncation round towards zero, round to nearest integer, round down, round up	IntegerPart Round Floor Ceiling	truncate round floor ceiling	truncate round floor ceil
absolute value and signum	Abs Sign	abs sign	abs sign
integer overflow	none, has arbitrary length integer type	none, has arbitrary length integer type	none, has arbitrary length integer type
float overflow	none	error	error
rational construction	use integer division: 1 / 7	use integer division: 1 / 7	use integer division: 1 / 7
rational decomposition	Numerator Denominator	ratnumer ratdenom	numerator denominator
complex construction	1 + 3I	1 + 3 * %i	1 + 3 * I
complex decomposition real and imaginary part, argument and modulus, conjugate	Re Im Arg Abs Conjugate	realpart imagpart carg abs conjugate	real imag ?? abs conj
random number uniform integer, uniform float	RandomInteger[{0, 99}] RandomReal[]	random(100) random(1.0)	random(100) ??
random seed set, get	SeedRandom[17] ??	set_random_state(make_random_state(17)); ??	setrand(17) getrand()
binary, octal, and hex literals	2^^101010 8^^52 16^^2a
base conversion	BaseForm[42, 7] BaseForm[7^^60, 10]		\\ 42 as powers of 7 up to 9th power: 42 + O(7^10)
strings
	mathematica	maxima	pari/gp
string literals	"don't say \"no\""	"don't say \"no\""	"don't say \"no\""
newline in literal	yes		no; use \n escape
string literal escapes	\\ \" \b \f \n \r \t \ooo		\n \t \" \\
character access	Characters["hello"][[1]]	charat("hello", 1)
chr and ord	FromCharacterCode[{65}] ToCharacterCode["A"][[1]]
length	StringLength["hello"]	slength("hello")	length("hello")
concatenate	"one " <> "two " <> "three"	concat("one", "two", "three");	Str("one", "two", "three")
index of substring	StringPosition["hello", "el"][[1]][[1]] StringPosition returns an array of pairs, one for each occurrence of the substring. Each pair contains the index of the first and last character of the occurrence.	ssearch("el", "hello") counts from one, returns false if not found
extract substring	StringTake["hello", {1, 4}]	substring("hello", 1, 5)
split	StringSplit["foo,bar,baz", ","]	split("foo,bar,baz",",")
join	StringJoin[Riffle[{"foo", "bar", "baz"}, ","]]	simplode(["foo","bar","baz"],",")
trim	StringTrim[" foo "]	strim(" ", " foo ") striml(" ", " foo") strimr(" ", "foo ")
convert from string, to string	7 + ToExpression["12"] 73.9 + ToExpression[".037"] "value: " <> ToString[8]	7 + parse_string("12") 73.9 + parse_string(".037")
case manipulation	ToUpperCase["foo"] ToLowerCase["FOO"]	supcase("foo") sdowncase("FOO")
regular expressions
	mathematica	maxima	pari/gp
regex test	re = RegularExpression["[a-z]+"] sc = StringCases["hello", re] Length[sc] > 0
regex substitution	s = "foo bar bar" re = RegularExpression["bar"] StringReplace[s, re -> "baz", 1] StringReplace[s, re -> "baz"]
arrays
	mathematica	maxima	pari/gp
literal	{1, 2, 3} List[1, 2, 3]	[1, 2, 3]	\\ [1, 2, 3] is a vector literal: List([1, 2, 3])
size	Length[{1, 2, 3}]	length([1, 2, 3])	length(List([1, 2, 3]))
empty test	Length[{}] == 0	emptyp([]);	length(List([])) == 0
lookup	(* access time is O(1) ) ( indices start at one: *) {1, 2, 3}[[1]] Part[{1, 2, 3}, 1]	/* access time is O(n) / / indices start at one: */ [1, 2, 3][1];	\\ access time is O(1). \\ indices start at one: List([1, 2, 3])[1]
update	a[[1]] = 7	a[1]: 7;	listput(a, 7, 1)
out-of-bounds behavior	left as unevaluated Part[] expression	invalid index error	out of allowed range error
element index	(* returns list of all positions: *) First /@ Position[{7, 8, 9, 9}, 9]	/* returns list of all positions: */ sublist_indices([7, 8, 9, 9], lambda([x], x = 9));	none
slice	{1, 2, 3}[[1 ;; 2]]	none	none
array of integers as index	(* evaluates to {7, 9, 9} *) {7, 8, 9}[[{1, 3, 3}]]	none	none
manipulate back	a = {6,7,8} AppendTo[a, 9] elem = a[[Length[a]]] a = Delete[a, Length[a]] elem	a: [6, 7, 8]; a: endcons(9, a); last(a); /* no easy way to delete last element */	a = List([6, 7, 8]) listput(a, 9) elem = listpop(a)
manipulate front	a = {6,7,8} PrependTo[a, 5] elem = a[[1]] a = Delete[a, 1] elem	load("basic"); a: [6, 7, 8]; push(5, a); elem: pop(a);	a = List([6, 7, 8]); listinsert(a, 5, 1); elem = a[1]; listpop(a, 1);
head	First[{1, 2, 3}]	first([1, 2, 3]);	List([1, 2, 3])[1]
tail	Rest[{1, 2, 3}]	rest([1, 2, 3]);	none
cons	(* first arg must be an array *) Prepend[{2, 3}, 1]	/* second arg must be an array */ cons(1, [2, 3]);	a = List([1, 2, 3]); listinsert(a, 1, 1);
concatenate	Join[{1, 2, 3}, {4, 5, 6}]	append([1, 2, 3], [4, 5, 6])	concat(List([1, 2, 3]), List([4, 5, 6]))
replicate	ten_zeros = Table[0, {i, 0, 9}]	ten_zeros: makelist(0, 10);
copy	a2 = a	a2: copylist(a);	a2 = a
iterate	Function[x, Print[x]] /@ {1, 2, 3}	a: [1, 2, 3]; for i from 1 thru length(a) do print(a[i]);	a = List([1, 2, 3]) for(i=1, length(a), print(a[i]))
reverse	Reverse[{1, 2, 3}]	reverse([1, 2, 3]);	a = List([1, 2, 3]) a2 = listcreate() while(i > 0, listput(a2, a[i]); i—)
sort	Sort[{3, 1, 4, 2}]	sort([3, 1, 4, 2]);	a = List([3,1,4,2]) listsort(a) a
dedupe	Union[{1, 2, 2, 3}]	unique([1, 2, 2, 3]);
membership	MemberQ[{1, 2, 3}, 2]	member(2, [1, 2, 3]);	/* The Set() constructor takes an array or vector as an argument. It converts the elements to strings and sorts them, discarding duplicates. setsearch() returns the index of the element or zero if not in the set. */ setsearch(Set([1, 2, 3]), 2)
intersection	Intersect[{1, 2}, {2, 3, 4}]	/* { } is literal notation for a set; use setify() to convert array to set */ intersect({1, 2}, {2, 3, 4});	setintersect(Set([1, 2]), Set([2, 3, 4]))
union	Union[{1, 2}, {2, 3, 4}]	union({1, 2}, {2, 3, 4});	setunion(Set([1, 2]), Set([2, 3, 4]))
relative complement, symmetric difference	Complement[{1, 2, 3}, {2}] none	setdifference({1, 2, 3}, {2}); symmdifference({1, 2}, {2, 3, 4});	setminus(Set([1, 2, 3]), Set([2]))
map	Function[x, x x] /@ {1, 2, 3} Map[Function[x, x x], {1, 2, 3}]	map(lambda([x], x * x), [1, 2, 3])
filter	Select[{1, 2, 3}, # > 2 &]	sublist([1, 2, 3], lambda([x], x > 2))
reduce	Fold[Plus, 0, {1, 2, 3}]	/* returns -4: / lreduce(lambda([x,y], x - y), [1, 2, 3]) / returns 2: */ rreduce(lambda([x,y], x - y), [1, 2, 3])
universal and existential tests	none	every(evenp, [1, 2, 3]); some(evenp, [1, 2, 3]);
min and max element	Min[{1, 2, 3}] Max[{1, 2, 3}]	apply(min, [1, 2, 3]); apply(max, [1, 2, 3]);
shuffle and sample	x = {3, 7, 5, 12, 19, 8, 4} RandomSample[x] RandomSample[x, 3]	random_permutation([3, 7, 5, 12, 19, 8, 4]);
zip	(* list of six elements: *) Riffle[{1, 2, 3}, {"a", "b", "c"}]	/* list of six elements: */ join([1, 2, 3], ["a", "b", "c"])
sequences
	mathematica	maxima	pari/gp
range	Range[1, 100]	makelist(i, i, 1, 100)
arithmetic sequence of integers with difference 10	Range[1, 100, 10]
arithmetic sequence of floats with difference 0.1	Range[1, 100, .1]
multidimensional-arrays
	mathematica	maxima	pari/gp
dictionaries
	mathematica	maxima	pari/gp
record literal	r = { n -> 10, avg -> 3.7, sd -> 0.4}	defstruct(point(x, y, z)); p: point(2, 3, 5);
record member access	n /. r	p@x
functions
	mathematica	maxima	pari/gp
definition	add[a_, b_] := a + b (* alternate syntax: *) add = Function[{a, b}, a + b]	add(a, b) := a + b;	add(x, y) = x + y
invocation	add[3, 7] add @@ {3, 7}	add(3, 7);	add(3, 7)
return value
function value
anonymous function	Function[{a, b}, a + b] (#1 + #2) &	lambda([a, b], a + b)
missing argument		error
extra argument		error
default argument
variable number of arguments		f(x, [L]) := if emptyp(L) then x else [x, apply("+", L)]; f(1); f(1, 2, 3);
execution control
	mathematica	maxima	pari/gp
if	If[x > 0, Print["positive"], If[x < 0, Print["negative"], Print["zero"]]]	if (is(x > 0)) then print("positive") elseif (is(x < 0)) then print("negative") else print("zero")	if(x > 0, \ print("positive"), \ if(x < 0, \ print("negative"), \ print("zero")))
while	i = 0 While[i < 10, Print[i]; i++]		i = 0 while(i < 10, print(i); i++)
for	For[i = 0, i < 10, i++, Print[i]]	for i from 0 thru 9 do print(i);	for(i=0, 9, print(i))
break/continue	Break[] Continue[]		break continue
raise exception	Throw["failed"]	throw("failed"); error("failed");	error("failed")
handle exception	Print[Catch[Throw["failed"]]]	catch(throw("failed")); errcatch(error("failed"));
finally block	none
files
	mathematica	maxima	pari/gp
write to stdout	Print["hello"]	print("hello")	print("hello")
read entire file into string or array	s = Import["/etc/hosts"] a = StringSplit[s, "\n"]
redirect to file
libraries and namespaces
	mathematica	maxima	pari/gp
load
reflection
	mathematica	maxima	pari/gp
list function documentation		??	?
get function documentation	?Tan Information[Tan]	? tan describe(tan)	? tan
grep documentation		?? tan
query data type	Head[x]	bigfloatp(x) floatnump(x) integerp(x) numberp(x) ratnump(x) stringp(x) listp(x)	type(x)
list variables in scope			? 0
algebra
	mathematica	maxima	pari/gp
solution to an equation	Solve[x^3 + x + 3 == 0, x]	solve(x^3 + x + 3, x)
solution to two equations	Solve[x + y == 3 && x == 2y, {x, y}]	solve([x+y=3, x=2*y], [x, y])
numerical approximation	N[Exp[1]] Exp[1] + 0. N[Exp[1], 10]	float(exp(1))	1/7 + 0.
expand polynomial	Expand[(1 + x)^5]	expand((1+x)^5)
factor polynomial	Factor[3 + 10 x + 9 x^2 + 2 x^3]	factor(3 + 10x + 9x^2 + 2*x^3)
add fractions	Together[a/b + c/d]	ratsimp(a/b + c/d)
decompose fraction	Apart[(b c + a d)/(b d)]
calculus
	mathematica	maxima	pari/gp
differentation	D[x^3 + x + 3, x]	diff(x^3 + x + 3, x) diff(sin(x), x)	P = x^3 + x + 3 P' sin(x)'
higher order differentiation	D[Log[x], {x, 3}]	diff(log(x), x, 3);
integration	Integrate[x^3 + x + 3, x] Integrate[x^3 + x + 3, {x, 0, 1}]	integrate(x^3 + 3x + 3, x); integrate(x^3 + 3x + 3, x, 0, 1);
find minimal value	Minimize[Sqrt[a^2 + x^2] + Sqrt[(b - x)^2 + c^2], x]
number theory
	mathematica	maxima	pari/gp
number tests	IntegerQ[7] PrimeQ[7] rational test? real test?	integerp(7) primep(7) ratnump(1/7)
solve diophantine equation	Solve[a^2 + b^2 == c^2 && a > 0 && a < 10 && b > 0 && b < 10 && c > 0 && c < 10, {a, b, c}, Integers]
factorial	10!	10!	10!
binomial coefficient	Binomial[10,3]	binomial(10,3)
greatest common divisor	GCD[14, 21]	gcd(14, 21)	gcd(14, 21)
prime factors	returns {{2, 2}, {3, 1}, {7, 1}} FactorInteger[84]	factor(84)	returns [2,2; 3,1; 7,1] factor(84)
Euler totient	EulerPhi[256]	totient(256)
vectors
	mathematica	maxima	pari/gp
vector literal	(* same as array: *) {1, 2, 3}	/* same as list: */ [1, 2, 3]	[1, 2, 3]
vector coordinate	indices start at one: {1,v2, 3}[[1]]	indices start at one: [1, 2, 3][1]	indices start at one: [1, 2, 3][1]
vector dimension	Length[{1, 2, 3}]	length([1, 2, 3])	length([1, 2, 3])
element-wise arithmetic operators	+ - * / adjacent lists are multiplied element-wise	+ - * /	+ -
vector length mismatch	error	error	error
scalar multiplication	3 {1, 2, 3} {1, 2, 3} 3 * may also be used	3 * [1, 2, 3] [1, 2, 3] * 3	3 * [1, 2, 3] [1, 2, 3] * 3
dot product	{1, 1, 1} . {2, 2, 2} Dot[{1, 1, 1}, {2, 2, 2}]	[1,1,1] . [2,2,2]
cross product	Cross[{1, 0, 0}, {0, 1, 0}]	load("vect"); express([1, 0, 0] ~ [0, 1, 0]);
norms	Norm[{1, 2, 3}, 1] Norm[{1, 2, 3}] Norm[{1, 2, 3}, Infinity]
matrices
	mathematica	maxima	pari/gp
literal or constructor	A = {{1, 2}, {3, 4}} B = {{4, 3}, {2, 1}}	A: matrix([1, 2], [3, 4]); B: matrix([4, 3], [2, 1]);	A = [1, 2; 3, 4] B = [4, 3; 2, 1]
zero, identity, ones, diagonal matrix	ConstantArray[0, {3, 3}] IdentityMatrix[3] ConstantArray[1, {3, 3}] DiagonalMatrix[{1, 2, 3}]	zeromatrix(3, 3) ident(3) 1 + zeromatrix(3, 3) diag_matrix(1, 2, 3)
dimensions rows, columns	Length[A] Length[A[[1]]]	matrix_size(A)[1] matrix_size(A)[2]
element access	A[[1, 1]]	A[1, 1]	A[1, 1]
row access	A[[1]]	as list: A[1] as 1xn matrix: row(A, 1)
column access		col(A, 1)
submatrix access	# [[1]] & /@ A
scalar multiplication	3 A A 3 * can also be used	3 * A A * 3
element-wise operators	+ - * / adjacent matrices are multiplied element-wise	+ - * /
multiplication	A . B	A . B
kronecker product	KroneckerProduct[A, B]	kronecker_product(A, B);
comparison	A == B A != B	is(A = B) is(A # B)
norms	Norm[A, 1] Norm[A] Norm[A, Infinity] Norm[A, "Frobenius"]
transpose	Transpose[A]
conjugate transpose	A = {{I, 2 I}, {3 I, 4 I}} ConjugateTranspose[A]
inverse	Inverse[A]
determinant	Det[A]		matdet(A)
trace	Tr[A]		trace(A)
eigenvalues	Eigenvalues[A]
eigenvectors	Eigenvectors[A]
system of equations	Solve[A. {x, y} == { 2, 3}, {x, y}]
distributions
	mathematica	maxima	pari/gp
normal	nd = NormalDistribution[0,1] RandomVariate[nd]	load(distrib); random_normal(0,1);
exponential	ed = ExponentialDistribution[1] RandomVariate[ed]	load(distrib); random_exp(1);
poisson	pd = PoissonDistribution[1] RandomVariate[pd]	load(distrib); random_poisson(1);
univariate charts
	mathematica	maxima	pari/gp
vertical bar chart	BarChart[{7, 3, 8, 5, 5}, ChartLegends-> {"a","b","c","d","e"}]
horizontal bar chart	BarChart[{7, 3, 8, 5, 5}, BarOrigin -> Left]
pie chart	PieChart[{7, 3, 8, 5, 5}]
stem-and-leaf plot	Needs["StatisticalPlots`"] nd = NormalDistribution[0, 1] n100 = RandomVariate[nd, 100] StemLeafPlot[20 * n100]
histogram	nd = NormalDistribution[0, 1] Histogram[RandomReal[nd, 100], 10]
box-and-whisker plot	nd = NormalDistribution[0, 1] n100 = RandomVariate[nd, 100] BoxWhiskerChart[d] ed = ExponentialDistribution[1] e100 = RandomVariate[ed, 100] u100 = RandomReal[1, 100] d = {n100, e100, u100} BoxWhiskerChart[d]
set chart title	BoxWhiskerChart[data, PlotLabel -> "chart example"]
chart options	PlotLabel -> "an example" AxisLabel -> {"time", "distance"}
bivariate charts
	mathematica	maxima	pari/gp
stacked bar chart	d = {{7, 1}, {3, 2}, {8, 1}, {5, 3}, {5, 1}} BarChart[d, ChartLayout -> "Stacked"]
scatter plot	nd = NormalDistribution[0, 1] rn = Function[RandomReal[nd]] d = {rn[],rn[]} & /@ Range[1,50] ListPlot[d]
linear regression line	d = Table[{i, 2 i + RandomReal[{-5, 5}]}, {i, 0, 20}] model = LinearModelFit[d, x, x] Show[ListPlot[d], Plot[model["BestFit"], {x, 0, 20}]]
polygonal line plot	f = Function[i, {i, rn[]}] d = f /@ Range[1, 20] ListLinePlot[d]
area chart	d = {{7, 1, 3, 2, 8}, {1, 5, 3, 5, 1}} sd = {d[[1]], d[[1]] + d[[2]]} ListLinePlot[sd, Filling -> {1 -> {Axis, LightBlue}, 2 -> {{1}, LightRed}}]
cubic spline	d = Table[{i, RandomReal[nd]}, {i, 0, 20}] f = Interpolation[d, InterpolationOrder -> 3] Show[ListPlot[d], Plot[f[x], {x, 0, 20}]]
function plot	Plot[Sin[x], {x, -4, 4}]	plot2d(sin(x),[x,-4,4]);	ploth(x=-4, 4, sin(x))
quantile-quantile plot	nd = NormalDistribution[0, 1] d1 = RandomReal[1, 50] d2 = RandomReal[nd, 50] QuantilePlot[d1, d2]
axis label	d = Table[{i, i^2}, {i, 1, 20}] ListLinePlot[d, AxesLabel -> {x, x^2}]	plot2d(sin(x), [x,-4,4], [ylabel,"sine function"]);
logarithmic y-axis	LogPlot[{x^2, x^3, x^4, x^5}, {x, 0, 20}]
trivariate charts
	mathematica	maxima	pari/gp
3d scatter plot	nd = NormalDistribution[0,1] d = RandomReal[nd, {50, 3}] ListPointPlot3D[d]
additional data set	nd = NormalDistribution[0, 1] x1 = RandomReal[nd, 20] x2 = RandomReal[nd, 20] ListLinePlot[{x1, x2}]
bubble chart	nd = NormalDistribution[0,1] d = RandomReal[nd, {50, 3}] BubbleChart[d]
surface plot	Plot3D[Sinc[Sqrt[x^2 + y^2]], {x, -25, 25}, {y, -25, 25}]	sinc(x) := sin(%pix)/(%pix);sinc(x) := sin(%pix)/(%pix); plot3d(sinc(sqrt(x^2+y^2)),[x,-25,25],[y,-25,25]);
	______________________________________________________	______________________________________________________	______________________________________________________

version used

The version of software used to check the examples in the reference sheet.

show version

How to determine the version of an installation.

Grammar and Invocation

interpreter

How to execute a script.

pari/gp

The shebang style notation doesn't work because GP doesn't recognize the hash tag # as the start of a comment.

The -q option suppresses the GP startup message.

After the script finishes it will drop the user into the REPL unless there is a quit statement in the script:

print("Hello, World!")

quit

repl

How to launch a command line read-eval-print loop for the language.

mathematica:

One can create a REPL called math on Mac OS X with the following command:

$ sudo ln -s /Applications/Mathematica.app/Contents/MacOS/MathKernel /usr/local/bin/math

$ math

end-of-line comment

Character used to start a comment that goes to the end of the line.

Variables and Expressions

assignment

How to perform assignment.

In all three languages an assignment is an expression that evaluates to the right side of the expression. Assignments can be chained to assign the same value to multiple variables.

mathematica:

The Set function behaves identically to assignment and can be nested:

Set[a, Set[b, 3]]

parallel assignment

How to assign values in parallel.

Parallel assignment can be used to swap the values held in two variables.

remove variable binding

How to remove a variable. Subsequent references to the variable will be treated as if the variable were undefined.

Arithmetic and Logic

falsehoods

Values which evaluate to false in a conditional test.

integer division

How to compute the quotient of two integers.

integer division by zero

The result of dividing an integer by zero.

float division

How to perform float division, even if the arguments are integers.

power

How to compute exponentiation.

Note that zero to a negative power is equivalent to division by zero, and negative numbers to a fractional power may have multiple complex solutions.

sqrt

The square root function.

For positive arguments the positive square root is returned.

sqrt -1

How the square root function handles negative arguments.

mathematica:

An uppercase I is used to enter the imaginary unit, but Mathematica displays it as a lowercase i.

transcendental functions

The standard transcendental functions such as one might find on a scientific calculator.

The functions are the exponential (not to be confused with exponentiation), natural logarithm, sine, cosine, tangent, arcsine, arccosine, arctangent, and the two argument arctangent.

transcendental constants

The transcendental constants pi and e.

The transcendental functions can used to computed to compute the transcendental constants:

pi = acos(-1)
pi = 4 * atan(1)
e = exp(1)

float truncation

Ways to convert a float to a nearby integer.

absolute value

How to get the absolute value and signum of a number.

integer overflow

What happens when the value of an integer expression cannot be stored in an integer.

The languages in this sheet all support arbitrary length integers so the situation does not happen.

float overflow

What happens when the value of a floating point expression cannot be stored in a float.

rational decomposition

How to extract the numerator and denominator from a rational number.

complex decomposition

How to extract the real and imaginary part from a complex number; how to extract the argument and modulus; how to get the complex conjugate.

random number

How to generate a random integer or a random float.

random seed

How to set or get the random seed.

maxima:

The seed is set to a fixed value at start up.

pari/gip:

The seed is set to a fixed value at start up.

mathematica:

The seed is not set to the same value at start up.

binary, octal, and hex literals

Binary, octal, and hex integer literals.

mathematica:

The notation works for any base from 2 to 36.

Strings

string literals

newline in literal

character access

chr and ord

length

concatenate

index of substring

extract substring

split

convert from string, to string

How to convert strings to numbers and vice versa.

join

trim

case manipulation

sprintf

Regular Expressions

regex test

How to test whether a string matches a regular expression.

regex substitution

How to replace substrings which match a regular expression.

Arrays

lookup

How to access an array element by its index.

update

How to change the value stored at an array index.

out-of-bounds behavior

What happens when an attempt is made to access an element at an out-of-bounds index.

element index

How to get the index of an element in an array.

slice

How to extract a subset of the elements. The indices for the elements must be contiguous.

copy

How to copy an array. Updating the copy will not alter the original.

membership

How to test whether a value is an element of a list.

intersection

How to to find the intersection of two lists.

relative complement, symmetric difference

How to find all elements in one list which are not in another; how to find all elements which are in one of two lists but not both.

universal and existential tests

shuffle and sample

How to shuffle an array. How to extract a random sample from an array without replacement.

Sequences

Multidimensional Arrays

Dictionaries

record literal

record member access

Functions

definition

invocation

function value

Execution Control

if

How to write a branch statement.

mathematica:

The 3rd argument (the else clause) of an If expression is optional.

while

How to write a conditional loop.

mathematica:

Do can be used for a finite unconditional loop:

Do[Print[foo], {10}]

for

How to write a C-style for statement.

break/continue

How to break out of a loop. How to jump to the next iteration of a loop.

raise exception

How to raise an exception.

handle exception

How to handle an exception.

finally block

How to write code that executes even if an exception is raised.

Files

Libraries and Namespaces

Reflection

]function-documentation]

function documentation

How to get the documentation for a function.

Algebra

Calculus

Number Theory

Vectors

vector coordinate

How to get one of the coordinates of a vector.

vector dimension

How to get the number of coordinates of a vector.

element-wise arithmetic operators

How to perform an element-wise arithmetic operatio on vectors.

vector length mismatch

What happens when an element-wise arithmetic operation is performed on vectors of different dimension.

dot product

How to compute the dot product of two vectors.

cross product

How to compute the cross product of two three-dimensional vectors.

Matrices

literal or constructor

Literal syntax or constructor for creating a matrix.

mathematica:

Matrices are represented as lists of lists. No error is generated if one of the rows contains too many or two few elements. The MatrixQ predicate can be used to test whether a list of lists is matrix: i.e. all of the sublists contain numbers and are of the same length.

Matrices are displayed by Mathematica using list notation. To see a matrix as it would be displayed in mathematical notation, use the MatrixForm function.

dimensions

How to get the dimensions of a matrix.

element access

How to access an element of a matrix. All languages described here follow the convention from mathematics of specifying the row index before the column index.

row access

How to access a row.

column access

How to access a column.

submatrix access

How to access a submatrix.

scalar multiplication

How to multiply a matrix by a scalar.

element-wise operators

Operators which act on two identically sized matrices element by element. Note that element-wise multiplication of two matrices is used less frequently in mathematics than matrix multiplication.

multiplication

How to multiply matrices. Matrix multiplication should not be confused with element-wise multiplication of matrices. Matrix multiplication in non-commutative and only requires that the number of columns of the matrix on the left match the number of rows of the matrix. Element-wise multiplication, by contrast, is commutative and requires that the dimensions of the two matrices be equal.

kronecker product

The Kronecker product is a non-commutative operation defined on any two matrices. If A is m x n and B is p x q, then the Kronecker product is a matrix with dimensions mp x nq.

comparison

How to test two matrices for equality.

norms

How to compute the 1-norm, the 2-norm, the infinity norm, and the frobenius norm.

Distributions

univariate-charts Univariate Charts

A univariate chart can be used to display a list or array of numerical values. Univariate data can be displayed in a table with a single column or two columns if each numerical value is given a name. A multivariate chart, by contrast, is used to display a list or array of tuples of numerical values.

In order for a list of numerical values to be meaningfully displayed in a univariate chart, it must be meaningful to perform comparisons (<, >, =) on the values. Hence the values should have the same unit of measurement.

vertical bar chart

A chart which represents values with rectangular bars which line up on the bottom. It is a deceptive practice for the bottom not to represent zero, even if a y-axis with labelled tick marks or grid lines is provided. A cut in the vertical axis and one of the bars may be desirable if the cut value is a large outlier. Putting such a cut all of the bars near the bottom is a deceptive practice similar not taking to the base of the bars to be zero, however.

Another bad practice is the 3D bar chart. In such a chart heights are represented by the height of what appear to be three dimensional blocks. Such charts impress an undiscriminating audience but make it more difficult to make a visual comparison of the charted quantities.

mathematica

horizontal bar chart

A bar chart in which zero is the y-axis and the bars extend to the right.

pie chart

A bar chart displays values using the areas of circular sectors or equivalently, the lengths of the arcs of those sectors. A pie chart implies that the values are percentages of a whole. The viewer is likely to make an assumption about what the whole circle represents. Thus, using a pie chart to show the revenue of some companies in a line of business could be regarded as deceptive if there are other companies in the same line of business which are left out. The viewer may mistakenly assume the whole circle represents the total market.

If two values are close in value, people cannot determine visually which of the corresponding sectors in a pie chart is larger without the aid of a protractor. For this reason many consider bar charts superior to pie charts.

Many software packages make 3D versions of pie charts which communicate no additional information and in fact make it harder to interpret the data.

stem-and-leaf plot

histogram

box-and-whisker plot

set chart title

Bivariate Charts

stacked bar chart

Trivariate Charts

Mathematica

Mathematica Documentation Center
WolframAlpha

Maxima

Maxima Manual

Pari/GP

A Tutorial for Pari/GP (pdf)
Pari/GP Functions by Category
Pari/GP Reference Card (pdf)