Function -- the class of all functions

Function -- the class of all functions

The type Function is a member of the class Type. Each object of class Function is called a function.

Common ways to make a function:

-> -- make a function

Returning from functions:

return -- return from a function

See also:

functions -- an overview

Functions and methods returning a a function :

{...} ==> Function -- attaching options to a function

Function @@ Function

Function _ Thing -- attach the first argument to a function of two or more arguments

OptionTable ==> Function -- attaching options to a function

profile Function

Methods for using a function :

List / Function

accumulate(Function,Thing,VisibleList) -- apply binary operator repeatedly

accumulate(Function,VisibleList) -- apply binary operator repeatedly

accumulate(VisibleList,Function) -- apply binary operator repeatedly

accumulate(VisibleList,Thing,Function) -- apply binary operator repeatedly

all(BasicList,Function)

all(HashTable,Function)

any(BasicList,Function)

any(HashTable,Function)

apply(BasicList,BasicList,Function) -- apply function to elements in pairs

apply(BasicList,Function) -- apply function to each element

apply(HashTable,Function) -- apply function to each value

apply(ZZ,Function) -- apply function to 0 .. n-1

applyKeys(HashTable,Function)

applyPairs(HashTable,Function)

applyValues(HashTable,Function)

autoload(Function,String)

code Function

documentation Function

expression Function

fold(Function,Thing,VisibleList) -- apply binary operator repeatedly

fold(Function,VisibleList) -- apply binary operator repeatedly

fold(VisibleList,Function) -- apply binary operator repeatedly

fold(VisibleList,Thing,Function) -- apply binary operator repeatedly

formatDocumentTag Function

Function ## Sequence

Function \ VisibleList -- apply a function to each member of a list

Function Thing -- function application

locate Function

map(ChainComplex,ChainComplex,Function) -- make a map of chain complexes

map(GradedModule,GradedModule,Function)

map(Module,Module,Function) -- make a map

map(Module,ZZ,Function) -- make a map

memoize Function

memoize(Function,List)

merge(HashTable,HashTable,Function)

mergePairs(BasicList,BasicList,Function)

net Function

new Command from Function

new Manipulator from Function

options Function -- get optional arguments and defaults

position(VisibleList,Function)

positions(VisibleList,Function)

precedence Function

product(Sequence,Sequence,Function)

product(Sequence,List,Function)

product(Sequence,Function)

product(List,Sequence,Function)

product({...},{...},Function) -- product of results of applying a function pairwise

product({...},Function) -- product of values of a function

product(ZZ,Function) -- product of consecutive values of a function

profile(String,Function)

scan(BasicList,Function) -- apply a function to each element of a list

scan(ZZ,Function) -- apply a function to 0 .. n-1

scanKeys(Database,Function)

scanKeys(HashTable,Function)

scanPairs(HashTable,Function) -- apply a function to pairs in a hash table

scanValues(HashTable,Function)

select(BasicList,Function) -- select elements from a list

select(HashTable,Function) -- select pairs from a hash table

select(ZZ,BasicList,Function) -- select a limited number of elements from a list

select(ZZ,HashTable,Function) -- select a limited number of pairs from a hash table

sum(Sequence,Sequence,Function)

sum(Sequence,List,Function)

sum(Sequence,Function)

sum(List,Sequence,Function)

sum({...},{...},Function) -- sum results of applying a function pairwise

sum({...},Function) -- sum results of applying a function

sum(ZZ,Function) -- sum consecutive values of a function

tex Function

texMath Function

toString Function

ultimate(Function,Thing)

VisibleList / Function -- apply a function to each member of a list

Fixed objects of class Function :

abs -- absolute value function

accumulate -- apply binary operator repeatedly

acos -- arccosine function

addEndFunction -- add an ending function

addStartFunction -- add a startup function

adjoint -- the adjoint map

adjoint1 -- the adjoint map

alarm -- set an alarm

all -- whether all elements satisfy a condition

allGenerators

ambient -- ambient free module of a subquotient, or ambient ring

ancestor -- whether one type is an ancestor of another

ann

ann'

annihilator -- the annihilator ideal

any -- whether an element satisfies a condition

append -- add to the end of a list

apply -- apply a function to each element

applyKeys -- apply a function to each key in a hash table

applyPairs -- apply a function to pairs in a hash table

applyTable -- apply a function to elements of a table

applyValues -- apply a function to each value

apropos -- symbols matching a pattern

ascii -- ASCII character conversion

asin -- arcsine function

ass

assert -- assert something is true

assign -- assign a value

atan -- arctangent function

atEndOfFile -- test for end of file

autoload -- arrange for a function to be loaded automatically

autoReduce

backtrace -- trace back through evaluations after an error

baseName -- the base name of a generator

basictype -- the basic type of an object

basis

benchmark -- accurate timing of execution

betti -- display degrees

between -- insert something between elements of a list

BinaryPowerMethod -- compute powers by squaring

binomial -- binomial coefficient

binomialCD

borel -- make a Borel fixed submodule

briefDocumentation -- get brief documentation

browse -- browse the contents of an object

buckets -- list the buckets in a hash table

cacheFileName

cacheFileNameKeys

callgg -- call the engine

CAVEAT

ceiling -- ceiling function

chainComplex -- make a chain complex

char -- the characteristic of a field or ring

characters -- get characters from a string

class -- class of an object

clearEcho -- turn off echoing

clone

code -- display source code

codim -- calculate the codimension

coefficientRing -- get the coefficient ring

coefficients -- the coefficients

cohomology -- general cohomology functor

coimage -- coimage of a map

coker

cokernel -- cokernel of a map

collectGarbage -- collect the garbage in memory

columnate -- arrange strings in columns

combine -- combine hash tables

complement -- find the minimal generators for cokernel of a matrix (low level form)

complete

components -- list the components of a direct sum

compress -- remove columns which are zero

concatenate -- join strings

conductor

cone -- mapping cone of a chain map

conjugate -- complex conjugate

connectionCount -- the number of connections

content -- the content of a polynomial

contract -- contract one matrix by another

contract'

ConvertApply -- apply a function after specified engine format conversions

ConvertFixedRepeat -- convert a fixed length sequence of items from engine format

ConvertJoin -- convert several items from engine format

ConvertList -- convert a sequence of items from engine format

ConvertRepeat -- convert a sequence of items from engine format

convert -- convert to engine format

copy -- copy an object

cos -- cosine function

cosh -- hyperbolic cosine function

cotangentSheaf -- cotangent sheaf of a variety

cover -- get the covering free module

coverMap -- get the map to the module given by the generators of a module

currentDirectory -- current working directory

decompose -- irreducible components of an ideal

deepSplice -- remove subsequences

degree -- the degree

degreeLength -- the number of degrees

degrees -- degrees of generators

degreesMonoid -- get the monoid of degrees

degreesRing -- the ring of degrees

delete -- delete elements of a list

demark -- insert a string between elements of a list of strings

denominator -- denominator of a fraction

depth -- depth of a net

describe -- real description

det -- determinant of a matrix

diff -- differentiate

diff'

difference -- difference

dim -- calculate the dimension

directSum -- direct sum of modules or maps

divideByVariable -- divide all columns by a (power of a) variable

document -- install documentation

documentation -- get documentation, unformatted

drop -- drop some elements

dual -- dual module or map

dumpdata -- dump state of the system to a file

echoOff -- turn off echoing

echoOn -- turn on echoing

eeLift -- call engine to lift a ring element

eePop -- pop object from engine stack

eePopBool -- pop Boolean value from engine stack

eePopInt -- pop integer from engine stack

eePopIntarray -- pop array of integers from engine stack

eePromote -- call engine to promote a ring element

EHVprimaryDecomposition

elements

End -- module of endomorphisms

entries -- list the entries of a matrix

erase -- remove a global symbol

error -- deliver error message

errorDepth -- set the error printing depth

euler -- list the sectional Euler characteristics

evaluate

even -- tell whether an integer is even

examine -- examine internal information about functions or symbols

examples -- list the examples in documentation

exec -- execute another program

exp -- exponential function

expand

exponents -- list the exponents in a polynomial

expression -- convert to an expression

extend -- extend a partial map of chain complexes

exteriorPower -- exterior power

extract

factor -- factor a ring element

factors

Fano -- Fano scheme

fastequal

fastgb

fastmingens

fastquotient

fileExists

findNonMember

first -- first element of a list

firstkey -- get the first key

fittingIdeal -- Fitting ideal of a module

flag -- flag a symbol

flatten -- flatten a list by unnesting lists

flattener

flip

floor -- floor function

fold -- apply binary operator repeatedly

forceGB -- declare that the columns of a matrix are a Groebner basis

fork -- fork the process

format -- format a string

formatDocumentTag -- format documentation tags

frac -- construct a fraction field

fraction

frame -- the frame of a function

fromDual

gb -- compute a Groebner basis

gbTrace -- provide tracing output during various computations in the engine.

gcd -- greatest common divisor

gcdCoefficients -- gcd with coefficients

genera -- list the sectional arithmetic genera

generators -- matrix of generators

genericMatrix -- make a generic matrix of variables

genericSkewMatrix -- make a generic skew symmetric matrix

genericSymmetricMatrix -- make a generic symmetric matrix

gens -- matrix of generators

genus

get -- get an entire file

getChangeMatrix -- get the change of basis matrix

getc -- get a byte

getenv -- get value of environment variable

getMatrix -- get a matrix from the engine's stack

getNonUnit -- retrieve a previously discovered non-unit

getParsing

getWWW -- get a web page

getZeroDivisor -- retrieve a previously discovered zero divisor

GF -- make a finite field

gg -- convert to engine communication format

ggPush -- convert a push command to engine communication format

globalAssignFunction -- the standard method for the global assignment hook

globalReleaseFunction -- the standard method for the global variable release hook

gradedModule -- make a graded module

gradedModuleMap -- make a map of graded modules

graphIdeal -- the ideal of the graph of a ring map

graphRing -- the ring of the graph of a ring map

Grassmannian -- find the ideal of a Grassmannian

group -- make a group (monoid with inverses)

handle -- get the handle of an object

hash -- hash code of an object

hashTable -- make a hash table

headline

height -- height of a net

help -- get help

hilbertFunction -- Hilbert function of a module

hilbertPolynomial -- compute Hilbert polynomial

hilbertSeries -- compute Hilbert series

hold -- hold something in a holder expression

Hom -- module of homomorphisms

homogenize -- homogenize with respect to a variable

homology -- general homology functor

homomorphism -- get the homomorphism from element of Hom

horizontalJoin -- join nets or strings horizontally

HprimaryDecomposition

html -- convert hypertext to html format

ICfractions

ICfractionsLong

ICmap

ideal -- make an ideal

identity -- the identity function

image -- image of a map

imaginaryPart -- imaginary part

independentSets

index -- yields the numeric index of a ring variable

indices -- specify keys for components of a direct sum

inducedMap -- compute an induced map

inducesWellDefinedMap -- whether a map is well defined

input -- read Macaulay 2 commands and echo

installMethod -- install methods

instance -- whether something has a certain type

integralClosure -- compute the integral closure of a ring

integrate -- numerical integration

intersect -- compute an intersection

irreducibleCharacteristicSeries -- compute the irreducible characteristic series

isAffineRing -- whether something is an affine ring

isBorel -- whether an ideal is Borel fixed

isCommutative -- whether a ring is commutative

isConstant -- test whether something is constant

isDirectSum -- whether something is a direct sum

isField -- whether something is a field

isFreeModule -- whether something is a free module

isHomogeneous -- test for homogeneity

isIdeal -- whether something is an ideal

isInjective -- whether a map is injective

isInputFile -- whether a file is open for input

isIsomorphism -- whether a map is an isomorphism

isListener -- whether a file is open for listening

isModule -- whether something is a module.

isMonomialIdeal

isNormal

isOpenFile -- whether a file is open

isOutputFile -- whether a file is open for output

isPolynomialRing -- whether something is a polynomial ring

isPrime -- tell whether an integer is a prime

isPrimitive -- whether an element is a primitive element of a finite field

isQuotientModule -- whether a module is evidently a quotient of a free module

isQuotientOf -- whether one thing is a quotient of another

isQuotientRing -- whether something is a quotient ring

isReady -- whether a file has data available for reading

isRing -- whether something is a ring

isSquareFree

isSubmodule -- whether a module is evidently a submodule of a free module

isSubset -- whether something is a subset of another

isSurjective -- whether a map is surjective

isTable -- whether something is a rectangular list of lists

isUnit -- whether a ring element is a unit

isWellDefined -- whether a map is well defined

jacobian -- the Jacobian matrix of partial derivatives

join -- join lists

ker

kernel -- kernel of a map

keys -- keys used in a hash table

kill -- kill a process

koszul -- a differential in a Koszul complex

last -- last element of a list

leadCoefficient -- the leading coefficient

leadComponent -- the leading component of a vector

leadMonomial -- the leading monomial

leadTerm -- get the leading term

length -- length

lift -- lift to another ring

liftable -- whether a ring element can be lifted to another ring

lineNumber -- current line number

lines -- split a string into lines

listForm -- convert to list form

load -- read Macaulay 2 commands

loaddata -- load state of the system from a file

localize

locate -- locate source code

log -- logarithm function

lookup -- look up methods

lookupCount -- reference count for a symbol

map -- make a map

match -- whether a string matches a pattern

mathML -- convert to MathML format

matrix -- make a matrix

max -- maximum of elements of a list

maxPosition -- position of largest element

member -- test membership in a list

memoize -- record results of function evaluation for future use

merge -- merge hash tables

mergePairs -- merge sorted lists of pairs

method -- make a new method function

methods -- list methods

min -- minimum of elements of a list

mingens -- returns a minimal generatoring set of a graded module

mingle -- mingle elements of several lists

minors -- ideal generated by minors

minPosition -- position of smallest element

minPres

minPresIdeal

minPresMap

minPresMapIdeal

minprimes -- compute the minimal primes

minSat

minSatPPD

minSatPPDold

minus -- additive inverse

mod -- reduce modulo an integer

modifyRing -- make a copy of a ring, with some features changed

module

modulo -- find the pre-image of a map (low level version)

monoid -- make a monoid

monomialCurve

monomialCurveIdeal -- make the ideal of a monomial curve

monomialIdeal -- make a monomial ideal

monomialOrdering

monomialSubideal

mutable -- whether something may be modified

needs -- read Macaulay 2 commands if necessary

net -- convert to net

netRows -- list the rows of a net

newClass -- copy an object, changing the class

newCoordinateSystem -- change variables

newdecompose

newDegreesMonoid

newDegreesRing

newHandle -- make a new handle

nextkey -- the next key in a database

notImplemented -- print an 'not implemented' error message

nullhomotopy -- make a null homotopy

number -- count how many elements of a list satisfy a condition

numerator -- numerator of a fraction

Numeric

numeric -- convert to floating point

numgens -- the number of generators

odd -- tell whether an integer is odd

oldCoefficients

on -- trace a function each time it's run

openDatabase -- open a database file

openDatabaseOut -- open a database file for writing

openFiles -- list the open files

openIn -- open an input file

openInOut -- open an input outpuf file

openListener -- open a port for listening

openOut -- open an output file

options -- get options

override -- override default values for optional arguments

pack -- pack elements of a list into shorter ones

pad -- pad a string with spaces

pairs -- list the pairs in a hash table

parent

partitions -- list the partitions of an integer

PDdonode

pdim -- calculate the projective dimension

PDinitialize

PDnext

peek -- examine contents of an object

peek2 -- examine contents of an object

pfaffians -- ideal generated by Pfaffians

plus -- addition

poincare -- assemble degrees into polynomial

poincareN -- assemble degrees into polynomial

position -- find first element of a list satisfying a condition

positions -- which elements of a list satisfy a condition

power -- power

PPD

PPDcharSets

PPDSpecialCharSets

precedence -- parsing precedence

prepend -- add to the beginning of a list

presentation -- presentation of a module or ring

primaryComponent

primaryDecomposition

print -- print something

printExamples -- print examples of use from documentation

printString -- lowlevel function to print a string

processID -- the process identifier

product

profile -- profile a function

Proj -- make a projective variety

projectiveHilbertPolynomial -- Hilbert polynomial of projective space

promote -- promote to another ring

protect -- protect a symbol

prune -- minimize generators and relations

pseudoRemainder -- compute the pseudo-remainder

pushForward

pushForward1 -- the main computational component of pushForward

quote

quotient -- ideal or submodule quotient

quotMin

quotMinold

radical -- compute the radical of an ideal

radicalContainment

random -- get a random element

rank -- compute the rank

read -- read from a file

readlink

realPart -- real part

regularity -- compute the regularity

relations -- the defining relations

remove -- remove an entry from a hash table

removeLowestDimension -- remove components of lower dimension

reorganize -- reorganize a database file

res

reshape -- reshape a matrix

resolution -- make a projective resolution

reverse -- reverse a list

ring -- get the associated ring

rsort -- sort a list in reverse order

run -- run an external command

runEndFunctions -- run the ending functions

runLengthEncoding

runStartFunctions -- run the start up functions

same -- whether everything in a list is the same

sameFunctionBody

saturate -- saturation of ideal or submodule

scan -- apply a function to each element

scanKeys -- apply a function to each key in a hash table or database

scanPairs -- apply a function to pairs in a hash table

scanValues -- apply a function to each value in a hash table

Schur -- make a Schur ring

SEEALSO -- crossreferences in documentation

see -- display engine object

select -- select elements from a list or hash table

selectInSubring -- select columns in a subring

sendgg -- send commands to engine

sendToEngine -- send a command string to the engine

separate -- split a string into pieces

seq

sequence -- make a sequence

set -- make a set

setEcho -- turn on echoing

setRandomSeed -- set starting point for random number generator

setrecursionlimit

setSpin -- set interval for spinning the cursor

sheaf -- make a coherent sheaf

sheafHom

showHtml

showTex -- convert to TeX and display on screen

SimplePowerMethod -- compute powers by multiplying

sin -- sine function

singleton -- make a sequence of length one

singularLocus -- singular locus

sinh -- hyperbolic sine function

size -- the size of an object

sleep -- sleep for a while

someTerms -- select some terms of a polynomial

sort -- sort a list

sortByDegree

sortColumns -- sort the columns of a matrix

source -- source of a map

Spec -- make an affine variety

splice -- remove subsequences

sqrt -- square root function

stack -- join nets or string vertically

standardForm -- convert to standard form

standardPairs

stats

status -- status of a resolution computation

sub

submatrix -- select part of a matrix

submodule

subquotient -- make a subquotient module

subsets -- produce all the subsets

substitute -- substitute values for variables

substring -- extract part of a string

subtable -- extract a subtable from a table

sum -- compute the sum

summary

super -- get the ambient module

symbolTable -- the global symbols

symmetricAlgebra -- the symmetric algebra of a module

symmetricPower -- symmetric power

SYprimaryDecomposition

syz -- the syzygy matrix

syzygyScheme -- construct a syzygy scheme

table -- make a table (nested list)

take -- take some elements from a list

tally -- tally the elements of a list

tan -- tangent function

tanh -- hyperbolic tangent function

target -- target of a map

temporaryFileName -- make a temporary file name

tensor -- tensor product

tensorAssociativity -- associativity isomorphisms for tensor products

terms -- provide a list of terms of a polynomial

TEST -- commands for testing later

tex -- convert to TeX

texMath -- convert to TeX math

text -- convert hypertext to text

times -- multiplication

toDual

toExternalString -- convert to a readable string

toField -- declare that a ring is a field

toFilename

toHandle -- convert to handle

toList -- list of elements

top -- compute the top dimensional components

topCoefficients -- list of top coefficients of a matrix

topicList -- list of help topics

toSequence -- convert to sequence

toString -- convert to a string

trace -- trace of a matrix

transnet -- assemble bytes into 4-byte integers

transpose -- transpose

trim -- simplify the presentation

truncate -- truncate the module at a specified degree

truncateOutput

ultimate -- ultimate value for an iteration

unhex -- translate a URL

uniform -- test whether elements of a list are of the same class

unique -- eliminate duplicates from a list

unlist

unSingleton

use -- install defaults

userSymbols -- a list of the user's symbols

value -- evaluate

values -- values in a hash table

variables

variety -- get the variety

vars -- variables

vector -- make a vector

verticalJoin

wait -- wait for child process

wedgeProduct -- the exterior multiplication map

width -- width of a file or net

xor -- logical exclusive-or

youngest -- the youngest member of a sequence

zero