Computer Algebra II: Magma, GAP, Singular, Pari/GP

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magmagapsingularpari/gp
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2.214.74.02.7
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$gap -h$ singular -vb$gp --version grammar and invocation magmagapsingularpari/gp interpreter$ singular -b foo.sing$cat hello.gp print("Hello, World!") quit$ gp -q hello.gp
Hello, World!
repl

online calculator$gap$ singular$gp block delimiters function( ) end if then elif then else fi while do od for do od { }{ } braces cannot be nested statement separator; A line can be broken anywhere, even inside a numeric literal or string, if the newline is preceded by a backslash: \ ; Two trailing semicolons ;; suppress echoing value of previous expression. ;newline or ; Newlines don't separate statements inside braces. A semicolon suppresses echoing value of previous expression. end-of-line comment 1 + 1; // addition1 + 1; # addition// comment1 + 1 \\ addition multiple line comment 1 + /* addition */ 1;none/* comment line another comment */ 1 + /* addition */ 1 variables and expressions magmagapsingularpari/gp assignmenta := 3;a := 3;int a; a = 3; // It is an error to assign to undeclared variable. x = 3.14 parallel assignmentnoneint a, b; a, b = 3, 4; [a, b] = [3, 4] compound assignmentnonenone+= -= *= /= \= \/= %= \\ bit operations: <<= >>= increment and decrementnonex++ x--postmodifiers: x++ x-- non-referential identifiernonenonenoneany unassigned identifier is non-referential identifier as valuex = 3 y = 'x global variablevariables are global by default local variabletmp = 19 add(x, y, z) = { \\ don't overwrite global tmp: my(tmp = x + y); tmp + z } \\ local keyword declares dynamic scope null none null test none undefined variable access errortreated as an unknown number remove variable bindingdelete x;kill(x) conditional expression if x lt 0 then -x; else x; end if;if(x > 0, x, -x) arithmetic and logic magmagapsingularpari/gp true and false true falsetrue false1 01 0 falsehoods falsefalse00 0.0 Mod(0, 5) Pol([0]) [0, 0, 0] [0, 0; 0, 0] [[0, 0], 0] logical operatorsnot true or (true and false);not true or (true and false)! 1 || (1 && 0)&& || ! relational operatorseq ne lt gt le ge eq raises an error when the operands are of different type. cmpeq does not. = <> < > <= >=== != < > <= >= <> is a synonym for != == != > < >= <= arithmetic operators+ - * / mod+ - * / mod the operators + - * / are overloaded for integers, rationals, and floats; other arithmetic functions aren't and there are no implicit conversions; use constructors to convert: Rat(3.1) Float(3) Float(31/10) Operators are for integers only unless base ring with coefficient field is declared. + - * / % mod is synonym for % + - * / % integer division a := 7; b := 3; div(a, b); QuoInt(a, b);int a, b = 7, 3; a div b; // a / b also performs integer division when // no base ring is declared. a \ b divrem(a, b)[1] \\ rounded integer division: a \/ b integer division by zeroUser errorerrorerrorerror float divisiondepending upon the types of a and b, the value can be an exact rational, a machine float, or an arbitrary precision float: a / b ring r = real,(x,y,z),(dp); 3.1 / 7.2; 7 / 3 float division by zeroRuntime errorerrorerrorerror power2 ^ 32;2 ^ 322 ^ 16 2 ** 16 2 ^ 32 sqrt Sqrt(2);2.0 ^ 0.5nonesqrt(2) sqrt -1 -1.0 ^ 0.5 evaluates to -1.none1.000 * I transcendental functionsExp Log Sin Cos Tan Arcsin Arccos Arctan Arctan2 arguments must be floats; no implicit conversion of integers to floats: Exp Log Sin Cos Tan Asin Acos Atan Atan2(y, x) noneexp log none sin cos tan asin acos atan none transcendental constants π and Euler's number FLOAT.PI FLOAT.ELIB "general.lib"; // print specified number of digits: number_pi(100); number_e(100); Pi exp(1) float truncation round towards zero, round to nearest integer, round down, round up Trunc Round Floor Ceiltruncate(x) round(x) floor(x) ceil(x) absolute value and signum AbsInt no absolute value for floats? SignInt SignFloat LIB "general.lib"; absValue(-7); ring r = real,(x,y,z),(dp); absValue(-7.1); abs(x) sign(x) integer overflow none, has arbitrary length integer typenone, has arbitrary length integer typemodular arithmetic with warningnone, has arbitrary length integer type float overflow # prints as inf: FLOAT.INFINTY error rational construction2 / 7;2 / 72 / 7 rational decomposition Numerator(2 / 7); Denominator(2 / 7 ); x := 2 / 7; NumeratorRat(x); DenominatorRat(x); x = 2 / 7 numerator(x) denominator(x) decimal approximation2 / 7 + 0. \\ change precision to 100: \p 100 2 / 7 + 0. complex construction none1 + 3 * I complex decomposition real and imaginary part, argument and modulus, conjugate nonereal(z) imag(z) arg(z) abs(z) conj(z) random number uniform integer, uniform float Random(0, 99); ?? rs := RandomSource(IsMersenneTwister); Random(rs, 0, 99); ?? random(100) random(1.0) random seed set, get SetSeed(42); seed := GetSeed((); rs := RandomSource(IsMersenneTwister, 17); State(rs); setrand(17) getrand() bit operatorsnone\\ left shift: 5 << 1 \\ right shift: 5 >> 1 binary, octal, and hex literalsnone radixIntegerToString(42, 7);none\\ 42 as powers of 7 up to 9th power: 42 + O(7^10) to array of digitsListOfDigits(1234); # other bases? \\ base 10: digits(1234) \\ base 2: digits(1234, 2) \\ number of digits in base 10: sizedigits(1234) strings magmagapsingularpari/gp string literal "don't say \"no\"""don't say \"no\""string s = "don't say \"no\"";"don't say \"no\"" newline in literal A line break in a string literal results in a newline in the string unless preceded by a backslash.noYes; runaway strings are possible; there is a predefined string variable newline which contains a single newline character.no; use \n escape literal escapes\" \\ \n \r \t\b \c \n \r \" \' \\ \ooo when writing to a buffered output stream, encountering a \c causes a flush of output. \" \\\n \t \" \\ concatenate "one " cat "two " cat "three"; "one " * "two " * "three"; &cat ["one ", "two ", "three "]; &* ["one ", "two ", "three "]; Concatenation("one ", "two ", "three");string s = "one" + "two" + "three";Str("one ", "two ", "three") concat("one ", "two ", "three") replicatehbar := "-" ^ 80; translate caseUppercaseString("foo"); LowercaseString("FOO"); trim none number to string "value: " * IntegerToString(8);Concatenation("value: ", String(8));"value: " + string(8)Str(8) \\ implicit conversion to string: concat("value: ", 8) string to number7 + Int("12"); 73.9 + Float(".037"); 7 + eval("12") 73.9 + eval(".037") string joina := ["foo", "bar", "baz"]; JoinStringsWithSeparator(a, ","); split Split("foo,bar,baz", ",");SplitString("foo,bar,baz", ","); substitute# replace all occurrences: ReplacedString("do re mi mi", "mi", "ma"); length # "hello";Length("hello");size("hello");length("hello") #"hello" index of substringIndex("hello", "el"); Position("hello", "el"); /* both return 0 if substring not found */ // evaluates to 2: find("hello", "el"); extract substringSubstring("hello", 2, 2);s := "hello"; s{[2..3]}; // start index and substring length: "hello"[2, 2] character literal 'h' character lookups := "hello"; # the character 'h': s[1]; # cannot use index notation on string literal // the character h: "hello"[1] chr and ordCharInt(65) IntChar('A') Strchr([65]) Vecsmall("A") delete characterss := "disemvowel me"; # no retval; modifies s in place: RemoveCharacters(s, "aeiou"); arrays magmagapsingularpari/gp literala := [1, 2, 3];[1, 2, 3]; # creates array with gap at fourth index; # reading a[4] causes an error: a := [1, 2, 3, , 5]; list a= 1, 2, 3; list a = list(1, 2, 3); // Singular lists are fixed length. \\ [1, 2, 3] is a vector literal: List([1, 2, 3]) size # [1, 2, 3];Length([1, 2, 3]);size(a);length(List([1, 2, 3])) #List([1, 2, 3]) lookup// indices start at one: [6, 7, 8][1]; # indices start at one: a := [1, 2, 3]; a[1]; // indices start at one: a[1]; \\ access time is O(1). \\ indices start at one: List([1, 2, 3])[1] update a[1] := 7;a[1] := 7;a[1] = 7;listput(a, 7, 1) out-of-bounds behaviorRuntime error on lookup. For update, size of array is increased if necessary; runtime error to look up unassigned slot in between assigned slots. Lookups result in errors; arrays can have gaps which also cause lookup errors. An update will expand the array, possibly creating gaps. errorout of allowed range error element index# returns 3: Position([7, 8, 9, 9], 9); # returns [3, 4]: Positions([7, 8, 9, 9], 9); none slice none array of integers as indexnone manipulate backa := [6, 7, 8]; // a not modified: a2 := Append(a, 9); // a2 not modified: a3 := Prune(a2); // a modified: Append(~a, 9); Prune(~a); a = [6, 7, 8]; Add(a, 9); elem := Remove(a); list a = list(6, 7, 8); list a2 = insert(a, 9, size(a)); int popme = a2[size(a2)]; list a3 = delete(a2, size(a2)); a = List([6, 7, 8]) listput(a, 9) elem = listpop(a) manipulate fronta = List([6, 7, 8]); listinsert(a, 5, 1); elem = a[1]; listpop(a, 1); head List([1, 2, 3])[1] tail none consa = List([1, 2, 3]); listinsert(a, 1, 1); concatenate Concatenation([1, 2, 3], [4, 5, 6]);list a1 = 1, 2, 3; list a2 = 4, 5, 6; list a3 = a1 + a2; concat(List([1, 2, 3]), List([4, 5, 6])) replicate copy a2 = a iterate Perform([1, 2, 3], function(x) Print(x); Print("\n"); end);a = List([1, 2, 3]) for(i=1, length(a), print(a[i])) reverse Reversed([1, 2, 3])a = List([1, 2, 3]) a2 = listcreate() while(i > 0, listput(a2, a[i]); i—) sortA := [3, 1, 4, 2] Sort(A); a = List([3,1,4,2]) listsort(a) a dedupe Set([1, 2, 2, 3]); Unique([1, 2, 2, 3]); Set([1, 2, 2, 3]) membership 2 in [1, 2, 3]\\ returns 1-based index of first occurrence \\ or 0 if not found: setsearch([1, 2, 3], 2) intersection Intersection(Set([1, 2]), Set([2, 3, 4]));setintersect([1, 2], [2, 3, 4]) union Union(Set([1, 2]), Set([2, 3, 4]));setunion([1, 2], [2, 3, 4]) relative complement, symmetric differencesetminus([1, 2, 3], [2]) ?? mapA := [1, 2, 3]; # modifies A: Apply(A, x -> x * x); apply(x -> x * x, [1, 2, 3]) filter select(x -> x > 2, [1, 2, 3]) reduce universal and existential tests min and max elementMinimum([1, 2, 3]) Maximum([1, 2, 3]) vecmin([1, 2, 3]) vecmax([1, 2, 3]) shuffle and sampleShuffle([1, 2, 3, 4]) flatten one level, completely # completely: Flat([1, [2, [3, 4]]]) zip cartesian productCartesian([1, 2, 3], ["a", "b", "c"]) sets magmagapsingularpari/gp literal{1, 2, 3}; size# {1, 2, 3}; add elements := {1, 2, 3}; Inlude(~s, 4); remove elements := {1, 2, 3}; Exclude(~s, 1); membership test7 in {6, 7, 8}; disjoint testIsDisjoint({1, 2, 3}, {2, 3, 4}); union// {1, 2, 3, 4}: s := {1, 2, 3} join {2, 3, 4}; intersection// {2, 3}: s := {1, 2, 3} meet {2, 3, 4}; relative complement// {1}: s := {1, 2, 3} diff {2, 3, 4}; arithmetic sequences magmagapsingularpari/gp unit difference[1 .. 100];[1 .. 100][1 .. 100] vector(100, i, i) difference of 10[1 .. 100 by 10];[1,11 .. 91]vector(10, i, 10 * i - 9) difference of 0.1[0.1 * i: i in [1 .. 1000]];nonevector(1000 - 9, i, i / 10 + 9 / 10) dictionaries magmagapsingularpari/gp literal // None, just empty constructor: d := AssociativeArray(); d["t"] := 1; d["f"] := 0; # dictionary constructor; type of keys derived # from first arg: d := NewDictionary("", true); AddDictionary(d, "t", 1); AddDictionary(d, "f", 0); # record literal with identifier keys: r := rec(t := 1, f := 0); # record literal with string keys: r2 := rec(("t") := 1, ("f") := 0); size # d;# no way to get size of dictionary? Length(RecNames(r)); lookup d["t"];LookupDictionary(d, "t"); # the same key can be looked up with identifier # or string notation: r.t; r.("t"); updated["f"] := -1;AddDictionary(d, "f", -1); r.f := -1; r2.("f") := -1; missing key behavior Runtime error# returns special object "fail": LookupDictionary(d, "not_a_key"); # raises an error: r.not_a_key; is key present IsDefined(d, "t");KnowsDictionary(d, "t"); # RecNames returns keys as strings: "t" in RecNames(r); deleteRemove(~d, "t"); iterate # no way to iterate over dictionary? for i in RecNames(r) do Print(r.(i)); od; keys and values as arraysKeys(d); ?? RecNames(r); functions magmagapsingularpari/gp define functionadd := function(a, b) return a + b; end function; // no return value: show := procedure(s) print(s); end procedure; add := function(x, y) return x + y; end; proc add(int x, int y) { return(x + y); } add(x, y) = x + y \\ function body w/ sequence of statements: say(s1, s2, s3) = print(s1); print(s2); print(s3) \\ function body w/ newlines: dire(s1, s2, s3) = { print(s1); print(s2); print(s3); } invoke functionadd(3, 7);add(3, 7);add(3, 7);add(3, 7) undefine functionkill(add) redefine functionadd(x, y, z) = x + y + z overload functionnone missing function behavior"not a function" error missing argument behaviorset to zero extra argument behavior"too many parameters" error default argumentmylog(x = 1, base = 10) = log(x) / log(base) \\ log10(3): mylog(3) \\ ln(3): mylog(3, exp(1)) \\ ln(1): mylog(, exp(1)) \\ If neither caller nor function definition \\ provide a value, zero is used. return value anonymous function# unary functions only? f := x -> x * x; f2 := function(x, y) return 2 * x + 3 * y; end; f = (x, y) -> x + y f(1, 2) variable number of arguments pass array elements as separate arguments execution control magmagapsingularpari/gp ifif n gt 0 then print "positive"; else if n lt 0 then print "negative"; else print "zero"; end if; end if; if x > 0 then Print("positive\n"); elif x < 0 then Print("negative\n"); else Print("zero\n"); fi; int x = -3; if (x > 0) { print("positive"); } else { if (x < 0) { print("negative"); } else { print("zero"); } }; if(x > 0, \ print("positive"), \ if(x < 0, \ print("negative"), \ print("zero"))) whilei := 0; while i lt 10 do print i; i := i + 1; end while; i := 0; while i < 10 do Print(i, "\n"); i := i + 1; od; int i = 0; while (i < 10) { print(i); i++; }; i = 0 while(i < 10, print(i); i++) forfor i := 0 to 9 by 1 do print i; end for; for i in [0..9] do Print(i, "\n"); od; int i; for (i = 0; i < 10; i++) { print(i); }; for(i = 0, 9, print(i)) breakbreakbreakbreak continuecontinuecontinuenext exceptions magmagapsingularpari/gp raise exceptionerror "failed";Error("failed");error("failed") handle exceptiontry error "failed"; catch e print "caught error"; end try; iferr(error("failed"), E, \ print(errname(E), ": ", component(E, 1))) uncaught exception behaviorError() invokes the GAP debugger. Type Quit; to return to REPL. streams magmagapsingularpari/gp write line to stdoutprint "hello";Print("hello");print("hello");print("hello") read file into array of stringsa = readstr("/etc/hosts") processes and environment magmagapsingularpari/gp environment variablegetenv("HOME") [[#external-cmd]]external commandsystem("ls /etc") command substitution\\ array of output lines: lines = externstr("ls /etc") libraries and namespaces magmagapsingularpari/gp load libraryRead('foo.g');LIB "add.sing"; define library$ cat add.sing
version="1.0"
category="misc"
info="an add function"

proc add(int x, int y) {
return(x + y);
}
library pathSearches current directory; additional directories can be added to the search path by adding them separated by colons to the/ SINGULARPATH //environment variable.
reflection
magmagapsingularpari/gp
list function documentation?
get function documentationhelp killall;? tan
query data typeType(x);typeof(x);type(x)
list types\t
list variables in scopelistvar();variable()
list built-in functions?*
list metacommands?\
search documentation??DirectProduct??? modulus
vectors
magmagapsingularpari/gp
vector literal# row vector is same as array:
[1, 2, 3]
// A ring must be declared. The elements of the vector are not
// limited to the coefficient field, but can be any element of
// the ring.

ring r = read, (x), dp;

vector v= [1.5, 3.2, 7.1];
[1, 2, 3]
constant vector

all zeros, all ones
vector(100, i, 0)
vector(100, i, 1)
vector coordinatevec := [1, 2, 3];
# indices start at one:
v[1];
v[1];\\ indices start at one:
[1, 2, 3][1]
vector dimension

Length([1, 2, 3])nrows(v);length([1, 2, 3])
#[1, 2, 3]
element-wise arithmetic operators+ - * /+ -+ -
vector length mismatch

shorter vector is zero-paddedshorter vector is zero-paddederror
scalar multiplication3 * [1, 2, 3];
[1, 2, 3] * 3;
3 * v;
v * 3;

// Scalar can be any ring element:
(1 + x) * v;
3 * [1, 2, 3]
[1, 2, 3] * 3
dot product[1, 1, 1] * [2, 2, 2][1, 1, 1] * [2, 2, 2] ~
cross product
normsvec = [1, 2, 3]

normlp(vec, 1)
normlp(vec, 2)
normlp(vec)
orthonormal basis
matrices
magmagapsingularpari/gp
literal[[1, 2], [3, 4]]// A ring must be declared. The elements of the matrix are not
// limited to the coefficient field, but can be any element of
// the ring.

ring r = read, (x), dp;

matrix m[2][2] = 1, 2, 3, 4;
[1, 2; 3, 4]

\\ from rows:
row1 = [1, 2]
row2 = [3, 4]
matconcat([row1; row2])
construct from columnscol1 = [1, 3]~
col2 = [2, 4]~
matconcat([col1, col2])
construct from submatricesA = [1, 2; 3, 4]
B = [4, 3; 2, 1]
\\ 4x4 matrix:
C = matconcat([A, B; B, A])
constant matricesmatrix(3, 3, i, j, 0)
matrix(3, 3, i, j, 1)

\\ 3x3 Hilbert matrix:
matrix(3, 3, i, j, 1 / (i + j - 1))
diagonal matrices
and identity
DiagonalMat([1, 2, 3])
IdentityMat(3)
matdiagonal([1, 2, 3])
matid(3)
dimensions# returns [3, 2]:
DimensionsMat([[1, 2], [3, 4], [5, 6]])
nrows(m);
ncols(m);
\\ [3, 2]:
matsize([1, 2; 3, 4; 5, 6])
element lookupA := [[1, 2], [3, 4]];

# top left corner:
A[1][1]
matrix m[2][2] = 1, 2, 3, 4;

m[1][1];
\\ top left corner:
A[1, 1]
extract row\\ first row:
[1, 2; 3, 4][1, ]
extract column\\ first column:
[1, 2; 3, 4][, 1]
extract submatrixA = [1, 2, 3; 4, 5, 6; 7, 8, 9]

vecextract(A, "1..2", "1..2")
element-wise operators+ -+ -+ -
productA := [[1, 2], [3, 4]];
B := [[4, 3], [2, 1]];
A * B;
matrix m[2][2] = 1, 2, 3, 4;
matrix m2[2][2] = 4, 3, 2, 1;

m * m2;
A = [1, 2; 3, 4]
B = [4, 3; 2, 1]
A * B
power[[1, 2], [3, 4]] ^ 3[1, 2; 3, 4] ^ 3
exponential
log
kronecker productA := [[1, 2], [3, 4]];
B := [[4, 3], [2, 1]];
KroneckerProduct(A, B);
norms
transposeA~
mattranspose(A)
conjugate transposeconj([1, I; 2, -I] ~)
inverseInverse([[1, 2], [3, 4]])[1, 2; 3, 4] ^ -1
1 / [1, 2; 3, 4]
row echelon form
pseudoinverse
determinantDeterminant([[1, 2], [3, 4]])matrix m[2][2] = 1, 2, 3, 4;

det(m);
matdet([1, 2; 3, 4])
traceTrace([[1, 2], [3, 4]])matrix m[2][2] = 1, 2, 3, 4;

trace(m);
trace([1, 2; 3, 4])
rankRankMat([[1, 1], [0, 0]])matrank([1, 1; 0, 0])
nullspace basismatker([1, 1; 0, 0])
range basismatimage([1, 1; 0, 0])
eigenvalues[vals, vecs] = mateigen([1, 2; 3, 4], flag=1)
eigenvectorsmateigen([1, 2; 3, 4])
singular value decomposition
qr decompositionmatqr([1, 2; 3, 4])
solve system of equationsA = [1, 2; 3, 4]
matsolve(A, [2, 3]~)
combinatorics
magmagapsingularpari/gp
factorialFactorial(10);Factorial(10);LIB "general.lib";

factorial(10);
10!
binomial coefficientBinomial(10, 3);Binomial(10, 3);LIB "general.lib";

binomial(10, 3);
binomial(10, 3)
multinomial coefficientMultinomial(12, [3, 4, 5]);
integer partitions

and count
Partitions(10);

NumberOfPartitions(10);
Partitions(10);

NrPartitions(10);
partitions(10)

length(partitions(10))
set partitions

and Bell number
StirlingSecond(10, 3);

Bell(10);
stirling(10, 3, 2)
sum(i=1, 10, stirling(10, i, 2))
permutations with k disjoint cyclesAbs(StirlingFirst(n, k));abs(stirling(n, k, 1))
fibonacci number

and lucas number
Fibonacci(10);
Lucas(10);
fibonacci(10)
bernoulli numberBernoulliNumber(100);bernfrac(100)
catalan numberCatalan(10);
number theory
magmagapsingularpari/gp
pseudoprime testIsProbablyPrime(7);IsPrimeInt(7);ispseudoprime(7)
true prime test

IsPrime(7);isprime(7)
divisors// [1, 2, 4, 5, 10, 20, 25, 50, 100]:
Divisors(100);
DivisorsInt(100);divisors(100)
prime factors// [2, 3, 7]:
PrimeDivisors(84);
# [ 2, 2, 3, 7 ]:
FactorsInt(84);
\\ [2,2; 3,1; 7,1]:
factor(84)
next prime

and preceding
NextPrime(1000);
PreviousPrime(1000);
NextPrimeInt(1000);
PrevPrimeInt(1000);
?
prime(1000);
nextprime(1000)
precprime(1000)
nth primeNthPrime(100);\\ first 100 primes: primes(100)
primes(100)[100]
prime counting functionLIB "general.lib";

size(primes(1, 100));
primepi(100)
divmod

divrem(7, 3)
greatest common divisor

and relatively prime test
Gcd(14, 21);
Gcd(Gcd(14, 21), 777);
GcdInt(14, 21);
GcdInt(GcdInt(14, 21), 777);
gcd(14, 21)
gcd(gcd(14, 21), 777)
extended euclidean algorithmret := Gcdex(3, 5);

# 2:
ret.coeff1;
# -1:
ret.coeff2;
# 1:
ret.gcd;
\\ [2, -1, 1]:
gcdext(3, 5)
least common multipleLcm(14, 21);LcmInt(14, 21);lcm(14, 21)
integer residuesr := ZmodnZ(5);
fam := ElementsFamily(FamilyObj(r));;

ZmodnZObj(fam, 2) + ZmodnZObj(fam, 3);
Mod(2, 5) + Mod(3, 5)
Mod(2, 5) - Mod(3, 5)
Mod(2, 5) * Mod(3, 5)
Mod(2, 5)^2
multiplicative inverser := ZmodnZ(7);
fam := ElementsFamily(FamilyObj(r));;

ZmodnZObj(2, 7)^-1;
Mod(2, 7)^-1

\\ raises error:
Mod(2, 4)^-1
chinese remainder theorem# 173:
ChineseRem([17, 11], [3, 8]);
\\ Mod(173, 187):
chinese(Mod(3, 17), Mod(8, 11))
lift integer residue\\ 7:
lift(-17, 12)
\\ -5:
centerlift(-17, 12)
euler totient

Phi(256);eulerphi(256)
multiplicative orderOrderMod(7, 108);znorder(Mod(7, 108))
primitive rootsPrimitiveRootMod(11);znprimroot(11)
discrete logarithm# arg: 10, base: 2, modulus: 11
LogMod(10, 2, 11);
znlog(10, Mod(2, 11))
znlog(Mod(10, 11), Mod(2, 11))
carmichael functionLambda(561);lcm(znstar(561)[2])
kronecker symbol

and jacobi symbol
Jacobi(3, 5);kronecker(3, 5)
moebius functionMoebiusMu(11);MoebiusMu(11);moebius(11)
riemann zeta functionzeta(2)
mangoldt lambda
dirichlet character
elliptic curves
magmagapsingularpari/gp
elliptic curve from coefficients\\ ellinit([a, b, c, d, e]) where
\\
\\   y^2 + axy + by = x^3 + cx^2 + dx + e
\\

e0 = ellinit([0,0,1,-7,6])

\\ ellinit([a, b]) where
\\
\\   y^2 = x^3 + ax + b
\\

e1 = ellinit([-1, 0])
discriminante0.disc
conductorellglobalred(e0)[1]
singularity teste0.disc == 0
convert to minimal modele0 = ellinit([6, -3, 9, -16, -14])
e = ellminimalmodel(e0)
coordinate transformation on pointe0 = ellinit([6, -3, 9, -16, -14])
e = ellminimalmodel(e0, &v)
\\ minimal to original:
ellchangepointinv([0, 0], v)
\\ original to minimal:
ellchangepoint([-2, 2], v)
coordinate transformation on curve: ellchangecurvee0 = ellinit([6, -3, 9, -16, -14])
e = ellminimalmodel(e0, &v)
\\ same as e0:
ellchangecurve(e, v)
point on curve testellisoncurve(e, [0, 2])
abscissa to ordinates\\ vector of size 0, 1, or 2:
ellordinate(e, 0)
group identity[0]
group operationelladd(e, [0, 2], [1, -1])
group inverseellneg(e, [0, 2])
group multiplicationellmul(e, [0, 2], 3)
canonical height of pointellheight(e, [0, -3])
order of point\\ returns 0 for infinite order:
ellorder(e, [0, 2])

ellorder(e1, [0, 0])
torsion subgroupe1 = ellinit([-1, 0])

\\ returns [t, v1, v2]:
\\
\\   t: order of torsion group
\\   v1: orders of component cyclic groups
\\   v2: generators of same cyclic groups
\\

elltors(e1)
analytic rank\\ first value is rank:
[a, b] = ellanalyticrank(e)

\\ recompute second value to higher precision:
\p 100
b = ellL1(e, a)
L-function valueelllseries(e, 1 + I)
L-function coefficients\\ tenth coefficient:
ellak(e, 10)
\\ first ten coefficients:
ellan(e, 10)
rational and algebraic numbers
magmagapsingularpari/gp
to continued fraction\p 100
contfrac(Pi)
from continued fraction
p-adic number\\ p is 2 and precision in powers of 2 is 100:
1/2 + O(2^100)
lift p-adic to rationallift(1/2 + O(2^100))
gaussian integer normnorm(1 + I)
quadratic extension\\ make w equal to sqrt(D)/4:
D = -4
w = quadgen(D)
quadratic number(1 + w)^2
polynomials
magmagapsingularpari/gp
from expression with indeterminates// A ring must be declared before a polynomial can be defined.
// To define a ring, one specifies (1) the coefficient field
// (or ring), (2) the indeterminate variables, and (3) the term
// ordering.

ring r = integer, (x, y), dp;

poly p1 = (x - 1) * (x - 2);
poly p2 = (1 + x)^2 * (2 + y)^3;
(x - 1) * (x - 2)

(1+x)^2 * (2+y)^3
from coefficient arrayPol([1, -3, 2])

@@\\@ zero-degree coefficient first:
Polrev([2, -3, 1])
to coefficient arraypoly p = (1 + x)^10;
coeffs(p, x);
Vec((x+1)^10)
lookup coefficientpolcoeff((x+1)^10, 3)
substitute indeterminatesubst((x - 1) * (x - 2), x, 3);

subst((x - 1) * (x - 2), x, (x - 1));
\\ replace x with 3:
subst((x-1)*(x-2), x, 3)
\\ replace x with (x-1):
subst((x-1)*(x-2), x, (x-1))
degreepoly p = (1 + x)^10;
deg(p);
poldegree((x-1)^10)
operations+ - * /
division and remainder
expand polynomialPolynomials are displayed in expanded form.
factor polynomialring r = 7, (x), dp;

// Doesn't work in ring with real or int. coefficients:
factorize(x^2 + 3*x + 2);
collect termsring r = integer, (x, y), dp;

coeffs((x + 2*y + 1)^10, x);
factorfactor(x^2-1)
rootspolroots(x^3+3*x^2+2*x-1)
greatest common divisorp1 = x^3 + 2*x^2 -x -2
p2 = x^3 -7*x + 6
gcd(p1, p2)
resultantpolresultant((x-1)*(x-2), (x-3)^2)
discriminantpoldisc((x+1)*(x-2))
homogenity testhomog((x + y)^2);
groebner basisnone
specify orderingnone
symmetric polynomialnone
cyclotomic polynomialpolcyclo(10)
hermite polynomialpolhermite(4)
chebyshev polynomial

first and second kind
polchebyshev(4, 1)
polychebyshev(4, 2)
interpolation polynomialpolinterpolate([1, 2, 3], [2, 4, 7])
characteristic polynomialcharpoly([1, 2; 3, 4])
minimal polynomial
piecewise polynomial
rational function(x - 1) / (x - 2)^2
add fractions
partial fraction decomposition
special functions
magmagapsingularpari/gp
gammagamma(1/2)
hyperbolicsinh cosh tanh
elliptic integerals
bessel functionsbesselh1 besselh2 besseli besselj besseljh besselk besseln
Riemann zetazeta(2)
permutations
magmagapsingularpari/gp
permutation from disjoint cyclesS4 := Sym(4);
p := S4!(1, 2)(3, 4);
p := (1, 2)(3, 4);
permutation from listS4 := Sym(4);
p2 := elt<S4 | 2, 1, 4, 3>;
p2 := PermList([2, 1, 4, 3]);
permutation from two lists# must be positive integers:
p := MappingPermListList([6, 8, 4, 2], [2, 4, 6, 8])
act on element1 ^ p;

# preimage of 1 under p:
1 / p;
act on list
composeS4 := Sym(4);

S4!(1,2)(3,4) * S4!(1,3);
(1, 2)(3, 4) * (1, 3);
invertS3 := Sym(3);

S3!(1,2,3) ^ -1;
Inverse(S3!(1,2,3));
(1, 2, 3) ^ -1;
powerS5 := Sym(5);

S5!(1,2,3,4,5) ^ 3;
(1, 2, 3, 4, 5) ^ 3;
orderS3 := Sym(3);

Order(S3!(1,2,3));
Order((1, 2, 3));
supportMovedPoints((1, 3, 5)(7, 8));
number of inversions
parity
to inversion vector
from inversion vector
all permutations
random permutation
groups
magmagapsingularpari/gp
named groups

symmetric, alternating, cyclic, dihedral
S4 := Sym(4);
A4 := AlternatingGroup(4);
Z5 := CyclicGroup(5);
D10 := DihedralGroup(10);
S4 := SymmetricGroup(4);
A4 := AlternatingGroup(4);
Z5 := CyclicGroup(5);
# argument is order of group, not vertices
# of polygon:

D10 := DihedralGroup(2 * 10);
group by orderNumberOfSmallGroups(8);

// first group of order 8:
G := SmallGroup(8, 1);
# number of groups of order 8:
Length(AllSmallGroups(8));

# first group of order 8:
G := AllSmallGroups(8)[1];
group from permutation generatorsS5 := Sym(5);
p1 := S5!(1, 3, 5, 2);
p2 := S5!(1, 2);
G := PermutationGroup<5 | p1, p2>;
G := Group((1, 3, 5, 2), (1, 2));

# or
G := GroupWithGenerators([(1, 3, 5, 2), (1, 2)]);
direct productZ3 := CyclicGroup(3);
A4 := AlternatingGroup(4);
G := DirectProduct(z3, a4);
Z3 := CyclicGroup(3);
A4 := AlternatingGroup(4);
G := DirectProduct(Z3, A4);
free product

F := FreeProduct(CyclicGroup(3), CyclicGroup(2));
free group# integers under addition:
Z := FreeGroup(1);

# free group with 2 generators:
F := FreeGroup("a", "b");
group from presentationF := FreeGroup( "a", "b" );
G := F / [ F.1^2, F.2^3, (F.1 * F.2)^5 ];
all elementsPerform(Enumerator(Z3), function(e)
Print(e);
Print("\n");
end);
generatorsS10 := Sym(10);
Generators(s10);

// notation for individual generators:
S10.1;
S10.2;
S10 := SymmetricGroup(10);
# return generators in an array:
GeneratorsOfGroup(S10);

# notation for individual generators:
S10.1;
S10.2;
identity elementId(G);Identity(G);
One(G);
random elementRandom(G);Random(G)
group operatione1 := Random(G);
e2 := Random(G);
e1 * e2;
e1 := Random(G);
e2 := Random(G);
e1 * e2;
inverse elementInverse(e1);
// or:
e1 ^ -1;
Inverse(e1);
# or:
e1^-1;
generator word for elementS10 := SymmetricGroup(10);
Factorization(S10, (1,3,8,10,5,9,2,7));
elements by generator word lengthS6 := SymmetricGroup(6);
GrowthFunctionOfGroup(S6);
element orderD10 := DihedralGroup(10);
Order(D10.1);
Order(D10.2);
D10 := DihedralGroup(2 * 10);

Order(D10.1);
Order(D10.2);
identify group// C2*C4:
GroupName(SmallGroup(8, 2));

// <8, 1>:
IdentifyGroup(CyclicGroup(8));
# C4 x C2:
StructureDescription(AllSmallGroups(8)[2]);
group to presentationFPGroup(AlternatingGroup(10));
group orderOrder(Sym(4));Size(SymmetricGroup(4));
cyclic testIsCyclic(AlternatingGroup(10));IsCyclic(AlternatingGroup(10));
abelian testIsAbelian(CyclicGroup(10));IsAbelian(CyclicGroup(10));
subgroups
magmagapsingularpari/gp
all subgroupsSubgroups(Sym(4));AllSubgroups(SymmetricGroup(4));
subgroup latticeSubgroupLattice(Sym(4));S4 := SymmetricGroup(4);
lat := LatticeSubgroups(S4);
DotFileLatticeSubgroups(lat, "lattice.dot");
# dot -Tpng < lattice.dot > lattice.png
maximal subgroupsMaximalSubgroups(Sym(4));MaximalSubgroups(S4);
frattini subgroupD4 := DihedralGroup(4);
FrattiniSubgroup(D4);
FrattiniSubgroup(DihedralGroup(2 * 4));
subgroup from generatorsG := Group((1, 3, 5, 7), (2, 4));
H := Subgroup(G, [(2, 4)]);
normal subgroupsNormalSubgroups(Sym(4));NormalSubgroups(S4);
cosetsRightCoset()
CanonicalRightCosetElement()
CosetDecomposition()
RightTraversal(G, U)
quotient group
centerZ4 := CyclicGroup(4);
D6 := DihedralGroup(6);
G := DirectProduct(Z4, D6);
center := Centre(G);
g := DirectProduct(CyclicGroup(4), DihedralGroup(2 * 6));
Center(g);
centralizerg := SymmetricGroup(5);
h := Centralizer(g, (1, 3)(4, 5));
normalizerS4 := SymmetricGroup(4);
G := Group([(1,2)(3,4)]);
Normalizer(S4, G);
commutator# e1^-1 * e2^-1 * e1 * e2:
Comm(e1, e2);
commutator subgroupG1 := Group((1,2,3),(1,2));
G2 := Group((2,3,4),(3,4));
CommutatorSubgroup(G1, G2);
subgroup test
subgroup indexIndex(Sym(4), AlternatingGroup(4));Index(G, H);
normal testIsNormal(G, H);
subnormal testIsSubnormal(G, H);
nonabelian simple groups# argument is list of orders:
AllSmallNonabelianSimpleGroups([1..10000]);
simple testIsSimple(SymmetricGroup(4));
solvable testIsSolvable(SymmetricGroup(4));
derived seriesDerivedSeriesOfGroup(SymmetricGroup(4));
characteristic testS4 := SymmetricGroup(4);
H := Subgroup(s4, [(1,4)(2,3), (1,3)(2,4), (2,4,3)]);
IsCharacteristicSubgroup(S4, H);
semidirect product
group homomorphisms
magmagapsingularpari/gp
all homomorphismsS4 := SymmetricGroup(4);
S3 := SymmetricGroup(3);
AllHomomorphisms(S3, S4);
all homomorphims classesAllHomomorphismClasses(S3, S4);
endomorphisms and automorphismsAllEndomorphisms(S4);
AllAutomorphisms(S4);
homomorphism from generator imageshom := GroupHomomorphismByImages(S3, S4,
[(1,2,3), (1,2)],
[(2,3,4), (2,3)]);

# uses generators of S3:
hom := GroupHomomorphismByImages(S3, S4,
[(2,3,4), (2,3)]);
surjective testIsSurjective(hom);
injective testIsInjective(hom);
bijective testIsBijective(hom);
kernelKernel(AllHomomorphisms(S3, S4)[1]);
imageImage(AllHomomorphisms(S3, S4)[1]);
actions
magmagapsingularpari/gp
conjugate element# (1,2,3)^-1 * (1,2) * (1,2,3):
(1,2)^(1,2,3)
conjugate setS3 := SymmetricGroup(3);
S3^(3,4);
(3,4)^S3;
conjugacy classS4 := SymmetricGroup(4);
AsList(ConjugacyClass(S4, (1,2,3)));
conjugate groupConjugateGroup(SymmetricGroup(4), (4, 5));
conjugacy classesConjugacyClasses(SymmetricGroup(4));
stabilizer
orbit
transitive test
____________________________________________________________________________________________________________________________________________________________________________________________________________________________

# Combinatorics

## factorial

n! is the product of the integers 1 through n.

n! is the number of permutations or bijections on a set of n elements.

## integer partitions

The number of ways of writing a positive integer n as a sum of positive integers.

When counting integer partitions, 2 + 1 and 1 + 2 are not considered to be distinct ways of summing to 3. In other words, one is counting the multisets which sum to n.

# Groups

A group is a set G and a binary operator—written here as*—which takes elements of the set as operands and obeys the following axioms:

• Closure: For every g and h in G, g * h is also in G.
• Identity: There exists e in G such that for all g in G, e * g = g * e = g.
• Associativity: For all f, g and h in G, (f * g) * h = f * (g * h).
• Inverse: For every g in G, there exists g' in G such that g * g' = g' * g = e.

Abelian groups obey an additional axiom:

• Commutativity: For all g and h in G, g * h = h * g.

The order of a group is the number elements in the set. The smallest group is the trivial group, which contains a single element which is the identity. It has order 1.

The integers, rationals, real numbers, and complex numbers are Abelian groups under addition; zero is the identity element. The integers and rationals are countably infinite. The real numbers and complex numbers are uncountably infinite.

The integers modulo n are an Abelian group under addition; zero is the identity number. The group is finite and has order n.

The non-zero rationals, non-zero real numbers, and non-zero complex numbers are Abelian groups under multiplication; one is the identity element.

A permutation is a bijection on a set of n elements. The permutations of size n form a group under composition; the group is non-Abelian when n > 2. The identity permutation which maps each element of the set to itself is the group identity. The order of the group is n!.

The classical Lie groups are examples of infinite, non-Abelian groups. In all cases the group operation is matrix multiplication:

 GL(n, ℝ) general linear group of degree n invertible n×n matrices SL(n, ℝ) special linear group of degree n n×n matrices with determinant one O(n, ℝ) orthogonal group of degree n n×n orthogonal matrices; i.e. MMT = I SO(n, ℝ) special orthogonal group of degree n n×n orthogonal matrices with determinant one U(n, ℂ) unitary group of degree n n×n unitary matrices; i.e. MM* = I SU(n, ℂ) special unitary group of degree n n×n unitary matrices with determinant one

## named groups

Some well-known groups can be specified by name.

The symmetric group Sn is the group of permutations on a set of n elements.

The alternating group An is the group of even permutations on a set of n elements.

The cyclic group Z/n is a group generated by a single element of order n. It is Abelian group and is isomorphic to the integers modulo n under addition.

The dihedral group Dn is the symmetry group of the regular n-polygon.

## group from permutation generators

gap:

When a group is created using GroupByGenerators, the generators returned by GeneratorsOfGroup will not necessarily be the same as the generators provided to the constructor.

If the group is created using GroupWithGenerators, then the generators returned by GeneratorsOfGroup will be the same.

## direct product

The direct product is a group defined on the Cartesian product of two groups.

Given groups G and H, elements g, g' in G, and elements h, h' in H, the group operation is (g, h) * (g', h') = (g * g', h * h').

## free product

A free product is group defined on equivalence classes of words consisting of elements from two groups G and H.

The equivalence relation is defined in terms of reductions. Two adjacent elements in the word from the same group can be replaced by their product, and the identity element of either group can be removed.

The group operation is concatenation.

The free product is an infinite group when both groups are non-trivial.

gap

Some functions do not appear to work on free products: Order() and Random().

## all elements

How to iterate through all elements in a group.

## identity element

The group identity e.

For all g in G, e * g = g * e = g.

## random element

How to select an element from the group randomly.

How

## inverse element

Each element g in a group has an inverse g-1 such that g * g-1 = g-1 * g = e where e is the identity element.

## element order

For an element g in a finite group, there must be a positive integer n such that gn is the identity element. The order of an element is the smallest such n.

## group to presentation

Provide a presentation for the group.

## group order

How many elements are in the group.

## cyclic test

Is the group generated by single element.

## abelian test

Is the group operation commutative.

# Subroups

A subgroup is a subset of a group which is itself a group.

A nontrivial group always has at least two subgroups: the group itself and the trivial subgroup. A proper subgroup is a subgroup which is not equal to the group itself.

One test whether a subset H of a group G is a subgroup is verify that H is nonempty and for every x and y in H, xy-1 is in H.

Lagrange's theorem and Sylow Theorems

## all subgroups

A list of all subgroups in a group.

## subgroup lattice

The subgroups arranged by inclusion in a lattice.

## maximal subgroups

A maximal subgroup is a proper subgroup which is not contained in any other proper subgroup.

## frattini subgroup

The Frattini subgroup is the intersection of all maximal subgroups.

## subgroup from generators

gap:

ClosureGroup finds the smallest group containing a group and a list of elements, some of which might not be in the group which is the first argument.

## normal subgroups

The normal subgroups of a group.

A subgroup H of a group G is normal if its left and right cosets coincide. That is gH = Hg for all g in G. Another way to express this is that H is invariant under conjugation, or that H = g-1Hg for all g in G.

The significance of normal subgroups is that we can define a group operation on the cosets using representatives if and only if the subgroup is normal.

The first group isomorphism theorem states that the kernel of a group homomorphism is a normal subgroup.

For n ≥ 5, the only proper non-trivial normal subgroup of Sn is An.

# Group Homomorphisms

A homomorphism is a function φ from (G, *) to (H, *') such that

• φ(x * y) = φ(x) *' φ(y) for all x, y ∈ G.

An isomorphism is a bijective homomorphism.

If an isomorphism exists between two groups, they are said to be isomorphic. Isomorphic groups are in a sense the same; group theory is the study of properties which are invariant under isomorphism. Elsewhere we may speak of two isomorphic groups as being the same group.

# Actions

A group G is said to act on a set A if there is an operation ⋅: G × A → A such that

• g₁⋅(g₂⋅a) = (g₁*g₂)⋅a for all g₁, g₂ ∈ G and a ∈ A
• e⋅a = a for all a ∈ A where e is the identity in A

# GAP

GAP - Reference Manual

Singular Manual

# Pari/GP

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