was@form:~/129/getmagma$ was@form:~/129/getmagma$ Magma V2.11-10    Thu Feb 17 2005 11:07:42 on form     [Seed = 4272455044]
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> A := Matrix(2,2,[1,2,3,4]);
> A;
[1 2]
[3 4]
> SmithForm(A);
> SmithForm(A);
[1 0]
[0 2]

[ 1  0]
[-1  1]

[-1  2]
[ 1 -1]
> SmithForm;
Intrinsic 'SmithForm'

Signatures:

    (<Mtrx> X) -> Mtrx, AlgMatElt, AlgMatElt
    [
        k: RngIntElt, 
        NormType: MonStgElt, 
        Partial: BoolElt, 
        RightInverse: BoolElt
    ]

        The smith form S of X, together with unimodular matrices 
        P and Q such that P * X * Q = S.


> n := 50; A := Matrix(n,n,[Random(100) : i in [1..n^2]]); time B:= SmithForm(A);
Time: 0.030
> n := 100; A := Matrix(n,n,[Random(100) : i in [1..n^2]]); time B:= SmithForm(A);
Time: 0.420
> n := 150; A := Matrix(n,n,[Random(100) : i in [1..n^2]]); time B:= SmithForm(A);
Time: 2.510
> n := 200; A := Matrix(n,n,[Random(100) : i in [1..n^2]]); time B:= SmithForm(A);
Time: 9.600
> 
>Q := RationalField();
> Q;
Rational Field
> Type(Q);
FldRat
> R<x> := PolynomialRing(Q);
> x;
x
> x in R;
true
> x^3 + 1;
x^3 + 1
> (x^3 + 1)^5;
x^15 + 5*x^12 + 10*x^9 + 10*x^6 + 5*x^3 + 1
> K<a> := NumberField(x^3-2);
> a;
a
> a in K;
true
> x in K;

>> x in K;
     ^
Runtime error in 'in': Bad argument types
> a in R;

>> a in R;
     ^
Runtime error in 'in': Bad argument types
> a^3;
2
> a^2;
a^2
> (a^2+a+10)^100;
1597545916784017656835196222164318853777480198287220038244483474\
    34237714547208741787168948657103837040069312512*a^2 + 
    201278172872978720642893096426089114904360174248554584523278\
    378810002098794754359916854560500609472348163145728*a + 
    253594606887045126582752821019824261365197098233169658224970\
    924663014477823176413602486220432339643744947011584
> R := RingOfIntegers(K);
Maximal Equation Order with defining polynomial x^3 - 2 over ZZ
> R := RingOfIntegers(K);
> R;
Maximal Equation Order with defining polynomial x^3 - 2 over ZZ
> Basis(R);
[
    R.1,
    R.2,
    R.3
]
> B := Basis(R);
> B;
[
    R.1,
    R.2,
    R.3
]
> K!B[1];
1
> K!B[2];
a
> K!B[3];
a^2
> K<a> := NumberField(x^2-5);
> R := RingOfIntegers(K);
> Basis(R);
[
    R.1,
    R.2
]
> K!R.1;
1
> K!R.2;
1/2*(a + 1)
> MinimalPolynomial($1);
x^2 - x - 1
>  /*  !   <-- coerce into something
     (int) 5.6
     SphericalCow(5) */
> > > 
> 
> R<x> := PolynomialRing(Q);
> R![1,2,3,4];
4*x^3 + 3*x^2 + 2*x + 1
> procedure ints(f) 
procedure>   K<a> := NumberField(f);
procedure>    R := MaximalOrder(K);
procedure>    print [K!z : z in Basis(R)];
procedure> end procedure;
> load "file";

>> load "file";
   ^
User error: Could not open file "file" (No such file or 
directory)
> ints(x^3-2);
[
    1,
    a,
    a^2
]
> ints(x^2-5);
[
    1,
    1/2*(a + 1)
]
> d := 10; time ints(&+[i*x^i + 2*x+ 1 : i in [0..d]]);
[
    1,
    10*a,
    10*a^2 + 9*a,
    10*a^3 + 9*a^2 + 8*a,
    10*a^4 + 9*a^3 + 8*a^2 + 7*a,
    10*a^5 + 9*a^4 + 8*a^3 + 7*a^2 + 6*a,
    10*a^6 + 9*a^5 + 8*a^4 + 7*a^3 + 6*a^2 + 5*a,
    10*a^7 + 9*a^6 + 8*a^5 + 7*a^4 + 6*a^3 + 5*a^2 + 4*a,
    10*a^8 + 9*a^7 + 8*a^6 + 7*a^5 + 6*a^4 + 5*a^3 + 4*a^2 + 3*a,
    10*a^9 + 9*a^8 + 8*a^7 + 7*a^6 + 6*a^5 + 5*a^4 + 4*a^3 + 
        3*a^2 + 2*a
]
Time: 0.090
> d := 10; time ints(&+[i*x^i + 2*x+ 1 : i in [0..d]]);
[
    1,
    10*a,
    10*a^2 + 9*a,
    10*a^3 + 9*a^2 + 8*a,
    10*a^4 + 9*a^3 + 8*a^2 + 7*a,
    10*a^5 + 9*a^4 + 8*a^3 + 7*a^2 + 6*a,
    10*a^6 + 9*a^5 + 8*a^4 + 7*a^3 + 6*a^2 + 5*a,
    10*a^7 + 9*a^6 + 8*a^5 + 7*a^4 + 6*a^3 + 5*a^2 + 4*a,
    10*a^8 + 9*a^7 + 8*a^6 + 7*a^5 + 6*a^4 + 5*a^3 + 4*a^2 + 3*a,
    10*a^9 + 9*a^8 + 8*a^7 + 7*a^6 + 6*a^5 + 5*a^4 + 4*a^3 + 
        3*a^2 + 2*a
]
Time: 0.110
> d := 20; time ints(&+[i*x^i + 2*x+ 1 : i in [0..d]]);
[
    1,
    20*a,
    20*a^2 + 19*a,
    20*a^3 + 19*a^2 + 18*a,
    20*a^4 + 19*a^3 + 18*a^2 + 17*a,
    20*a^5 + 19*a^4 + 18*a^3 + 17*a^2 + 16*a,
    20*a^6 + 19*a^5 + 18*a^4 + 17*a^3 + 16*a^2 + 15*a,
    20*a^7 + 19*a^6 + 18*a^5 + 17*a^4 + 16*a^3 + 15*a^2 + 14*a,
    20*a^8 + 19*a^7 + 18*a^6 + 17*a^5 + 16*a^4 + 15*a^3 + 14*a^2 
        + 13*a,
    20*a^9 + 19*a^8 + 18*a^7 + 17*a^6 + 16*a^5 + 15*a^4 + 14*a^3 
        + 13*a^2 + 12*a,
    20*a^10 + 19*a^9 + 18*a^8 + 17*a^7 + 16*a^6 + 15*a^5 + 14*a^4
        + 13*a^3 + 12*a^2 + 11*a,
    20*a^11 + 19*a^10 + 18*a^9 + 17*a^8 + 16*a^7 + 15*a^6 + 
        14*a^5 + 13*a^4 + 12*a^3 + 11*a^2 + 10*a,
    20*a^12 + 19*a^11 + 18*a^10 + 17*a^9 + 16*a^8 + 15*a^7 + 
        14*a^6 + 13*a^5 + 12*a^4 + 11*a^3 + 10*a^2 + 9*a,
    20*a^13 + 19*a^12 + 18*a^11 + 17*a^10 + 16*a^9 + 15*a^8 + 
        14*a^7 + 13*a^6 + 12*a^5 + 11*a^4 + 10*a^3 + 9*a^2 + 8*a,
    20*a^14 + 19*a^13 + 18*a^12 + 17*a^11 + 16*a^10 + 15*a^9 + 
        14*a^8 + 13*a^7 + 12*a^6 + 11*a^5 + 10*a^4 + 9*a^3 + 
        8*a^2 + 7*a,
    20*a^15 + 19*a^14 + 18*a^13 + 17*a^12 + 16*a^11 + 15*a^10 + 
        14*a^9 + 13*a^8 + 12*a^7 + 11*a^6 + 10*a^5 + 9*a^4 + 
        8*a^3 + 7*a^2 + 6*a,
    20*a^16 + 19*a^15 + 18*a^14 + 17*a^13 + 16*a^12 + 15*a^11 + 
        14*a^10 + 13*a^9 + 12*a^8 + 11*a^7 + 10*a^6 + 9*a^5 + 
        8*a^4 + 7*a^3 + 6*a^2 + 5*a,
    20*a^17 + 19*a^16 + 18*a^15 + 17*a^14 + 16*a^13 + 15*a^12 + 
        14*a^11 + 13*a^10 + 12*a^9 + 11*a^8 + 10*a^7 + 9*a^6 + 
        8*a^5 + 7*a^4 + 6*a^3 + 5*a^2 + 4*a,
    20*a^18 + 19*a^17 + 18*a^16 + 17*a^15 + 16*a^14 + 15*a^13 + 
        14*a^12 + 13*a^11 + 12*a^10 + 11*a^9 + 10*a^8 + 9*a^7 + 
        8*a^6 + 7*a^5 + 6*a^4 + 5*a^3 + 4*a^2 + 3*a,
    1/7*(20*a^19 + 99*a^18 + 134*a^17 + 7*a^16 + 86*a^15 + 
        121*a^14 + 134*a^13 + 66*a^12 + 101*a^11 + 114*a^10 + 
        46*a^9 + 81*a^8 + 94*a^7 + 26*a^6 + 61*a^5 + 74*a^4 + 
        6*a^3 + 41*a^2 + 54*a)
]
Time: 4.850
> GF(NextPrime(10^20));
Finite field of size 100000000000000000039
> GF(NextPrime(10^100));
Finite field of size 1000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000267
> GF(NextPrime(10^100)^2);
Finite field of size 1000000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000000267^2
>