Presentation 2008-01-09 (SAGE)

var('x')
sol = integrate(sin(x^2)); sol

sqrt(pi)*((sqrt(2)*I + sqrt(2))*erf((sqrt(2)*I + sqrt(2))*x/2) +
(sqrt(2)*I - sqrt(2))*erf((sqrt(2)*I - sqrt(2))*x/2))/8

sqrt(pi)*((sqrt(2)*I + sqrt(2))*erf((sqrt(2)*I + sqrt(2))*x/2) + (sqrt(2)*I - sqrt(2))*erf((sqrt(2)*I - sqrt(2))*x/2))/8

latex(sol)

\frac{{\sqrt{ \pi } \cdot \left( {\left( {\sqrt{ 2 } \cdot i} +
\sqrt{ 2 } \right) \cdot \left( \text{erf} \left( \frac{{\left(
{\sqrt{ 2 } \cdot i} + \sqrt{ 2 } \right) \cdot x}}{2} \right)
\right)} + {\left( {\sqrt{ 2 } \cdot i} - \sqrt{ 2 } \right) \cdot
\left( \text{erf} \left( \frac{{\left( {\sqrt{ 2 } \cdot i} - \sqrt{
2 } \right) \cdot x}}{2} \right) \right)} \right)}}{8}

\frac{{\sqrt{ \pi } \cdot \left( {\left( {\sqrt{ 2 } \cdot i} + \sqrt{ 2 } \right) \cdot \left( \text{erf} \left( \frac{{\left( {\sqrt{ 2 } \cdot i} + \sqrt{ 2 } \right) \cdot x}}{2} \right) \right)} + {\left( {\sqrt{ 2 } \cdot i} - \sqrt{ 2 } \right) \cdot \left( \text{erf} \left( \frac{{\left( {\sqrt{ 2 } \cdot i} - \sqrt{ 2 } \right) \cdot x}}{2} \right) \right)} \right)}}{8}

show(sol)

8pÙÁÒ°p2Ái+p2ÑÁÒerfÒ2(p2Ái+p2)ÁxÓÓ+°p2ÁiÀp2ÑÁÒerfÒ2(p2ÁiÀp2)ÁxÓÓÓ

8pÙÁÒ°p2Ái+p2ÑÁÒerfÒ2(p2Ái+p2)ÁxÓÓ+°p2ÁiÀp2ÑÁÒerfÒ2(p2ÁiÀp2)ÁxÓÓÓ

V = RR^3; V

Vector space of dimension 3 over Real Field with 53 bits of
precision

Vector space of dimension 3 over Real Field with 53 bits of precision

V.is_finite()

False

False

(QQ^3).random_element()

(1/28, -3, -1/2)

(1/28, -3, -1/2)

var('x y z');
sol = solve([x^3+2*y^2==3+z, x-y^2+z==2],x,y,z)
for s in sol:
    show(s)

[x=r1; y=ÀpÀr13Àr1+5; z=Àr13À2Ár1+7];  
[x=r2; y=pÀr23Àr2+5; z=Àr23À2Ár2+7];

[x=r1; y=ÀpÀr13Àr1+5; z=Àr13À2Ár1+7];  
[x=r2; y=pÀr23Àr2+5; z=Àr23À2Ár2+7];

f(x) = (2*x+5*x^2+x^3)/(2-3*x^3); f

x |--> (x^3 + 5*x^2 + 2*x)/(2 - 3*x^3)

x |--> (x^3 + 5*x^2 + 2*x)/(2 - 3*x^3)

f.limit(x=oo)

x |--> -1/3

x |--> -1/3

show(f.partial_fraction())

x 7! 3Á(3Áx3À2)À15Áx2+6Áx+2À31

x 7! 3Á(3Áx3À2)À15Áx2+6Áx+2À31

show(f.derivative(x).rational_simplify())

x 7! 9Áx6À12Áx3+415Áx4+12Áx3+6Áx2+20Áx+4

x 7! 9Áx6À12Áx3+415Áx4+12Áx3+6Áx2+20Áx+4

f(3)

-78/79

-78/79

f.subs(x=3).n(digits=150)

-0.98734177215189873417721518987341772151898734177215189873417721518\
98734177215189873417721518987341772151898734177215189873417721518987\
34177215189873417722

-0.987341772151898734177215189873417721518987341772151898734177215189873417721518987341772151898734177215189873417721518987341772151898734177215189873417722

plot(chebyshev_T(8,PolynomialRing(QQ, 'x').gen()) * sin(x) + x/3).show()

RIF = RealIntervalField(150); RIF

Real Interval Field with 150 bits of precision

Real Interval Field with 150 bits of precision

def wrap(n,f):
    for _ in range(20):
        n = f(n)
    return n

shift=10^7

a = RIF(2+1/shift); a

[2.0000000999999999999999999999999999999999999984 ..
2.0000001000000000000000000000000000000000000013]

[2.0000000999999999999999999999999999999999999984 .. 2.0000001000000000000000000000000000000000000013]

b = wrap(a,lambda(x): x^2); b

[7.1039990175447649168126806044014828532186237369e315652 ..
7.1039990175447649168126806044014828532394989976e315652]

[7.1039990175447649168126806044014828532186237369e315652 .. 7.1039990175447649168126806044014828532394989976e315652]

c = wrap(b,sqrt)-2
c.center()-1/shift

-1.9244198246303974986110659184666764220799913e-46

-1.9244198246303974986110659184666764220799913e-46

1/shift in c

True

True

c.diameter()

8.4077907859489024255423774997394967876977518e-38

8.4077907859489024255423774997394967876977518e-38

P.<x> = PolynomialRing(GF(5)); P

Univariate Polynomial Ring in x over Finite Field of size 5

Univariate Polynomial Ring in x over Finite Field of size 5

14*x+100

4*x

4*x

poly1 = 3*x^4-x^2-14
poly1.roots()

[(3, 1), (2, 1)]

[(3, 1), (2, 1)]

poly1.subs(x=3)

var('x')
poly1 = 3*x^4-x^2-14
poly1roots = poly1.roots(); poly1roots

[(-sqrt(7)/sqrt(3), 1), (sqrt(7)/sqrt(3), 1), (-sqrt(2)*I, 1),
(sqrt(2)*I, 1)]

[(-sqrt(7)/sqrt(3), 1), (sqrt(7)/sqrt(3), 1), (-sqrt(2)*I, 1), (sqrt(2)*I, 1)]

[poly1.subs(x=poly1roots[i][0]) for i in range(len(poly1roots))]

[0, 0, 0, 0]

[0, 0, 0, 0]

P.<x,y,z> = QQ[]
time p = (x+y+z+1)^20
q = p+1
time r = p*q

Time: CPU 0.00 s, Wall: 0.02 s
Time: CPU 2.74 s, Wall: 5.77 s

Time: CPU 0.00 s, Wall: 0.02 s
Time: CPU 2.74 s, Wall: 5.77 s

P.<x,y,z> = GF(997)[]
time p = (x+y+z+1)^20
q = p+1
time r = p*q

Time: CPU 0.00 s, Wall: 0.04 s
Time: CPU 0.28 s, Wall: 0.35 s

Time: CPU 0.00 s, Wall: 0.04 s
Time: CPU 0.28 s, Wall: 0.35 s

var('x')
poly = x^3-2*x^2+3
show(poly.roots())

[°23Àp3Ái;1Ñ; °2p3Ái+3;1Ñ; (À1;1)];

[°23Àp3Ái;1Ñ; °2p3Ái+3;1Ñ; (À1;1)];

P.<x> = GF(7)[]
poly = x^3-2*x^2+3
show(poly.roots())

[(4;1);(6;2)]

[(4;1);(6;2)]

show(poly.factor())

(x+3)Á(x+1)2

(x+3)Á(x+1)2

0

P.<x> = QQ['x']; P

Univariate Polynomial Ring in x over Rational Field

Univariate Polynomial Ring in x over Rational Field

Pext = P.extension(x^2+5,'a'); Pext

Univariate Quotient Polynomial Ring in a over Univariate Polynomial
Ring in x over Rational Field with modulus a^2 + 5

Univariate Quotient Polynomial Ring in a over Univariate Polynomial Ring in x over Rational Field with modulus a^2 + 5

R.<x,y> = PolynomialRing(QQ, 2); R

Multivariate Polynomial Ring in x, y over Rational Field

Multivariate Polynomial Ring in x, y over Rational Field

S = R.quotient(x^2 + y^2); S

Quotient of Multivariate Polynomial Ring in x, y over Rational Field
by the ideal (x^2 + y^2)

Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)

pi = S.cover(); pi

Ring morphism:
  From: Multivariate Polynomial Ring in x, y over Rational Field
  To:   Quotient of Multivariate Polynomial Ring in x, y over
Rational Field by the ideal (x^2 + y^2)
  Defn: Natural quotient map

Ring morphism:
  From: Multivariate Polynomial Ring in x, y over Rational Field
  To:   Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
  Defn: Natural quotient map

L = S.lift(); L

Set-theoretic ring morphism:
  From: Quotient of Multivariate Polynomial Ring in x, y over
Rational Field by the ideal (x^2 + y^2)
  To:   Multivariate Polynomial Ring in x, y over Rational Field
  Defn: Choice of lifting map

Set-theoretic ring morphism:
  From: Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
  To:   Multivariate Polynomial Ring in x, y over Rational Field
  Defn: Choice of lifting map

L(S.0)

L(S.1)

L( x^2+y^2 )

L( x^3 )

-x*y^2

-x*y^2

L( 4*x^2+4*y^2+x^3 )

-x*y^2

-x*y^2

var('x')

K.<a> = NumberField(x^4+1); K

Number Field in a with defining polynomial x^4 + 1

Number Field in a with defining polynomial x^4 + 1

V, from_V, to_V  = K.vector_space()
V; from_V; to_V;

Vector space of dimension 4 over Rational Field
Isomorphism from Vector space of dimension 4 over Rational Field to
Number Field in a with defining polynomial x^4 + 1
Isomorphism from Number Field in a with defining polynomial x^4 + 1
to Vector space of dimension 4 over Rational Field

Vector space of dimension 4 over Rational Field
Isomorphism from Vector space of dimension 4 over Rational Field to Number Field in a with defining polynomial x^4 + 1
Isomorphism from Number Field in a with defining polynomial x^4 + 1 to Vector space of dimension 4 over Rational Field

from_V([4,0,0,5])

5*a^3 + 4

5*a^3 + 4

to_V(2*a-1/2*a^3)

(0, 2, 0, -1/2)

(0, 2, 0, -1/2)

KembCC = K.embeddings(CC); KembCC

[Ring morphism:
  From: Number Field in a with defining polynomial x^4 + 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.707106781186548 - 0.707106781186548*I, Ring
morphism:
  From: Number Field in a with defining polynomial x^4 + 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.707106781186547 + 0.707106781186547*I, Ring
morphism:
  From: Number Field in a with defining polynomial x^4 + 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 0.707106781186548 - 0.707106781186547*I, Ring
morphism:
  From: Number Field in a with defining polynomial x^4 + 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 0.707106781186548 + 0.707106781186548*I]

[Ring morphism:
  From: Number Field in a with defining polynomial x^4 + 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.707106781186548 - 0.707106781186548*I, Ring morphism:
  From: Number Field in a with defining polynomial x^4 + 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> -0.707106781186547 + 0.707106781186547*I, Ring morphism:
  From: Number Field in a with defining polynomial x^4 + 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 0.707106781186548 - 0.707106781186547*I, Ring morphism:
  From: Number Field in a with defining polynomial x^4 + 1
  To:   Complex Field with 53 bits of precision
  Defn: a |--> 0.707106781186548 + 0.707106781186548*I]

S = SymmetricGroup(4)

S.random_element()

(2,3,4)

(2,3,4)

S.order()

S.cayley_graph().show()

S.character_table()

[ 1 -1  1  1 -1]
[ 3 -1 -1  0  1]
[ 2  0  2 -1  0]
[ 3  1 -1  0 -1]
[ 1  1  1  1  1]

[ 1 -1  1  1 -1]
[ 3 -1 -1  0  1]
[ 2  0  2 -1  0]
[ 3  1 -1  0 -1]
[ 1  1  1  1  1]

A5 = AlternatingGroup(5); A5

Alternating group of order 5!/2 as a permutation group

Alternating group of order 5!/2 as a permutation group

G = A5.cayley_graph()
G.show3d(color_by_label=True, edge_size=0.01, edge_size2=0.02, vertex_size=0.03)

P.<x,y,z> = PolynomialRing(QQ,3); P

Multivariate Polynomial Ring in x, y, z over Rational Field

Multivariate Polynomial Ring in x, y, z over Rational Field

p = (x + y + z + 1)^20
q=p+1
time p=p*q

Time: CPU 2.86 s, Wall: 5.39 s

Time: CPU 2.86 s, Wall: 5.39 s

P.<x,y,z> = QQ[]
p1 = 2*x^2+4*y
p2 = 2*x^2-y+z
Id = ideal(p1,p2); Id

Ideal (2*x^2 + 4*y, 2*x^2 - y + z) of Multivariate Polynomial Ring
in x, y, z over Rational Field

Ideal (2*x^2 + 4*y, 2*x^2 - y + z) of Multivariate Polynomial Ring in x, y, z over Rational Field

IdG = Id.groebner_fan()
IdG.buchberger()

[y - 1/5*z, x^2 + 2/5*z]

[y - 1/5*z, x^2 + 2/5*z]

G = SL(2,ZZ)
print G

Special Linear Group of degree 2 over Integer Ring

Special Linear Group of degree 2 over Integer Ring

SL_gens = G.gens(); SL_gens

[
[ 0  1]
[-1  0],
[1 1]
[0 1]
]

[
[ 0  1]
[-1  0],
[1 1]
[0 1]
]

type(SL_gens[0])

<class
'sage.groups.matrix_gps.matrix_group_element.MatrixGroupElement'>

<class 'sage.groups.matrix_gps.matrix_group_element.MatrixGroupElement'>

G = GU(3, GF(7)); G

General Unitary Group of degree 3 over Finite Field of size 7

General Unitary Group of degree 3 over Finite Field of size 7

G.random()

[5*a + 5 5*a + 2 4*a + 2]
[  a + 4 6*a + 4 4*a + 1]
[5*a + 5       0     2*a]

[5*a + 5 5*a + 2 4*a + 2]
[  a + 4 6*a + 4 4*a + 1]
[5*a + 5       0     2*a]

G.center()

Matrix group over Finite Field in a of size 7^2 with 2 generators: 
 [[[6, 0, 0], [0, 6, 0], [0, 0, 6]], [[2*a + 4, 0, 0], [0, 2*a + 4,
0], [0, 0, 2*a + 4]]]

Matrix group over Finite Field in a of size 7^2 with 2 generators: 
 [[[6, 0, 0], [0, 6, 0], [0, 0, 6]], [[2*a + 4, 0, 0], [0, 2*a + 4, 0], [0, 0, 2*a + 4]]]

G.order()

45308928

45308928

M = random_matrix(GF(3),40,40)
M.image()

Vector space of degree 40 and dimension 39 over Finite Field of size
3
Basis matrix:
39 x 40 dense matrix over Finite Field of size 3

Vector space of degree 40 and dimension 39 over Finite Field of size 3
Basis matrix:
39 x 40 dense matrix over Finite Field of size 3

k=M.kernel(); k

Vector space of degree 40 and dimension 1 over Finite Field of size
3
Basis matrix:
[0 1 2 2 0 1 1 0 0 0 1 0 1 0 0 2 1 0 0 1 1 1 0 2 2 0 1 2 1 1 0 0 2 0
2 0 0 0 0 0]

Vector space of degree 40 and dimension 1 over Finite Field of size 3
Basis matrix:
[0 1 2 2 0 1 1 0 0 0 1 0 1 0 0 2 1 0 0 1 1 1 0 2 2 0 1 2 1 1 0 0 2 0 2 0 0 0 0 0]

k.gen() * M

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

K = CyclotomicField(7); a=K.0
M = MatrixSpace(K,4,2)([2*a^3,-1, 0,-a, 0,-a^5-1, 1,a^3-1])
M

[   2*zeta7^3           -1]
[           0       -zeta7]
[           0 -zeta7^5 - 1]
[           1  zeta7^3 - 1]

[   2*zeta7^3           -1]
[           0       -zeta7]
[           0 -zeta7^5 - 1]
[           1  zeta7^3 - 1]

M.kernel()

Vector space of degree 4 and dimension 2 over Cyclotomic Field of
order 7 and degree 6
Basis matrix:
[                                    1                              
0 zeta7^5 + zeta7^4 + 2*zeta7^3 - zeta7                           
-2*zeta7^3]
[                                    0                              
1        -zeta7^4 - zeta7^3 - zeta7 - 1                             
0]

Vector space of degree 4 and dimension 2 over Cyclotomic Field of order 7 and degree 6
Basis matrix:
[                                    1                                     0 zeta7^5 + zeta7^4 + 2*zeta7^3 - zeta7                            -2*zeta7^3]
[                                    0                                     1        -zeta7^4 - zeta7^3 - zeta7 - 1                                     0]

show(matrix(M.kernel().gens()))

ï1 0 0 1 °75+°74+2°73À°7 À°74À°73À°7À1 À2°73 0  !

ï1 0 0 1 °75+°74+2°73À°7 À°74À°73À°7À1 À2°73 0  !

M.kernel().gen() * M

(0, 0)

(0, 0)

M = matrix(RDF,4,range(36)); M

[ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  7.0  8.0]
[ 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0]
[18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0]
[27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0]

[ 0.0  1.0  2.0  3.0  4.0  5.0  6.0  7.0  8.0]
[ 9.0 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0]
[18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0 26.0]
[27.0 28.0 29.0 30.0 31.0 32.0 33.0 34.0 35.0]

Q,R = M.QR()

matrix_plot(Q,cmap='Oranges')

M = MatrixSpace(GF(3),39).random_element()
M

39 x 39 dense matrix over Finite Field of size 3

39 x 39 dense matrix over Finite Field of size 3

Me = M.echelon_form(); matrix_plot(Me).show()

Mh = M.hessenberg_form(); matrix_plot(Mh,cmap='Blues').show()

s     = 7
s2    = 2^s
P.<x> = GF(2)[]
M     = matrix(parent(x),s2)
for i in range(s2):
    p  = (1+x)^i
    pc = p.coeffs()
    a  = pc.count(1)
    for j in range(a):
        idx        = pc.index(1)
        M[i,idx+j] = pc.pop(idx)

matrix_plot(M,cmap='Greys').show()

s=10
G_cycle = graphs.CycleGraph(s)
G_cycle.add_edges([[i,j] for (i,j) in tuples(range(G_cycle.size()),2) if is_prime(mod(i-j+1,s))])
G_cycle.plot().show()

G = Graph([GF(17), lambda i,j: i!=j and (i-j).is_square() and (i+j)>8])
G.vertices()

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

matrix_plot(G.adjacency_matrix()).show()

G.plot().show()

G.plot3d().show()

G.cliques_containing_vertex(0)

[[0, 15, 13], [0, 15, 16], [0, 9]]

[[0, 15, 13], [0, 15, 16], [0, 9]]

msp=G.min_spanning_tree(); msp

[(0, 9, None), (0, 13, None), (0, 15, None), (0, 16, None), (1, 9,
None), (1, 10, None), (1, 14, None), (2, 10, None), (2, 11, None),
(3, 7, None), (3, 11, None), (3, 12, None), (4, 5, None), (4, 6,
None), (4, 8, None), (4, 12, None)]

[(0, 9, None), (0, 13, None), (0, 15, None), (0, 16, None), (1, 9, None), (1, 10, None), (1, 14, None), (2, 10, None), (2, 11, None), (3, 7, None), (3, 11, None), (3, 12, None), (4, 5, None), (4, 6, None), (4, 8, None), (4, 12, None)]

G.delete_edges(msp)
G.connected_components()

[[5, 6, 7, 8, 9, 10], [11, 12, 13, 14, 15, 16], [0], [1], [2], [3],
[4]]

[[5, 6, 7, 8, 9, 10], [11, 12, 13, 14, 15, 16], [0], [1], [2], [3], [4]]

graphs_list.show_graphs(GraphDatabase().get_list(num_vertices=7,diameter=5))

graphs.RandomGNP(15,.3).show()

G3 = graphs.FlowerSnark()
G3.plot3d(iterations=100).show()

G3.diameter()

G3.show()

G4 = DiGraph({1:[2,2,3,5], 2:[3,4], 3:[4], 4:[5,7], 5:[6]}, multiedges=True)
G4.show(talk=true,layout='circular')

G4_path = GraphPaths(G4); G4_path

Paths in Multi-digraph on 7 vertices

Paths in Multi-digraph on 7 vertices

G4_path.incoming_paths(5)

[[5], [1, 5], [4, 5], [2, 4, 5], [1, 2, 4, 5], [1, 2, 4, 5], [3, 4,
5], [1, 3, 4, 5], [2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5]]

[[5], [1, 5], [4, 5], [2, 4, 5], [1, 2, 4, 5], [1, 2, 4, 5], [3, 4, 5], [1, 3, 4, 5], [2, 3, 4, 5], [1, 2, 3, 4, 5], [1, 2, 3, 4, 5]]

for p in G4_path.incoming_paths(4):
    print "%s has length %d" % (repr(p).rjust(20), len(p))

[4] has length 1
              [2, 4] has length 2
           [1, 2, 4] has length 3
           [1, 2, 4] has length 3
              [3, 4] has length 2
           [1, 3, 4] has length 3
           [2, 3, 4] has length 3
        [1, 2, 3, 4] has length 4
        [1, 2, 3, 4] has length 4

[4] has length 1
              [2, 4] has length 2
           [1, 2, 4] has length 3
           [1, 2, 4] has length 3
              [3, 4] has length 2
           [1, 3, 4] has length 3
           [2, 3, 4] has length 3
        [1, 2, 3, 4] has length 4
        [1, 2, 3, 4] has length 4

E = EllipticCurve('5077a')
print E
E.plot().show()

Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational
Field

Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field

E.rational_points(bound=5)

[(1 : 0 : 1), (2 : 0 : 1), (-3 : 0 : 1), (1 : -1 : 1), (2 : -1 : 1),
(-3 : -1 : 1), (0 : 2 : 1), (-1 : 3 : 1), (-2 : 3 : 1), (3 : 3 : 1),
(0 : -3 : 1), (-1 : -4 : 1), (-2 : -4 : 1), (3 : -4 : 1), (0 : 1 :
0)]

[(1 : 0 : 1), (2 : 0 : 1), (-3 : 0 : 1), (1 : -1 : 1), (2 : -1 : 1), (-3 : -1 : 1), (0 : 2 : 1), (-1 : 3 : 1), (-2 : 3 : 1), (3 : 3 : 1), (0 : -3 : 1), (-1 : -4 : 1), (-2 : -4 : 1), (3 : -4 : 1), (0 : 1 : 0)]

E2 = E.change_ring(GF(997)); E2

Elliptic Curve defined by y^2 + y = x^3 + 990*x + 6 over Finite
Field of size 997

Elliptic Curve defined by y^2 + y = x^3 + 990*x + 6 over Finite Field of size 997

E2.plot().show()

E2.cardinality()

ecurve = EllipticCurve(GF(next_prime(10^20)),[0,-1,2765,765465,-5645])
print "%s and it's cardinality is: %s" % (ecurve,ecurve.cardinality())

Elliptic Curve defined by y^2 + 2765*y = x^3 +
100000000000000000038*x^2 + 765465*x + 99999999999999994394 over
Finite Field of size 100000000000000000039 and it's cardinality is:
99999999993684546596

Elliptic Curve defined by y^2 + 2765*y = x^3 + 100000000000000000038*x^2 + 765465*x + 99999999999999994394 over Finite Field of size 100000000000000000039 and it's cardinality is: 99999999993684546596

G2 = E2.abelian_group(); G2

(Multiplicative Abelian Group isomorphic to C949, ((324 : 505 :
1),))

(Multiplicative Abelian Group isomorphic to C949, ((324 : 505 : 1),))

G2[0].random()

f^713

f^713

x,y,z = MPolynomialRing(GF(7), 3, 'xyz').gens()
C = Curve(y^2*z^7 - x^8*z - x*z^8 ); C

Projective Curve over Finite Field of size 7 defined by -x^8*z +
y^2*z^7 - x*z^8

Projective Curve over Finite Field of size 7 defined by -x^8*z + y^2*z^7 - x*z^8

show(C.rational_points())

[(0:0:1); (0:1:0); (1:0:0); (1:1:0); (1:3:1); (1:4:1); (2:1:0); (3:1:0); (4:1:0); (5:1:0); (5:3:1); (5:4:1); (6:0:1); (6:1:0)];

[(0:0:1); (0:1:0); (1:0:0); (1:1:0); (1:3:1); (1:4:1); (2:1:0); (3:1:0); (4:1:0); (5:1:0); (5:3:1); (5:4:1); (6:0:1); (6:1:0)];

(y^2*z^7 - x^8*z - x*z^8).subs(x=5,y=3,z=1)

C = HammingCode(2,GF(5)); C

Linear code of length 6, dimension 4 over Finite Field of size 5

Linear code of length 6, dimension 4 over Finite Field of size 5

C.minimum_distance()

C.random()

(0, 3, 4, 1, 3, 4)

(0, 3, 4, 1, 3, 4)

C.gen_mat()

[4 4 1 0 0 0]
[3 4 0 1 0 0]
[1 4 0 0 1 0]
[2 4 0 0 0 1]

[4 4 1 0 0 0]
[3 4 0 1 0 0]
[1 4 0 0 1 0]
[2 4 0 0 0 1]

C.check_mat()

[1 2 3 4 1 0]
[4 4 3 2 0 1]

[1 2 3 4 1 0]
[4 4 3 2 0 1]

C_perp = C.dual_code()
C_perp.gen_mat()

[1 2 3 4 1 0]
[4 4 3 2 0 1]

[1 2 3 4 1 0]
[4 4 3 2 0 1]

C_perp.check_mat()

[4 4 1 0 0 0]
[3 4 0 1 0 0]
[1 4 0 0 1 0]
[2 4 0 0 0 1]

[4 4 1 0 0 0]
[3 4 0 1 0 0]
[1 4 0 0 1 0]
[2 4 0 0 0 1]

v = matrix(GF(5),[[0,0,1,2,1,2]]); v

[0 0 1 2 1 2]

[0 0 1 2 1 2]

C.decode(v)

(0, 4, 1, 2, 1, 2)

(0, 4, 1, 2, 1, 2)

def test_mod_2(n):
    """this is the interpreted python code function test_mod_2"""
    ret = 0
    for i in range(n):
        if i%9997==0: ret += i
    return ret

test_mod_2?

time test_mod_2(10^7)

5003498500
CPU time: 26.39 s,  Wall time: 52.70 s

5003498500
CPU time: 26.39 s,  Wall time: 52.70 s

%cython
def test_mod_2_c(unsigned int n):
    """this is the built-in test_mod_2 function"""
    ret = 0
    for i in range(n):
        if i%9997==0: ret += i
    return ret

__home_harri__sage_sage_notebook_worksheets_harri_5_code_sage183_spyx.c

test_mod_2_c?

<built-in function test_mod_2_c>

<built-in function test_mod_2_c>

time test_mod_2_c(10^7)

5003498500L
CPU time: 1.70 s,  Wall time: 2.93 s

5003498500L
CPU time: 1.70 s,  Wall time: 2.93 s

Presentation 2008-01-09

SAGE in action

basic calculus

Arbitrary Precision Interval Arithmetics

polynomial rings

ring extension

benchmarking calculation time in Q

ideals and gröbner basis

matrix groups

different calculations with matrix visualization

rule based calculation of a matrix with graph

included library of graphs

Directed Graphs

included library of elliptic curves

change base ring and plot over finite field

coding theory

embedded compilation in cython