Intro to Sage -- Sage

3 + 2 + 1

factorial(30)

265252859812191058636308480000000

265252859812191058636308480000000

Ellip

EllipticCurve

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/constructor.py

Type: <type ‘function’>

Definition: EllipticCurve(x=None, y=None, j=None)

Docstring:

There are several ways to construct an elliptic curve:

y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.

EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given a-invariants. The invariants are coerced into the parent of the first element. If all are integers, they are coerced into the rational numbers.
EllipticCurve([a4,a6]): Same as above, but a1=a2=a3=0.
EllipticCurve(label): Returns the elliptic curve over Q from the Cremona database with the given label. The label is a string, such as “11a” or “37b2”. The letters in the label must be lower case (Cremona’s new labeling).
EllipticCurve(R, [a1,a2,a3,a4,a6]): Create the elliptic curve over R with given a-invariants. Here R can be an arbitrary ring. Note that addition need not be defined.
EllipticCurve(j): Return an elliptic curve with j-invariant j. Warning: this is deprecated. Use EllipticCurve_from_j(j) or EllipticCurve(j=j) instead.

EXAMPLES: We illustrate creating elliptic curves.

sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

We create a curve from a Cremona label:

sage: EllipticCurve('37b2')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field
sage: EllipticCurve('5077a')
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: EllipticCurve('389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field

Unicode labels are allowed:

sage: EllipticCurve(u'389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field

We create curves over a finite field as follows:

sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve(GF(5), [0, 0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5

Elliptic curves over \ZZ/N\ZZ with N prime are of type “elliptic curve over a finite field”:

sage: F = Zmod(101)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field'>

In contrast, elliptic curves over \ZZ/N\ZZ with N composite are of type “generic elliptic curve”:

sage: F = Zmod(95)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic'>

The following is a curve over the complex numbers:

sage: E = EllipticCurve(CC, [0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x over Complex Field with 53 bits of precision
sage: E.j_invariant()
2988.97297297297

We can also create elliptic curves by giving the Weierstrass equation:

sage: x, y = var('x,y')
sage: EllipticCurve(y^2 + y ==  x^3 + x - 9)
Elliptic Curve defined by y^2 + y = x^3 + x - 9 over Rational Field

sage: R.<x,y> = GF(5)[]
sage: EllipticCurve(x^3 + x^2 + 2 - y^2 - y*x)
Elliptic Curve defined by y^2 + x*y  = x^3 + x^2 + 2 over Finite Field of size 5

We can explicitly specify the j-invariant:

sage: E = EllipticCurve(j=1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'

sage: E = EllipticCurve(j=GF(5)(2)); E; E.j_invariant()
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
2

See trac #6657:

sage: EllipticCurve(GF(144169),j=1728)
Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 144169

TESTS:

sage: R = ZZ['u', 'v']
sage: EllipticCurve(R, [1,1])
Elliptic Curve defined by y^2 = x^3 + x + 1 over Multivariate Polynomial Ring in u, v
over Integer Ring

We create a curve and a point over QQbar (see #6879):

sage: E = EllipticCurve(QQbar,[0,1])
sage: E(0)
(0 : 1 : 0)
sage: E.base_field()
Algebraic Field

sage: E = EllipticCurve(RR,[1,2]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: EllipticCurve(CC,[3,4]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 3.00000000000000*x + 4.00000000000000 over Complex Field with 53 bits of precision
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: E = EllipticCurve(QQbar,[5,6]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 5*x + 6 over Algebraic Field
Algebraic Field

See trac #6657:

sage: EllipticCurve(3,j=1728)
Traceback (click to the left of this block for traceback)
...

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/schemes/elliptic_curves/constructor.py

Type: <type ‘function’>

Definition: EllipticCurve(x=None, y=None, j=None)

Docstring:

There are several ways to construct an elliptic curve:

y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.

EllipticCurve([a1,a2,a3,a4,a6]): Elliptic curve with given a-invariants. The invariants are coerced into the parent of the first element. If all are integers, they are coerced into the rational numbers.
EllipticCurve([a4,a6]): Same as above, but a1=a2=a3=0.
EllipticCurve(label): Returns the elliptic curve over Q from the Cremona database with the given label. The label is a string, such as “11a” or “37b2”. The letters in the label must be lower case (Cremona’s new labeling).
EllipticCurve(R, [a1,a2,a3,a4,a6]): Create the elliptic curve over R with given a-invariants. Here R can be an arbitrary ring. Note that addition need not be defined.
EllipticCurve(j): Return an elliptic curve with j-invariant j. Warning: this is deprecated. Use EllipticCurve_from_j(j) or EllipticCurve(j=j) instead.

EXAMPLES: We illustrate creating elliptic curves.

sage: EllipticCurve([0,0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field

We create a curve from a Cremona label:

sage: EllipticCurve('37b2')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 1873*x - 31833 over Rational Field
sage: EllipticCurve('5077a')
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 over Rational Field
sage: EllipticCurve('389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field

Unicode labels are allowed:

sage: EllipticCurve(u'389a')
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field

We create curves over a finite field as follows:

sage: EllipticCurve([GF(5)(0),0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5
sage: EllipticCurve(GF(5), [0, 0,1,-1,0])
Elliptic Curve defined by y^2 + y = x^3 + 4*x over Finite Field of size 5

Elliptic curves over \ZZ/N\ZZ with N prime are of type “elliptic curve over a finite field”:

sage: F = Zmod(101)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 101
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_finite_field.EllipticCurve_finite_field'>

In contrast, elliptic curves over \ZZ/N\ZZ with N composite are of type “generic elliptic curve”:

sage: F = Zmod(95)
sage: EllipticCurve(F, [2, 3])
Elliptic Curve defined by y^2 = x^3 + 2*x + 3 over Ring of integers modulo 95
sage: E = EllipticCurve([F(2), F(3)])
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic'>

The following is a curve over the complex numbers:

sage: E = EllipticCurve(CC, [0,0,1,-1,0])
sage: E
Elliptic Curve defined by y^2 + 1.00000000000000*y = x^3 + (-1.00000000000000)*x over Complex Field with 53 bits of precision
sage: E.j_invariant()
2988.97297297297

We can also create elliptic curves by giving the Weierstrass equation:

sage: x, y = var('x,y')
sage: EllipticCurve(y^2 + y ==  x^3 + x - 9)
Elliptic Curve defined by y^2 + y = x^3 + x - 9 over Rational Field

sage: R.<x,y> = GF(5)[]
sage: EllipticCurve(x^3 + x^2 + 2 - y^2 - y*x)
Elliptic Curve defined by y^2 + x*y  = x^3 + x^2 + 2 over Finite Field of size 5

We can explicitly specify the j-invariant:

sage: E = EllipticCurve(j=1728); E; E.j_invariant(); E.label()
Elliptic Curve defined by y^2 = x^3 - x over Rational Field
1728
'32a2'

sage: E = EllipticCurve(j=GF(5)(2)); E; E.j_invariant()
Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 5
2

See trac #6657:

sage: EllipticCurve(GF(144169),j=1728)
Elliptic Curve defined by y^2 = x^3 + x over Finite Field of size 144169

TESTS:

sage: R = ZZ['u', 'v']
sage: EllipticCurve(R, [1,1])
Elliptic Curve defined by y^2 = x^3 + x + 1 over Multivariate Polynomial Ring in u, v
over Integer Ring

We create a curve and a point over QQbar (see #6879):

sage: E = EllipticCurve(QQbar,[0,1])
sage: E(0)
(0 : 1 : 0)
sage: E.base_field()
Algebraic Field

sage: E = EllipticCurve(RR,[1,2]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: EllipticCurve(CC,[3,4]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 3.00000000000000*x + 4.00000000000000 over Complex Field with 53 bits of precision
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 2.00000000000000 over Real Field with 53 bits of precision
Real Field with 53 bits of precision
sage: E = EllipticCurve(QQbar,[5,6]); E; E.base_field()
Elliptic Curve defined by y^2 = x^3 + 5*x + 6 over Algebraic Field
Algebraic Field

See trac #6657:

sage: EllipticCurve(3,j=1728)
Traceback (most recent call last):
...
ValueError: First parameter (if present) must be a ring when j is specified

sage: EllipticCurve(GF(5),j=3/5)
Traceback (most recent call last):
...
ValueError: First parameter must be a ring containing 3/5

If the universe of the coefficients is a general field, the object constructed has type EllipticCurve_field. Otherwise it is EllipticCurve_generic. See trac #9816:

sage: E = EllipticCurve([QQbar(1),3]); E
Elliptic Curve defined by y^2 = x^3 + x + 3 over Algebraic Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field'>

sage: E = EllipticCurve([RR(1),3]); E
Elliptic Curve defined by y^2 = x^3 + 1.00000000000000*x + 3.00000000000000 over Real Field with 53 bits of precision
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field'>

sage: E = EllipticCurve([i,i]); E
Elliptic Curve defined by y^2 = x^3 + I*x + I over Symbolic Ring
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field'>
sage: is_field(SR)
True

sage: F = FractionField(PolynomialRing(QQ,'t'))
sage: t = F.gen()
sage: E = EllipticCurve([t,0]); E
Elliptic Curve defined by y^2 = x^3 + t*x over Fraction Field of Univariate Polynomial Ring in t over Rational Field
sage: type(E)
<class 'sage.schemes.elliptic_curves.ell_field.EllipticCurve_field'>

R.<x> = PolynomialRing(ZZ)

R.completion?

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/rings/polynomial/polynomial_ring.py

Type: <type ‘instancemethod’>

Definition: R.completion(p, prec=20, extras=None)

Docstring:

Return the completion of self with respect to the irreducible polynomial p. Currently only implemented for p=R.gen(), i.e. you can only complete R[x] with respect to x, the result being a ring of power series in x. The prec variable controls the precision used in the power series ring.

EXAMPLES:
sage: P.<x>=PolynomialRing(QQ)
sage: P
Univariate Polynomial Ring in x over Rational Field
sage: PP=P.completion(x)
sage: PP
Power Series Ring in x over Rational Field
sage: f=1-x
sage: PP(f)
1 - x
sage: 1/f
1/(-x + 1)
sage: 1/PP(f)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + O(x^20)

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/rings/polynomial/polynomial_ring.py

Type: <type ‘instancemethod’>

Definition: R.completion(p, prec=20, extras=None)

Docstring:

Return the completion of self with respect to the irreducible polynomial p. Currently only implemented for p=R.gen(), i.e. you can only complete R[x] with respect to x, the result being a ring of power series in x. The prec variable controls the precision used in the power series ring.

EXAMPLES:
sage: P.<x>=PolynomialRing(QQ)
sage: P
Univariate Polynomial Ring in x over Rational Field
sage: PP=P.completion(x)
sage: PP
Power Series Ring in x over Rational Field
sage: f=1-x
sage: PP(f)
1 - x
sage: 1/f
1/(-x + 1)
sage: 1/PP(f)
1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12 + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + O(x^20)

R.cyclotomic_polynomial??

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/rings/polynomial/polynomial_ring.py

Source Code (starting at line 867):

def cyclotomic_polynomial(self, n):
    """
    Return the nth cyclotomic polynomial as a polynomial in this
    polynomial ring. For details of the implementation, see the
    documentation for
    :func:`sage.rings.polynomial.cyclotomic.cyclotomic_coeffs`.

    EXAMPLES::

        sage: R = ZZ['x']
        sage: R.cyclotomic_polynomial(8)
        x^4 + 1
        sage: R.cyclotomic_polynomial(12)
        x^4 - x^2 + 1
        sage: S = PolynomialRing(FiniteField(7), 'x')
        sage: S.cyclotomic_polynomial(12)
        x^4 + 6*x^2 + 1
        sage: S.cyclotomic_polynomial(1)
        x + 6

    TESTS:

    Make sure it agrees with other systems for the trivial case::

        sage: ZZ['x'].cyclotomic_polynomial(1)
        x - 1
        sage: gp('polcyclo(1)')
        x - 1
    """
    if n <= 0:
        raise ArithmeticError, "n=%s must be positive"%n
    elif n == 1:
        return self.gen() - 1
    else:
        return self(cyclotomic.cyclotomic_coeffs(n), check=True)

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/rings/polynomial/polynomial_ring.py

Source Code (starting at line 867):

def cyclotomic_polynomial(self, n):
    """
    Return the nth cyclotomic polynomial as a polynomial in this
    polynomial ring. For details of the implementation, see the
    documentation for
    :func:`sage.rings.polynomial.cyclotomic.cyclotomic_coeffs`.

    EXAMPLES::

        sage: R = ZZ['x']
        sage: R.cyclotomic_polynomial(8)
        x^4 + 1
        sage: R.cyclotomic_polynomial(12)
        x^4 - x^2 + 1
        sage: S = PolynomialRing(FiniteField(7), 'x')
        sage: S.cyclotomic_polynomial(12)
        x^4 + 6*x^2 + 1
        sage: S.cyclotomic_polynomial(1)
        x + 6

    TESTS:

    Make sure it agrees with other systems for the trivial case::

        sage: ZZ['x'].cyclotomic_polynomial(1)
        x - 1
        sage: gp('polcyclo(1)')
        x - 1
    """
    if n <= 0:
        raise ArithmeticError, "n=%s must be positive"%n
    elif n == 1:
        return self.gen() - 1
    else:
        return self(cyclotomic.cyclotomic_coeffs(n), check=True)

sage.rings.polynomial.cyclotomic.cyclotomic_coeffs??


Traceback (click to the left of this block for traceback)
...


Traceback (most recent call last):
  File "", line 1, in 
  File "_sage_input_13.py", line 10, in 
    exec compile(u'print _support_.source_code("", globals(), system="sage")
  File "", line 1, in 
    
  File "/sagenb/sage_install/sage-4.7.2/devel/sagenb-flask/sagenb/misc/support.py", line 320, in source_code
    obj = eval(s, globs)
  File "", line 0
    
    ^
SyntaxError: unexpected EOF while parsing

magma_free?

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/interfaces/magma_free.py

Type: <type 'instance'>

Definition: magma_free(code, strip=True, columns=0)

Docstring:


    Evaluate MAGMA code without requiring that MAGMA be installed
    on your computer by using the free online MAGMA calculator.

    EXAMPLES:
        sage: magma_free("Factorization(9290348092384)")  # optional - internet
        [ <2, 5>, <290323377887, 1> ]

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/interfaces/magma_free.py

Type: <type 'instance'>

Definition: magma_free(code, strip=True, columns=0)

Docstring:


    Evaluate MAGMA code without requiring that MAGMA be installed
    on your computer by using the free online MAGMA calculator.

    EXAMPLES:
        sage: magma_free("Factorization(9290348092384)")  # optional - internet
        [ <2, 5>, <290323377887, 1> ]

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/rings/polynomial/polynomial_ring.py

Source Code (starting at line 867):

def cyclotomic_polynomial(self, n):
    """
    Return the nth cyclotomic polynomial as a polynomial in this
    polynomial ring. For details of the implementation, see the
    documentation for
    :func:`sage.rings.polynomial.cyclotomic.cyclotomic_coeffs`.

    EXAMPLES::

        sage: R = ZZ['x']
        sage: R.cyclotomic_polynomial(8)
        x^4 + 1
        sage: R.cyclotomic_polynomial(12)
        x^4 - x^2 + 1
        sage: S = PolynomialRing(FiniteField(7), 'x')
        sage: S.cyclotomic_polynomial(12)
        x^4 + 6*x^2 + 1
        sage: S.cyclotomic_polynomial(1)
        x + 6

    TESTS:

    Make sure it agrees with other systems for the trivial case::

        sage: ZZ['x'].cyclotomic_polynomial(1)
        x - 1
        sage: gp('polcyclo(1)')
        x - 1
    """
    if n <= 0:
        raise ArithmeticError, "n=%s must be positive"%n
    elif n == 1:
        return self.gen() - 1
    else:
        return self(cyclotomic.cyclotomic_coeffs(n), check=True)

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/rings/polynomial/polynomial_ring.py

Source Code (starting at line 867):

def cyclotomic_polynomial(self, n):
    """
    Return the nth cyclotomic polynomial as a polynomial in this
    polynomial ring. For details of the implementation, see the
    documentation for
    :func:`sage.rings.polynomial.cyclotomic.cyclotomic_coeffs`.

    EXAMPLES::

        sage: R = ZZ['x']
        sage: R.cyclotomic_polynomial(8)
        x^4 + 1
        sage: R.cyclotomic_polynomial(12)
        x^4 - x^2 + 1
        sage: S = PolynomialRing(FiniteField(7), 'x')
        sage: S.cyclotomic_polynomial(12)
        x^4 + 6*x^2 + 1
        sage: S.cyclotomic_polynomial(1)
        x + 6

    TESTS:

    Make sure it agrees with other systems for the trivial case::

        sage: ZZ['x'].cyclotomic_polynomial(1)
        x - 1
        sage: gp('polcyclo(1)')
        x - 1
    """
    if n <= 0:
        raise ArithmeticError, "n=%s must be positive"%n
    elif n == 1:
        return self.gen() - 1
    else:
        return self(cyclotomic.cyclotomic_coeffs(n), check=True)

preparse("n = 5")

'n = Integer(5)'

'n = Integer(5)'

n.divisors()

[1, 5]

[1, 5]

a = float(2^64); a

1.8446744073709552e+19

1.8446744073709552e+19

math.sin(a)

0.023598509904439558

0.023598509904439558

a = sin(2^64); a

sin(18446744073709551616)

sin(18446744073709551616)

type(a)

<type 'sage.symbolic.expression.Expression'>

<type 'sage.symbolic.expression.Expression'>

N(a, digits=1000)

0.0235985099044395586343659228761347751887971250470813595052082575127870\
367195714805353268175666623235437729751383647524917627871288325844860738\
774780555533262957487022395763401807856585404052810226503822554355095528\
028310586788053564221509928173361541766973214641915324432036026757696249\
788261667623001542396934455889240433419116118797637288382051695173924449\
181581804029311144948276644785620162298683748498605311140905034937489190\
912440501083085575276067441836443904976066614700407935079718694250167780\
060282736897308702214472463724083609963439955539184511211192964943428037\
445515558183613980406937553891459323066921748683325795142578242683676201\
615993420650011493920650606656966134469661959900199302403661301884174196\
373458046150133147119106730368104179014971275456878541846106129727392370\
953571120197087441563843741360076443085933594082080125509731439475852770\
875606175218014179445677770819225855418035844180363843045553659301208637\
1338941013691045701740737164131815074217522503906093271710053729660

0.02359850990443955863436592287613477518879712504708135950520825751278703671957148053532681756666232354377297513836475249176278712883258448607387747805555332629574870223957634018078565854040528102265038225543550955280283105867880535642215099281733615417669732146419153244320360267576962497882616676230015423969344558892404334191161187976372883820516951739244491815818040293111449482766447856201622986837484986053111409050349374891909124405010830855752760674418364439049760666147004079350797186942501677800602827368973087022144724637240836099634399555391845112111929649434280374455155581836139804069375538914593230669217486833257951425782426836762016159934206500114939206506066569661344696619599001993024036613018841741963734580461501331471191067303681041790149712754568785418461061297273923709535711201970874415638437413600764430859335940820801255097314394758527708756061752180141794456777708192258554180358441803638430455536593012086371338941013691045701740737164131815074217522503906093271710053729660

a = [bernoulli(2*n) for n in [1..16]]; a

[1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, 43867/798,
-174611/330, 854513/138, -236364091/2730, 8553103/6, -23749461029/870,
8615841276005/14322, -7709321041217/510]

[1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510, 43867/798, -174611/330, 854513/138, -236364091/2730, 8553103/6, -23749461029/870, 8615841276005/14322, -7709321041217/510]

B._latex_()

'\\left(\\begin{array}{rrrr}\n\\frac{1}{6} & -\\frac{1}{30} &
\\frac{1}{42} & -\\frac{1}{30} \\\\\n\\frac{5}{66} &
-\\frac{691}{2730} & \\frac{7}{6} & -\\frac{3617}{510}
\\\\\n\\frac{43867}{798} & -\\frac{174611}{330} &
\\frac{854513}{138} & -\\frac{236364091}{2730}
\\\\\n\\frac{8553103}{6} & -\\frac{23749461029}{870} &
\\frac{8615841276005}{14322} &
-\\frac{7709321041217}{510}\n\\end{array}\\right)'

'\\left(\\begin{array}{rrrr}\n\\frac{1}{6} & -\\frac{1}{30} & \\frac{1}{42} & -\\frac{1}{30} \\\\\n\\frac{5}{66} & -\\frac{691}{2730} & \\frac{7}{6} & -\\frac{3617}{510} \\\\\n\\frac{43867}{798} & -\\frac{174611}{330} & \\frac{854513}{138} & -\\frac{236364091}{2730} \\\\\n\\frac{8553103}{6} & -\\frac{23749461029}{870} & \\frac{8615841276005}{14322} & -\\frac{7709321041217}{510}\n\\end{array}\\right)'

B = matrix(4,4,a); show(B)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
\frac{1}{6} & -\frac{1}{30} & \frac{1}{42} & -\frac{1}{30} \\
\frac{5}{66} & -\frac{691}{2730} & \frac{7}{6} & -\frac{3617}{510} \\
\frac{43867}{798} & -\frac{174611}{330} & \frac{854513}{138} & -\frac{236364091}{2730} \\
\frac{8553103}{6} & -\frac{23749461029}{870} & \frac{8615841276005}{14322} & -\frac{7709321041217}{510}
\end{array}\right)

\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrrr}
\frac{1}{6} & -\frac{1}{30} & \frac{1}{42} & -\frac{1}{30} \\
\frac{5}{66} & -\frac{691}{2730} & \frac{7}{6} & -\frac{3617}{510} \\
\frac{43867}{798} & -\frac{174611}{330} & \frac{854513}{138} & -\frac{236364091}{2730} \\
\frac{8553103}{6} & -\frac{23749461029}{870} & \frac{8615841276005}{14322} & -\frac{7709321041217}{510}
\end{array}\right)

f = B.charpoly(); f

x^4 + 8067801164850701/533715*x^3 -
58285217560700198615240633/1403923430910*x^2 +
8132809004828014213156640/4258567740427*x +
55574407859795565506560000/73612385227381

x^4 + 8067801164850701/533715*x^3 - 58285217560700198615240633/1403923430910*x^2 + 8132809004828014213156640/4258567740427*x + 55574407859795565506560000/73612385227381

plot(f, -5000, 5000)

f.roots(CC)

[(-1.51163123214893e10, 1), (-0.113796311512103, 1), (0.159804146544381,
1), (2746.38784246979, 1)]

[(-1.51163123214893e10, 1), (-0.113796311512103, 1), (0.159804146544381, 1), (2746.38784246979, 1)]

RDF

Real Double Field

Real Double Field

a = sqrt(2) + 5/3; a

sqrt(2) + 5/3

sqrt(2) + 5/3

R.<x> = ZZ[]

x.parent()

Univariate Polynomial Ring in x over Integer Ring

Univariate Polynomial Ring in x over Integer Ring

x + 1/3

x + 1/3

x + 1/3

S = QQ['x'] S.coerce('1/3')

Traceback (click to the left of this block for traceback)
...
TypeError: no canonical coercion from <type 'str'> to Univariate
Polynomial Ring in x over Rational Field

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_53.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("UyA9IFFRWyd4J10KUy5jb2VyY2UoJzEvMycp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmp89sEzk/___code___.py", line 3, in <module>
    exec compile(u"S.coerce('1/3')" + '\n', '', 'single')
  File "", line 1, in <module>
    
  File "parent.pyx", line 1067, in sage.structure.parent.Parent.coerce (sage/structure/parent.c:8016)
  File "parent.pyx", line 1094, in sage.structure.parent.Parent.coerce (sage/structure/parent.c:7958)
TypeError: no canonical coercion from <type 'str'> to Univariate Polynomial Ring in x over Rational Field

y = x + 1/3 y.parent()

Univariate Polynomial Ring in x over Rational Field

Univariate Polynomial Ring in x over Rational Field

parent(5/3)

Rational Field

Rational Field

a = 5/3

a.parent()

Rational Field

Rational Field

parent(sqrt(2))

Symbolic Ring

Symbolic Ring

C = ZZ.category(); C

Category of euclidean domains

Category of euclidean domains

type(C)

<class 'sage.categories.euclidean_domains.EuclideanDomains'>

<class 'sage.categories.euclidean_domains.EuclideanDomains'>

type(ZZ)

<type 'sage.rings.integer_ring.IntegerRing_class'>

<type 'sage.rings.integer_ring.IntegerRing_class'>

ZZ.func

C.all_super_categories()

[Category of euclidean domains, Category of principal ideal domains,
Category of unique factorization domains, Category of gcd domains,
Category of integral domains, Category of commutative rings, Category of
domains, Category of rings, Category of rngs, Category of commutative
additive groups, Category of semirings, Category of commutative additive
monoids, Category of commutative additive semigroups, Category of
additive magmas, Category of monoids, Category of semigroups, Category
of magmas, Category of sets, Category of sets with partial maps,
Category of objects]

[Category of euclidean domains, Category of principal ideal domains, Category of unique factorization domains, Category of gcd domains, Category of integral domains, Category of commutative rings, Category of domains, Category of rings, Category of rngs, Category of commutative additive groups, Category of semirings, Category of commutative additive monoids, Category of commutative additive semigroups, Category of additive magmas, Category of monoids, Category of semigroups, Category of magmas, Category of sets, Category of sets with partial maps, Category of objects]

a = sin(2^64) v = RealIntervalField(100)(a); v

0.0235985099044395586343659228762?

0.0235985099044395586343659228762?

v.lower()

0.023598509904439558634365922876

0.023598509904439558634365922876

v.upper()

0.023598509904439558634365922877

0.023598509904439558634365922877

f.roots(RealIntervalField(100))

[(-1.5116312321489265648220103115283?e10, 1),
(-0.11379631151210255188658154882301?, 1),
(0.15980414654438143948220086785611?, 1),
(2746.3878424697929411908606679704?, 1)]

[(-1.5116312321489265648220103115283?e10, 1), (-0.11379631151210255188658154882301?, 1), (0.15980414654438143948220086785611?, 1), (2746.3878424697929411908606679704?, 1)]

v = B.eigenvalues(); v

[-1.511631232148927?e10, -0.1137963115121026?, 0.1598041465443815?,
2746.387842469793?]

[-1.511631232148927?e10, -0.1137963115121026?, 0.1598041465443815?, 2746.387842469793?]

B = matrix(RDF,2,2,[0,1,-1,0]) B.eigenvalues()

[1.0*I, -1.0*I]

[1.0*I, -1.0*I]

alpha = v[0]; alpha

-1.511631232148927?e10

-1.511631232148927?e10

matrix?

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/matrix/constructor.py

Type: <type ‘function’>

Definition: matrix(*args, **kwds)

Docstring:

Create a matrix.

INPUT: The matrix command takes the entries of a matrix, optionally preceded by a ring and the dimensions of the matrix, and returns a matrix.

The entries of a matrix can be specified as a flat list of elements, a list of lists (i.e., a list of rows), a list of Sage vectors, a callable object, or a dictionary having positions as keys and matrix entries as values (see the examples). If you pass in a callable object, then you must specify the number of rows and columns. You can create a matrix of zeros by passing an empty list or the integer zero for the entries. To construct a multiple of the identity (cI), you can specify square dimensions and pass in c. Calling matrix() with a Sage object may return something that makes sense. Calling matrix() with a NumPy array will convert the array to a matrix.

The ring, number of rows, and number of columns of the matrix can be specified by setting the ring, nrows, or ncols parameters or by passing them as the first arguments to the function in the order ring, nrows, ncols. The ring defaults to ZZ if it is not specified or cannot be determined from the entries. If the numbers of rows and columns are not specified and cannot be determined, then an empty 0x0 matrix is returned.

ring - the base ring for the entries of the matrix.
nrows - the number of rows in the matrix.
ncols - the number of columns in the matrix.
sparse - create a sparse matrix. This defaults to True when the entries are given as a dictionary, otherwise defaults to False.

OUTPUT:

a matrix

EXAMPLES:

sage: m=matrix(2); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

sage: m=matrix(2,3); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

sage: m=matrix(QQ,[[1,2,3],[4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

sage: m = matrix(QQ, 3, 3, lambda i, j: i+j); m
[0 1 2]
[1 2 3]
[2 3 4]
sage: m = matrix(3, lambda i,j: i-j); m
[ 0 -1 -2]
[ 1  0 -1]
[ 2  1  0]

sage: matrix(QQ,2,3,lambda x, y: x+y)
[0 1 2]
[1 2 3]
sage: matrix(QQ,3,2,lambda x, y: x+y)
[0 1]
[1 2]
[2 3]

sage: v1=vector((1,2,3))
sage: v2=vector((4,5,6))
sage: m=matrix([v1,v2]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

sage: m=matrix(QQ,2,[1,2,3,4,5,6]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

sage: m=matrix(QQ,2,3,[1,2,3,4,5,6]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

sage: m=matrix({(0,1): 2, (1,1):2/5}); m; m.parent()
[  0   2]
[  0 2/5]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

sage: m=matrix(QQ,2,3,{(1,1): 2}); m; m.parent()
[0 0 0]
[0 2 0]
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

sage: import numpy
sage: n=numpy.array([[1,2],[3,4]],float)
sage: m=matrix(n); m; m.parent()
[1.0 2.0]
[3.0 4.0]
Full MatrixSpace of 2 by 2 dense matrices over Real Double Field

sage: v = vector(ZZ, [1, 10, 100])
sage: m=matrix(v); m; m.parent()
[  1  10 100]
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
sage: m=matrix(GF(7), v); m; m.parent()
[1 3 2]
Full MatrixSpace of 1 by 3 dense matrices over Finite Field of size 7

sage: g = graphs.PetersenGraph()
sage: m = matrix(g); m; m.parent()
[0 1 0 0 1 1 0 0 0 0]
[1 0 1 0 0 0 1 0 0 0]
[0 1 0 1 0 0 0 1 0 0]
[0 0 1 0 1 0 0 0 1 0]
[1 0 0 1 0 0 0 0 0 1]
[1 0 0 0 0 0 0 1 1 0]
[0 1 0 0 0 0 0 0 1 1]
[0 0 1 0 0 1 0 0 0 1]
[0 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 0]
Full MatrixSpace of 10 by 10 dense matrices over Integer Ring

sage: matrix(ZZ, 10, 10, range(100), sparse=True).parent()
Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring

sage: R = PolynomialRing(QQ, 9, 'x')
sage: A = matrix(R, 3, 3, R.gens()); A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
sage: det(A)
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

TESTS:

sage: m=matrix(); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring
sage: m=matrix(QQ); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Rational Field
sage: m=matrix(QQ,2); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: m=matrix(QQ,2,3); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: m=matrix([]); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring
sage: m=matrix(QQ,[]); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Rational Field
sage: m=matrix(2,2,1); m; m.parent()
[1 0]
[0 1]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: m=matrix(QQ,2,2,1); m; m.parent()
[1 0]
[0 1]
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: m=matrix(2,3,0); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
sage: m=matrix(QQ,2,3,0); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: m=matrix([[1,2,3],[4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
sage: m=matrix(QQ,2,[[1,2,3],[4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: m=matrix(QQ,3,[[1,2,3],[4,5,6]]); m; m.parent()
Traceback (click to the left of this block for traceback)
...

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/matrix/constructor.py

Type: <type ‘function’>

Definition: matrix(*args, **kwds)

Docstring:

Create a matrix.

INPUT: The matrix command takes the entries of a matrix, optionally preceded by a ring and the dimensions of the matrix, and returns a matrix.

The entries of a matrix can be specified as a flat list of elements, a list of lists (i.e., a list of rows), a list of Sage vectors, a callable object, or a dictionary having positions as keys and matrix entries as values (see the examples). If you pass in a callable object, then you must specify the number of rows and columns. You can create a matrix of zeros by passing an empty list or the integer zero for the entries. To construct a multiple of the identity (cI), you can specify square dimensions and pass in c. Calling matrix() with a Sage object may return something that makes sense. Calling matrix() with a NumPy array will convert the array to a matrix.

The ring, number of rows, and number of columns of the matrix can be specified by setting the ring, nrows, or ncols parameters or by passing them as the first arguments to the function in the order ring, nrows, ncols. The ring defaults to ZZ if it is not specified or cannot be determined from the entries. If the numbers of rows and columns are not specified and cannot be determined, then an empty 0x0 matrix is returned.

ring - the base ring for the entries of the matrix.
nrows - the number of rows in the matrix.
ncols - the number of columns in the matrix.
sparse - create a sparse matrix. This defaults to True when the entries are given as a dictionary, otherwise defaults to False.

OUTPUT:

a matrix

EXAMPLES:

sage: m=matrix(2); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

sage: m=matrix(2,3); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

sage: m=matrix(QQ,[[1,2,3],[4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

sage: m = matrix(QQ, 3, 3, lambda i, j: i+j); m
[0 1 2]
[1 2 3]
[2 3 4]
sage: m = matrix(3, lambda i,j: i-j); m
[ 0 -1 -2]
[ 1  0 -1]
[ 2  1  0]

sage: matrix(QQ,2,3,lambda x, y: x+y)
[0 1 2]
[1 2 3]
sage: matrix(QQ,3,2,lambda x, y: x+y)
[0 1]
[1 2]
[2 3]

sage: v1=vector((1,2,3))
sage: v2=vector((4,5,6))
sage: m=matrix([v1,v2]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring

sage: m=matrix(QQ,2,[1,2,3,4,5,6]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

sage: m=matrix(QQ,2,3,[1,2,3,4,5,6]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field

sage: m=matrix({(0,1): 2, (1,1):2/5}); m; m.parent()
[  0   2]
[  0 2/5]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field

sage: m=matrix(QQ,2,3,{(1,1): 2}); m; m.parent()
[0 0 0]
[0 2 0]
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field

sage: import numpy
sage: n=numpy.array([[1,2],[3,4]],float)
sage: m=matrix(n); m; m.parent()
[1.0 2.0]
[3.0 4.0]
Full MatrixSpace of 2 by 2 dense matrices over Real Double Field

sage: v = vector(ZZ, [1, 10, 100])
sage: m=matrix(v); m; m.parent()
[  1  10 100]
Full MatrixSpace of 1 by 3 dense matrices over Integer Ring
sage: m=matrix(GF(7), v); m; m.parent()
[1 3 2]
Full MatrixSpace of 1 by 3 dense matrices over Finite Field of size 7

sage: g = graphs.PetersenGraph()
sage: m = matrix(g); m; m.parent()
[0 1 0 0 1 1 0 0 0 0]
[1 0 1 0 0 0 1 0 0 0]
[0 1 0 1 0 0 0 1 0 0]
[0 0 1 0 1 0 0 0 1 0]
[1 0 0 1 0 0 0 0 0 1]
[1 0 0 0 0 0 0 1 1 0]
[0 1 0 0 0 0 0 0 1 1]
[0 0 1 0 0 1 0 0 0 1]
[0 0 0 1 0 1 1 0 0 0]
[0 0 0 0 1 0 1 1 0 0]
Full MatrixSpace of 10 by 10 dense matrices over Integer Ring

sage: matrix(ZZ, 10, 10, range(100), sparse=True).parent()
Full MatrixSpace of 10 by 10 sparse matrices over Integer Ring

sage: R = PolynomialRing(QQ, 9, 'x')
sage: A = matrix(R, 3, 3, R.gens()); A
[x0 x1 x2]
[x3 x4 x5]
[x6 x7 x8]
sage: det(A)
-x2*x4*x6 + x1*x5*x6 + x2*x3*x7 - x0*x5*x7 - x1*x3*x8 + x0*x4*x8

TESTS:

sage: m=matrix(); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring
sage: m=matrix(QQ); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Rational Field
sage: m=matrix(QQ,2); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: m=matrix(QQ,2,3); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: m=matrix([]); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring
sage: m=matrix(QQ,[]); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Rational Field
sage: m=matrix(2,2,1); m; m.parent()
[1 0]
[0 1]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: m=matrix(QQ,2,2,1); m; m.parent()
[1 0]
[0 1]
Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: m=matrix(2,3,0); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
sage: m=matrix(QQ,2,3,0); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: m=matrix([[1,2,3],[4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
sage: m=matrix(QQ,2,[[1,2,3],[4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: m=matrix(QQ,3,[[1,2,3],[4,5,6]]); m; m.parent()
Traceback (most recent call last):
...
ValueError: Number of rows does not match up with specified number.
sage: m=matrix(QQ,2,3,[[1,2,3],[4,5,6]]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Rational Field
sage: m=matrix(QQ,2,4,[[1,2,3],[4,5,6]]); m; m.parent()
Traceback (most recent call last):
...
ValueError: Number of columns does not match up with specified number.
sage: m=matrix([(1,2,3),(4,5,6)]); m; m.parent()
[1 2 3]
[4 5 6]
Full MatrixSpace of 2 by 3 dense matrices over Integer Ring
sage: m=matrix([1,2,3,4,5,6]); m; m.parent()
[1 2 3 4 5 6]
Full MatrixSpace of 1 by 6 dense matrices over Integer Ring
sage: m=matrix((1,2,3,4,5,6)); m; m.parent()
[1 2 3 4 5 6]
Full MatrixSpace of 1 by 6 dense matrices over Integer Ring
sage: m=matrix(QQ,[1,2,3,4,5,6]); m; m.parent()
[1 2 3 4 5 6]
Full MatrixSpace of 1 by 6 dense matrices over Rational Field
sage: m=matrix(QQ,3,2,[1,2,3,4,5,6]); m; m.parent()
[1 2]
[3 4]
[5 6]
Full MatrixSpace of 3 by 2 dense matrices over Rational Field
sage: m=matrix(QQ,2,4,[1,2,3,4,5,6]); m; m.parent()
Traceback (most recent call last):
...
ValueError: entries has the wrong length
sage: m=matrix(QQ,5,[1,2,3,4,5,6]); m; m.parent()
Traceback (most recent call last):
...
TypeError: entries has the wrong length
sage: m=matrix({(1,1): 2}); m; m.parent()
[0 0]
[0 2]
Full MatrixSpace of 2 by 2 sparse matrices over Integer Ring
sage: m=matrix(QQ,{(1,1): 2}); m; m.parent()
[0 0]
[0 2]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
sage: m=matrix(QQ,3,{(1,1): 2}); m; m.parent()
[0 0 0]
[0 2 0]
[0 0 0]
Full MatrixSpace of 3 by 3 sparse matrices over Rational Field
sage: m=matrix(QQ,3,4,{(1,1): 2}); m; m.parent()
[0 0 0 0]
[0 2 0 0]
[0 0 0 0]
Full MatrixSpace of 3 by 4 sparse matrices over Rational Field
sage: m=matrix(QQ,2,{(1,1): 2}); m; m.parent()
[0 0]
[0 2]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
sage: m=matrix(QQ,1,{(1,1): 2}); m; m.parent()
Traceback (most recent call last):
...
IndexError: invalid entries list
sage: m=matrix({}); m; m.parent()
[]
Full MatrixSpace of 0 by 0 sparse matrices over Integer Ring
sage: m=matrix(QQ,{}); m; m.parent()
[]
Full MatrixSpace of 0 by 0 sparse matrices over Rational Field
sage: m=matrix(QQ,2,{}); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 sparse matrices over Rational Field
sage: m=matrix(QQ,2,3,{}); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 sparse matrices over Rational Field
sage: m=matrix(2,{}); m; m.parent()
[0 0]
[0 0]
Full MatrixSpace of 2 by 2 sparse matrices over Integer Ring
sage: m=matrix(2,3,{}); m; m.parent()
[0 0 0]
[0 0 0]
Full MatrixSpace of 2 by 3 sparse matrices over Integer Ring
sage: m=matrix(0); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring
sage: m=matrix(0,2); m; m.parent()
[]
Full MatrixSpace of 0 by 2 dense matrices over Integer Ring
sage: m=matrix(2,0); m; m.parent()
[]
Full MatrixSpace of 2 by 0 dense matrices over Integer Ring
sage: m=matrix(0,[1]); m; m.parent()
Traceback (most recent call last):
...
ValueError: entries has the wrong length
sage: m=matrix(1,0,[]); m; m.parent()
[]
Full MatrixSpace of 1 by 0 dense matrices over Integer Ring
sage: m=matrix(0,1,[]); m; m.parent()
[]
Full MatrixSpace of 0 by 1 dense matrices over Integer Ring
sage: m=matrix(0,[]); m; m.parent()
[]
Full MatrixSpace of 0 by 0 dense matrices over Integer Ring
sage: m=matrix(0,{}); m; m.parent()
[]
Full MatrixSpace of 0 by 0 sparse matrices over Integer Ring
sage: m=matrix(0,{(1,1):2}); m; m.parent()
Traceback (most recent call last):
...
IndexError: invalid entries list
sage: m=matrix(2,0,{(1,1):2}); m; m.parent()
Traceback (most recent call last):
...
IndexError: invalid entries list
sage: import numpy
sage: n=numpy.array([[numpy.complex(0,1),numpy.complex(0,2)],[3,4]],complex)
sage: m=matrix(n); m; m.parent()
[1.0*I 2.0*I]
[  3.0   4.0]
Full MatrixSpace of 2 by 2 dense matrices over Complex Double Field
sage: n=numpy.array([[1,2],[3,4]],'int32')
sage: m=matrix(n); m; m.parent()
[1 2]
[3 4]
Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: n = numpy.array([[1,2,3],[4,5,6],[7,8,9]],'float32')
sage: m=matrix(n); m; m.parent()
[1.0 2.0 3.0]
[4.0 5.0 6.0]
[7.0 8.0 9.0]
Full MatrixSpace of 3 by 3 dense matrices over Real Double Field
sage: n=numpy.array([[1,2,3],[4,5,6],[7,8,9]],'float64')
sage: m=matrix(n); m; m.parent()
[1.0 2.0 3.0]
[4.0 5.0 6.0]
[7.0 8.0 9.0]
Full MatrixSpace of 3 by 3 dense matrices over Real Double Field
sage: n=numpy.array([[1,2,3],[4,5,6],[7,8,9]],'complex64')
sage: m=matrix(n); m; m.parent()
[1.0 2.0 3.0]
[4.0 5.0 6.0]
[7.0 8.0 9.0]
Full MatrixSpace of 3 by 3 dense matrices over Complex Double Field
sage: n=numpy.array([[1,2,3],[4,5,6],[7,8,9]],'complex128')
sage: m=matrix(n); m; m.parent()
[1.0 2.0 3.0]
[4.0 5.0 6.0]
[7.0 8.0 9.0]
Full MatrixSpace of 3 by 3 dense matrices over Complex Double Field
sage: a = matrix([[1,2],[3,4]])
sage: b = matrix(a.numpy()); b
[1 2]
[3 4]
sage: a == b
True
sage: c = matrix(a.numpy('float32')); c
[1.0 2.0]
[3.0 4.0]
sage: matrix(numpy.array([[5]]))
[5]
sage: v = vector(ZZ, [1, 10, 100])
sage: m=matrix(ZZ['x'], v); m; m.parent()
[  1  10 100]
Full MatrixSpace of 1 by 3 dense matrices over Univariate Polynomial Ring in x over Integer Ring
sage: matrix(ZZ, 10, 10, range(100)).parent()
Full MatrixSpace of 10 by 10 dense matrices over Integer Ring
sage: m = matrix(GF(7), [[1/3,2/3,1/2], [3/4,4/5,7]]); m; m.parent()
[5 3 4]
[6 5 0]
Full MatrixSpace of 2 by 3 dense matrices over Finite Field of size 7
sage: m = matrix([[1,2,3], [RDF(2), CDF(1,2), 3]]); m; m.parent()
[        1.0         2.0         3.0]
[        2.0 1.0 + 2.0*I         3.0]
Full MatrixSpace of 2 by 3 dense matrices over Complex Double Field
sage: m=matrix(3,3,1/2); m; m.parent()
[1/2   0   0]
[  0 1/2   0]
[  0   0 1/2]
Full MatrixSpace of 3 by 3 dense matrices over Rational Field
sage: matrix([[1],[2,3]])
Traceback (most recent call last):
...
ValueError: List of rows is not valid (rows are wrong types or lengths)
sage: matrix([[1],2])
Traceback (most recent call last):
...
ValueError: List of rows is not valid (rows are wrong types or lengths)
sage: matrix(vector(RR,[1,2,3])).parent()
Full MatrixSpace of 1 by 3 dense matrices over Real Field with 53 bits of precision
sage: matrix(ZZ, [[0] for i in range(10^5)]).is_zero() # see #10158
True

AUTHORS:

??: Initial implementation
Jason Grout (2008-03): almost a complete rewrite, with bits and pieces from the original implementation

alpha.minpoly()

x^4 + 8067801164850701/533715*x^3 -
58285217560700198615240633/1403923430910*x^2 +
8132809004828014213156640/4258567740427*x +
55574407859795565506560000/73612385227381

x^4 + 8067801164850701/533715*x^3 - 58285217560700198615240633/1403923430910*x^2 + 8132809004828014213156640/4258567740427*x + 55574407859795565506560000/73612385227381

a = gp("2")

b = a._sage_??

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/interfaces/gp.py

Source Code (starting at line 809):

def _sage_(self):
    """
    Convert this GpElement into a Sage object, if possible.

    EXAMPLES::

        sage: gp(I).sage()
        i
        sage: gp(I).sage().parent()
        Maximal Order in Number Field in i with defining polynomial x^2 + 1

    ::

        sage: M = Matrix(ZZ,2,2,[1,2,3,4]); M
        [1 2]
        [3 4]
        sage: gp(M)
        [1, 2; 3, 4]
        sage: gp(M).sage()
        [1 2]
        [3 4]
        sage: gp(M).sage() == M
        True
    """
    return pari(str(self)).python()

File: /sagenb/sage_install/sage-4.7.2/local/lib/python2.6/site-packages/sage/interfaces/gp.py

Source Code (starting at line 809):

def _sage_(self):
    """
    Convert this GpElement into a Sage object, if possible.

    EXAMPLES::

        sage: gp(I).sage()
        i
        sage: gp(I).sage().parent()
        Maximal Order in Number Field in i with defining polynomial x^2 + 1

    ::

        sage: M = Matrix(ZZ,2,2,[1,2,3,4]); M
        [1 2]
        [3 4]
        sage: gp(M)
        [1, 2; 3, 4]
        sage: gp(M).sage()
        [1 2]
        [3 4]
        sage: gp(M).sage() == M
        True
    """
    return pari(str(self)).python()

type(b)

<type 'sage.rings.integer.Integer'>

<type 'sage.rings.integer.Integer'>

factor(2839479287342398) # uses Pari

2 * 4017089 * 353424991

2 * 4017089 * 353424991

c = factorial(25); c

15511210043330985984000000

15511210043330985984000000

[c.valuation(p) for p in prime_range(2, 25)]

[22, 10, 6, 3, 2, 1, 1, 1, 1]

[22, 10, 6, 3, 2, 1, 1, 1, 1]

L = divisors(c); len(L)

L[17000], factor(L[17000])

(10680579, 3^6 * 7^2 * 13 * 23)

(10680579, 3^6 * 7^2 * 13 * 23)

a = mod(4/7,2301); a

a.rational_reconstruction()

4/7

4/7

time number_of_partitions(10^7)

920271755026045466855962781668256054307294052810239793953285763517412985\
262323501978822916547103339332198764311128926694429965192014469337180574\
258854255101965669713692722439368861237049443900118466267242229358838809\
496460215546742114497122936318794382420922229797018587870350451317915617\
184990942766797810155029441933075045772129188981041614489343545384206438\
995186836592262593125170223430127680062490663477743842242002004914231357\
899486287124678626100600066102278733540933447719703464029124680195026177\
412964857500689657276787365748796835192363570613191348609145244276270764\
465804777408575949440508551447566418811489630464191115045309280131652547\
731782793747141150484980319367430611463990946023472819466195667158678183\
681130408875817996838721729445775753916663228712954510481120704923859087\
275241592392223665876910286300131474621294645735699401736284697581755151\
900016413451408993670931908590802671856117921704294651978579681174352991\
410791557056622496173369118595092855785844134417338169649142692589187435\
301742615221458864914330678814703958326474085781955660425664604942849123\
726120485051272439872545976606903238042522371480831216853001064737946906\
898017475046619462994720059814429480949267645638531717278315386109140567\
421507374975384275021240249642090800336572787686941682199463463099066866\
702204042926855402821066676615841472011557402435218180206292340119241672\
141006748199826744593298451761269206891815323035747468238859771982766512\
554781263236924319788525197851803361705413947686093663891147724061366833\
494127467087092245969359401102159299184733830143890656327648052676575908\
655135939782442443116808542177355365928232190011995697704315727188881084\
528176142957772556227153624081222658008489322761962675453544265567233476\
857358975517683731089773908740330164186182660688614126800469264172995743\
679921180655438244719647115856821466186517301168907732792274321379544884\
573350740004120158204881150331945483918135267242867162717503464701181877\
399958004314416087296985163324282563458624844066236399562733354725457411\
347627828347256593033627186674151973687980402192233716371343312564266298\
874305195901960277032311918990723575237081597193774377829718464719281814\
341332773477437283460627207869478917449115162489982657581564116298023459\
732495270075351700956243598913730840133454971118265085585706676644886436\
331834764264219387054191767502487300255270790939231320204783038516420542\
008888313103784838110884255487585083733240938734797302734376936793420810\
867020185209718779670927269908884747550412550772822753523086679809536245\
524746452957745548238178302717520159051754876562006809869627325035982116\
186671112647935832871470323287962444698515988848621901187755088113589185\
457016501136471407349024158013999313263656227842088073986822774216895311\
418550769982898380687461265543773217506522881285332749712463895138180510\
378178497047690889118184846592767949868064656342862993393444237184906674\
722873849811294966439997475566061223014020274110963671727713854372115721\
224489228469570767118124546455721433201289716309063821673453815865659403\
167483160827068775845409484961883982333993500194238336824233008140594931\
590134175918949793856506343243088419472771218165592793359389269115106835\
490432906902871027335713152227618464826154317860618134546336344159741794\
139420247061299867257344715523461677386135094707607583386378705799210071\
685144173415484815139532963734550586141746926780137597372467246969311252\
404574068882891540550303875935489428805492623836212595940806996986432453\
55453826567378500963781681659096276126857969078217677288980
Time: CPU 0.32 s, Wall: 0.32 s

92027175502604546685596278166825605430729405281023979395328576351741298526232350197882291654710333933219876431112892669442996519201446933718057425885425510196566971369272243936886123704944390011846626724222935883880949646021554674211449712293631879438242092222979701858787035045131791561718499094276679781015502944193307504577212918898104161448934354538420643899518683659226259312517022343012768006249066347774384224200200491423135789948628712467862610060006610227873354093344771970346402912468019502617741296485750068965727678736574879683519236357061319134860914524427627076446580477740857594944050855144756641881148963046419111504530928013165254773178279374714115048498031936743061146399094602347281946619566715867818368113040887581799683872172944577575391666322871295451048112070492385908727524159239222366587691028630013147462129464573569940173628469758175515190001641345140899367093190859080267185611792170429465197857968117435299141079155705662249617336911859509285578584413441733816964914269258918743530174261522145886491433067881470395832647408578195566042566460494284912372612048505127243987254597660690323804252237148083121685300106473794690689801747504661946299472005981442948094926764563853171727831538610914056742150737497538427502124024964209080033657278768694168219946346309906686670220404292685540282106667661584147201155740243521818020629234011924167214100674819982674459329845176126920689181532303574746823885977198276651255478126323692431978852519785180336170541394768609366389114772406136683349412746708709224596935940110215929918473383014389065632764805267657590865513593978244244311680854217735536592823219001199569770431572718888108452817614295777255622715362408122265800848932276196267545354426556723347685735897551768373108977390874033016418618266068861412680046926417299574367992118065543824471964711585682146618651730116890773279227432137954488457335074000412015820488115033194548391813526724286716271750346470118187739995800431441608729698516332428256345862484406623639956273335472545741134762782834725659303362718667415197368798040219223371637134331256426629887430519590196027703231191899072357523708159719377437782971846471928181434133277347743728346062720786947891744911516248998265758156411629802345973249527007535170095624359891373084013345497111826508558570667664488643633183476426421938705419176750248730025527079093923132020478303851642054200888831310378483811088425548758508373324093873479730273437693679342081086702018520971877967092726990888474755041255077282275352308667980953624552474645295774554823817830271752015905175487656200680986962732503598211618667111264793583287147032328796244469851598884862190118775508811358918545701650113647140734902415801399931326365622784208807398682277421689531141855076998289838068746126554377321750652288128533274971246389513818051037817849704769088911818484659276794986806465634286299339344423718490667472287384981129496643999747556606122301402027411096367172771385437211572122448922846957076711812454645572143320128971630906382167345381586565940316748316082706877584540948496188398233399350019423833682423300814059493159013417591894979385650634324308841947277121816559279335938926911510683549043290690287102733571315222761846482615431786061813454633634415974179413942024706129986725734471552346167738613509470760758338637870579921007168514417341548481513953296373455058614174692678013759737246724696931125240457406888289154055030387593548942880549262383621259594080699698643245355453826567378500963781681659096276126857969078217677288980
Time: CPU 0.32 s, Wall: 0.32 s

@interact def _(n=(25..10000)): P = plot(prime_pi, 0, n, gridlines='minor') Q = plot(x/log(x),2,n,color=(1,0,0)) R = plot(Li,2,n,color=(0,1,0)) (P+Q+R).show()

Click to the left again to hide and once more to show the dynamic interactive window

R.<x> = ZZ[] K.<a> = NumberField(x^6 + 40*x^3 + 1372) K

Number Field in a with defining polynomial x^6 + 40*x^3 + 1372

Number Field in a with defining polynomial x^6 + 40*x^3 + 1372

a^6

-40*a^3 - 1372

-40*a^3 - 1372

G = K.galois_group() G

Galois group of Number Field in a with defining polynomial x^6 + 40*x^3
+ 1372

Galois group of Number Field in a with defining polynomial x^6 + 40*x^3 + 1372

G.order()

f = G.1 f(a)

1/36*a^4 + 1/18*a

1/36*a^4 + 1/18*a

f(a)^6 + 40*f(a)^3 + 1372

P = K.primes_above(2)[0] H = G.inertia_group(P); H

Subgroup [(), (1,4,6)(2,5,3), (1,6,4)(2,3,5)] of Galois group of Number
Field in a with defining polynomial x^6 + 40*x^3 + 1372

Subgroup [(), (1,4,6)(2,5,3), (1,6,4)(2,3,5)] of Galois group of Number Field in a with defining polynomial x^6 + 40*x^3 + 1372

H.order()

[G.artin_symbol(Q) for Q in K.primes_above(5)]

[(1,5)(2,4)(3,6), (1,2)(3,4)(5,6), (1,3)(2,6)(4,5)]

[(1,5)(2,4)(3,6), (1,2)(3,4)(5,6), (1,3)(2,6)(4,5)]

A = NumberField(x^5 - x + 2, 'a').galois_group(type="pari"); A

Galois group PARI group [120, -1, 5, "S5"] of degree 5 of the Number
Field in a with defining polynomial x^5 - x + 2

Galois group PARI group [120, -1, 5, "S5"] of degree 5 of the Number Field in a with defining polynomial x^5 - x + 2

L.<a> = NumberField(x^2 + 23) L.class_number()

CyclotomicField(23).class_number(proof=False)

proof.number_field(False)

K.<a> = QuadraticField(-30) C = K.class_group() C

Class group of order 4 with structure C2 x C2 of Number Field in a with
defining polynomial x^2 + 30

Class group of order 4 with structure C2 x C2 of Number Field in a with defining polynomial x^2 + 30

C.gens()

[Fractional ideal class (5, a), Fractional ideal class (3, a)]

[Fractional ideal class (5, a), Fractional ideal class (3, a)]

E = EllipticCurve('37a')

E.rank()

v = E.integral_points(both_signs=True); v

[(-1 : -1 : 1), (-1 : 0 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 1),
(1 : 0 : 1), (2 : -3 : 1), (2 : 2 : 1), (6 : -15 : 1), (6 : 14 : 1)]

[(-1 : -1 : 1), (-1 : 0 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 1), (1 : 0 : 1), (2 : -3 : 1), (2 : 2 : 1), (6 : -15 : 1), (6 : 14 : 1)]

plot(E, xmax=7, gridlines=True) + points([z[:2] for z in v], color='red', pointsize=50)

E.change_ring(GF(next_prime(10^50))).cardinality()

100000000000000000000000001917684156174529696959920

100000000000000000000000001917684156174529696959920

set_random_seed(1) G = graphs.RandomLobster(8, .6, .3)

show(G, figsize=7)

G.automorphism_group()

Permutation Group with generators [(12,26)]

Permutation Group with generators [(12,26)]

G.chromatic_number()

G.shortest_path(13,20)

[13, 0, 1, 2, 3, 4, 5, 19, 20]

[13, 0, 1, 2, 3, 4, 5, 19, 20]

G = Graph({0:[1,13],1:[0,2,14],2:[1,3],3:[2,4,15],4:[3,5,17],5:[4,6,19],6:[5,7,21],7:[6,8],8:[7,9,22],9:[8,10],10:[9,11,24],11:[10,12,26],12:[11],13:[0],14:[1,27],15:[3,16],16:[15],17:[4,18],18:[17],19:[5,20],20:[19],21:[6],22:[8,23],23:[22],24:[10,25],25:[24],26:[11],27:[14]}); G.set_pos({0:[59.01325118963035,295.75251934606877],1:[94.36463578086565,286.04432694721777],2:[133.30963998762118,258.9301868699391],3:[179.99545918994187,230.9185690558532],4:[220.94569638353892,189.5405491129016],5:[255.53780531460455,193.66006066064193],6:[241.42300483955069,220.64477638661037],7:[207.24688263349074,216.294217659962],8:[171.93156299963962,207.8112323413829],9:[166.0508845760556,156.6569713830302],10:[167.145549092765,106.64751472831455],11:[181.78743153620886,64.52861700322893],12:[177.9915397622531,35],13:[35,303.3680516143165],14:[81,313],15:[193.38498733723105,265.60960759890276],16:[203.7527488812743,293.81972078992794],17:[244.4047669028139,148.9760323769727],18:[262.7735147030095,118.7511936438419],19:[289.4417315188959,186.0479380736545],20:[315,179.87133655914585],21:[263.8398063810127,240.815465279789],22:[159.42976668674544,254.61374301786665],23:[151.36632083811443,293.0030774459344],24:[144.22651404070498,86.34019199645445],25:[126.0053532049252,70.73123588302923],26:[198.49018000485754,37.72078507164241],27:[61,336]}); graph_editor(G);

show(G, figsize=7)

G.automorphism_group()

Permutation Group with generators [(13,27)(14,28), (12,26)]

Permutation Group with generators [(13,27)(14,28), (12,26)]

class RSAPrivateKey: def __init__(self, p, q, e): assert p.is_prime() and q.is_prime() self.p = p self.q = q self.n = p * q self.e = e self.d = (~mod(e, (p-1)*(q-1))).lift() self.length = RR(p * q).log(256).floor() assert self.length > 0 def decrypt(self, ciphertext): blocks = ciphertext.digits(self.n) plaintext = "" for b in blocks: pt = (mod(b, self.n)^self.d).lift() for c in pt.digits(256): plaintext += chr(c) return plaintext class RSAPublicKey: def __init__(self, private_key): self.n = private_key.n self.e = private_key.e self.length = private_key.length def encrypt(self, plaintext): blocks = [plaintext[i*self.length:(i+1)*self.length] for i in range(ceil(len(plaintext)/self.length))] ciphertext = 0 for i, b in enumerate(blocks): pt = sum([256^j * ord(ch) for j, ch in enumerate(b)]) ct = (mod(pt, self.n)^self.e).lift() ciphertext += ct * self.n^i return ciphertext

Priv = RSAPrivateKey(next_prime(120309), next_prime(1200), 31) Pub = RSAPublicKey(Priv)

ct = Pub.encrypt("hello world");ct

121827446329895738728628753088007

121827446329895738728628753088007

Priv.decrypt(ct)

'hello world'

'hello world'

def python_sum2(n): s = 0 for i in xrange(1, n+1): s += i*i return s

python_sum2(3)

n = int(60)

n.divisors()

Traceback (click to the left of this block for traceback)
...
AttributeError: 'int' object has no attribute 'divisors'

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_169.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("bi5kaXZpc29ycygp"),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpQ4JLm8/___code___.py", line 2, in <module>
    exec compile(u'n.divisors()
  File "", line 1, in <module>
    
AttributeError: 'int' object has no attribute 'divisors'

divisors(n)

[1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]

[1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]

python_sum2(2*10^6)

2666668666667000000

2666668666667000000

timeit("python_sum2(2*10^6)")

5 loops, best of 3: 2.44 s per loop

5 loops, best of 3: 2.44 s per loop

srange(1/3, 99/5,20/7)

[1/3, 67/21, 127/21, 187/21, 247/21, 307/21, 367/21]

[1/3, 67/21, 127/21, 187/21, 247/21, 307/21, 367/21]

%cython def cython_sum2(long n): cdef long i, s = 0 for i in range(1, n+1): s += i*i return s

__sagenb_s..._code_sage145_spyx.c __sagenb_s...de_sage145_spyx.html

cython_sum2(3)

range(2,6)

[2, 3, 4, 5]

[2, 3, 4, 5]

cython_sum2(2*10^6)

2666668666667000000

2666668666667000000

timeit("cython_sum2(2*10^6)")

125 loops, best of 3: 2.16 ms per loop

125 loops, best of 3: 2.16 ms per loop

204/2.16

94.4444444444444

94.4444444444444

var('k, n') show(factor(sum(k^2, k, 1, n)))

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{6} \, {\left(n + 1\right)} {\left(2 \, n + 1\right)} n

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{6} \, {\left(n + 1\right)} {\left(2 \, n + 1\right)} n

def sum2(n): return n*(2*n+1)*(n+1)/6

timeit("sum2(2*10^6)")

625 loops, best of 3: 2.75 µs per loop

625 loops, best of 3: 2.75 µs per loop

%cython def c_sum2(long n): return n*(2*n+1)*(n+1)/6

__sagenb_s..._code_sage160_spyx.c __sagenb_s...de_sage160_spyx.html

c_sum2(3)

timeit("c_sum2(2*10^6)")

625 loops, best of 3: 741 ns per loop

625 loops, best of 3: 741 ns per loop

c_sum2(2*10^6)

-407788678951258603

-407788678951258603

n = 2*10^6; n*(2*n+1)*(n+1) > 2^63

True

True

x = var('x'); show(solve(x^3 + x - 1==0, x)[0])

\newcommand{\Bold}[1]{\mathbf{#1}}x = -\frac{1}{2} \, {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{18} \, \sqrt{3} \sqrt{31} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} + \frac{-i \, \sqrt{3} + 1}{6 \, {\left(\frac{1}{18} \, \sqrt{3} \sqrt{31} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}}

\newcommand{\Bold}[1]{\mathbf{#1}}x = -\frac{1}{2} \, {\left(i \, \sqrt{3} + 1\right)} {\left(\frac{1}{18} \, \sqrt{3} \sqrt{31} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)} + \frac{-i \, \sqrt{3} + 1}{6 \, {\left(\frac{1}{18} \, \sqrt{3} \sqrt{31} + \frac{1}{2}\right)}^{\left(\frac{1}{3}\right)}}

var('alpha, y') show(solve([3*x + 7*y == 2, alpha*x + 3*y == 8], x,y)[0])

\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = \frac{50}{7 \, \alpha - 9}, y = \frac{2 \, {\left(\alpha - 12\right)}}{7 \, \alpha - 9}\right]

\newcommand{\Bold}[1]{\mathbf{#1}}\left[x = \frac{50}{7 \, \alpha - 9}, y = \frac{2 \, {\left(\alpha - 12\right)}}{7 \, \alpha - 9}\right]

n = 500 A = random_matrix(QQ,n,n,num_bound=10, den_bound=10) v = random_matrix(QQ,n,1,num_bound=10, den_bound=10) A.visualize_structure()

# Using IML -- http://www.cs.uwaterloo.ca/~astorjoh/iml.html time w = A \ v

Time: CPU 5.03 s, Wall: 5.02 s

Time: CPU 5.03 s, Wall: 5.02 s

w.ncols(), w.nrows()

(1, 500)

(1, 500)

A[0,0]

-4/5

-4/5

w[0]

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var('x, t') f = sqrt(1 - x^2) assume(t+1 > 0) g = integral(f, x, -1, t); show(g)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{4} \, \pi + \frac{1}{2} \, \sqrt{-t^{2} + 1} t + \frac{1}{2} \, \arcsin\left(t\right)

\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{4} \, \pi + \frac{1}{2} \, \sqrt{-t^{2} + 1} t + \frac{1}{2} \, \arcsin\left(t\right)

y = function('y',x) show(desolve(diff(y,x)==(x+y)^2,y,contrib_ode=True))

\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[x = c - \arctan\left(\sqrt{t}\right), y\left(x\right) = -x - \sqrt{t}\right], \left[x = c + \arctan\left(\sqrt{t}\right), y\left(x\right) = -x + \sqrt{t}\right]\right]

\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[x = c - \arctan\left(\sqrt{t}\right), y\left(x\right) = -x - \sqrt{t}\right], \left[x = c + \arctan\left(\sqrt{t}\right), y\left(x\right) = -x + \sqrt{t}\right]\right]

plot(x*sin(1/x^2),x,.01,.4)

points([(cos(theta), sin(theta)) for theta in srange(0, 2*pi, pi/37)], color=(1,0,0), aspect_ratio=1)

x,y = var('x,y') a=plot_vector_field((x,y), (x,-3,3),(y,-3,3),color='blue') b=plot_vector_field((y,-x),(x,-3,3),(y,-3,3),color='red') show(a+b)

f(x,y) = sin(x - y) * y * cos(x) plot3d(f, (x,-3,3), (y,-3,3), opacity=.9, color='red')

t6 = Tachyon(camera_center=(0,-4,1), xres = 800, yres = 600, raydepth = 12, aspectratio=.75, antialiasing = True) t6.light((0.02,0.012,0.001), 0.01, (1,0,0)) t6.light((0,0,10), 0.01, (0,0,1)) t6.texture('s', color = (.8,1,1), opacity = .9, specular = .95, diffuse = .3, ambient = 0.05) t6.texture('p', color = (0,0,1), opacity = 1, specular = .2) t6.sphere((-1,-.57735,-0.7071),1,'s') t6.sphere((1,-.57735,-0.7071),1,'s') t6.sphere((0,1.15465,-0.7071),1,'s') t6.sphere((0,0,0.9259),1,'s') t6.plane((0,0,-1.9259),(0,0,1),'p') t6.show()

P1 = Zp(3).plot(rgbcolor=(0,1,0)) P2 = Zp(7).plot(rgbcolor=(1,0,0)) P3 = text("Have a great break!\n\nQuestions?",(0.0,0.0),fontsize=20) (P1+P2+P3).show(axes=False)

search_src("modular polynomial")

Search Source Code: "modular polynomial"databases/db_modular_polynomials.py
modular/ssmod/ssmod.py

Search Source Code: "modular polynomial"databases/db_modular_polynomials.py
modular/ssmod/ssmod.py

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Search Source Code: "modular polynomial"

Search Source Code: "modular polynomial"