Functional notation¶

These are function so that you can write foo(x) instead of x.foo() in certain common cases.

AUTHORS:

William Stein: Initial version
David Joyner (2005-12-20): More Examples

sage.misc.functional.N(x, prec=None, digits=None)¶

Returns a numerical approximation of x with at least prec bits of precision.

Note

Both upper case N and lower case n are aliases for numerical_approx().

INPUT:

x - an object that has a numerical_approx method, or can be coerced into a real or complex field
prec (optional) - an integer (bits of precision)
digits (optional) - an integer (digits of precision)

If neither the prec or digits are specified, the default is 53 bits of precision.

EXAMPLES:

sage: numerical_approx(pi, 10)
3.1
sage: numerical_approx(pi, digits=10)
3.141592654
sage: numerical_approx(pi^2 + e, digits=20)
12.587886229548403854
sage: n(pi^2 + e)        
12.5878862295484
sage: N(pi^2 + e)
12.5878862295484
sage: n(pi^2 + e, digits=50)
12.587886229548403854194778471228813633070946500941
sage: a = CC(-5).n(prec=100)
sage: b = ComplexField(100)(-5)
sage: a == b
True
sage: type(a) == type(b)
True

You can also usually use method notation:

sage: (pi^2 + e).n()
12.5878862295484

TESTS:

sage: numerical_approx(I)
1.00000000000000*I
sage: x = QQ['x'].gen()
sage: F.<k> = NumberField(x^2+2, embedding=sqrt(CC(2))*CC.0)
sage: numerical_approx(k)
1.41421356237309*I

sage: type(numerical_approx(CC(1/2)))
<type 'sage.rings.complex_number.ComplexNumber'>

sage.misc.functional.acos(x)¶

Returns the arc cosine of x.

EXAMPLES:

sage: acos(.5)
1.04719755119660
sage: acos(sin(pi/3))
arccos(1/2*sqrt(3))
sage: acos(sin(pi/3)).simplify_full()
1/6*pi

sage.misc.functional.additive_order(x)¶

Returns the additive order of $x$ .

EXAMPLES:

sage: additive_order(5)
+Infinity
sage: additive_order(Mod(5,11))
11
sage: additive_order(Mod(4,12))
3

sage.misc.functional.arg(x)¶

Returns the argument of a complex number $x$ .

EXAMPLES:

sage: z = CC(1,2)
sage: theta = arg(z)
sage: cos(theta)*abs(z)   
1.00000000000000
sage: sin(theta)*abs(z)
2.00000000000000

sage.misc.functional.asin(x)¶

Returns the arc sine of x.

EXAMPLES:

sage: asin(.5)
0.523598775598299
sage: asin(sin(pi/3))
arcsin(1/2*sqrt(3))
sage: asin(sin(pi/3)).simplify_full()
1/3*pi

sage.misc.functional.atan(x)¶

Returns the arc tangent of x.

EXAMPLES:

sage: z = atan(3);z
arctan(3)
sage: n(z)
1.24904577239825
sage: atan(tan(pi/4))
1/4*pi

sage.misc.functional.base_field(x)¶

Returns the base field over which x is defined.

EXAMPLES:

sage: R = PolynomialRing(GF(7), 'x')
sage: base_ring(R)
Finite Field of size 7
sage: base_field(R)
Finite Field of size 7

This catches base rings which are fields as well, but does not implement a base_field method for objects which do not have one:

sage: R.base_field()
...
AttributeError: 'PolynomialRing_dense_mod_p' object has no attribute 'base_field'

sage.misc.functional.base_ring(x)¶

Returns the base ring over which x is defined.

EXAMPLES:

sage: R = PolynomialRing(GF(7), 'x')
sage: base_ring(R)
Finite Field of size 7

sage.misc.functional.basis(x)¶

Returns the fixed basis of x.

EXAMPLES:

sage: V = VectorSpace(QQ,3)
sage: S = V.subspace([[1,2,0],[2,2,-1]])
sage: basis(S)
[
(1, 0, -1),
(0, 1, 1/2)
]

sage.misc.functional.category(x)¶

Returns the category of x.

EXAMPLES:

sage: V = VectorSpace(QQ,3)
sage: category(V)
Category of vector spaces over Rational Field

sage.misc.functional.ceil(x)¶

Returns the ceiling (least integer) function of x.

EXAMPLES:

sage: ceil(3.5)
4
sage: ceil(7/2)
4
sage: ceil(-3.5)
-3
sage: ceil(RIF(1.3,2.3))
3.?

sage.misc.functional.characteristic_polynomial(x, var='x')¶

Returns the characteristic polynomial of x in the given variable.

EXAMPLES:

sage: M = MatrixSpace(QQ,3,3)
sage: A = M([1,2,3,4,5,6,7,8,9])
sage: charpoly(A)
x^3 - 15*x^2 - 18*x
sage: charpoly(A, 't')
t^3 - 15*t^2 - 18*t

sage: k.<alpha> = GF(7^10); k
Finite Field in alpha of size 7^10
sage: alpha.charpoly('T')
T^10 + T^6 + T^5 + 4*T^4 + T^3 + 2*T^2 + 3*T + 3
sage: characteristic_polynomial(alpha, 'T')
T^10 + T^6 + T^5 + 4*T^4 + T^3 + 2*T^2 + 3*T + 3

sage.misc.functional.charpoly(x, var='x')¶

Returns the characteristic polynomial of x in the given variable.

EXAMPLES:

sage: M = MatrixSpace(QQ,3,3)
sage: A = M([1,2,3,4,5,6,7,8,9])
sage: charpoly(A)
x^3 - 15*x^2 - 18*x
sage: charpoly(A, 't')
t^3 - 15*t^2 - 18*t

sage: k.<alpha> = GF(7^10); k
Finite Field in alpha of size 7^10
sage: alpha.charpoly('T')
T^10 + T^6 + T^5 + 4*T^4 + T^3 + 2*T^2 + 3*T + 3
sage: characteristic_polynomial(alpha, 'T')
T^10 + T^6 + T^5 + 4*T^4 + T^3 + 2*T^2 + 3*T + 3

sage.misc.functional.coerce(P, x)¶

Attempts to coerce x to type P if possible.

EXAMPLES:

sage: type(5)
<type 'sage.rings.integer.Integer'>
sage: type(coerce(QQ,5))
<type 'sage.rings.rational.Rational'>

sage.misc.functional.cyclotomic_polynomial(n, var='x')¶

Returns the $n^{th}$ cyclotomic polynomial.

EXAMPLES:

sage: cyclotomic_polynomial(3)
x^2 + x + 1
sage: cyclotomic_polynomial(4)
x^2 + 1
sage: cyclotomic_polynomial(9)
x^6 + x^3 + 1
sage: cyclotomic_polynomial(10)
x^4 - x^3 + x^2 - x + 1
sage: cyclotomic_polynomial(11)
x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1

sage.misc.functional.decomposition(x)¶

Returns the decomposition of x.

EXAMPLES:

sage: M = matrix([[2, 3], [3, 4]])
sage: M.decomposition()
[
(Ambient free module of rank 2 over the principal ideal domain Integer Ring, True)
]

sage: G.<a,b> = DirichletGroup(20)
sage: c = a*b
sage: d = c.decomposition(); d
[Dirichlet character modulo 4 of conductor 4 mapping 3 |--> -1,
Dirichlet character modulo 5 of conductor 5 mapping 2 |--> zeta4]
sage: d[0].parent()
Group of Dirichlet characters of modulus 4 over Cyclotomic Field of order 4 and degree 2

sage.misc.functional.denominator(x)¶

Returns the denominator of x.

EXAMPLES:

sage: denominator(17/11111)
11111
sage: R.<x> = PolynomialRing(QQ)
sage: F = FractionField(R)
sage: r = (x+1)/(x-1)
sage: denominator(r)
x - 1

sage.misc.functional.det(x)¶

Returns the determinant of x.

EXAMPLES:

sage: M = MatrixSpace(QQ,3,3)
sage: A = M([1,2,3,4,5,6,7,8,9])
sage: det(A)
0

sage.misc.functional.dim(x)¶

Returns the dimension of x.

EXAMPLES:

sage: V = VectorSpace(QQ,3)
sage: S = V.subspace([[1,2,0],[2,2,-1]])
sage: dimension(S)
2

sage.misc.functional.dimension(x)¶

Returns the dimension of x.

EXAMPLES:

sage: V = VectorSpace(QQ,3)
sage: S = V.subspace([[1,2,0],[2,2,-1]])
sage: dimension(S)
2

sage.misc.functional.disc(x)¶

Returns the discriminant of x.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^29 - 17*x - 1, 'alpha')
sage: K = S.number_field()
sage: discriminant(K)
-15975100446626038280218213241591829458737190477345113376757479850566957249523

sage.misc.functional.discriminant(x)¶

Returns the discriminant of x.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^29 - 17*x - 1, 'alpha')
sage: K = S.number_field()
sage: discriminant(K)
-15975100446626038280218213241591829458737190477345113376757479850566957249523

sage.misc.functional.eta(x)¶

Returns the value of the eta function at $x$ , which must be in the upper half plane.

The $\eta$ function is

$\eta(z) = e^{\pi i z / 12} \prod_{n=1}^{\infty}(1-e^{2\pi inz})$

EXAMPLES:

sage: eta(1+I)
0.742048775837 + 0.19883137023*I

sage.misc.functional.exp(x)¶

Returns the value of the exponentiation function at x.

EXAMPLES:

sage: exp(3)
e^3
sage: exp(0)
1
sage: exp(2.5)
12.1824939607035
sage: exp(pi*i)
-1

sage.misc.functional.factor(x, *args, **kwds)¶

Returns the (prime) factorization of x.

EXAMPLES:

sage: factor(factorial(10))
2^8 * 3^4 * 5^2 * 7
sage: n = next_prime(10^6); n
1000003
sage: factor(n)
1000003

Note that this depends on the type of x::

sage: factor(55)
5 * 11
sage: factor(x^2+2*x+1)
(x + 1)^2
sage: factor(55*x^2+110*x+55)
55*(x + 1)^2

sage.misc.functional.factorisation(x, *args, **kwds)¶

Returns the (prime) factorization of x.

EXAMPLES:

sage: factor(factorial(10))
2^8 * 3^4 * 5^2 * 7
sage: n = next_prime(10^6); n
1000003
sage: factor(n)
1000003

Note that this depends on the type of x::

sage: factor(55)
5 * 11
sage: factor(x^2+2*x+1)
(x + 1)^2
sage: factor(55*x^2+110*x+55)
55*(x + 1)^2

sage.misc.functional.factorization(x, *args, **kwds)¶

Returns the (prime) factorization of x.

EXAMPLES:

sage: factor(factorial(10))
2^8 * 3^4 * 5^2 * 7
sage: n = next_prime(10^6); n
1000003
sage: factor(n)
1000003

Note that this depends on the type of x::

sage: factor(55)
5 * 11
sage: factor(x^2+2*x+1)
(x + 1)^2
sage: factor(55*x^2+110*x+55)
55*(x + 1)^2

sage.misc.functional.fcp(x, var='x')¶

Returns the factorization of the characteristic polynomial of x.

EXAMPLES:

sage: M = MatrixSpace(QQ,3,3)
sage: A = M([1,2,3,4,5,6,7,8,9])
sage: fcp(A, 'x')
x * (x^2 - 15*x - 18)

sage.misc.functional.gen(x)¶

Returns the generator of x.

EXAMPLES:

sage: R.<x> = QQ[]; R
Univariate Polynomial Ring in x over Rational Field
sage: gen(R)
x
sage: gen(GF(7))
1
sage: A = AbelianGroup(1, [23])
sage: gen(A)
f

sage.misc.functional.gens(x)¶

Returns the generators of x.

EXAMPLES:

sage: R.<x,y> = SR[]
sage: R
Multivariate Polynomial Ring in x, y over Symbolic Ring
sage: gens(R)
(x, y)
sage: A = AbelianGroup(5, [5,5,7,8,9])
sage: gens(A)
(f0, f1, f2, f3, f4)

sage.misc.functional.hecke_operator(x, n)¶

Returns the n-th Hecke operator T_n acting on x.

EXAMPLES:

sage: M = ModularSymbols(1,12)
sage: hecke_operator(M,5)
Hecke operator T_5 on Modular Symbols space of dimension 3 for Gamma_0(1) of weight 12 with sign 0 over Rational Field

sage.misc.functional.image(x)¶

Returns the image of x.

EXAMPLES:

sage: M = MatrixSpace(QQ,3,3)
sage: A = M([1,2,3,4,5,6,7,8,9])
sage: image(A)
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[ 1  0 -1]
[ 0  1  2]

sage.misc.functional.integral(x, *args, **kwds)¶

Returns an indefinite or definite integral of an object x.

First call x.integrate() and if that fails make an object and integrate it using Maxima, maple, etc, as specified by algorithm.

For symbolic expression calls sage.calculus.calculus.integral - see this function for available options.

EXAMPLES:

sage: f = cyclotomic_polynomial(10)
sage: integral(f)
1/5*x^5 - 1/4*x^4 + 1/3*x^3 - 1/2*x^2 + x

sage: integral(sin(x),x)
-cos(x)

sage: y = var('y')
sage: integral(sin(x),y)
y*sin(x)

sage: integral(sin(x), x, 0, pi/2)
1
sage: sin(x).integral(x, 0,pi/2)
1
sage: integral(exp(-x), (x, 1, oo))
e^(-1)

Numerical approximation:

sage: h = integral(sin(x)/x^2, (x, 1, pi/2)); h
integrate(sin(x)/x^2, x, 1, 1/2*pi)
sage: h.n()
0.339447940978915...

Specific algorithm can be used for integration:

sage: integral(sin(x)^2, x, algorithm='maxima')
1/2*x - 1/4*sin(2*x)
sage: integral(sin(x)^2, x, algorithm='sympy')
-1/2*sin(x)*cos(x) + 1/2*x

sage.misc.functional.integral_closure(x)¶

Returns the integral closure of x.

EXAMPLES:

sage: integral_closure(QQ)
Rational Field
sage: K.<a> = QuadraticField(5)
sage: O2 = K.order(2*a); O2
Order in Number Field in a with defining polynomial x^2 - 5
sage: integral_closure(O2)
Maximal Order in Number Field in a with defining polynomial x^2 - 5

sage.misc.functional.integrate(x, *args, **kwds)¶

Returns an indefinite or definite integral of an object x.

First call x.integrate() and if that fails make an object and integrate it using Maxima, maple, etc, as specified by algorithm.

For symbolic expression calls sage.calculus.calculus.integral - see this function for available options.

EXAMPLES:

sage: f = cyclotomic_polynomial(10)
sage: integral(f)
1/5*x^5 - 1/4*x^4 + 1/3*x^3 - 1/2*x^2 + x

sage: integral(sin(x),x)
-cos(x)

sage: y = var('y')
sage: integral(sin(x),y)
y*sin(x)

sage: integral(sin(x), x, 0, pi/2)
1
sage: sin(x).integral(x, 0,pi/2)
1
sage: integral(exp(-x), (x, 1, oo))
e^(-1)

Numerical approximation:

sage: h = integral(sin(x)/x^2, (x, 1, pi/2)); h
integrate(sin(x)/x^2, x, 1, 1/2*pi)
sage: h.n()
0.339447940978915...

Specific algorithm can be used for integration:

sage: integral(sin(x)^2, x, algorithm='maxima')
1/2*x - 1/4*sin(2*x)
sage: integral(sin(x)^2, x, algorithm='sympy')
-1/2*sin(x)*cos(x) + 1/2*x

sage.misc.functional.interval(a, b)¶

Integers between a and b inclusive (a and b integers).

EXAMPLES:

sage: I = interval(1,3)
sage: 2 in I
True
sage: 1 in I
True
sage: 4 in I
False

sage.misc.functional.is_commutative(x)¶

Returns whether or not x is commutative.

EXAMPLES:

sage: R = PolynomialRing(QQ, 'x')
sage: is_commutative(R)
True

sage.misc.functional.is_even(x)¶

Returns whether or not an integer x is even, e.g., divisible by 2.

EXAMPLES:

sage: is_even(-1)
False
sage: is_even(4)
True
sage: is_even(-2)
True

sage.misc.functional.is_field(x)¶

Returns whether or not x is a field.

EXAMPLES:

sage: R = PolynomialRing(QQ, 'x')
sage: F = FractionField(R)
sage: is_field(F)
True

sage.misc.functional.is_integrally_closed(x)¶

Returns whether x is integrally closed.

EXAMPLES:

sage: is_integrally_closed(QQ)
True
sage: K.<a> = NumberField(x^2 + 189*x + 394)
sage: R = K.order(2*a)
sage: is_integrally_closed(R)
False

sage.misc.functional.is_noetherian(x)¶

Returns whether or not x is a Noetherian object (has ascending chain condition).

EXAMPLES:

sage: from sage.misc.functional import is_noetherian
sage: is_noetherian(ZZ)
True
sage: is_noetherian(QQ)
True
sage: A = SteenrodAlgebra(3)
sage: is_noetherian(A)
False

sage.misc.functional.is_odd(x)¶

Returns whether or not x is odd. This is by definition the complement of is_even.

EXAMPLES:

sage: is_odd(-2)
False
sage: is_odd(-3)
True
sage: is_odd(0)
False
sage: is_odd(1)
True

sage.misc.functional.isqrt(x)¶

Returns an integer square root, i.e., the floor of a square root.

EXAMPLES:

sage: isqrt(10)
3
sage: isqrt(10r)
3

sage.misc.functional.kernel(x)¶

Returns the left kernel of x.

EXAMPLES:

sage: M = MatrixSpace(QQ,3,2)
sage: A = M([1,2,3,4,5,6])
sage: kernel(A)
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 -2  1]
sage: kernel(A.transpose())
Vector space of degree 2 and dimension 0 over Rational Field
Basis matrix:
[]

Here are two corner cases:: sage: M=MatrixSpace(QQ,0,3) sage: A=M([]) sage: kernel(A) Vector space of degree 0 and dimension 0 over Rational Field Basis matrix: [] sage: kernel(A.transpose()).basis() [ (1, 0, 0), (0, 1, 0), (0, 0, 1) ]

sage.misc.functional.krull_dimension(x)¶

Returns the Krull dimension of x.

EXAMPLES:

sage: krull_dimension(QQ)
0
sage: krull_dimension(ZZ)
1
sage: krull_dimension(ZZ[sqrt(5)])
1
sage: U.<x,y,z> = PolynomialRing(ZZ,3); U
Multivariate Polynomial Ring in x, y, z over Integer Ring
sage: U.krull_dimension()
4

sage.misc.functional.lift(x)¶

Lift an object of a quotient ring $R/I$ to $R$ .

EXAMPLES: We lift an integer modulo $3$ .

sage: Mod(2,3).lift()
2

We lift an element of a quotient polynomial ring.

sage: R.<x> = QQ['x']
sage: S.<xmod> = R.quo(x^2 + 1)
sage: lift(xmod-7)
x - 7

sage.misc.functional.log(x, b=None)¶

Returns the log of x to the base b. The default base is e.

INPUT:

x - number
b - base (default: None, which means natural log)

OUTPUT: number

Note

In Magma, the order of arguments is reversed from in Sage, i.e., the base is given first. We use the opposite ordering, so the base can be viewed as an optional second argument.

EXAMPLES:

sage: log(e^2)
2
sage: log(16,2)
4
sage: log(3.)
1.09861228866811

sage.misc.functional.minimal_polynomial(x, var='x')¶

Returns the minimal polynomial of x.

EXAMPLES:

sage: a = matrix(ZZ, 2, [1..4])
sage: minpoly(a)
x^2 - 5*x - 2
sage: minpoly(a,'t')
t^2 - 5*t - 2
sage: minimal_polynomial(a)
x^2 - 5*x - 2
sage: minimal_polynomial(a,'theta')
theta^2 - 5*theta - 2

sage.misc.functional.minpoly(x, var='x')¶

Returns the minimal polynomial of x.

EXAMPLES:

sage: a = matrix(ZZ, 2, [1..4])
sage: minpoly(a)
x^2 - 5*x - 2
sage: minpoly(a,'t')
t^2 - 5*t - 2
sage: minimal_polynomial(a)
x^2 - 5*x - 2
sage: minimal_polynomial(a,'theta')
theta^2 - 5*theta - 2

sage.misc.functional.multiplicative_order(x)¶

Returns the multiplicative order of self, if self is a unit, or raise ArithmeticError otherwise.

EXAMPLES:

sage: a = mod(5,11)
sage: multiplicative_order(a)
5
sage: multiplicative_order(mod(2,11))
10
sage: multiplicative_order(mod(2,12))
...
ArithmeticError: multiplicative order of 2 not defined since it is not a unit modulo 12

sage.misc.functional.n(x, prec=None, digits=None)¶

Returns a numerical approximation of x with at least prec bits of precision.

Note

Both upper case N and lower case n are aliases for numerical_approx().

INPUT:

x - an object that has a numerical_approx method, or can be coerced into a real or complex field
prec (optional) - an integer (bits of precision)
digits (optional) - an integer (digits of precision)

If neither the prec or digits are specified, the default is 53 bits of precision.

EXAMPLES:

sage: numerical_approx(pi, 10)
3.1
sage: numerical_approx(pi, digits=10)
3.141592654
sage: numerical_approx(pi^2 + e, digits=20)
12.587886229548403854
sage: n(pi^2 + e)        
12.5878862295484
sage: N(pi^2 + e)
12.5878862295484
sage: n(pi^2 + e, digits=50)
12.587886229548403854194778471228813633070946500941
sage: a = CC(-5).n(prec=100)
sage: b = ComplexField(100)(-5)
sage: a == b
True
sage: type(a) == type(b)
True

You can also usually use method notation:

sage: (pi^2 + e).n()
12.5878862295484

TESTS:

sage: numerical_approx(I)
1.00000000000000*I
sage: x = QQ['x'].gen()
sage: F.<k> = NumberField(x^2+2, embedding=sqrt(CC(2))*CC.0)
sage: numerical_approx(k)
1.41421356237309*I

sage: type(numerical_approx(CC(1/2)))
<type 'sage.rings.complex_number.ComplexNumber'>

sage.misc.functional.ngens(x)¶

Returns the number of generators of x.

EXAMPLES:

sage: R.<x,y> = SR[]; R
Multivariate Polynomial Ring in x, y over Symbolic Ring
sage: ngens(R)
2
sage: A = AbelianGroup(5, [5,5,7,8,9])
sage: ngens(A)
5
sage: ngens(ZZ)
1

sage.misc.functional.norm(x)¶

Returns the norm of x.

EXAMPLES:

sage: z = 1+2*I
sage: norm(z)
5
sage: norm(CDF(z))
5.0
sage: norm(CC(z))
5.00000000000000

sage.misc.functional.numerator(x)¶

Returns the numerator of x.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: F = FractionField(R)
sage: r = (x+1)/(x-1)
sage: numerator(r)
x + 1
sage: numerator(17/11111)
17

sage.misc.functional.numerical_approx(x, prec=None, digits=None)¶

Returns a numerical approximation of x with at least prec bits of precision.

Note

Both upper case N and lower case n are aliases for numerical_approx().

INPUT:

x - an object that has a numerical_approx method, or can be coerced into a real or complex field
prec (optional) - an integer (bits of precision)
digits (optional) - an integer (digits of precision)

If neither the prec or digits are specified, the default is 53 bits of precision.

EXAMPLES:

sage: numerical_approx(pi, 10)
3.1
sage: numerical_approx(pi, digits=10)
3.141592654
sage: numerical_approx(pi^2 + e, digits=20)
12.587886229548403854
sage: n(pi^2 + e)        
12.5878862295484
sage: N(pi^2 + e)
12.5878862295484
sage: n(pi^2 + e, digits=50)
12.587886229548403854194778471228813633070946500941
sage: a = CC(-5).n(prec=100)
sage: b = ComplexField(100)(-5)
sage: a == b
True
sage: type(a) == type(b)
True

You can also usually use method notation:

sage: (pi^2 + e).n()
12.5878862295484

TESTS:

sage: numerical_approx(I)
1.00000000000000*I
sage: x = QQ['x'].gen()
sage: F.<k> = NumberField(x^2+2, embedding=sqrt(CC(2))*CC.0)
sage: numerical_approx(k)
1.41421356237309*I

sage: type(numerical_approx(CC(1/2)))
<type 'sage.rings.complex_number.ComplexNumber'>

sage.misc.functional.objgen(x)¶

EXAMPLES:

sage: R, x = objgen(FractionField(QQ['x']))
sage: R
Fraction Field of Univariate Polynomial Ring in x over Rational Field
sage: x
x

sage.misc.functional.objgens(x)¶

EXAMPLES:

sage: R, x = objgens(PolynomialRing(QQ,3, 'x'))
sage: R
Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
sage: x
(x0, x1, x2)

sage.misc.functional.one(R)¶

Returns the one element of the ring R.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: one(R)*x == x
True
sage: one(R) in R
True

sage.misc.functional.order(x)¶

Returns the order of x. If x is a ring or module element, this is the additive order of x.

EXAMPLES:

sage: C = CyclicPermutationGroup(10)
sage: order(C)
10
sage: F = GF(7)
sage: order(F)
7

sage.misc.functional.parent(x)¶

Returns x.parent() if defined, or type(x) if not.

EXAMPLE:

sage: Z = parent(int(5))
sage: Z(17)
17
sage: Z
<type 'int'>

sage.misc.functional.quo(x, y, *args, **kwds)¶

Returns the quotient object x/y, e.g., a quotient of numbers or of a polynomial ring x by the ideal generated by y, etc.

EXAMPLES:

sage: quotient(5,6)
5/6
sage: quotient(5.,6.)
0.833333333333333
sage: R.<x> = ZZ[]; R
Univariate Polynomial Ring in x over Integer Ring
sage: I = Ideal(R, x^2+1)
sage: quotient(R, I)
Univariate Quotient Polynomial Ring in xbar over Integer Ring with modulus x^2 + 1

sage.misc.functional.quotient(x, y, *args, **kwds)¶

Returns the quotient object x/y, e.g., a quotient of numbers or of a polynomial ring x by the ideal generated by y, etc.

EXAMPLES:

sage: quotient(5,6)
5/6
sage: quotient(5.,6.)
0.833333333333333
sage: R.<x> = ZZ[]; R
Univariate Polynomial Ring in x over Integer Ring
sage: I = Ideal(R, x^2+1)
sage: quotient(R, I)
Univariate Quotient Polynomial Ring in xbar over Integer Ring with modulus x^2 + 1

sage.misc.functional.rank(x)¶

Returns the rank of x.

EXAMPLES: We compute the rank of a matrix:

sage: M = MatrixSpace(QQ,3,3)
sage: A = M([1,2,3,4,5,6,7,8,9])
sage: rank(A)
2

We compute the rank of an elliptic curve:

sage: E = EllipticCurve([0,0,1,-1,0])
sage: rank(E)
1

sage.misc.functional.regulator(x)¶

Returns the regulator of x.

EXAMPLES:

sage: regulator(NumberField(x^2-2, 'a'))
0.881373587019543
sage: regulator(EllipticCurve('11a'))
1.00000000000000

sage.misc.functional.round(x, ndigits=0)¶

round(number[, ndigits]) - double-precision real number

Round a number to a given precision in decimal digits (default 0 digits). This always returns a real double field element.

EXAMPLES:

sage: round(sqrt(2),2)
1.41
sage: round(sqrt(2),5)
1.41421
sage: round(pi)
3.0
sage: b = 5.4999999999999999
sage: round(b)
5.0

Since we use floating-point with a limited range, some roundings can’t be performed:

sage: round(sqrt(Integer('1'*1000)))
+infinity

IMPLEMENTATION: If ndigits is specified, it calls Python’s builtin round function, and converts the result to a real double field element. Otherwise, it tries the argument’s .round() method, and if that fails, it falls back to the builtin round function.

Note

This is currently slower than the builtin round function, since it does more work - i.e., allocating an RDF element and initializing it. To access the builtin version do import __builtin__; __builtin__.round.

sage.misc.functional.show(x, *args, **kwds)¶

Show a graphics object x.

For additional ways to show objects in the notebook, look at the methods on the html object. For example, html.table will produce an HTML table from a nested list.

OPTIONAL INPUT:

filename - (default: None) string

SOME OF THESE MAY APPLY:

dpi - dots per inch
figsize- [width, height] (same for square aspect)
axes - (default: True)
fontsize - positive integer
frame - (default: False) draw a MATLAB-like frame around the image

EXAMPLES:

sage: show(graphs(3))
sage: show(list(graphs(3)))

sage.misc.functional.sqrt(x)¶

Returns a square root of x.

This function (numerical_sqrt) is deprecated. Use sqrt(x, prec=n) instead.

EXAMPLES:

sage: numerical_sqrt(10.1)
doctest:1: DeprecationWarning: numerical_sqrt is deprecated, use sqrt(x, prec=n) instead
3.17804971641414
sage: numerical_sqrt(9)
3

sage.misc.functional.squarefree_part(x)¶

Returns the square free part of $x$ , i.e., a divisor $z$ such that $x = z y^2$ , for a perfect square $y^2$ .

EXAMPLES:

sage: squarefree_part(100)
1
sage: squarefree_part(12)
3
sage: squarefree_part(10)
10

sage: x = QQ['x'].0
sage: S = squarefree_part(-9*x*(x-6)^7*(x-3)^2); S
-9*x^2 + 54*x
sage: S.factor()
(-9) * (x - 6) * x

sage: f = (x^3 + x + 1)^3*(x-1); f
x^10 - x^9 + 3*x^8 + 3*x^5 - 2*x^4 - x^3 - 2*x - 1
sage: g = squarefree_part(f); g
x^4 - x^3 + x^2 - 1
sage: g.factor()
(x - 1) * (x^3 + x + 1)

sage.misc.functional.symbolic_sum(expression, *args, **kwds)¶

Returns the symbolic sum $\sum_{v = a}^b expression$ with respect to the variable $v$ with endpoints $a$ and $b$ .

INPUT:

expression - a symbolic expression
v - a variable or variable name
a - lower endpoint of the sum
b - upper endpoint of the sum
algorithm - (default: ‘maxima’) one of - ‘maxima’ - use Maxima (the default) - ‘maple’ - (optional) use Maple - ‘mathematica’ - (optional) use Mathematica

EXAMPLES:

sage: k, n = var('k,n')
sage: sum(k, k, 1, n).factor()
1/2*(n + 1)*n

sage: sum(1/k^4, k, 1, oo)
1/90*pi^4

sage: sum(1/k^5, k, 1, oo)
zeta(5)

A well known binomial identity:

sage: sum(binomial(n,k), k, 0, n)
2^n

The binomial theorem:

sage: x, y = var('x, y')
sage: sum(binomial(n,k) * x^k * y^(n-k), k, 0, n)
(x + y)^n

sage: sum(k * binomial(n, k), k, 1, n)
n*2^(n - 1)

sage: sum((-1)^k*binomial(n,k), k, 0, n)
0

sage: sum(2^(-k)/(k*(k+1)), k, 1, oo)
-log(2) + 1

Summing a hypergeometric term:

sage: sum(binomial(n, k) * factorial(k) / factorial(n+1+k), k, 0, n)
1/2*sqrt(pi)/factorial(n + 1/2)

We check a well known identity:

sage: bool(sum(k^3, k, 1, n) == sum(k, k, 1, n)^2)
True

A geometric sum:

sage: a, q = var('a, q')
sage: sum(a*q^k, k, 0, n)
(a*q^(n + 1) - a)/(q - 1)

The geometric series:

sage: assume(abs(q) < 1)
sage: sum(a*q^k, k, 0, oo)
-a/(q - 1)

A divergent geometric series. Don’t forget to forget your assumptions:

sage: forget()
sage: assume(q > 1)
sage: sum(a*q^k, k, 0, oo)
...
ValueError: Sum is divergent.

This summation only Mathematica can perform:

sage: sum(1/(1+k^2), k, -oo, oo, algorithm = 'mathematica')     # optional  -- requires mathematica
pi*coth(pi)

Use Maple as a backend for summation:

sage: sum(binomial(n,k)*x^k, k, 0, n, algorithm = 'maple')      # optional  -- requires maple
(x + 1)^n

Note

Sage can currently only understand a subset of the output of Maxima, Maple and Mathematica, so even if the chosen backend can perform the summation the result might not be convertable into a Sage expression.

sage.misc.functional.transpose(x)¶

Returns the transpose of x.

EXAMPLES:

sage: M = MatrixSpace(QQ,3,3)
sage: A = M([1,2,3,4,5,6,7,8,9])
sage: transpose(A)
[1 4 7]
[2 5 8]
[3 6 9]

sage.misc.functional.xinterval(a, b)¶

Iterator over the integers between a and b, inclusive.

EXAMPLES:

sage: I = xinterval(2,5); I
xrange(2, 6)
sage: 5 in I
True
sage: 6 in I
False

sage.misc.functional.zero(R)¶

Returns the zero element of the ring R.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: zero(R) in R
True
sage: zero(R)*x == zero(R)
True

Functional notation¶

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