Functional notation¶

These are functions 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, algorithm=None)

Returns a numerical approximation of an object x with at least prec bits (or decimal digits) of precision.

Note

Both upper case N and lower case n are aliases for numerical_approx(), and all three may be used as methods.

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)
• algorithm (optional) - a string specifying the algorithm to use for functions that implement more than one

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

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=40)
sage: b = ComplexField(40)(-5)
sage: a == b
True
sage: parent(a) is parent(b)
True
sage: numerical_approx(9)
9.00000000000000


You can also usually use method notation.

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


Vectors and matrices may also have their entries approximated.

sage: v = vector(RDF, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)

sage: v = vector(CDF, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Complex Field with 53 bits of precision
sage: v.n(prec=20)
(1.0000, 2.0000, 3.0000)

sage: u = vector(QQ, [1/2, 1/3, 1/4])
sage: n(u, prec=15)
(0.5000, 0.3333, 0.2500)
sage: n(u, digits=5)
(0.50000, 0.33333, 0.25000)

sage: v = vector(QQ, [1/2, 0, 0, 1/3, 0, 0, 0, 1/4], sparse=True)
sage: u = v.numerical_approx(digits=4)
sage: u.is_sparse()
True
sage: u
(0.5000, 0.0000, 0.0000, 0.3333, 0.0000, 0.0000, 0.0000, 0.2500)

sage: A = matrix(QQ, 2, 3, range(6))
sage: A.n()
[0.000000000000000  1.00000000000000  2.00000000000000]
[ 3.00000000000000  4.00000000000000  5.00000000000000]

sage: B = matrix(Integers(12), 3, 8, srange(24))
sage: N(B, digits=2)
[0.00  1.0  2.0  3.0  4.0  5.0  6.0  7.0]
[ 8.0  9.0  10.  11. 0.00  1.0  2.0  3.0]
[ 4.0  5.0  6.0  7.0  8.0  9.0  10.  11.]


Internally, numerical approximations of real numbers are stored in base-2. Therefore, numbers which look the same in their decimal expansion might be different:

sage: x=N(pi, digits=3); x
3.14
sage: y=N(3.14, digits=3); y
3.14
sage: x==y
False
sage: x.str(base=2)
'11.001001000100'
sage: y.str(base=2)
'11.001000111101'


Increasing the precision of a floating point number is not allowed:

sage: CC(-5).n(prec=100)
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 100 bits, use at most 53 bits
sage: n(1.3r, digits=20)
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 70 bits, use at most 53 bits
sage: RealField(24).pi().n()
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 53 bits, use at most 24 bits


As an exceptional case, digits=1 usually leads to 2 digits (one significant) in the decimal output (see trac ticket #11647):

sage: N(pi, digits=1)
3.2
sage: N(pi, digits=2)
3.1
sage: N(100*pi, digits=1)
320.
sage: N(100*pi, digits=2)
310.


In the following example, pi and 3 are both approximated to two bits of precision and then subtracted, which kills two bits of precision:

sage: N(pi, prec=2)
3.0
sage: N(3, prec=2)
3.0
sage: N(pi - 3, prec=2)
0.00


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'>


The following tests trac ticket #10761, in which n() would break when called on complex-valued algebraic numbers.

sage: E = matrix(3, [3,1,6,5,2,9,7,3,13]).eigenvalues(); E
[18.16815365088822?, -0.08407682544410650? - 0.2190261484802906?*I, -0.08407682544410650? + 0.2190261484802906?*I]
sage: E[1].parent()
Algebraic Field
sage: [a.n() for a in E]
[18.1681536508882, -0.0840768254441065 - 0.219026148480291*I, -0.0840768254441065 + 0.219026148480291*I]


Make sure we’ve rounded up log(10,2) enough to guarantee sufficient precision (trac ticket #10164):

sage: ks = 4*10**5, 10**6
sage: check_str_length = lambda k: len(str(numerical_approx(1+10**-k,digits=k+1)))-1 >= k+1
sage: check_precision = lambda k: numerical_approx(1+10**-k,digits=k+1)-1 > 0
sage: all(check_str_length(k) and check_precision(k) for k in ks)
True


Testing we have sufficient precision for the golden ratio (trac ticket #12163), note that the decimal point adds 1 to the string length:

sage: len(str(n(golden_ratio, digits=5000)))
5001
sage: len(str(n(golden_ratio, digits=5000000)))  # long time (4s on sage.math, 2012)
5000001


Check that trac ticket #14778 is fixed:

sage: n(0, algorithm='foo')
0.000000000000000


Returns the additive order of $$x$$.

EXAMPLES:

sage: additive_order(5)
+Infinity
11
3

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()
Traceback (most recent call last):
...
AttributeError: 'PolynomialRing_dense_mod_p_with_category' 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.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


Ensure the variable name of the polynomial does not conflict with variables used within the matrix, and that non-integral powers of variables don’t confuse the computation (trac ticket #14403):

sage: y = var('y')
sage: a = matrix([[x,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: characteristic_polynomial(a).list()
[x, -3*x - 1, 3*x + 3, -x - 3, 1]
sage: b = matrix([[y^(1/2),0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: charpoly(b).list()
[sqrt(y), -3*sqrt(y) - 1, 3*sqrt(y) + 3, -sqrt(y) - 3, 1]

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


Ensure the variable name of the polynomial does not conflict with variables used within the matrix, and that non-integral powers of variables don’t confuse the computation (trac ticket #14403):

sage: y = var('y')
sage: a = matrix([[x,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: characteristic_polynomial(a).list()
[x, -3*x - 1, 3*x + 3, -x - 3, 1]
sage: b = matrix([[y^(1/2),0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]])
sage: charpoly(b).list()
[sqrt(y), -3*sqrt(y) - 1, 3*sqrt(y) + 3, -sqrt(y) - 3, 1]

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)
0.7420487758365647 + 0.1988313702299107*I

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.integral() 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(tan(x)/x, (x, 1, pi/3)); h
integrate(tan(x)/x, x, 1, 1/3*pi)
sage: h.n()
0.07571599101...


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*cos(x)*sin(x) + 1/2*x


TESTS:

A symbolic integral from trac ticket #11445 that was incorrect in earlier versions of Maxima:

sage: f = abs(x - 1) + abs(x + 1) - 2*abs(x)
sage: integrate(f, (x, -Infinity, Infinity))
2


Another symbolic integral, from trac ticket #11238, that used to return zero incorrectly; with Maxima 5.26.0 one gets 1/2*sqrt(pi)*e^(1/4), whereas with 5.29.1, and even more so with 5.33.0, the expression is less pleasant, but still has the same value. Unfortunately, the computation takes a very long time with the default settings, so we temporarily use the Maxima setting domain: real:

sage: sage.calculus.calculus.maxima('domain: real')
real
sage: f = exp(-x) * sinh(sqrt(x))
sage: t = integrate(f, x, 0, Infinity); t            # long time
1/4*sqrt(pi)*(erf(1) - 1)*e^(1/4) - 1/4*(sqrt(pi)*(erf(1) - 1) - sqrt(pi) + 2*e^(-1) - 2)*e^(1/4) + 1/4*sqrt(pi)*e^(1/4) - 1/2*e^(1/4) + 1/2*e^(-3/4)
1/2*sqrt(pi)*e^(1/4)
sage: sage.calculus.calculus.maxima('domain: complex')
complex


An integral which used to return -1 before maxima 5.28. See trac ticket #12842:

sage: f = e^(-2*x)/sqrt(1-e^(-2*x))
sage: integrate(f, x, 0, infinity)
1


This integral would cause a stack overflow in earlier versions of Maxima, crashing sage. See trac ticket #12377. We don’t care about the result here, just that the computation completes successfully:

sage: y = (x^2)*exp(x) / (1 + exp(x))^2
sage: _ = integrate(y, x, -1000, 1000)

sage.misc.functional.integral_closure(x)

Returns the integral closure of x.

EXAMPLES:

sage: integral_closure(QQ)
Rational Field
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.integral() 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(tan(x)/x, (x, 1, pi/3)); h
integrate(tan(x)/x, x, 1, 1/3*pi)
sage: h.n()
0.07571599101...


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*cos(x)*sin(x) + 1/2*x


TESTS:

A symbolic integral from trac ticket #11445 that was incorrect in earlier versions of Maxima:

sage: f = abs(x - 1) + abs(x + 1) - 2*abs(x)
sage: integrate(f, (x, -Infinity, Infinity))
2


Another symbolic integral, from trac ticket #11238, that used to return zero incorrectly; with Maxima 5.26.0 one gets 1/2*sqrt(pi)*e^(1/4), whereas with 5.29.1, and even more so with 5.33.0, the expression is less pleasant, but still has the same value. Unfortunately, the computation takes a very long time with the default settings, so we temporarily use the Maxima setting domain: real:

sage: sage.calculus.calculus.maxima('domain: real')
real
sage: f = exp(-x) * sinh(sqrt(x))
sage: t = integrate(f, x, 0, Infinity); t            # long time
1/4*sqrt(pi)*(erf(1) - 1)*e^(1/4) - 1/4*(sqrt(pi)*(erf(1) - 1) - sqrt(pi) + 2*e^(-1) - 2)*e^(1/4) + 1/4*sqrt(pi)*e^(1/4) - 1/2*e^(1/4) + 1/2*e^(-3/4)
1/2*sqrt(pi)*e^(1/4)
sage: sage.calculus.calculus.maxima('domain: complex')
complex


An integral which used to return -1 before maxima 5.28. See trac ticket #12842:

sage: f = e^(-2*x)/sqrt(1-e^(-2*x))
sage: integrate(f, x, 0, infinity)
1


This integral would cause a stack overflow in earlier versions of Maxima, crashing sage. See trac ticket #12377. We don’t care about the result here, just that the computation completes successfully:

sage: y = (x^2)*exp(x) / (1 + exp(x))^2
sage: _ = integrate(y, x, -1000, 1000)

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_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))
Traceback (most recent call last):
...
ArithmeticError: multiplicative order of 2 not defined since it is not a unit modulo 12

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

Returns a numerical approximation of an object x with at least prec bits (or decimal digits) of precision.

Note

Both upper case N and lower case n are aliases for numerical_approx(), and all three may be used as methods.

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)
• algorithm (optional) - a string specifying the algorithm to use for functions that implement more than one

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

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=40)
sage: b = ComplexField(40)(-5)
sage: a == b
True
sage: parent(a) is parent(b)
True
sage: numerical_approx(9)
9.00000000000000


You can also usually use method notation.

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


Vectors and matrices may also have their entries approximated.

sage: v = vector(RDF, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)

sage: v = vector(CDF, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Complex Field with 53 bits of precision
sage: v.n(prec=20)
(1.0000, 2.0000, 3.0000)

sage: u = vector(QQ, [1/2, 1/3, 1/4])
sage: n(u, prec=15)
(0.5000, 0.3333, 0.2500)
sage: n(u, digits=5)
(0.50000, 0.33333, 0.25000)

sage: v = vector(QQ, [1/2, 0, 0, 1/3, 0, 0, 0, 1/4], sparse=True)
sage: u = v.numerical_approx(digits=4)
sage: u.is_sparse()
True
sage: u
(0.5000, 0.0000, 0.0000, 0.3333, 0.0000, 0.0000, 0.0000, 0.2500)

sage: A = matrix(QQ, 2, 3, range(6))
sage: A.n()
[0.000000000000000  1.00000000000000  2.00000000000000]
[ 3.00000000000000  4.00000000000000  5.00000000000000]

sage: B = matrix(Integers(12), 3, 8, srange(24))
sage: N(B, digits=2)
[0.00  1.0  2.0  3.0  4.0  5.0  6.0  7.0]
[ 8.0  9.0  10.  11. 0.00  1.0  2.0  3.0]
[ 4.0  5.0  6.0  7.0  8.0  9.0  10.  11.]


Internally, numerical approximations of real numbers are stored in base-2. Therefore, numbers which look the same in their decimal expansion might be different:

sage: x=N(pi, digits=3); x
3.14
sage: y=N(3.14, digits=3); y
3.14
sage: x==y
False
sage: x.str(base=2)
'11.001001000100'
sage: y.str(base=2)
'11.001000111101'


Increasing the precision of a floating point number is not allowed:

sage: CC(-5).n(prec=100)
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 100 bits, use at most 53 bits
sage: n(1.3r, digits=20)
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 70 bits, use at most 53 bits
sage: RealField(24).pi().n()
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 53 bits, use at most 24 bits


As an exceptional case, digits=1 usually leads to 2 digits (one significant) in the decimal output (see trac ticket #11647):

sage: N(pi, digits=1)
3.2
sage: N(pi, digits=2)
3.1
sage: N(100*pi, digits=1)
320.
sage: N(100*pi, digits=2)
310.


In the following example, pi and 3 are both approximated to two bits of precision and then subtracted, which kills two bits of precision:

sage: N(pi, prec=2)
3.0
sage: N(3, prec=2)
3.0
sage: N(pi - 3, prec=2)
0.00


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'>


The following tests trac ticket #10761, in which n() would break when called on complex-valued algebraic numbers.

sage: E = matrix(3, [3,1,6,5,2,9,7,3,13]).eigenvalues(); E
[18.16815365088822?, -0.08407682544410650? - 0.2190261484802906?*I, -0.08407682544410650? + 0.2190261484802906?*I]
sage: E[1].parent()
Algebraic Field
sage: [a.n() for a in E]
[18.1681536508882, -0.0840768254441065 - 0.219026148480291*I, -0.0840768254441065 + 0.219026148480291*I]


Make sure we’ve rounded up log(10,2) enough to guarantee sufficient precision (trac ticket #10164):

sage: ks = 4*10**5, 10**6
sage: check_str_length = lambda k: len(str(numerical_approx(1+10**-k,digits=k+1)))-1 >= k+1
sage: check_precision = lambda k: numerical_approx(1+10**-k,digits=k+1)-1 > 0
sage: all(check_str_length(k) and check_precision(k) for k in ks)
True


Testing we have sufficient precision for the golden ratio (trac ticket #12163), note that the decimal point adds 1 to the string length:

sage: len(str(n(golden_ratio, digits=5000)))
5001
sage: len(str(n(golden_ratio, digits=5000000)))  # long time (4s on sage.math, 2012)
5000001


Check that trac ticket #14778 is fixed:

sage: n(0, algorithm='foo')
0.000000000000000

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.

For matrices and vectors, this returns the L2-norm. The L2-norm of a vector $$\textbf{v} = (v_1, v_2, \dots, v_n)$$, also called the Euclidean norm, is defined as

$|\textbf{v}| = \sqrt{\sum_{i=1}^n |v_i|^2}$

where $$|v_i|$$ is the complex modulus of $$v_i$$. The Euclidean norm is often used for determining the distance between two points in two- or three-dimensional space.

For complex numbers, the function returns the field norm. If $$c = a + bi$$ is a complex number, then the norm of $$c$$ is defined as the product of $$c$$ and its complex conjugate

$\text{norm}(c) = \text{norm}(a + bi) = c \cdot \overline{c} = a^2 + b^2.$

The norm of a complex number is different from its absolute value. The absolute value of a complex number is defined to be the square root of its norm. A typical use of the complex norm is in the integral domain $$\ZZ[i]$$ of Gaussian integers, where the norm of each Gaussian integer $$c = a + bi$$ is defined as its complex norm.

EXAMPLES:

The norm of vectors:

sage: z = 1 + 2*I
sage: norm(vector([z]))
sqrt(5)
sage: v = vector([-1,2,3])
sage: norm(v)
sqrt(14)
sage: _ = var("a b c d")
sage: v = vector([a, b, c, d])
sage: norm(v)
sqrt(abs(a)^2 + abs(b)^2 + abs(c)^2 + abs(d)^2)


The norm of matrices:

sage: z = 1 + 2*I
sage: norm(matrix([[z]]))
2.23606797749979
sage: M = matrix(ZZ, [[1,2,4,3], [-1,0,3,-10]])
sage: norm(M)  # abs tol 1e-14
10.690331129154467
sage: norm(CDF(z))
5.0
sage: norm(CC(z))
5.00000000000000


The norm of complex numbers:

sage: z = 2 - 3*I
sage: norm(z)
13
sage: a = randint(-10^10, 100^10)
sage: b = randint(-10^10, 100^10)
sage: z = a + b*I
sage: bool(norm(z) == a^2 + b^2)
True


The complex norm of symbolic expressions:

sage: a, b, c = var("a, b, c")
sage: assume((a, 'real'), (b, 'real'), (c, 'real'))
sage: z = a + b*I
sage: bool(norm(z).simplify() == a^2 + b^2)
True
sage: norm(a + b).simplify()
a^2 + 2*a*b + b^2
sage: v = vector([a, b, c])
sage: bool(norm(v).simplify() == sqrt(a^2 + b^2 + c^2))
True
sage: forget()

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, algorithm=None)

Returns a numerical approximation of an object x with at least prec bits (or decimal digits) of precision.

Note

Both upper case N and lower case n are aliases for numerical_approx(), and all three may be used as methods.

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)
• algorithm (optional) - a string specifying the algorithm to use for functions that implement more than one

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

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=40)
sage: b = ComplexField(40)(-5)
sage: a == b
True
sage: parent(a) is parent(b)
True
sage: numerical_approx(9)
9.00000000000000


You can also usually use method notation.

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


Vectors and matrices may also have their entries approximated.

sage: v = vector(RDF, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)

sage: v = vector(CDF, [1,2,3])
sage: v.n()
(1.00000000000000, 2.00000000000000, 3.00000000000000)
sage: _.parent()
Vector space of dimension 3 over Complex Field with 53 bits of precision
sage: v.n(prec=20)
(1.0000, 2.0000, 3.0000)

sage: u = vector(QQ, [1/2, 1/3, 1/4])
sage: n(u, prec=15)
(0.5000, 0.3333, 0.2500)
sage: n(u, digits=5)
(0.50000, 0.33333, 0.25000)

sage: v = vector(QQ, [1/2, 0, 0, 1/3, 0, 0, 0, 1/4], sparse=True)
sage: u = v.numerical_approx(digits=4)
sage: u.is_sparse()
True
sage: u
(0.5000, 0.0000, 0.0000, 0.3333, 0.0000, 0.0000, 0.0000, 0.2500)

sage: A = matrix(QQ, 2, 3, range(6))
sage: A.n()
[0.000000000000000  1.00000000000000  2.00000000000000]
[ 3.00000000000000  4.00000000000000  5.00000000000000]

sage: B = matrix(Integers(12), 3, 8, srange(24))
sage: N(B, digits=2)
[0.00  1.0  2.0  3.0  4.0  5.0  6.0  7.0]
[ 8.0  9.0  10.  11. 0.00  1.0  2.0  3.0]
[ 4.0  5.0  6.0  7.0  8.0  9.0  10.  11.]


Internally, numerical approximations of real numbers are stored in base-2. Therefore, numbers which look the same in their decimal expansion might be different:

sage: x=N(pi, digits=3); x
3.14
sage: y=N(3.14, digits=3); y
3.14
sage: x==y
False
sage: x.str(base=2)
'11.001001000100'
sage: y.str(base=2)
'11.001000111101'


Increasing the precision of a floating point number is not allowed:

sage: CC(-5).n(prec=100)
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 100 bits, use at most 53 bits
sage: n(1.3r, digits=20)
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 70 bits, use at most 53 bits
sage: RealField(24).pi().n()
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 53 bits, use at most 24 bits


As an exceptional case, digits=1 usually leads to 2 digits (one significant) in the decimal output (see trac ticket #11647):

sage: N(pi, digits=1)
3.2
sage: N(pi, digits=2)
3.1
sage: N(100*pi, digits=1)
320.
sage: N(100*pi, digits=2)
310.


In the following example, pi and 3 are both approximated to two bits of precision and then subtracted, which kills two bits of precision:

sage: N(pi, prec=2)
3.0
sage: N(3, prec=2)
3.0
sage: N(pi - 3, prec=2)
0.00


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'>


The following tests trac ticket #10761, in which n() would break when called on complex-valued algebraic numbers.

sage: E = matrix(3, [3,1,6,5,2,9,7,3,13]).eigenvalues(); E
[18.16815365088822?, -0.08407682544410650? - 0.2190261484802906?*I, -0.08407682544410650? + 0.2190261484802906?*I]
sage: E[1].parent()
Algebraic Field
sage: [a.n() for a in E]
[18.1681536508882, -0.0840768254441065 - 0.219026148480291*I, -0.0840768254441065 + 0.219026148480291*I]


Make sure we’ve rounded up log(10,2) enough to guarantee sufficient precision (trac ticket #10164):

sage: ks = 4*10**5, 10**6
sage: check_str_length = lambda k: len(str(numerical_approx(1+10**-k,digits=k+1)))-1 >= k+1
sage: check_precision = lambda k: numerical_approx(1+10**-k,digits=k+1)-1 > 0
sage: all(check_str_length(k) and check_precision(k) for k in ks)
True


Testing we have sufficient precision for the golden ratio (trac ticket #12163), note that the decimal point adds 1 to the string length:

sage: len(str(n(golden_ratio, digits=5000)))
5001
sage: len(str(n(golden_ratio, digits=5000000)))  # long time (4s on sage.math, 2012)
5000001


Check that trac ticket #14778 is fixed:

sage: n(0, algorithm='foo')
0.000000000000000

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: one(RR)
doctest:...: DeprecationWarning: one(R) is deprecated, use R.one() or R(1) instead
See http://trac.sagemath.org/17158 for details.
1.00000000000000
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.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). If no precision is specified this just calls the element’s .round() method.

EXAMPLES:

sage: round(sqrt(2),2)
1.41
sage: q = round(sqrt(2),5); q
1.41421
sage: type(q)
<type 'sage.rings.real_double.RealDoubleElement'>
sage: q = round(sqrt(2)); q
1
sage: type(q)
<type 'sage.rings.integer.Integer'>
sage: round(pi)
3
sage: b = 5.4999999999999999
sage: round(b)
5


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

sage: round(sqrt(Integer('1'*1000)),2)
+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; if that fails, it reverts to the builtin round function, converted to a real double field element.

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.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: squarefree_part(216r) # see #8976
6

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
• 'giac' - (optional) use Giac

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)


Warning

This function only works with symbolic expressions. To sum any other objects like list elements or function return values, please use python summation, see http://docs.python.org/library/functions.html#sum

In particular, this does not work:

sage: n = var('n')
sage: list=[1,2,3,4,5]
sage: sum(list[n],n,0,3)
Traceback (most recent call last):
...
TypeError: unable to convert n to an integer


sage: sum(list[n] for n in range(4))
10


Also, only a limited number of functions are recognized in symbolic sums:

sage: sum(valuation(n,2),n,1,5)
Traceback (most recent call last):
...
AttributeError: 'sage.symbolic.expression.Expression' object has no attribute 'valuation'


Again, use python sum():

sage: sum(valuation(n+1,2) for n in range(5))
3


(now back to the Sage sum examples)

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)
2^(n - 1)*n

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

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


Another binomial identity (trac ticket #7952):

sage: t,k,i = var('t,k,i')
sage: sum(binomial(i+t,t),i,0,k)
binomial(k + t + 1, t + 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)
Traceback (most recent call last):
...
ValueError: Sum is divergent.


This summation only Mathematica can perform:

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


Use Maple as a backend for summation:

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


Python ints should work as limits of summation (trac ticket #9393):

sage: sum(x, x, 1r, 5r)
15


Note

1. 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: zero(RR)
doctest:...: DeprecationWarning: zero(R) is deprecated, use R.zero() or R(0) instead
See http://trac.sagemath.org/17158 for details.
0.000000000000000
sage: R.<x> = PolynomialRing(QQ)
sage: zero(R) in R
True
sage: zero(R)*x == zero(R)
True


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