# 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)#

Return a numerical approximation of self with prec bits (or decimal digits) of precision.

Note

Lower case n() is an alias for numerical_approx() and may be used as a method.

INPUT:

• prec – precision in bits

• digits – precision in decimal digits (only used if prec is not given)

• algorithm – which algorithm to use to compute this approximation (the accepted algorithms depend on the object)

If neither prec nor digits is given, the default precision is 53 bits (roughly 16 digits).

EXAMPLES:

sage: # needs sage.symbolic
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: # needs sage.rings.real_mpfr
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()                                                            # needs sage.symbolic
12.5878862295484
sage: (pi^2 + e).numerical_approx()                                             # needs sage.symbolic
12.5878862295484

Vectors and matrices may also have their entries approximated:

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

sage: # needs sage.modules
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])                                           # needs sage.modules
sage: n(u, prec=15)                                                             # needs sage.modules
(0.5000, 0.3333, 0.2500)
sage: n(u, digits=5)                                                            # needs sage.modules
(0.50000, 0.33333, 0.25000)

sage: # needs sage.modules
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: # needs sage.modules
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                                                    # needs sage.symbolic
3.14
sage: y = N(3.14, digits=3); y                                                  # needs sage.rings.real_mpfr
3.14
sage: x == y                                                                    # needs sage.rings.real_mpfr sage.symbolic
False
sage: x.str(base=2)                                                             # needs sage.symbolic
'11.001001000100'
sage: y.str(base=2)                                                             # needs sage.rings.real_mpfr
'11.001000111101'

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

sage: CC(-5).n(prec=100)                                                        # needs sage.rings.real_mpfr
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)                                                        # needs sage.rings.real_mpfr
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 70 bits, use at most 53 bits
sage: RealField(24).pi().n()                                                    # needs sage.rings.real_mpfr
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 github issue #11647):

sage: # needs sage.symbolic
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)                                                             # needs sage.symbolic
3.0
sage: N(3, prec=2)                                                              # needs sage.rings.real_mpfr
3.0
sage: N(pi - 3, prec=2)                                                         # needs sage.symbolic
0.00

Return the additive order of x.

EXAMPLES:

+Infinity
11
3
sage.misc.functional.base_field(x)#

Return 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)#

Return 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)#

Return the fixed basis of x.

EXAMPLES:

sage: V = VectorSpace(QQ, 3)                                                    # needs sage.modules
sage: S = V.subspace([[1,2,0], [2,2,-1]])                                       # needs sage.modules
sage: basis(S)                                                                  # needs sage.modules
[
(1, 0, -1),
(0, 1, 1/2)
]
sage.misc.functional.category(x)#

Return the category of x.

EXAMPLES:

sage: V = VectorSpace(QQ, 3)                                                    # needs sage.modules
sage: category(V)                                                               # needs sage.modules
Category of finite dimensional vector spaces with basis over
(number fields and quotient fields and metric spaces)
sage.misc.functional.characteristic_polynomial(x, var='x')#

Return the characteristic polynomial of x in the given variable.

EXAMPLES:

sage: # needs sage.libs.pari sage.modules
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                                                   # needs sage.rings.finite_rings
Finite Field in alpha of size 7^10
sage: alpha.charpoly('T')                                                       # needs sage.rings.finite_rings
T^10 + T^6 + T^5 + 4*T^4 + T^3 + 2*T^2 + 3*T + 3
sage: characteristic_polynomial(alpha, 'T')                                     # needs sage.rings.finite_rings
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 do not confuse the computation (github issue #14403):

sage: # needs sage.libs.pari sage.symbolic
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')#

Return the characteristic polynomial of x in the given variable.

EXAMPLES:

sage: # needs sage.libs.pari sage.modules
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                                                   # needs sage.rings.finite_rings
Finite Field in alpha of size 7^10
sage: alpha.charpoly('T')                                                       # needs sage.rings.finite_rings
T^10 + T^6 + T^5 + 4*T^4 + T^3 + 2*T^2 + 3*T + 3
sage: characteristic_polynomial(alpha, 'T')                                     # needs sage.rings.finite_rings
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 do not confuse the computation (github issue #14403):

sage: # needs sage.libs.pari sage.symbolic
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)#

Coerce x to type P if possible.

EXAMPLES:

sage: type(5)
<class 'sage.rings.integer.Integer'>
sage: type(coerce(QQ,5))
<class 'sage.rings.rational.Rational'>
sage.misc.functional.cyclotomic_polynomial(n, var='x')#

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

EXAMPLES:

sage: # needs sage.libs.pari
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)#

Return the decomposition of x.

EXAMPLES:

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

sage: # needs sage.groups
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 modulo 4
with values in Cyclotomic Field of order 4 and degree 2
sage.misc.functional.denominator(x)#

Return 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)#

Return the determinant of x.

EXAMPLES:

sage: M = MatrixSpace(QQ, 3, 3)                                                 # needs sage.modules
sage: A = M([1,2,3, 4,5,6, 7,8,9])                                              # needs sage.modules
sage: det(A)                                                                    # needs sage.modules
0
sage.misc.functional.dim(x)#

Return the dimension of x.

EXAMPLES:

sage: V = VectorSpace(QQ, 3)                                                    # needs sage.modules
sage: S = V.subspace([[1,2,0], [2,2,-1]])                                       # needs sage.modules
sage: dimension(S)                                                              # needs sage.modules
2
sage.misc.functional.dimension(x)#

Return the dimension of x.

EXAMPLES:

sage: V = VectorSpace(QQ, 3)                                                    # needs sage.modules
sage: S = V.subspace([[1,2,0], [2,2,-1]])                                       # needs sage.modules
sage: dimension(S)                                                              # needs sage.modules
2
sage.misc.functional.disc(x)#

Return the discriminant of x.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^29 - 17*x - 1, 'alpha')                                  # needs sage.libs.pari
sage: K = S.number_field()                                                      # needs sage.libs.pari sage.rings.number_field
sage: discriminant(K)                                                           # needs sage.libs.pari sage.rings.number_field
-15975100446626038280218213241591829458737190477345113376757479850566957249523
sage.misc.functional.discriminant(x)#

Return the discriminant of x.

EXAMPLES:

sage: R.<x> = PolynomialRing(QQ)
sage: S = R.quotient(x^29 - 17*x - 1, 'alpha')                                  # needs sage.libs.pari
sage: K = S.number_field()                                                      # needs sage.libs.pari sage.rings.number_field
sage: discriminant(K)                                                           # needs sage.libs.pari sage.rings.number_field
-15975100446626038280218213241591829458737190477345113376757479850566957249523
sage.misc.functional.eta(x)#

Return 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)                                                                # needs sage.symbolic
0.7420487758365647 + 0.1988313702299107*I
sage.misc.functional.fcp(x, var='x')#

Return the factorization of the characteristic polynomial of x.

EXAMPLES:

sage: M = MatrixSpace(QQ, 3, 3)                                                 # needs sage.modules
sage: A = M([1,2,3, 4,5,6, 7,8,9])                                              # needs sage.modules
sage: fcp(A, 'x')                                                               # needs sage.libs.pari sage.modules
x * (x^2 - 15*x - 18)
sage.misc.functional.gen(x)#

Return 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])                                                 # needs sage.groups
sage: gen(A)                                                                    # needs sage.groups
f
sage.misc.functional.gens(x)#

Return the generators of x.

EXAMPLES:

sage: R.<x,y> = SR[]                                                            # needs sage.symbolic
sage: R                                                                         # needs sage.symbolic
Multivariate Polynomial Ring in x, y over Symbolic Ring
sage: gens(R)                                                                   # needs sage.symbolic
(x, y)

sage: A = AbelianGroup(5, [5,5,7,8,9])                                          # needs sage.groups
sage: gens(A)                                                                   # needs sage.groups
(f0, f1, f2, f3, f4)
sage.misc.functional.hecke_operator(x, n)#

Return the $$n$$-th Hecke operator $$T_n$$ acting on x.

EXAMPLES:

sage: M = ModularSymbols(1,12)                                                  # needs sage.modular
sage: hecke_operator(M,5)                                                       # needs sage.modular
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)#

Return the image of x.

EXAMPLES:

sage: M = MatrixSpace(QQ, 3, 3)                                                 # needs sage.modules
sage: A = M([1,2,3, 4,5,6, 7,8,9])                                              # needs sage.modules
sage: image(A)                                                                  # needs sage.modules
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)#

Return 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)                                                       # needs sage.symbolic
-cos(x)
sage: y = var('y')                                                              # needs sage.symbolic
sage: integral(sin(x), y)                                                       # needs sage.symbolic
y*sin(x)
sage: integral(sin(x), x, 0, pi/2)                                              # needs sage.symbolic
1
sage: sin(x).integral(x, 0, pi/2)                                               # needs sage.symbolic
1
sage: integral(exp(-x), (x, 1, oo))                                             # needs sage.symbolic
e^(-1)

Numerical approximation:

sage: h = integral(tan(x)/x, (x, 1, pi/3))                                      # needs sage.symbolic
...
sage: h                                                                         # needs sage.symbolic
integrate(tan(x)/x, x, 1, 1/3*pi)
sage: h.n()                                                                     # needs sage.symbolic
0.07571599101...

Specific algorithm can be used for integration:

sage: integral(sin(x)^2, x, algorithm='maxima')                                 # needs sage.symbolic
1/2*x - 1/4*sin(2*x)
sage: integral(sin(x)^2, x, algorithm='sympy')                                  # needs sage.symbolic
-1/2*cos(x)*sin(x) + 1/2*x
sage.misc.functional.integral_closure(x)#

Return the integral closure of x.

EXAMPLES:

sage: integral_closure(QQ)
Rational Field
sage: K.<a> = QuadraticField(5)                                                 # needs sage.rings.number_field
sage: O2 = K.order(2 * a); O2                                                   # needs sage.rings.number_field
Order in Number Field in a
with defining polynomial x^2 - 5 with a = 2.236067977499790?
sage: integral_closure(O2)                                                      # needs sage.rings.number_field
Maximal Order in Number Field in a
with defining polynomial x^2 - 5 with a = 2.236067977499790?
sage.misc.functional.integrate(x, *args, **kwds)#

Return 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)                                                       # needs sage.symbolic
-cos(x)
sage: y = var('y')                                                              # needs sage.symbolic
sage: integral(sin(x), y)                                                       # needs sage.symbolic
y*sin(x)
sage: integral(sin(x), x, 0, pi/2)                                              # needs sage.symbolic
1
sage: sin(x).integral(x, 0, pi/2)                                               # needs sage.symbolic
1
sage: integral(exp(-x), (x, 1, oo))                                             # needs sage.symbolic
e^(-1)

Numerical approximation:

sage: h = integral(tan(x)/x, (x, 1, pi/3))                                      # needs sage.symbolic
...
sage: h                                                                         # needs sage.symbolic
integrate(tan(x)/x, x, 1, 1/3*pi)
sage: h.n()                                                                     # needs sage.symbolic
0.07571599101...

Specific algorithm can be used for integration:

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

Return whether or not x is commutative.

EXAMPLES:

sage: R = PolynomialRing(QQ, 'x')
sage: is_commutative(R)
doctest:...DeprecationWarning: use X.is_commutative() or X in Rings().Commutative()
See https://github.com/sagemath/sage/issues/32347 for details.
True
sage.misc.functional.is_even(x)#

Return 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, proof=True)#

Return whether or not x is a field.

Alternatively, one can use x in Fields().

EXAMPLES:

sage: R = PolynomialRing(QQ, 'x')
sage: F = FractionField(R)
sage: is_field(F)
doctest:...DeprecationWarning: use X.is_field() or X in Fields()
See https://github.com/sagemath/sage/issues/32347 for details.
True
sage.misc.functional.is_integrally_closed(x)#

Return whether x is integrally closed.

EXAMPLES:

sage: is_integrally_closed(QQ)
doctest:...DeprecationWarning: use X.is_integrally_closed()
See https://github.com/sagemath/sage/issues/32347 for details.
True
sage: x = polygen(ZZ, 'x')
sage: K.<a> = NumberField(x^2 + 189*x + 394)                                    # needs sage.rings.number_field
sage: R = K.order(2*a)                                                          # needs sage.rings.number_field
sage: is_integrally_closed(R)                                                   # needs sage.rings.number_field
False
sage.misc.functional.is_odd(x)#

Return 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)#

Return 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)#

Return the left kernel of x.

EXAMPLES:

sage: # needs sage.modules
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: # needs sage.modules
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)#

Return the Krull dimension of x.

EXAMPLES:

sage: krull_dimension(QQ)
0
sage: krull_dimension(ZZ)
1
sage: krull_dimension(ZZ[sqrt(5)])                                              # needs sage.rings.number_field sage.symbolic
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)                                                 # needs sage.libs.pari
sage: lift(xmod - 7)                                                            # needs sage.libs.pari
x - 7
sage.misc.functional.log(*args, **kwds)#

Return the logarithm of the first argument to the base of the second argument which if missing defaults to e.

It calls the log method of the first argument when computing the logarithm, thus allowing the use of logarithm on any object containing a log method. In other words, log works on more than just real numbers.

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)                                                                  # needs sage.symbolic
2

To change the base of the logarithm, add a second parameter:

sage: log(1000,10)
3

The synonym ln can only take one argument:

sage: # needs sage.symbolic
sage: ln(RDF(10))
2.302585092994046
sage: ln(2.718)
0.999896315728952
sage: ln(2.0)
0.693147180559945
sage: ln(float(-1))
3.141592653589793j

sage: ln(complex(-1))
3.141592653589793j

You can use RDF, RealField or n to get a numerical real approximation:

sage: log(1024, 2)
10
sage: RDF(log(1024, 2))
10.0

sage: # needs sage.symbolic
sage: log(10, 4)
1/2*log(10)/log(2)
sage: RDF(log(10, 4))
1.6609640474436813
sage: log(10, 2)
log(10)/log(2)
sage: n(log(10, 2))
3.32192809488736
sage: log(10, e)
log(10)
sage: n(log(10, e))
2.30258509299405

The log function works for negative numbers, complex numbers, and symbolic numbers too, picking the branch with angle between $$-\pi$$ and $$\pi$$:

sage: log(-1+0*I)                                                               # needs sage.symbolic
I*pi
sage: log(CC(-1))                                                               # needs sage.rings.real_mpfr
3.14159265358979*I
sage: log(-1.0)                                                                 # needs sage.symbolic
3.14159265358979*I

Small integer powers are factored out immediately:

sage: # needs sage.symbolic
sage: log(4)
2*log(2)
sage: log(1000000000)
9*log(10)
sage: log(8) - 3*log(2)
0
sage: bool(log(8) == 3*log(2))
True

The hold parameter can be used to prevent automatic evaluation:

sage: # needs sage.symbolic
sage: log(-1, hold=True)
log(-1)
sage: log(-1)
I*pi
sage: I.log(hold=True)
log(I)
sage: I.log(hold=True).simplify()
1/2*I*pi

For input zero, the following behavior occurs:

sage: log(0)                                                                    # needs sage.symbolic
-Infinity
sage: log(CC(0))                                                                # needs sage.rings.real_mpfr
-infinity
sage: log(0.0)                                                                  # needs sage.symbolic
-infinity

The log function also works in finite fields as long as the argument lies in the multiplicative group generated by the base:

sage: # needs sage.libs.pari
sage: F = GF(13); g = F.multiplicative_generator(); g
2
sage: a = F(8)
sage: log(a, g); g^log(a, g)
3
8
sage: log(a, 3)
Traceback (most recent call last):
...
ValueError: no logarithm of 8 found to base 3 modulo 13
sage: log(F(9), 3)
2

sage: R = Zp(5); R                                                              # needs sage.rings.padics
5-adic Ring with capped relative precision 20
sage: a = R(16); a                                                              # needs sage.rings.padics
1 + 3*5 + O(5^20)
3*5 + 3*5^2 + 3*5^4 + 3*5^5 + 3*5^6 + 4*5^7 + 2*5^8 + 5^9 +
5^11 + 2*5^12 + 5^13 + 3*5^15 + 2*5^16 + 4*5^17 + 3*5^18 +
3*5^19 + O(5^20)
sage.misc.functional.minimal_polynomial(x, var='x')#

Return the minimal polynomial of x.

EXAMPLES:

sage: # needs sage.libs.pari sage.modules
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')#

Return the minimal polynomial of x.

EXAMPLES:

sage: # needs sage.libs.pari sage.modules
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)#

Return the multiplicative order of x, if x is a unit, or raise ArithmeticError otherwise.

EXAMPLES:

sage: a = mod(5, 11)
sage: multiplicative_order(a)                                                   # needs sage.libs.pari
5
sage: multiplicative_order(mod(2, 11))                                          # needs sage.libs.pari
10
sage: multiplicative_order(mod(2, 12))                                          # needs sage.libs.pari
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)#

Return a numerical approximation of self with prec bits (or decimal digits) of precision.

Note

Lower case n() is an alias for numerical_approx() and may be used as a method.

INPUT:

• prec – precision in bits

• digits – precision in decimal digits (only used if prec is not given)

• algorithm – which algorithm to use to compute this approximation (the accepted algorithms depend on the object)

If neither prec nor digits is given, the default precision is 53 bits (roughly 16 digits).

EXAMPLES:

sage: # needs sage.symbolic
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: # needs sage.rings.real_mpfr
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()                                                            # needs sage.symbolic
12.5878862295484
sage: (pi^2 + e).numerical_approx()                                             # needs sage.symbolic
12.5878862295484

Vectors and matrices may also have their entries approximated:

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

sage: # needs sage.modules
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])                                           # needs sage.modules
sage: n(u, prec=15)                                                             # needs sage.modules
(0.5000, 0.3333, 0.2500)
sage: n(u, digits=5)                                                            # needs sage.modules
(0.50000, 0.33333, 0.25000)

sage: # needs sage.modules
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: # needs sage.modules
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                                                    # needs sage.symbolic
3.14
sage: y = N(3.14, digits=3); y                                                  # needs sage.rings.real_mpfr
3.14
sage: x == y                                                                    # needs sage.rings.real_mpfr sage.symbolic
False
sage: x.str(base=2)                                                             # needs sage.symbolic
'11.001001000100'
sage: y.str(base=2)                                                             # needs sage.rings.real_mpfr
'11.001000111101'

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

sage: CC(-5).n(prec=100)                                                        # needs sage.rings.real_mpfr
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)                                                        # needs sage.rings.real_mpfr
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 70 bits, use at most 53 bits
sage: RealField(24).pi().n()                                                    # needs sage.rings.real_mpfr
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 github issue #11647):

sage: # needs sage.symbolic
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)                                                             # needs sage.symbolic
3.0
sage: N(3, prec=2)                                                              # needs sage.rings.real_mpfr
3.0
sage: N(pi - 3, prec=2)                                                         # needs sage.symbolic
0.00
sage.misc.functional.ngens(x)#

Return the number of generators of x.

EXAMPLES:

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

Return 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.

For vector fields on a pseudo-Riemannian manifold $$(M,g)$$, the function returns the norm with respect to the metric $$g$$:

$|v| = \sqrt{g(v,v)}$

EXAMPLES:

The norm of vectors:

sage: # needs sage.modules sage.symbolic
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", domain='real')
sage: v = vector([a, b, c, d])
sage: norm(v)
sqrt(a^2 + b^2 + c^2 + d^2)

The norm of matrices:

sage: # needs sage.modules sage.symbolic
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: # needs sage.symbolic
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: # needs sage.symbolic
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)#

Return 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)#

Return a numerical approximation of self with prec bits (or decimal digits) of precision.

Note

Lower case n() is an alias for numerical_approx() and may be used as a method.

INPUT:

• prec – precision in bits

• digits – precision in decimal digits (only used if prec is not given)

• algorithm – which algorithm to use to compute this approximation (the accepted algorithms depend on the object)

If neither prec nor digits is given, the default precision is 53 bits (roughly 16 digits).

EXAMPLES:

sage: # needs sage.symbolic
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: # needs sage.rings.real_mpfr
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()                                                            # needs sage.symbolic
12.5878862295484
sage: (pi^2 + e).numerical_approx()                                             # needs sage.symbolic
12.5878862295484

Vectors and matrices may also have their entries approximated:

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

sage: # needs sage.modules
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])                                           # needs sage.modules
sage: n(u, prec=15)                                                             # needs sage.modules
(0.5000, 0.3333, 0.2500)
sage: n(u, digits=5)                                                            # needs sage.modules
(0.50000, 0.33333, 0.25000)

sage: # needs sage.modules
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: # needs sage.modules
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                                                    # needs sage.symbolic
3.14
sage: y = N(3.14, digits=3); y                                                  # needs sage.rings.real_mpfr
3.14
sage: x == y                                                                    # needs sage.rings.real_mpfr sage.symbolic
False
sage: x.str(base=2)                                                             # needs sage.symbolic
'11.001001000100'
sage: y.str(base=2)                                                             # needs sage.rings.real_mpfr
'11.001000111101'

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

sage: CC(-5).n(prec=100)                                                        # needs sage.rings.real_mpfr
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)                                                        # needs sage.rings.real_mpfr
Traceback (most recent call last):
...
TypeError: cannot approximate to a precision of 70 bits, use at most 53 bits
sage: RealField(24).pi().n()                                                    # needs sage.rings.real_mpfr
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 github issue #11647):

sage: # needs sage.symbolic
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)                                                             # needs sage.symbolic
3.0
sage: N(3, prec=2)                                                              # needs sage.rings.real_mpfr
3.0
sage: N(pi - 3, prec=2)                                                         # needs sage.symbolic
0.00
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.order(x)#

Return the order of x.

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

EXAMPLES:

sage: C = CyclicPermutationGroup(10)                                            # needs sage.groups
sage: order(C)                                                                  # needs sage.groups
10
sage: F = GF(7)
sage: order(F)
7
sage.misc.functional.quo(x, y, *args, **kwds)#

Return 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)                                                            # needs sage.libs.pari
Univariate Quotient Polynomial Ring in xbar over Integer Ring with modulus x^2 + 1
sage.misc.functional.quotient(x, y, *args, **kwds)#

Return 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)                                                            # needs sage.libs.pari
Univariate Quotient Polynomial Ring in xbar over Integer Ring with modulus x^2 + 1
sage.misc.functional.rank(x)#

Return the rank of x.

EXAMPLES:

We compute the rank of a matrix:

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

We compute the rank of an elliptic curve:

sage: E = EllipticCurve([0,0,1,-1,0])                                           # needs sage.schemes
sage: rank(E)                                                                   # needs sage.schemes
1
sage.misc.functional.regulator(x)#

Return the regulator of x.

EXAMPLES:

sage: x = polygen(ZZ, 'x')
sage: regulator(NumberField(x^2 - 2, 'a'))                                      # needs sage.rings.number_field
0.881373587019543
sage: regulator(EllipticCurve('11a'))                                           # needs sage.schemes
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: # needs sage.symbolic
sage: round(sqrt(2), 2)
1.41
sage: q = round(sqrt(2), 5); q
1.41421
sage: type(q)
<class 'sage.rings.real_double...RealDoubleElement...'>
sage: q = round(sqrt(2)); q
1
sage: type(q)
<class 'sage.rings.integer.Integer'>
sage: round(pi)
3

sage: b = 5.4999999999999999
sage: round(b)
5

This example addresses github issue #23502:

sage: n = round(6); type(n)
<class 'sage.rings.integer.Integer'>

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

sage: round(sqrt(Integer('1'*1000)), 2)                                         # needs sage.symbolic
+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 builtins; builtins.round.

sage.misc.functional.sqrt(x, *args, **kwds)#

INPUT:

• x – a number

• prec – integer (default: None): if None, returns an exact square root; otherwise returns a numerical square root if necessary, to the given bits of precision.

• extend – bool (default: True); this is a placeholder, and is always ignored or passed to the sqrt method of x, since in the symbolic ring everything has a square root.

• all – bool (default: False); if True, return all square roots of self, instead of just one.

EXAMPLES:

sage: sqrt(4)
2
sage: sqrt(4, all=True)
[2, -2]

sage: # needs sage.symbolic
sage: sqrt(-1)
I
sage: sqrt(2)
sqrt(2)
sage: sqrt(2)^2
2
sage: sqrt(x^2)
sqrt(x^2)

For a non-symbolic square root, there are a few options. The best is to numerically approximate afterward:

sage: sqrt(2).n()                                                               # needs sage.symbolic
1.41421356237310
sage: sqrt(2).n(prec=100)                                                       # needs sage.symbolic
1.4142135623730950488016887242

Or one can input a numerical type:

sage: sqrt(2.)
1.41421356237310
sage: sqrt(2.000000000000000000000000)
1.41421356237309504880169
sage: sqrt(4.0)
2.00000000000000

To prevent automatic evaluation, one can use the hold parameter after coercing to the symbolic ring:

sage: sqrt(SR(4), hold=True)                                                    # needs sage.symbolic
sqrt(4)
sage: sqrt(4, hold=True)
Traceback (most recent call last):
...
TypeError: ..._do_sqrt() got an unexpected keyword argument 'hold'

This illustrates that the bug reported in github issue #6171 has been fixed:

sage: a = 1.1
sage: a.sqrt(prec=100)  # this is supposed to fail
Traceback (most recent call last):
...
TypeError: ...sqrt() got an unexpected keyword argument 'prec'
sage: sqrt(a, prec=100)                                                         # needs sage.rings.real_mpfr
1.0488088481701515469914535137
sage: sqrt(4.00, prec=250)                                                      # needs sage.rings.real_mpfr
2.0000000000000000000000000000000000000000000000000000000000000000000000000

One can use numpy input as well:

sage: import numpy                                                              # needs numpy
sage: a = numpy.arange(2,5)                                                     # needs numpy
sage: sqrt(a)                                                                   # needs numpy
array([1.41421356, 1.73205081, 2.        ])
sage.misc.functional.squarefree_part(x)#

Return 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()                                                                # needs sage.libs.pari
(-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()                                                                # needs sage.libs.pari
(x - 1) * (x^3 + x + 1)
sage.misc.functional.symbolic_prod(expression, *args, **kwds)#

Return the symbolic product $$\prod_{v = a}^b expression$$ 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 product

• b - upper endpoint of the prduct

• algorithm - (default: 'maxima') one of

• 'maxima' - use Maxima (the default)

• 'giac' - (optional) use Giac

• 'sympy' - use SymPy

• hold - (default: False) if True don’t evaluate

EXAMPLES:

sage: # needs sage.symbolic
sage: i, k, n = var('i,k,n')
sage: product(k, k, 1, n)
factorial(n)
sage: product(x + i*(i+1)/2, i, 1, 4)
x^4 + 20*x^3 + 127*x^2 + 288*x + 180
sage: product(i^2, i, 1, 7)
25401600
sage: f = function('f')
sage: product(f(i), i, 1, 7)
f(7)*f(6)*f(5)*f(4)*f(3)*f(2)*f(1)
sage: product(f(i), i, 1, n)
product(f(i), i, 1, n)
sage: assume(k>0)
sage: product(integrate(x^k, x, 0, 1), k, 1, n)
1/factorial(n + 1)
sage: product(f(i), i, 1, n).log().log_expand()
sum(log(f(i)), i, 1, n)
sage.misc.functional.symbolic_sum(expression, *args, **kwds)#

Return 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

• 'sympy' - use SymPy

EXAMPLES:

sage: k, n = var('k,n')                                                         # needs sage.symbolic
sage: sum(k, k, 1, n).factor()                                                  # needs sage.symbolic
1/2*(n + 1)*n
sage: sum(1/k^4, k, 1, oo)                                                      # needs sage.symbolic
1/90*pi^4
sage: sum(1/k^5, k, 1, oo)                                                      # needs sage.symbolic
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')                                                          # needs sage.symbolic
sage: mylist = [1,2,3,4,5]
sage: sum(mylist[n], n, 0, 3)                                               # needs sage.symbolic
Traceback (most recent call last):
...
TypeError: unable to convert n to an integer

sage: sum(mylist[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)                                         # needs sage.symbolic
Traceback (most recent call last):
...
TypeError: unable to convert n to an integer

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)                                              # needs sage.symbolic
2^n

The binomial theorem:

sage: x, y = var('x, y')                                                        # needs sage.symbolic
sage: sum(binomial(n, k) * x^k * y^(n-k), k, 0, n)                              # needs sage.symbolic
(x + y)^n
sage: sum(k * binomial(n, k), k, 1, n)                                          # needs sage.symbolic
2^(n - 1)*n
sage: sum((-1)^k * binomial(n, k), k, 0, n)                                     # needs sage.symbolic
0
sage: sum(2^(-k)/(k*(k+1)), k, 1, oo)                                           # needs sage.symbolic
-log(2) + 1

Another binomial identity (github issue #7952):

sage: t, k, i = var('t,k,i')                                                    # needs sage.symbolic
sage: sum(binomial(i + t, t), i, 0, k)                                          # needs sage.symbolic
binomial(k + t + 1, t + 1)

Summing a hypergeometric term:

sage: sum(binomial(n, k) * factorial(k) / factorial(n+1+k), k, 0, n)            # needs sage.symbolic
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)                              # needs sage.symbolic
True

A geometric sum:

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

The geometric series:

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

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

sage: forget()                                                                  # needs sage.symbolic
sage: assume(q > 1)                                                             # needs sage.symbolic
sage: sum(a * q^k, k, 0, oo)                                                    # needs sage.symbolic
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, needs sage.symbolic
pi*coth(pi)

Use Maple as a backend for summation:

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

Python ints should work as limits of summation (github issue #9393):

sage: sum(x, x, 1r, 5r)                                                         # needs sage.symbolic
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 convertible into a Sage expression.

sage.misc.functional.transpose(x)#

Return the transpose of x.

EXAMPLES:

sage: M = MatrixSpace(QQ, 3, 3)                                                 # needs sage.modules
sage: A = M([1,2,3, 4,5,6, 7,8,9])                                              # needs sage.modules
sage: transpose(A)                                                              # needs sage.modules
[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
range(2, 6)
sage: 5 in I
True
sage: 6 in I
False