def numerical_approx(x, prec=None, digits=None, algorithm=None):
r"""
Return a numerical approximation of ``self`` with ``prec`` bits
(or decimal ``digits``) of precision.
No guarantee is made about the accuracy of the result.
.. NOTE::
Lower case :func:`n` is an alias for :func:`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: 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).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:`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:`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:`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:`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:`14778` is fixed::
sage: n(0, algorithm='foo')
0.000000000000000
"""
if prec is None:
from sage.arith.numerical_approx import digits_to_bits
prec = digits_to_bits(digits)
try:
n = x.numerical_approx
except AttributeError:
from sage.arith.numerical_approx import numerical_approx_generic
return numerical_approx_generic(x, prec)
else:
return n(prec, algorithm=algorithm)
def numerical_approx(x, prec=None, digits=None, algorithm=None):
r"""
Return a numerical approximation of ``self`` with ``prec`` bits
(or decimal ``digits``) of precision.
No guarantee is made about the accuracy of the result.
.. NOTE::
Lower case :func:`n` is an alias for :func:`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: 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).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:`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:`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:`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:`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:`14778` is fixed::
sage: n(0, algorithm='foo')
0.000000000000000
"""
if prec is None:
from sage.arith.numerical_approx import digits_to_bits
prec = digits_to_bits(digits)
try:
n = x.numerical_approx
except AttributeError:
from sage.arith.numerical_approx import numerical_approx_generic
return numerical_approx_generic(x, prec)
else:
return n(prec, algorithm=algorithm)