Numerical evaluation¶
Floatingpoint numbers¶
Floatingpoint numbers in Diofant are instances of the class Float
. A Float
can be created with a custom precision as second argument:
>>> Float(0.1)
0.100000000000000
>>> Float(0.1, 10)
0.1000000000
>>> Float(0.125, 30)
0.125000000000000000000000000000
>>> Float(0.1, 30)
0.100000000000000005551115123126
As the last example shows, some Python floats are only accurate to about 15
digits as inputs, while others (those that have a denominator that is a
power of 2, like 0.125 = 1/8) are exact. To create a Float
from a
highprecision decimal number, it is better to pass a string, Rational
,
or evalf
a Rational
:
>>> Float('0.1', 30)
0.100000000000000000000000000000
>>> Float(Rational(1, 10), 30)
0.100000000000000000000000000000
>>> Rational(1, 10).evalf(30)
0.100000000000000000000000000000
The precision of a number determines 1) the precision to use when performing arithmetic with the number, and 2) the number of digits to display when printing the number. When two numbers with different precision are used together in an arithmetic operation, the higher of the precisions is used for the result. The product of 0.1 +/ 0.001 and 3.1415 +/ 0.0001 has an uncertainty of about 0.003 and yet 5 digits of precision are shown.
>>> Float(0.1, 3)*Float(3.1415, 5)
0.31417
So the displayed precision should not be used as a model of error propagation or significance arithmetic; rather, this scheme is employed to ensure stability of numerical algorithms.
Function N()
(or
evalf()
method) can be used to
change the precision of existing floatingpoint numbers:
>>> N(3.5)
3.50000000000000
>>> N(3.5, 5)
3.5000
However, you can “increase” precision of the
Float
number only with it’s class
constructor:
>>> Float(3.5, 30)
3.50000000000000000000000000000
Accuracy and error handling¶
When the input to N
or evalf
is a complicated expression, numerical
error propagation becomes a concern. As an example, consider the 100’th
Fibonacci number and the excellent (but not exact) approximation \(\varphi^{100} / \sqrt{5}\)
where \(\varphi\) is the golden ratio. With ordinary floatingpoint arithmetic,
subtracting these numbers from each other erroneously results in a complete
cancellation:
>>> a, b = GoldenRatio**1000/sqrt(5), fibonacci(1000)
>>> float(a)
4.3466557686937455e+208
>>> float(b)
4.3466557686937455e+208
>>> float(a)  float(b)
0.0
N
and evalf
keep track of errors and automatically increase the
precision used internally in order to obtain a correct result:
>>> N(fibonacci(100)  GoldenRatio**100/sqrt(5))
5.64613129282185e22
Unfortunately, numerical evaluation cannot tell an expression that is exactly zero apart from one that is merely very small. The working precision is therefore capped, by default to around 100 digits. If we try with the 1000’th Fibonacci number, the following happens:
>>> N(fibonacci(1000)  (GoldenRatio)**1000/sqrt(5))
Traceback (most recent call last):
...
PrecisionExhausted: ...
The exception indicates that N
failed to achieve full accuracy. To force a
higher working precision, the maxn
keyword argument can be used:
>>> N(fibonacci(1000)  (GoldenRatio)**1000/sqrt(5), maxn=500)
4.60123853010113e210
Normally, maxn
can be set very high (thousands of digits), but be aware that
this may cause significant slowdown in extreme cases.
Also, you can set strict
keyword argument to False
to obtain imprecise
answer instead of exception. For example, if we add a term so that the
Fibonacci approximation becomes exact (the full form of Binet’s formula), we
get an expression that is exactly zero, but N
does not know this:
>>> f = fibonacci(100)  (GoldenRatio**100  (GoldenRatio1)**100)/sqrt(5)
>>> N(f, strict=False)
0.e126
>>> N(f, maxn=1000, strict=False)
0.e1336
In situations where such cancellations are known to occur, the chop
options
is useful. This basically replaces very small numbers in the real or
imaginary portions of a number with exact zeros:
>>> N(f, chop=True)
0
>>> N(3 + I*f, chop=True)
3.00000000000000
In situations where you wish to remove meaningless digits, reevaluation or
the use of the round
method are useful:
>>> Float('.1')*Float('.12345')
0.012297
>>> ans = _
>>> N(ans, 1)
0.01
>>> ans.round(2)
0.01
If you are dealing with a numeric expression that contains no floats, it can be evaluated to arbitrary precision. To round the result relative to a given decimal, the round method is useful:
>>> v = 10*pi + cos(1)
>>> N(v)
31.9562288417661
>>> v.round(3)
31.956
Sums and integrals¶
Sums (in particular, infinite series) and integrals can be used like regular closedform expressions, and support arbitraryprecision evaluation:
>>> Sum(1/n**n, (n, 1, oo)).evalf()
1.29128599706266
>>> Integral(x**(x), (x, 0, 1)).evalf()
1.29128599706266
>>> Sum(1/n**n, (n, 1, oo)).evalf(50)
1.2912859970626635404072825905956005414986193682745
>>> Integral(x**(x), (x, 0, 1)).evalf(50)
1.2912859970626635404072825905956005414986193682745
>>> (Integral(exp(x**2), (x, oo, oo)) ** 2).evalf(30)
3.14159265358979323846264338328
By default, the tanhsinh quadrature algorithm is used to evaluate integrals.
This algorithm is very efficient and robust for smooth integrands (and even
integrals with endpoint singularities), but may struggle with integrals that
are highly oscillatory or have midinterval discontinuities. In many cases,
evalf
/N
will correctly estimate the error. With the following integral,
the result is accurate but only good to four digits:
>>> f = abs(sin(x))
>>> Integral(abs(sin(x)), (x, 0, 4)).evalf()
Traceback (most recent call last):
...
PrecisionExhausted: ...
It is better to split this integral into two pieces:
>>> (Integral(f, (x, 0, pi)) + Integral(f, (x, pi, 4))).evalf()
2.34635637913639
A similar example is the following oscillatory integral:
>>> Integral(sin(x)/x**2, (x, 1, oo)).evalf()
Traceback (most recent call last):
...
PrecisionExhausted: ...
It can be dealt with much more efficiently by telling evalf
or N
to
use an oscillatory quadrature algorithm:
>>> Integral(sin(x)/x**2, (x, 1, oo)).evalf(quad='osc')
0.504067061906928
>>> Integral(sin(x)/x**2, (x, 1, oo)).evalf(20, quad='osc')
0.50406706190692837199
Oscillatory quadrature requires an integrand containing a factor cos(ax+b) or sin(ax+b). Note that many other oscillatory integrals can be transformed to this form with a change of variables:
>>> init_printing(pretty_print=True, use_unicode=False,
... wrap_line=False, no_global=True)
>>> intgrl = Integral(sin(1/x), (x, 0, 1)).transform(x, 1/x)
>>> intgrl
oo
/

 sin(x)
  dx
 2
 x

/
1
>>> N(intgrl, quad='osc')
0.504067061906928
Infinite series use direct summation if the series converges quickly enough. Otherwise, extrapolation methods (generally the EulerMaclaurin formula but also Richardson extrapolation) are used to speed up convergence. This allows highprecision evaluation of slowly convergent series:
>>> Sum(1/k**2, (k, 1, oo)).evalf(strict=False)
1.64493406684823
>>> zeta(2).evalf()
1.64493406684823
>>> Sum(1/klog(1+1/k), (k, 1, oo)).evalf()
0.577215664901533
>>> Sum(1/klog(1+1/k), (k, 1, oo)).evalf(50)
0.57721566490153286060651209008240243104215933593992
>>> EulerGamma.evalf(50)
0.57721566490153286060651209008240243104215933593992
The EulerMaclaurin formula is also used for finite series, allowing them to be approximated quickly without evaluating all terms:
>>> Sum(1/k, (k, 10000000, 20000000)).evalf()
0.693147255559946
Note that evalf
makes some assumptions that are not always optimal. For
finetuned control over numerical summation, it might be worthwhile to manually
use the method Sum.euler_maclaurin
.
Special optimizations are used for rational hypergeometric series (where the
term is a product of polynomials, powers, factorials, binomial coefficients and
the like). N
/evalf
sum series of this type very rapidly to high
precision. For example, this Ramanujan formula for pi can be summed to 10,000
digits in a fraction of a second with a simple command:
>>> f = factorial
>>> n = Symbol('n', integer=True)
>>> R = 9801/sqrt(8)/Sum(f(4*n)*(1103+26390*n)/f(n)**4/396**(4*n),
... (n, 0, oo))
>>> N(R, 10000, strict=False)
3.141592653589793238462643383279502884197169399375105820974944592307...
Numerical simplification¶
The function nsimplify
attempts to find a formula that is numerically equal
to the given input. This feature can be used to guess an exact formula for an
approximate floatingpoint input, or to guess a simpler formula for a
complicated symbolic input. The algorithm used by nsimplify
is capable of
identifying simple fractions, simple algebraic expressions, linear combinations
of given constants, and certain elementary functional transformations of any of
the preceding.
Optionally, nsimplify
can be passed a list of constants to include (e.g. pi)
and a minimum numerical tolerance. Here are some elementary examples:
>>> nsimplify(0.1)
1/10
>>> nsimplify(6.28, [pi], tolerance=0.01)
2*pi
>>> nsimplify(pi, tolerance=0.01)
22/7
>>> nsimplify(pi, tolerance=0.001)
355

113
>>> nsimplify(0.33333, tolerance=1e4)
1/3
>>> nsimplify(2.0**(1/3.), tolerance=0.001)
635

504
>>> nsimplify(2.0**(1/3.), tolerance=0.001, full=True)
3 ___
\/ 2
Here are several more advanced examples:
>>> nsimplify(Float('0.130198866629986772369127970337', 30), [pi, E])
1

5*pi
 + 2*E
7
>>> nsimplify(cos(atan('1/3')))
____
3*\/ 10

10
>>> nsimplify(4/(1+sqrt(5)), [GoldenRatio])
2 + 2*GoldenRatio
>>> nsimplify(2 + exp(2*atan('1/4')*I))
49 8*I
 + 
17 17
>>> nsimplify((1/(exp(3*pi*I/5)+1)))
___________
/ ___
1 / \/ 5 1
  I* /  + 
2 \/ 10 4
>>> nsimplify(I**I, [pi])
pi

2
E
>>> n = Symbol('n')
>>> nsimplify(Sum(1/n**2, (n, 1, oo)), [pi])
2
pi

6
>>> nsimplify(gamma('1/4')*gamma('3/4'), [pi])
___
\/ 2 *pi
uFuncify¶
While NumPy operations are very efficient for vectorized data they sometimes incur unnecessary costs when chained together. Consider the following operation
x = get_numpy_array(...)
y = sin(x)/x
The operators sin
and /
call routines that execute tight for loops in
C
. The resulting computation looks something like this
for(int i = 0; i < n; i++)
{
temp[i] = sin(x[i]);
}
for(int i = i; i < n; i++)
{
y[i] = temp[i] / x[i];
}
This is slightly suboptimal because
We allocate an extra
temp
arrayWe walk over
x
memory twice when once would have been sufficient
A better solution would fuse both elementwise operations into a single for loop
for(int i = i; i < n; i++)
{
y[i] = sin(x[i]) / x[i];
}
Statically compiled projects like NumPy are unable to take advantage of such
optimizations. Fortunately, Diofant is able to generate efficient lowlevel C
or Fortran code. It can then depend on projects like Cython
or f2py
to
compile and reconnect that code back up to Python. Fortunately this process is
well automated and a Diofant user wishing to make use of this code generation
should call the ufuncify
function
>>> expr = sin(x)/x
>>> from diofant.utilities.autowrap import ufuncify
>>> f = ufuncify((x,), expr)
This function f
consumes and returns a NumPy array. Generally ufuncify
performs at least as well as lambdify
. If the expression is complicated
then ufuncify
often significantly outperforms the NumPy backed solution.
Jensen has a good blog post
on this topic.