Here we discuss some of the most basic aspects of expression manipulation in Diofant.
The assumptions system allows users to declare certain mathematical properties on symbols, such as being positive, imaginary or integer.
By default, all symbols are complex valued. This assumption makes it easier to treat mathematical problems in full generality.
>>> sqrt(x**2) ____ ╱ 2 ╲╱ x
Yet obviously we can simplify above expression if some additional
mathematical properties on
x are assumed. This is where
assumptions system come into play.
Assumptions are set on
when they are created. For instance, we can create a symbol that is
assumed to be positive.
>>> p = symbols('p', positive=True)
And then, certain simplifications will be possible:
>>> sqrt(p**2) p
The assumptions system additionally has deductive capabilities. You
might check assumptions on any expression with
>>> p.is_positive True >>> (1 + p).is_positive True >>> (-p).is_positive False
False is returned also if certain assumption doesn’t make sense
for given object.
In a three-valued logic, used by system,
None represents the
>>> (p - 1).is_positive is None True
One of the most common things you might want to do with a mathematical
expression is substitution with
method. It replaces all instances of something in an expression with
>>> expr = cos(x) + 1 >>> expr.subs(x, y) cos(y) + 1 >>> expr cos(x) + 1
We see that performing substitution leaves original expression
Almost all Diofant expressions are immutable. No function (or method) will change them in-place.
To perform several substitutions in one shot, you can provide
Iterable sequence of pairs.
>>> x**y y x >>> _.subs(((y, x**y), (y, x**x))) ⎛ ⎛ x⎞⎞ ⎜ ⎝x ⎠⎟ ⎝x ⎠ x
simultaneous to do all substitutions at once.
>>> (x - y).subs(((x, y), (y, x))) 0 >>> (x - y).subs(((x, y), (y, x)), simultaneous=True) -x + y
To evaluate a numerical expression into a floating point number with
arbitrary precision, use
By default, 15 digits of precision are used.
>>> expr = sqrt(8) >>> expr.evalf() 2.82842712474619
But you can change that. Let’s compute the first 70 digits of \(\pi\).
>>> pi.evalf(70) 3.141592653589793238462643383279502884197169399375105820974944592307816
Sometimes there are roundoff errors smaller than the desired precision
that remain after an expression is evaluated. Such numbers can be
removed by setting the
>>> one = cos(1)**2 + sin(1)**2 >>> (one - 1).evalf(strict=False) -0.e-146 >>> (one - 1).evalf(chop=True) 0
Discussed above method is not effective enough if you intend to evaluate an expression at many points, there are better ways, especially if you only care about machine precision.
The easiest way to convert a Diofant expression to an expression that
can be numerically evaluated with libraries like
numpy — use
lambdify() function. It acts
lambda form, except it converts the Diofant names to
the names of the given numerical library.
>>> import numpy >>> a = numpy.arange(5) >>> expr = sin(x) >>> f = lambdify(x, expr, "numpy") >>> f(a) [ 0. 0.84147098 0.90929743 0.14112001 -0.7568025 ]
You can use other libraries than NumPy. For example, the standard
>>> f = lambdify(x, expr, "math") >>> f(0.1) 0.09983341664682815