2016-04-29.sagews

%md
# Math 480: Open Source Mathematical Software
### 2016-04-29
### William Stein

## **Lectures 15: Symbolic Calculus (part 3/3)**

Math 480: Open Source Mathematical Software

2016-04-29

William Stein

Lectures 15: Symbolic Calculus (part 3/3)

Plan:

• reminder: homework and peer grading due today at 6pm.

• start screencast

• talk for a few minutes about something

• finish up and polish your homework and peer grading, and ask questions.

1. Example involving rounding error

10e100 + 1 - 10e100

0.000000000000000

10^100 + 1 - 10^100

1

a = exp(100+10^(-20)) + 1 - exp(100)
show(a)

$\displaystyle e^{\frac{10000000000000000000001}{100000000000000000000}} - e^{100} + 1$

N(a)

0.000000000000000

F = RealIntervalField(200); F

Real Interval Field with 200 bits of precision

F(a)

2.6881171418161354484260761372890942647?e23

2. Assumptions - sometimes needed when integrating

E.g., when computing $$\int x^n dx$$ the answer is usually easy -- it's $\frac{x^{n+1}}{n+1}$. However, when $n=1$, it's $\log(x)$.

var('x, n')
integral(x^n, x)

(x, n)

Error in lines 2-2 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/misc/functional.py", line 664, in integral return x.integral(*args, **kwds) File "sage/symbolic/expression.pyx", line 11352, in sage.symbolic.expression.Expression.integral (/projects/sage/sage-6.10/src/build/cythonized/sage/symbolic/expression.cpp:60288) return integral(self, *args, **kwds) File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py", line 759, in integrate return indefinite_integral(expression, v, hold=hold) File "sage/symbolic/function.pyx", line 988, in sage.symbolic.function.BuiltinFunction.__call__ (/projects/sage/sage-6.10/src/build/cythonized/sage/symbolic/function.cpp:11343) res = super(BuiltinFunction, self).__call__( File "sage/symbolic/function.pyx", line 508, in sage.symbolic.function.Function.__call__ (/projects/sage/sage-6.10/src/build/cythonized/sage/symbolic/function.cpp:7211) res = g_function_eval2(self._serial, (<Expression>args[0])._gobj, File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py", line 85, in _eval_ res = integrator(f, x) File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/symbolic/integration/external.py", line 22, in maxima_integrator result = maxima.sr_integral(expression,v) File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py", line 784, in sr_integral self._missing_assumption(s) File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py", line 993, in _missing_assumption raise ValueError(outstr) ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(n>0)', see `assume?` for more details) Is n equal to -1?

forget()
assume(n==-1)
integral(x^n, x)

log(x)

forget(n==-1)
assume(n!=-1)
show(integral(x^n, x))

$\displaystyle \frac{x^{n + 1}}{n + 1}$

3. Sympy

Sympy is a Python library for symbolic calculus, which can be used independently from Sage, and is also in Sage. See http://www.sympy.org/en/index.html

Integration with Sage could be improved.

But being able to use Sympy without Sage is potentially very valuable (see Hamster). Or it can cause you to waste a lot of time (see Chris Swierczewski).

On Sage support list, here about sympy as follows:

1. I've been using Sympy, but switched to Sage since Sympy is too slow or missing something.

2. I've been using Sage to compute this integral (or series) and it's wrong! ... There is a bug in Maxima... Use algorithm='sympy', since Sympy is right.

from sympy import Limit, symbols, cos
x = symbols('x')
expr = Limit((cos(x) - 1)/x, x, 0)
print expr

Limit((cos(x) - 1)/x, x, 0)

expr.doit()   # really??

0

show(expr)   # this doesn't work.  sigh.

$\displaystyle \text{\texttt{Limit((cos(x){ }{-}{ }1)/x,{ }x,{ }0)}}$

reset()  # since we overwrote x above
m = integrate(sin(x)*cos(x)*tan(x), x); show(m)     # uses maxima

$\displaystyle \frac{1}{2} \, x - \frac{1}{4} \, \sin\left(2 \, x\right)$

s = integrate(sin(x)*cos(x)*tan(x), x, algorithm='sympy'); show(s)   # uses sympy instead under the hood!

$\displaystyle -\frac{1}{2} \, \cos\left(x\right) \sin\left(x\right) + \frac{1}{2} \, x$

show(s-m)

$\displaystyle -\frac{1}{2} \, \cos\left(x\right) \sin\left(x\right) + \frac{1}{4} \, \sin\left(2 \, x\right)$

(s - m).simplify_full()

0

bool(s==m)

True