2016-04-13-formula.sagews

Is there a formula?

From last time...

Throw out a mathematical question: Is there a simple formula for $\sum_{0< m\leq n} \sin(m)$ similar to the formula $\sum_{0< m< n} m = \frac{n(n-1)}{2}$?

YES!

k, n = var('k,n')
show(sum(k^2, k, 1, n).factor())

$\displaystyle \frac{1}{6} \, {\left(2 \, n + 1\right)} {\left(n + 1\right)} n$

k, n = var('k,n')
f = sum(sin(k), k, 1, n-1)
show(f)

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

%timeit float(f(n=10000))

625 loops, best of 3: 349 µs per loop

%timeit float(f(n=100000))

625 loops, best of 3: 315 µs per loop

g = fast_float(f, 'n')
g

<sage.ext.interpreters.wrapper_rdf.Wrapper_rdf object at 0x7f7586a69980>

%timeit g(10000)

625 loops, best of 3: 827 ns per loop

g(10000)

1.9395054106807146

︠fbd4b34a-5f02-4ddf-9e7c-848f8e3baa15s︠
315/.827

380.894800483676

500x speedup!

%time float(sum(sin(j) for j in [1..10000]))

1.6338910217924854 CPU time: 1.82 s, Wall time: 1.84 s

1.84 / (766*10^(-9))

2.40208877284595e6

So the net speedup from our one liner from day one is a factor of several million in speed.

Exercise now: Draw a plot of the function $f(n) = \sum_{0< m\leq n} \sin(m)$ for various values of $n$...

# one way
stats.TimeSeries([sin(n) for n in range(10000)]).sums().plot()

# Another -- use the formula above...