Maxima: How to factor out a finite sum? - math

Question

This question is about humane representation of Maxima output.
In short, how do I make
b*d4 + b*d3 + b*d2 + b*d1 + a*sin(c4 + alpha) + a*sin(c3 + alpha) + a*sin(c2 + alpha) + a*sin(c1 + alpha)
look like
b*sum_{i=1}^{4} d_i + a*sum_{j=1}^{4}sin(c_i + \alpha)
where sum_{*}^{*}* is a summation sign and an expression with subscripts ?
Or deeper, how to properly model a finite set of items here ?
Consider a finite set of entities $x_i$ (trying to speak tex here) that are numbered from 1 to n where n is known. Let a function $F$ depend on several characteristics of those entities $c_ji = c_j(x_i), j = 1..k$ (k - also known) so that $F = F(c_11,...,c_kn)$.
Now when I try to implement that in Maxima and do things with it, it would yield sums and products of all kinds, where the numbered items are represented something like $c_1*a + c_2*a + c_3*a + c_4*a + d_1*b + d_2*b + d_3*b + d_4*b$ which you would write down on paper as $a*\sum_{i=1}^{4}c_i + b*sum_{i=1}^{4}d_i$.
So how can I make Maxima do that sort of expression contraction ?
To be more specific, here is an actual code example:
(Maxima output marked as ">>>")
/* let's have 4 entities: */
n: 4 $
/* F is a sum of similar components corresponding to each entity F = F_1 + F_2 + F_3 + F_4 */
F_i: a*sin(alpha + c_i) + b*d_i;
>>> b*d_i + a*sin(c_i + alpha)
/* defining the characteristics */
c(i) := concat(c, i) $
d(i) := concat(d, i) $
/* now let's see what F looks like */
/* first, we should model the fact that we have 4 entities somehow: */
F_i(i) := subst(c(i), c_i, subst(d(i), d_i, F_i)) $
/* now we can evaluate F: */
F: sum(F_i(i), i, 1, 4);
>>> b*d4 + b*d3 + b*d2 + b*d1 + a*sin(c4 + alpha) + a*sin(c3 + alpha) + a*sin(c2 + alpha) + a*sin(c1 + alpha)
/* at this point it would be nice to do something like: */
/* pretty(F); */
/* and get an output of: */
/* $b*\sum_{i=1}^{4}d_i + a*\sum_{j=1}^4 sin(c_j + \alpha)$ */
/* not to mention having Maxima write things in the same order as I do */
So, to sum up, there are three quetions here:
How do I factor out a sum from an expression like the one on top of this post ?
How do I properly let Maxima know what I'm speaking about here ?
How to make Maxima preserve my order of things in output ?
Thanks in advance.

Answer 1

Here's a way to go about what I think you want.
(%i1) n: 4 $
(%i2) F(i) := a*sin(alpha + c[i]) + b*d[i];
(%o2) F(i) := a sin(alpha + c ) + b d
i i
(%i3) 'sum(F(i), i,1,4);
4
====
\
(%o3) > (a sin(c + alpha) + b d )
/ i i
====
i = 1
(%i4) declare (nounify(sum), linear);
(%o4) done
(%i5) 'sum(F(i), i,1,4);
4 4
==== ====
\ \
(%o5) a > sin(c + alpha) + b > d
/ i / i
==== ====
i = 1 i = 1
(%i6)
The most important thing here is that I've written what we would call "c sub i" and "d sub i" as c[i] and d[i] respectively. These are indexed variables named c and d, and i is the index. It's not necessary for there to actually exist arrays or lists named c or d, and i might or might not have a specific value.
I have written F as an ordinary function. I've avoided the construction of variable names via concat and avoided the substitution of those names into an expression. I would like to emphasize that such operations are almost certainly not the best way to go about it.
In %i3 note that I wrote the summation as 'sum(...) which makes it a so-called noun expression, which means it is maintained in a symbolic form and not evaluated.
By default, summations are not treated as linear, so in %i4 I declared summations as linear so that the result in %o5 is as expected.
Maxima doesn't have a way to collect expressions such as a1 + a2 + a3 back into a symbolic summation, but perhaps you don't need such an operation.