This twentieth article of the mathematical journey through open source, showcases the list manipulations in Maxima.

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Lists are the basic building blocks of Maxima. The fundamental reason being that Maxima is implemented in Lisp, the building blocks of which, are also lists.

Creation of lists

To begin with, let us walk through the ways to create a list. Simplest way to have a list in Maxima is to just define it using [ ]. So, [x, 5, 3, 2*y] is a list consisting of 4 members. However, Maxima provides two powerful functions for automatically generating lists: makelist(), create_list().

makelist() can take two forms. makelist(e, x, x0, xn) creates and returns a list using the expression e, evaluated for x using the values ranging from x0 to xn. makelist(e, x, L) – creates and returns a list using the expression e, evaluated for x using the members of the list L. Check out the example below for better clarity.

$ maxima -q
(%i1) makelist(2 * i, i, 1, 5);
(%o1)                        [2, 4, 6, 8, 10]
(%i2) makelist(concat(x, 2 * i - 1), i, 1, 5);
(%o2)                        [x1, x3, x5, x7, x9]
(%i3) makelist(concat(x, 2), x, [a, b, c, d]);
(%o3)                          [a2, b2, c2, d2]
(%i4) quit();

Note the interesting usage of concat() to just concatenate its arguments. Note that, makelist() is limited by the variation it can have – to be specific, just one – i in the first two examples, and x in the last one. If we want more, that is where the create_list() function comes into play.

create_list(f, x1, L1, …, xn, Ln) – creates and returns a list with members of the form f, evaluated for the variables x1, …, xn using the values from the corresponding lists L1, …, Ln. Here is a glimpse of its power:

$ maxima -q
(%i1) create_list(concat(x, y), x, [p, q], y, [1, 2]);
(%o1)                          [p1, p2, q1, q2]
(%i2) create_list(concat(x, y, z), x, [p, q], y, [1, 2], z, [a, b]);
(%o2)              [p1a, p1b, p2a, p2b, q1a, q1b, q2a, q2b]
(%i3) create_list(concat(x, y, z), x, [p, q], y, [1, 2, 3], z, [a, b]);
(%o3)    [p1a, p1b, p2a, p2b, p3a, p3b, q1a, q1b, q2a, q2b, q3a, q3b]
(%i4) quit();

Note the “all possible combinations” being created using the values for the variables x, y, z.

Once we have lists created, Maxima provides a host of functions to play around with them. Let’s take a walk through them.

Testing the lists

The following set of functions demonstrates the various checks on lists:

• atom(v) – returns true if v is an atomic element, false otherwise
• listp(L) – returns true if L is a list, false otherwise
• member(v, L) – returns true if v is a member of list L, false otherwise
• some(p, L) – returns true if predicate p is true for at least one member of list L, false otherwise
• every(p, L) – returns true if predicate p is true for all members of list L, false otherwise

$ maxima -q
(%i1) atom(5);
(%o1)                                true
(%i2) atom([5]);
(%o2)                                false
(%i3) listp(x);
(%o3)                                false
(%i4) listp([x]);
(%o4)                                true
(%i5) listp([x, 5]);
(%o5)                                true
(%i6) member(x, [a, b, c]);
(%o6)                                false
(%i7) member(x, [a, x, c]);
(%o7)                                true
(%i8) some(primep, [1, 4, 9]);
(%o8)                                false
(%i9) some(primep, [1, 2, 4, 9]);
(%o9)                                true
(%i10) every(integerp, [1, 2, 4, 9]);
(%o10)                               true
(%i11) every(integerp, [1, 2, 4, x]);
(%o11)                               false
(%i12) quit();

List recreations

Next, is a set of functions operating on list(s) to create and return new lists:

• cons(v, L) – returns a list with v followed by members of L
• endcons(v, L) – returns a list with members of L followed by v
• rest(L, n) – returns a list with members of L, except the first n members (if n is non-negative), otherwise except the last -n members. n is optional, in which case, it is taken as 1.
• join(L1, L2) – returns a list with members of L1 and L2 interspersed
• delete(v, L, n) – returns a list like L but with the first n occurences of v, deleted from it. n is optional, in which case all occurences of v are deleted
• append(L1, …, Ln) – returns a list with members of L1, …, Ln, one after the other
• unique(L) – returns a list obtained by removing the duplicate members in the list L
• reverse(L) – returns a list with members of the list L in reverse order

$ maxima -q
(%i1) L: makelist(i, i, 1, 10);
(%o1)                   [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
(%i2) cons(0, L);
(%o2)                 [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
(%i3) endcons(11, L);
(%o3)                 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]
(%i4) rest(L);
(%o4)                    [2, 3, 4, 5, 6, 7, 8, 9, 10]
(%i5) rest(L, 3);
(%o5)                       [4, 5, 6, 7, 8, 9, 10]
(%i6) rest(L, -3);
(%o6)                        [1, 2, 3, 4, 5, 6, 7]
(%i7) join(L, [a, b, c, d]);
(%o7)                      [1, a, 2, b, 3, c, 4, d]
(%i8) delete(6, L);
(%o8)                    [1, 2, 3, 4, 5, 7, 8, 9, 10]
(%i9) delete(4, delete(6, L));
(%o9)                      [1, 2, 3, 5, 7, 8, 9, 10]
(%i10) delete(4, delete(6, join(L, L)));
(%o10)        [1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 8, 8, 9, 9, 10, 10]
(%i11) L1: rest(L, 7);
(%o11)                            [8, 9, 10]
(%i12) L2: rest(rest(L, -3), 3);
(%o12)                           [4, 5, 6, 7]
(%i13) L3: rest(L, -7);
(%o13)                             [1, 2, 3]
(%i14) append(L1, L2, L3);
(%o14)                  [8, 9, 10, 4, 5, 6, 7, 1, 2, 3]
(%i15) reverse(L);
(%o15)                   [10, 9, 8, 7, 6, 5, 4, 3, 2, 1]
(%i16) join(reverse(L), L); 
(%o16)   [10, 1, 9, 2, 8, 3, 7, 4, 6, 5, 5, 6, 4, 7, 3, 8, 2, 9, 1, 10]
(%i17) unique(join(reverse(L), L)); 
(%o17)                   [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
(%i18) L;
(%o18)                   [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
(%i19) quit();

Note that the list L is still not modified. For that matter, even L1, L2, L3 are not modified. In fact, that is what is meant by that all these functions recreate new modified lists, rather than modifying the existing ones.

List extractions

Here goes a set of functions extracting the various members of a list. first(L), second(L), third(L), fourth(L), fifth(L), sixth(L), seventh(L), eight(L), ninth(L), tenth(L) – respectively return the first, second, … member of the list L. last(L) – returns the last member of the list L

$ maxima -q
(%i1) L: create_list(i * x, x, [a, b, c], i, [1, 2, 3, 4]);
(%o1)       [a, 2 a, 3 a, 4 a, b, 2 b, 3 b, 4 b, c, 2 c, 3 c, 4 c]
(%i2) first(L);
(%o2)                                  a
(%i3) seventh(L);
(%o3)                                 3 b
(%i4) last(L);
(%o4)                                 4 c
(%i5) third(L); last(L);
(%o5)                                 3 a
(%o6)                                 4 c
(%i7) L;
(%o7)       [a, 2 a, 3 a, 4 a, b, 2 b, 3 b, 4 b, c, 2 c, 3 c, 4 c]
(%i8) quit();

Again, note that the list L is still not modified. But, why have we been talking of that? To bring out the fact, that we may need to modify the existing lists, and none of the above functions would do that. Note that, we may achieve that by assigning the return values of the various list recreation functions back to the original list. However, there are few functions, which does that right away.

List manipulations

The following are the two list manipulating functions provided by Maxima:

• push(v, L) – inserts v at the beginning of the list L
• pop(L) – removes and returns the first element from list L

Note that L must be a symbol bound to a list, not the list itself, in both the above functions, for them to modify it. Also, these functionalities are not available by default, so we need to load the basic Maxima file. Check out the demonstration below.

We may display L after doing these operations, or even check the length of L to verify the actual modification of L. And, in case we need to preserve a copy of the list, function copylist() can be used, as such.

$ maxima -q
(%i1) L: makelist(2 * x, x, 1, 10);
(%o1)                [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]
(%i2) push(0, L); /* This doesn't work */
(%o2)            push(0, [2, 4, 6, 8, 10, 12, 14, 16, 18, 20])
(%i3) pop(L); /* Nor does this work */
(%o3)              pop([2, 4, 6, 8, 10, 12, 14, 16, 18, 20])
(%i4) load(basic); /* Loading the basic Maxima file */
(%o4)           /usr/share/maxima/5.24.0/share/macro/basic.mac
(%i5) push(0, L); /* Now, this works */
(%o5)               [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]
(%i6) L;
(%o6)               [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]
(%i7) pop(L); /* Even this works */
(%o7)                                  0
(%i8) L;
(%o8)                [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]
(%i9) K: copylist(L);
(%o9)                [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]
(%i10) length(L);
(%o10)                                10
(%i11) pop(L);
(%o11)                                 2
(%i12) length(L);
(%o12)                                 9
(%i13) K;
(%o13)               [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]
(%i14) L;
(%o14)                 [4, 6, 8, 10, 12, 14, 16, 18, 20]
(%i15) pop([1, 2, 3]); /* Actual list is not allowed */
arg must be a symbol [1, 2, 3]
#0: symbolcheck(x=[1,2,3])(basic.mac line 22)
#1: pop(l=[1,2,3])(basic.mac line 26)
 -- an error. To debug this try: debugmode(true);
(%i16) quit();

Advanced list operations

And finally, if you are still with me, here is a bonus of two sophisticated list operations:

• sublist_indices(L, p) – returns the list indices for the members of the list L, for which predicate p is true.
• assoc(k, L, d) – L must have all its members in the form of x op y, where op is some binary operator. Then, assoc() searches for k in the left operand of the members of L. If found, it returns the corresponding right operand, otherwise d, or false, if d is missing.

Check out the demonstration below for both the above operations.

$ maxima -q
(%i1) sublist_indices([12, 23, 57, 37, 64, 67], primep);
(%o1)                              [2, 4, 6]
(%i2) sublist_indices([12, 23, 57, 37, 64, 67], evenp);
(%o2)                               [1, 5]
(%i3) sublist_indices([12, 23, 57, 37, 64, 67], oddp);
(%o3)                            [2, 3, 4, 6]
(%i4) sublist_indices([2 > 0, -2 > 0, 1 = 1, x = y], identity);
(%o4)                               [1, 3]
(%i5) assoc(2, [2^r, x+y, 2=4, 5/6]);
(%o5)                                  r
(%i6) assoc(6, [2^r, x+y, 2=4, 5/6]);
(%o6)                               false
(%i7) assoc(6, [2^r, x+y, 2=4, 5/6], na);
(%o7)                                na
(%i8) quit();

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