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<h1>3 Lists and Records</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">Plain Lists</a>
<li> <A HREF="CHAP003.htm#SECT002">Identical Lists</a>
<li> <A HREF="CHAP003.htm#SECT003">Immutability</a>
<li> <A HREF="CHAP003.htm#SECT004">Sets</a>
<li> <A HREF="CHAP003.htm#SECT005">Ranges</a>
<li> <A HREF="CHAP003.htm#SECT006">For and While Loops</a>
<li> <A HREF="CHAP003.htm#SECT007">List Operations</a>
<li> <A HREF="CHAP003.htm#SECT008">Vectors and Matrices</a>
<li> <A HREF="CHAP003.htm#SECT009">Plain Records</a>
<li> <A HREF="CHAP003.htm#SECT010">Further Information about Lists</a>
</ol><p>
<p>
Modern mathematics, especially algebra, is based on set theory. When sets
are represented in a computer, they inadvertently turn into lists. That's
why we start our survey of the various objects <font face="Gill Sans,Helvetica,Arial">GAP</font> can handle with a
description of lists and their manipulation. <font face="Gill Sans,Helvetica,Arial">GAP</font> regards sets as a
special kind of lists, namely as lists without holes or duplicates whose
entries are ordered with respect to the precedence relation <code><</code>.
<p>
After the introduction of the basic manipulations with lists in <a href="CHAP003.htm#SECT001">Plain Lists</a>, some difficulties concerning identity and mutability of lists are
discussed in <a href="CHAP003.htm#SECT002">Identical Lists</a> and <a href="CHAP003.htm#SECT003">Immutability</a>. Sets, ranges, row
vectors, and matrices are introduced as special kinds of lists in <a href="CHAP003.htm#SECT004">Sets</a>,
<a href="CHAP003.htm#SECT005">Ranges</a>, <a href="CHAP003.htm#SECT008">Vectors and Matrices</a>. Handy list operations are
shown in <a href="CHAP003.htm#SECT007">List Operations</a>. Finally we explain how to use records
in <a href="CHAP003.htm#SECT009">Plain Records</a>.
<p>
<p>
<h2><a name="SECT001">3.1 Plain Lists</a></h2>
<p><p>
<a name = "I0"></a>
A <strong>list</strong> is a collection of objects separated by commas and enclosed in
brackets. Let us for example construct the list <code>primes</code> of the first 10
prime numbers.
<pre>
gap> primes:= [2, 3, 5, 7, 11, 13, 17, 19, 23, 29];
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ]
</pre>
The next two primes are 31 and 37. They may be appended to the existing
list by the function <code>Append</code> which takes the existing list as its first
and another list as a second argument. The second argument is appended
to the list <code>primes</code> and no value is returned. Note that by appending
another list the object <code>primes</code> is changed.
<pre>
gap> Append(primes, [31, 37]);
gap> primes;
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 ]
</pre>
You can as well add single new elements to existing lists by the function
<code>Add</code> which takes the existing list as its first argument and a new
element as its second argument. The new element is added to the list
<code>primes</code> and again no value is returned but the list <code>primes</code> is changed.
<pre>
gap> Add(primes, 41);
gap> primes;
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 ]
</pre>
Single elements of a list are referred to by their position in the list.
To get the value of the seventh prime, that is the seventh entry in our
list <code>primes</code>, you simply type
<pre>
gap> primes[7];
17
</pre>
This value can be handled like any other value, for example multiplied by 2
or assigned to a variable. On the other hand this mechanism allows one to
assign a value to a position in a list. So the next prime 43 may be
inserted in the list directly after the last occupied position of
<code>primes</code>. This last occupied position is returned by the function
<code>Length</code>.
<pre>
gap> Length(primes);
13
gap> primes[14]:= 43;
43
gap> primes;
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 ]
</pre>
Note that this operation again has changed the object <code>primes</code>. The
next position after the end of a list is not the only position capable
of taking a new value. If you know that 71 is the 20th prime, you can
enter it right now in the 20th position of <code>primes</code>. This will result
in a list with holes which is however still a list and now has length
20.
<pre>
gap> primes[20]:= 71;
71
gap> primes;
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,,,,,, 71 ]
gap> Length(primes);
20
</pre>
The list itself however must exist before a value can be assigned to a
position of the list. This list may be the empty list <code>[ ]</code>.
<pre>
gap> lll[1]:= 2;
Variable: 'lll' must have a value
</pre>
<p>
<pre>
gap> lll:= []; lll[1]:= 2;
[ ]
2
</pre>
Of course existing entries of a list can be changed by this mechanism,
too. We will not do it here because <code>primes</code> then may no longer be a list
of primes. Try for yourself to change the 17 in the list into a 9.
<p>
To get the position of 17 in the list <code>primes</code> use the function
<code>Position</code> which takes the list as its first argument and the element as
its second argument and returns the position of the first occurrence of
the element 17 in the list <code>primes</code>.
If the element is not contained in the list then <code>Position</code> will return
the special object <code>fail</code>.
<pre>
gap> Position(primes, 17);
7
gap> Position(primes, 20);
fail
</pre>
In all of the above changes to the list <code>primes</code>, the list has been
automatically resized. There is no need for you to tell <font face="Gill Sans,Helvetica,Arial">GAP</font> how big
you want a list to be. This is all done dynamically.
<p>
It is not necessary for the objects collected in a list to be of the same
type.
<pre>
gap> lll:= [true, "This is a String",,, 3];
[ true, "This is a String",,, 3 ]
</pre>
In the same way a list may be part of another list. A list may even be
part of itself.
<pre>
gap> lll[3]:= [4,5,6];; lll;
[ true, "This is a String", [ 4, 5, 6 ],, 3 ]
gap> lll[4]:= lll;
[ true, "This is a String", [ 4, 5, 6 ], ~, 3 ]
</pre>
Now the tilde in the fourth position of <code>lll</code> denotes the object that is
currently printed. Note that the result of the last operation is the
actual value of the object <code>lll</code> on the right hand side of the
assignment. In fact it is identical to the value of the whole list
<code>lll</code> on the left hand side of the assignment.
<p>
<a name = "I1"></a>
<a name = "I1"></a>
<a name = "I2"></a>
A <strong>string</strong> is a special type of list,
namely a dense list of <strong>characters</strong>, where <strong>dense</strong> means that the list has
no holes. Here, <strong>characters</strong> are special <font face="Gill Sans,Helvetica,Arial">GAP</font> objects representing an
element of the character set of the operating system. The input of printable
characters is by enclosing them in single quotes <code>'</code>. A string literal
can either be entered as the list of characters or by writing the characters
between doublequotes <code>"</code>. Strings are handled specially by <code>Print</code>. You can
learn much more about strings in the reference manual.
<p>
<pre>
gap> s1 := ['H','a','l','l','o',' ','w','o','r','l','d','.'];
"Hallo world."
gap> s1 = "Hallo world.";
true
gap> s1[7];
'w'
</pre>
<p>
Sublists of lists can easily be extracted and assigned using the operator
<code></code><var>list</var><code>{ </code><var>positions</var><code> }</code>.
<pre>
gap> sl := lll{ [ 1, 2, 3 ] };
[ true, "This is a String", [ 4, 5, 6 ] ]
gap> sl{ [ 2, 3 ] } := [ "New String", false ];
[ "New String", false ]
gap> sl;
[ true, "New String", false ]
</pre>
This way you get a new list whose <var>i</var>th entry is that element of the
original list whose position is the <var>i</var>th entry of the argument in the
curly braces.
<p>
<p>
<h2><a name="SECT002">3.2 Identical Lists</a></h2>
<p><p>
<a name = "I3"></a>
This second section about lists is dedicated to the subtle difference
between <strong>equality</strong> and <strong>identity</strong> of lists. It is really important to
understand this difference in order to understand how complex data
structures are realized in <font face="Gill Sans,Helvetica,Arial">GAP</font>. This section applies to all <font face="Gill Sans,Helvetica,Arial">GAP</font>
objects that have subobjects, e.g., to lists and to records. After
reading the section <a href="CHAP003.htm#SECT009">Plain Records</a> about records you should return to
this section and translate it into the record context.
<p>
Two lists are <strong>equal</strong> if all their entries are equal. This means that the
equality operator <code>=</code> returns <code>true</code> for the comparison of two lists if
and only if these two lists are of the same length and for each position
the values in the respective lists are equal.
<pre>
gap> numbers := primes;; numbers = primes;
true
</pre>
We assigned the list <code>primes</code> to the variable <code>numbers</code> and, of course
they are equal as they have both the same length and the same entries.
Now we will change the third number to 4 and compare the result again
with <code>primes</code>.
<pre>
gap> numbers[3]:= 4;; numbers = primes;
true
</pre>
You see that <code>numbers</code> and <code>primes</code> are still equal, check this by
printing the value of <code>primes</code>. The list <code>primes</code> is no longer a list of
primes! What has happened? The truth is that the lists <code>primes</code> and
<code>numbers</code> are not only equal but they are also <strong>identical</strong>. <code>primes</code> and
<code>numbers</code> are two variables pointing to the same list. If you change the
value of the subobject <code>numbers[3]</code> of <code>numbers</code> this will also change
<code>primes</code>. Variables do <strong>not</strong> point to a certain block of storage memory
but they do point to an object that occupies storage memory. So the
assignment <code>numbers := primes</code> did <strong>not</strong> create a new list in a different
place of memory but only created the new name <code>numbers</code> for the same old
list of primes.
<p>
From this we see that <strong>the same object can have several names.</strong>
<p>
If you want to change a list with the contents of <code>primes</code> independently
from <code>primes</code> you will have to make a copy of <code>primes</code> by the function
<code>ShallowCopy</code> which takes an object as its argument and returns a copy of
the argument. (We will first restore the old value of <code>primes</code>.)
<pre>
gap> primes[3]:= 5;; primes;
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,,,,,, 71 ]
gap> numbers:= ShallowCopy(primes);; numbers = primes;
true
gap> numbers[3]:= 4;; numbers = primes;
false
</pre>
Now <code>numbers</code> is no longer equal to <code>primes</code> and <code>primes</code> still is a list
of primes. Check this by printing the values of <code>numbers</code> and <code>primes</code>.
<p>
Lists and records can be changed this way because <font face="Gill Sans,Helvetica,Arial">GAP</font> objects of these
types have subobjects.
To clarify this statement consider the following example.
<pre>
gap> i:= 1;; j:= i;; i:= i+1;;
</pre>
By adding 1 to <code>i</code> the value of <code>i</code> has changed. What happens to <code>j</code>?
After the second statement <code>j</code> points to the same object as <code>i</code>, namely
to the integer 1. The addition does <strong>not</strong> change the object <code>1</code> but
creates a new object according to the instruction <code>i+1</code>. It is actually
the assignment that changes the value of <code>i</code>. Therefore <code>j</code> still points
to the object <code>1</code>. Integers (like permutations and booleans) have no
subobjects. Objects of these types cannot be changed but can only be
replaced by other objects. And a replacement does not change the values
of other variables. In the above example an assignment of a new value to
the variable <code>numbers</code> would also not change the value of <code>primes</code>.
<p>
Finally try the following examples and explain the results.
<pre>
gap> l:= [];; l:= [l];
[ [ ] ]
gap> l[1]:= l;
[ ~ ]
</pre>
Now return to Section <a href="CHAP003.htm#SECT001">Plain Lists</a> and find out whether the functions
<code>Add</code> and <code>Append</code> change their arguments.
<p>
<p>
<h2><a name="SECT003">3.3 Immutability</a></h2>
<p><p>
<font face="Gill Sans,Helvetica,Arial">GAP</font> has a mechanism that protects lists against changes like the ones
that have bothered us in Section <a href="CHAP003.htm#SECT002">Identical Lists</a>. The function
<code>Immutable</code> takes as argument a list and returns an immutable copy of it,
i.e., a list which looks exactly like the old one, but has two extra
properties:
(1) The new list is immutable, i.e., the list itself and its subobjects
cannot be changed.
(2) In constructing the copy, every part of the list that can be changed
has been copied, so that changes to the old list will not affect the
new one. In other words, the new list has no mutable subobjects in
common with the old list.
<pre>
gap> list := [ 1, 2, "three", [ 4 ] ];; copy := Immutable( list );;
gap> list[3][5] := 'w';; list; copy;
[ 1, 2, "threw", [ 4 ] ]
[ 1, 2, "three", [ 4 ] ]
gap> copy[3][5] := 'w';
Lists Assignment: <list> must be a mutable list
not in any function
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' and ignore the assignment to continue
brk> quit;
</pre>
As a consequence of these rules, in the immutable copy of a list which
contains an already immutable list as subobject, this immutable subobject
need not be copied, because it is unchangeable. Immutable lists are
useful in many complex <font face="Gill Sans,Helvetica,Arial">GAP</font> objects, for example as generator lists of
groups. By making them immutable, <font face="Gill Sans,Helvetica,Arial">GAP</font> ensures that no generators can
be added to the list, removed or exchanged. Such changes would of course
lead to serious inconsistencies with other knowledge that may already
have been calculated for the group.
<p>
A converse function to <code>Immutable</code> is <code>ShallowCopy</code>, which produces a
new mutable list whose <i>i</i>th entry is the <i>i</i>th entry of the old
list. The single entries are not copied, they are just placed in the
new list. If the old list is immutable, and hence the list entries
are immutable themselves, the result of <code>ShallowCopy</code> is mutable only
on the top level.
<p>
It should be noted that also other objects than lists can appear in
mutable or immutable form.
Records (see Section <a href="CHAP003.htm#SECT009">Plain Records</a>) provide another example.
<p>
<p>
<h2><a name="SECT004">3.4 Sets</a></h2>
<p><p>
<a name = "I4"></a>
<a name = "I4"></a>
<a name = "I5"></a>
<font face="Gill Sans,Helvetica,Arial">GAP</font> knows several special kinds of lists. A <strong>set</strong> in <font face="Gill Sans,Helvetica,Arial">GAP</font> is a
list that contains no holes (such a list is called <strong>dense</strong>) and whose
elements are strictly sorted w.r.t. <code><</code>; in particular, a set cannot
contain duplicates. (More precisely, the elements of a set in <font face="Gill Sans,Helvetica,Arial">GAP</font>
are required to lie in the same <strong>family</strong>, but roughly this means that
they can be compared using the <code><</code> operator.)
<p>
This provides a natural model for mathematical sets whose elements are
given by an explicit enumeration.
<p>
<font face="Gill Sans,Helvetica,Arial">GAP</font> also calls a set a <strong>strictly sorted list</strong>, and the function
<code>IsSSortedList</code> tests whether a given list is a set. It returns a
boolean value. For almost any list whose elements are contained in
the same family, there exists a corresponding set. This set is
constructed by the function <code>Set</code> which takes the list as its argument
and returns a set obtained from this list by ignoring holes and
duplicates and by sorting the elements.
<p>
The elements of the sets used in the examples of this section are
strings.
<pre>
gap> fruits:= ["apple", "strawberry", "cherry", "plum"];
[ "apple", "strawberry", "cherry", "plum" ]
gap> IsSSortedList(fruits);
false
gap> fruits:= Set(fruits);
[ "apple", "cherry", "plum", "strawberry" ]
</pre>
Note that the original list <code>fruits</code> is not changed by the function
<code>Set</code>. We have to make a new assignment to the variable <code>fruits</code> in
order to make it a set.
<p>
The operator <code>in</code> is used to test whether an object is an element of a
set. It returns a boolean value <code>true</code> or <code>false</code>.
<pre>
gap> "apple" in fruits;
true
gap> "banana" in fruits;
false
</pre>
The operator <code>in</code> can also be applied to ordinary lists. It is however
much faster to perform a membership test for sets since sets are
always sorted and a binary search can be used instead of a linear
search. New elements may be added to a set by the function <code>AddSet</code>
which takes the set <code>fruits</code> as its first argument and an element as
its second argument and adds the element to the set if it wasn't
already there. Note that the object <code>fruits</code> is changed.
<pre>
gap> AddSet(fruits, "banana");
gap> fruits; # The banana is inserted in the right place.
[ "apple", "banana", "cherry", "plum", "strawberry" ]
gap> AddSet(fruits, "apple");
gap> fruits; # fruits has not changed.
[ "apple", "banana", "cherry", "plum", "strawberry" ]
</pre>
Note that inserting new elements into a set with <code>AddSet</code> is usually more
expensive than simply adding new elements at the end of a list.
<p>
Sets can be intersected by the function <code>Intersection</code> and united by the
function <code>Union</code> which both take two sets as their arguments and return
the intersection resp. union of the two sets as a new object.
<pre>
gap> breakfast:= ["tea", "apple", "egg"];
[ "tea", "apple", "egg" ]
gap> Intersection(breakfast, fruits);
[ "apple" ]
</pre>
The arguments of the functions <code>Intersection</code> and <code>Union</code> could be
ordinary lists, while their result is always a set. Note that in the
preceding example at least one argument of <code>Intersection</code> was not a
set. The functions <code>IntersectSet</code> and <code>UniteSet</code> also form the
intersection resp. union of two sets. They will however not return the
result but change their first argument to be the result. Try them
carefully.
<p>
<p>
<h2><a name="SECT005">3.5 Ranges</a></h2>
<p><p>
A <strong>range</strong> is a finite arithmetic progression of integers. This is another
special kind of list. A range is described by the first two values and the last
value of the arithmetic progression which are given in the form
<code>[</code><var>first</var><code>,</code><var>second</var><code>..</code><var>last</var><code>]</code>.
In the usual case of an ascending list of
consecutive integers the second entry may be omitted.
<pre>
gap> [1..999999]; # a range of almost a million numbers
[ 1 .. 999999 ]
gap> [1, 2..999999]; # this is equivalent
[ 1 .. 999999 ]
gap> [1, 3..999999]; # here the step is 2
[ 1, 3 .. 999999 ]
gap> Length( last );
500000
gap> [ 999999, 999997 .. 1 ];
[ 999999, 999997 .. 1 ]
</pre>
This compact printed representation of a fairly long list corresponds to
a compact internal representation.
The function <code>IsRange</code> tests whether an object is a range,
the function <code>ConvertToRangeRep</code> changes the representation of a list
that is in fact a range to this compact internal representation.
<pre>
gap> a:= [-2,-1,0,1,2,3,4,5];
[ -2, -1, 0, 1, 2, 3, 4, 5 ]
gap> IsRange( a );
true
gap> ConvertToRangeRep( a );; a;
[ -2 .. 5 ]
gap> a[1]:= 0;; IsRange( a );
false
</pre>
Note that this change of representation does <strong>not</strong> change the value of
the list <code>a</code>. The list <code>a</code> still behaves in any context in the same way
as it would have in the long representation.
<p>
<p>
<h2><a name="SECT006">3.6 For and While Loops</a></h2>
<p><p>
<a name = "I6"></a>
<a name = "I7"></a>
Given a list <code>pp</code> of permutations we can form their product by means of a
<code>for</code> loop instead of writing down the product explicitly.
<p>
<pre>
gap> pp:= [ (1,3,2,6,8)(4,5,9), (1,6)(2,7,8), (1,5,7)(2,3,8,6),
> (1,8,9)(2,3,5,6,4), (1,9,8,6,3,4,7,2)];;
gap> prod:= ();
()
gap> for p in pp do
> prod:= prod*p;
> od;
gap> prod;
(1,8,4,2,3,6,5,9)
</pre>
<p>
First a new variable <code>prod</code> is initialized to the identity permutation
<code>()</code>. Then the loop variable <code>p</code> takes as its value one permutation after
the other from the list <code>pp</code> and is multiplied with the present value of
<code>prod</code> resulting in a new value which is then assigned to <code>prod</code>.
<p>
The <code>for</code> loop has the following syntax
<p>
<code> for </code><var>var</var><code> in </code><var>list</var><code> do </code><var>statements</var><code> od;</code>
<p>
The effect of the <code>for</code> loop is to execute the <var>statements</var> for every
element of the <var>list</var>. A <code>for</code> loop is a statement and therefore
terminated by a semicolon. The list of <var>statements</var> is enclosed by the
keywords <code>do</code> and <code>od</code> (reverse <code>do</code>). A <code>for</code> loop returns no value.
Therefore we had to ask explicitly for the value of <code>prod</code> in the
preceding example.
<p>
The <code>for</code> loop can loop over any kind of list, even a list with holes.
In many programming languages the <code>for</code> loop has the form
<p>
<code>for </code><var>var</var><code> from </code><var>first</var><code> to </code><var>last</var><code> do </code><var>statements</var><code> od;</code>
<p>
In <font face="Gill Sans,Helvetica,Arial">GAP</font> this is merely a special case of the general <code>for</code> loop as defined
above where the <var>list</var> in the loop body is a range (see <a href="CHAP003.htm#SECT005">Ranges</a>):
<p>
<code> for </code><var>var</var><code> in [</code><var>first</var><code>..</code><var>last</var><code>] do </code><var>statements</var><code> od;</code>
<p>
You can for instance loop over a range to compute the factorial 15!
of the number 15 in the following way.
<p>
<pre>
gap> ff:= 1;
1
gap> for i in [1..15] do
> ff:= ff * i;
> od;
gap> ff;
1307674368000
</pre>
<p>
The <code>while</code> loop has the following syntax
<p>
<code> while </code><var>condition</var><code> do </code><var>statements</var><code> od; </code>
<p>
The <code>while</code> loop loops over the <var>statements</var> as long as the
<var>condition</var> evaluates to <code>true</code>. Like the <code>for</code> loop the <code>while</code> loop
is terminated by the keyword <code>od</code> followed by a semicolon.
<p>
We can use our list <code>primes</code> to perform a very simple factorization. We
begin by initializing a list <code>factors</code> to the empty list. In this list
we want to collect the prime factors of the number 1333. Remember that a
list has to exist before any values can be assigned to positions of the
list. Then we will loop over the list <code>primes</code> and test for each prime
whether it divides the number. If it does we will divide the number by
that prime, add it to the list <code>factors</code> and continue.
<p>
<pre>
gap> n:= 1333;;
gap> factors:= [];;
gap> for p in primes do
> while n mod p = 0 do
> n:= n/p;
> Add(factors, p);
> od;
> od;
gap> factors;
[ 31, 43 ]
gap> n;
1
</pre>
<p>
As <code>n</code> now has the value 1 all prime factors of 1333 have been found and
<code>factors</code> contains a complete factorization of 1333. This can of course
be verified by multiplying 31 and 43.
<p>
This loop may be applied to arbitrary numbers in order to find prime
factors. But as <code>primes</code> is not a complete list of all primes this loop
may fail to find all prime factors of a number greater than 2000, say.
You can try to improve it in such a way that new primes are added to the
list <code>primes</code> if needed.
<p>
You have already seen that list objects may be changed. This of
course also holds for the list in a loop body. In most cases you have to be
careful not to change this list, but there are situations where this is
quite useful. The following example shows a quick way to determine the
primes smaller than 1000 by a sieve method. Here we will make use of the
function <code>Unbind</code> to delete entries from a list, and the 'if'
statement covered in <a href="CHAP004.htm#SECT002">If Statements</a>.
<pre>
gap> primes:= [];;
gap> numbers:= [2..1000];;
gap> for p in numbers do
> Add(primes, p);
> for n in numbers do
> if n mod p = 0 then
> Unbind(numbers[n-1]);
> fi;
> od;
> od;
</pre>
The inner loop removes all entries from <code>numbers</code> that are divisible by
the last detected prime <code>p</code>. This is done by the function <code>Unbind</code> which
deletes the binding of the list position <code>numbers[n-1]</code> to the value <code>n</code>
so that afterwards <code>numbers[n-1]</code> no longer has an assigned value. The
next element encountered in <code>numbers</code> by the outer loop necessarily is
the next prime.
<p>
In a similar way it is possible to enlarge the list which is looped over.
This yields a nice and short orbit algorithm for the action of a group,
for example.
<p>
More about <code>for</code> and <code>while</code> loops can be found in the
sections <a href="../ref/CHAP004.htm#SECT017">While</a> and <a href="../ref/CHAP004.htm#SECT019">For</a> of the reference manual.
<p>
<p>
<h2><a name="SECT007">3.7 List Operations</a></h2>
<p><p>
There is a more comfortable way than that given in the previous section to
compute the product of a list of numbers or permutations.
<pre>
gap> Product([1..15]);
1307674368000
gap> Product(pp);
(1,8,4,2,3,6,5,9)
</pre>
The function <code>Product</code> takes a list as its argument and computes the
product of the elements of the list. This is possible whenever a
multiplication of the elements of the list is defined. So <code>Product</code>
executes a loop over all elements of the list.
<p>
There are other often used loops available as functions. Guess what the
function <code>Sum</code> does. The function <code>List</code> may take a list and a function
as its arguments. It will then apply the function to each element of the
list and return the corresponding list of results. A list of cubes is
produced as follows with the function <code>cubed</code> from Section <a href="CHAP004.htm">Functions</a>.
<pre>
gap> cubed:= x -> x^3;;
gap> List([2..10], cubed);
[ 8, 27, 64, 125, 216, 343, 512, 729, 1000 ]
</pre>
To add all these cubes we might apply the function <code>Sum</code> to the last
list. But we may as well give the function <code>cubed</code> to <code>Sum</code> as an
additional argument.
<pre>
gap> Sum(last) = Sum([2..10], cubed);
true
</pre>
The primes less than 30 can be retrieved out of the list <code>primes</code> from
Section <a href="CHAP003.htm#SECT001">Plain Lists</a> by the function <code>Filtered</code>. This function takes the
list <code>primes</code> and a property as its arguments and will return the list of
those elements of <code>primes</code> which have this property. Such a property will
be represented by a function that returns a boolean value. In this
example the property of being less than 30 can be represented by the
function <code>x -> x < 30</code> since <code>x < 30</code> will evaluate to <code>true</code> for
values <code>x</code> less than 30 and to <code>false</code> otherwise.
<pre>
gap> Filtered(primes, x-> x < 30);
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ]
</pre>
We have already mentioned the operator <code>{ }</code> that forms sublists. It
takes a list of positions as its argument and will return the list of
elements from the original list corresponding to these positions.
<pre>
gap> primes{ [1 .. 10] };
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ]
</pre>
<p>
Finally we mention the function <code>ForAll</code> that checks whether a property
holds for all elements of a list. It takes as its arguments a list and a
function that returns a boolean value. <code>ForAll</code> checks whether the
function returns <code>true</code> for all elements of the list.
<p>
<pre>
gap> list:= [ 1, 2, 3, 4 ];;
gap> ForAll( list, x -> x > 0 );
true
gap> ForAll( list, x -> x in primes );
false
</pre>
<p>
You will find more predefined <code>for</code> loops in chapter <a href="../ref/CHAP021.htm">Lists</a> of the
reference manual.
<p>
<p>
<h2><a name="SECT008">3.8 Vectors and Matrices</a></h2>
<p><p>
<a name = "I8"></a>
<a name = "I8"></a>
<a name = "I9"></a>
This section describes how <font face="Gill Sans,Helvetica,Arial">GAP</font> uses lists to represent row vectors and
matrices. A <strong>row vector</strong> is a dense list of elements from a common field.
A <strong>matrix</strong> is a dense list of row vectors over a common field and of
equal length.
<pre>
gap> v:= [3, 6, 2, 5/2];; IsRowVector(v);
true
</pre>
Row vectors may be added and multiplied by scalars from their field.
Multiplication of row vectors of equal length results in their scalar
product.
<pre>
gap> 2 * v; v * 1/3;
[ 6, 12, 4, 5 ]
[ 1, 2, 2/3, 5/6 ]
gap> v * v; # the scalar product of `v' with itself
221/4
</pre>
Note that the expression <code>v * 1/3</code> is actually evaluated by first
multiplying <code>v</code> by 1 (which yields again <code>v</code>) and by then dividing by 3.
This is also an allowed scalar operation. The expression <code>v/3</code> would
result in the same value.
<p>
Such arithmetical operations (if the results are again vectors)
result in <strong>mutable</strong> vectors except if the operation is binary
and both operands are immutable;
thus the vectors shown in the examples above are all mutable.
<p>
So if you want to produce a mutable list with 100 entries equal to 25,
you can simply say <code>25 + 0 * [ 1 .. 100 ]</code>.
Note that ranges are also vectors (over the rationals),
and that <code>[ 1 .. 100 ]</code> is mutable.
<p>
A matrix is a dense list of row vectors of equal length.
<pre>
gap> m:= [[1,-1, 1],
> [2, 0,-1],
> [1, 1, 1]];
[ [ 1, -1, 1 ], [ 2, 0, -1 ], [ 1, 1, 1 ] ]
gap> m[2][1];
2
</pre>
Syntactically a matrix is a list of lists. So the number 2 in the second
row and the first column of the matrix <code>m</code> is referred to as the first
element of the second element of the list <code>m</code> via <code>m[2][1]</code>.
<p>
A matrix may be multiplied by scalars, row vectors and other matrices.
(If the row vectors and matrices involved in such a multiplication do not
have suitable dimensions then the ``missing'' entries are treated as zeros,
so the results may look unexpectedly in such cases.)
<p>
<pre>
gap> [1, 0, 0] * m;
[ 1, -1, 1 ]
gap> [1, 0, 0, 2] * m;
[ 1, -1, 1 ]
gap> m * [1, 0, 0];
[ 1, 2, 1 ]
gap> m * [1, 0, 0, 2];
[ 1, 2, 1 ]
</pre>
Note that multiplication of a row vector with a matrix will result in a
linear combination of the rows of the matrix, while multiplication of a
matrix with a row vector results in a linear combination of the columns
of the matrix. In the latter case the row vector is considered as a
column vector.
<p>
A vector or matrix of integers can also be multiplied
with a finite field scalar and vice versa.
Such products result in a matrix over the finite field with the integers
mapped into the finite field in the obvious way.
Finite field matrices are nicer to read when they are <code>Display</code>ed rather
than <code>Print</code>ed.
(Here we write <code>Z(q)</code> to denote a primitive root of the finite field
with <code>q</code> elements.)
<pre>
gap> Display( m * One( GF(5) ) );
1 4 1
2 . 4
1 1 1
gap> Display( m^2 * Z(2) + m * Z(4) );
z = Z(4)
z^1 z^1 z^2
1 1 z^2
z^1 z^1 z^2
</pre>
Submatrices can easily be extracted using the expression
<code></code><var>mat</var><code>{</code><var>rows</var><code>}{</code><var>columns</var><code>}</code>. They can also be assigned to, provided
the big matrix is mutable (which it is not if it is the result of an
arithmetical operation, see above).
<pre>
gap> sm := m{ [ 1, 2 ] }{ [ 2, 3 ] };
[ [ -1, 1 ], [ 0, -1 ] ]
gap> sm{ [ 1, 2 ] }{ [2] } := [[-2],[0]];; sm;
[ [ -1, -2 ], [ 0, 0 ] ]
</pre>
The first curly brackets contain the selection of rows,
the second that of columns.
<p>
Matrices appear not only in linear algebra, but also as group elements,
provided they are invertible.
Here we have the opportunity to meet a group-theoretical function,
namely <code>Order</code>, which computes the order of a group element.
<pre>
gap> Order( m * One( GF(5) ) );
8
gap> Order( m );
infinity
</pre>
For matrices whose entries are more complex objects, for example rational
functions, <font face="Gill Sans,Helvetica,Arial">GAP</font>'s <code>Order</code> methods might not be able to prove that the
matrix has infinite order, and one gets the following warning.
<pre>
|#I Order: warning, order of <mat> might be infinite
</pre>
In such a case, if the order of the matrix really is infinite, you will
have to interrupt <font face="Gill Sans,Helvetica,Arial">GAP</font> by pressing <code></code><var>ctl</var><code>-C</code> (followed by <code></code><var>ctl</var><code>-D</code> or
<code>quit;</code> to leave the break loop).
<p>
To prove that the order of <code>m</code> is infinite, we also could look at the
minimal polynomial of <code>m</code> over the rationals.
<pre>
gap> f:= MinimalPolynomial( Rationals, m );; Factors( f );
[ x_1-2, x_1^2+3 ]
</pre>
<code>Factors</code> returns a list of irreducible factors of the polynomial <code>f</code>.
The first irreducible factor <i>X</i>−2 reveals that 2 is an eigenvalue of
<code>m</code>, hence its order cannot be finite.
<p>
<p>
<h2><a name="SECT009">3.9 Plain Records</a></h2>
<p><p>
A record provides another way to build new data structures. Like a list
a record contains subobjects.
In a record the elements, the so-called <strong>record components</strong>,
are not indexed by numbers but by names.
<p>
In this section you will see how to define and how to use records.
Records are changed by assignments to record components.
<p>
Initially a record is defined as a comma separated list of assignments to
its record components.
<p>
<pre>
gap> date:= rec(year:= 1997,
> month:= "Jul",
> day:= 14);
rec( year := 1997, month := "Jul", day := 14 )
</pre>
<p>
The value of a record component is accessible by the record name and the
record component name separated by one dot as the record component
selector.
<p>
<pre>
gap> date.year;
1997
</pre>
<p>
Assignments to new record components are possible in the same way. The
record is automatically resized to hold the new component.
<p>
<pre>
gap> date.time:= rec(hour:= 19, minute:= 23, second:= 12);
rec( hour := 19, minute := 23, second := 12 )
gap> date;
rec( year := 1997, month := "Jul", day := 14,
time := rec( hour := 19, minute := 23, second := 12 ) )
</pre>
<p>
We may use the <code>Display</code> function to illustrate the hierarchy of the record
components.
<p>
<pre>
gap> Display( date );
rec(
year := 1997,
month := "Jul",
day := 14,
time := rec(
hour := 19,
minute := 23,
second := 12 ) )
</pre>
<p>
Records are objects that may be changed. An assignment to a record
component changes the original object.
The remarks made in Sections <a href="CHAP003.htm#SECT002">Identical Lists</a> and <a href="CHAP003.htm#SECT003">Immutability</a>
about identity and mutability of lists are also true for records.
<p>
Sometimes it is interesting to know which components of a certain record
are bound. This information is available from the function <code>RecNames</code>,
which takes a record as its argument and returns a list of names of
all bound components of this record as a list of strings.
<p>
<pre>
gap> RecNames(date);
[ "year", "month", "day", "time" ]
</pre>
<p>
Now return to Sections <a href="CHAP003.htm#SECT002">Identical Lists</a> and <a href="CHAP003.htm#SECT003">Immutability</a> and find out
what these sections mean for records.
<p>
<p>
<h2><a name="SECT010">3.10 Further Information about Lists</a></h2>
<p><p>
(The following cross-references point to the <font face="Gill Sans,Helvetica,Arial">GAP</font> Reference Manual.)
<p>
You will find more about lists, sets, and ranges in Chapter <a href="../ref/CHAP021.htm">Lists</a>,
in particular more about identical lists in Section <a href="../ref/CHAP021.htm#SECT006">Identical Lists</a>.
A more detailed description of strings is contained in
Chapter <a href="../ref/CHAP026.htm">Strings and Characters</a>.
Fields are described in Chapter <a href="../ref/CHAP056.htm">Fields and Division Rings</a>, some
known fields in <font face="Gill Sans,Helvetica,Arial">GAP</font> are described in Chapters <a href="../ref/CHAP016.htm">Rational Numbers</a>, <a href="../ref/CHAP058.htm">Abelian Number Fields</a>, and <a href="../ref/CHAP057.htm">Finite Fields</a>. Row
vectors and matrices are described in more detail in Chapters <a href="../ref/CHAP023.htm">Row Vectors</a> and <a href="../ref/CHAP024.htm">Matrices</a>. Vector spaces are described in
Chapter <a href="../ref/CHAP059.htm">Vector Spaces</a>, further matrix related structures are
described in Chapters <a href="../ref/CHAP042.htm">Matrix Groups</a>, <a href="../ref/CHAP060.htm">Algebras</a>,
and <a href="../ref/CHAP061.htm">Lie Algebras</a>.
<p>
You will find more list operations in Chapter <a href="../ref/CHAP021.htm">Lists</a>.
<p>
Records and functions for records are described in detail
in Chapter <a href="../ref/CHAP027.htm">Records</a>.
<p>
<p>
[<a href="../index.htm">Top</a>] [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Next</a>] [<a href = "theindex.htm">Index</a>]
<P>
<font face="Gill Sans,Helvetica,Arial">GAP 4 manual<br>December 2008
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