Updating table with max from another table

Well, I *thought* I knew my way around SQL a little bit, but I have been
beating my head on the following problem for a couple days now and I don't
have any idea where to turn next. If you could suggest any strategies or
places to look I would appreciate it very much. Thanks in advance.

Given two table defs :

create table a (
  id integer,
  maxtype varchar(8)
);

create table b (
  id integer,
  type varchar(8),
  val integer
);

and data rows as follows:

select * from a;
 id | maxtype
----+---------
  1 |
  2 |
(2 rows)

select * from b;
 id | type  | val
----+-------+-----
  1 | type1 |   5
  1 | type2 |   6
  2 | type1 |  19
  2 | type2 |   4
(4 rows)

And given the following task :

update a from b such that a.maxtype is set equal to the b.type whose val
number is the highest for that matching id, that is, the result :

select * from a;
 id | maxtype
----+---------
  1 | type2
  2 | type1
(2 rows)

is to be obtained, how can this be accomplished with SQL statements? I am
looking for a single (perhaps compound ) statement to do it, no procedural
stuff