The SQL index allows to find quickly a string which matches my query. Now, I have to search in a big table the strings which do not match. Of course, the normal index does not help and I have to do a slow sequential scan:

essais=> \d phone_idx
Index "public.phone_idx"
 Column | Type 
--------+------
 phone  | text
btree, for table "public.phonespersons"

essais=> EXPLAIN SELECT person FROM PhonesPersons WHERE phone = '+33 1234567';
                                  QUERY PLAN                                   
-------------------------------------------------------------------------------
 Index Scan using phone_idx on phonespersons  (cost=0.00..8.41 rows=1 width=4)
   Index Cond: (phone = '+33 1234567'::text)
(2 rows)

essais=> EXPLAIN SELECT person FROM PhonesPersons WHERE phone != '+33 1234567';
                              QUERY PLAN                              
----------------------------------------------------------------------
 Seq Scan on phonespersons  (cost=0.00..18621.00 rows=999999 width=4)
   Filter: (phone <> '+33 1234567'::text)
(2 rows)

I understand (see Mark Byers' very good explanations) that PostgreSQL can decide not to use an index when it sees that a sequential scan would be faster (for instance if almost all the tuples match). But, here, "not equal" searches are really slower.

Any way to make these "is not equal to" searches faster?

Here is another example, to address Mark Byers' excellent remarks. The index is used for the '=' query (which returns the vast majority of tuples) but not for the '!=' query:

essais=> EXPLAIN ANALYZE SELECT person FROM EmailsPersons WHERE tld(email) = 'fr';
                             QUERY PLAN                                                             
------------------------------------------------------------------------------------------------------------------------------------
 Index Scan using tld_idx on emailspersons  (cost=0.25..4010.79 rows=97033 width=4) (actual time=0.137..261.123 rows=97110 loops=1)
   Index Cond: (tld(email) = 'fr'::text)
 Total runtime: 444.800 ms
(3 rows)

essais=> EXPLAIN ANALYZE SELECT person FROM EmailsPersons WHERE tld(email) != 'fr';
                         QUERY PLAN                                                     
--------------------------------------------------------------------------------------------------------------------
 Seq Scan on emailspersons  (cost=0.00..27129.00 rows=2967 width=4) (actual time=1.004..1031.224 rows=2890 loops=1)
   Filter: (tld(email) <> 'fr'::text)
 Total runtime: 1037.278 ms
(3 rows)

DBMS is PostgreSQL 8.3 (but I can upgrade to 8.4).