xref: /dflybsd-src/contrib/gcc-4.7/gcc/doc/loop.texi (revision 04febcfb30580676d3e95f58a16c5137ee478b32)

@c Copyright (c) 2006, 2007, 2008 Free Software Foundation, Inc.
@c Free Software Foundation, Inc.
@c This is part of the GCC manual.
@c For copying conditions, see the file gcc.texi.

@c ---------------------------------------------------------------------
@c Loop Representation
@c ---------------------------------------------------------------------

@node Loop Analysis and Representation
@chapter Analysis and Representation of Loops

GCC provides extensive infrastructure for work with natural loops, i.e.,
strongly connected components of CFG with only one entry block.  This
chapter describes representation of loops in GCC, both on GIMPLE and in
RTL, as well as the interfaces to loop-related analyses (induction
variable analysis and number of iterations analysis).

@menu
* Loop representation::         Representation and analysis of loops.
* Loop querying::               Getting information about loops.
* Loop manipulation::           Loop manipulation functions.
* LCSSA::                       Loop-closed SSA form.
* Scalar evolutions::           Induction variables on GIMPLE.
* loop-iv::                     Induction variables on RTL.
* Number of iterations::        Number of iterations analysis.
* Dependency analysis::         Data dependency analysis.
* Lambda::                      Linear loop transformations framework.
* Omega::                       A solver for linear programming problems.
@end menu

@node Loop representation
@section Loop representation
@cindex Loop representation
@cindex Loop analysis

This chapter describes the representation of loops in GCC, and functions
that can be used to build, modify and analyze this representation.  Most
of the interfaces and data structures are declared in @file{cfgloop.h}.
At the moment, loop structures are analyzed and this information is
updated only by the optimization passes that deal with loops, but some
efforts are being made to make it available throughout most of the
optimization passes.

In general, a natural loop has one entry block (header) and possibly
several back edges (latches) leading to the header from the inside of
the loop.  Loops with several latches may appear if several loops share
a single header, or if there is a branching in the middle of the loop.
The representation of loops in GCC however allows only loops with a
single latch.  During loop analysis, headers of such loops are split and
forwarder blocks are created in order to disambiguate their structures.
Heuristic based on profile information and structure of the induction
variables in the loops is used to determine whether the latches
correspond to sub-loops or to control flow in a single loop.  This means
that the analysis sometimes changes the CFG, and if you run it in the
middle of an optimization pass, you must be able to deal with the new
blocks.  You may avoid CFG changes by passing
@code{LOOPS_MAY_HAVE_MULTIPLE_LATCHES} flag to the loop discovery,
note however that most other loop manipulation functions will not work
correctly for loops with multiple latch edges (the functions that only
query membership of blocks to loops and subloop relationships, or
enumerate and test loop exits, can be expected to work).

Body of the loop is the set of blocks that are dominated by its header,
and reachable from its latch against the direction of edges in CFG@.  The
loops are organized in a containment hierarchy (tree) such that all the
loops immediately contained inside loop L are the children of L in the
tree.  This tree is represented by the @code{struct loops} structure.
The root of this tree is a fake loop that contains all blocks in the
function.  Each of the loops is represented in a @code{struct loop}
structure.  Each loop is assigned an index (@code{num} field of the
@code{struct loop} structure), and the pointer to the loop is stored in
the corresponding field of the @code{larray} vector in the loops
structure.  The indices do not have to be continuous, there may be
empty (@code{NULL}) entries in the @code{larray} created by deleting
loops.  Also, there is no guarantee on the relative order of a loop
and its subloops in the numbering.  The index of a loop never changes.

The entries of the @code{larray} field should not be accessed directly.
The function @code{get_loop} returns the loop description for a loop with
the given index.  @code{number_of_loops} function returns number of
loops in the function.  To traverse all loops, use @code{FOR_EACH_LOOP}
macro.  The @code{flags} argument of the macro is used to determine
the direction of traversal and the set of loops visited.  Each loop is
guaranteed to be visited exactly once, regardless of the changes to the
loop tree, and the loops may be removed during the traversal.  The newly
created loops are never traversed, if they need to be visited, this
must be done separately after their creation.  The @code{FOR_EACH_LOOP}
macro allocates temporary variables.  If the @code{FOR_EACH_LOOP} loop
were ended using break or goto, they would not be released;
@code{FOR_EACH_LOOP_BREAK} macro must be used instead.

Each basic block contains the reference to the innermost loop it belongs
to (@code{loop_father}).  For this reason, it is only possible to have
one @code{struct loops} structure initialized at the same time for each
CFG@.  The global variable @code{current_loops} contains the
@code{struct loops} structure.  Many of the loop manipulation functions
assume that dominance information is up-to-date.

The loops are analyzed through @code{loop_optimizer_init} function.  The
argument of this function is a set of flags represented in an integer
bitmask.  These flags specify what other properties of the loop
structures should be calculated/enforced and preserved later:

@itemize
@item @code{LOOPS_MAY_HAVE_MULTIPLE_LATCHES}: If this flag is set, no
changes to CFG will be performed in the loop analysis, in particular,
loops with multiple latch edges will not be disambiguated.  If a loop
has multiple latches, its latch block is set to NULL@.  Most of
the loop manipulation functions will not work for loops in this shape.
No other flags that require CFG changes can be passed to
loop_optimizer_init.
@item @code{LOOPS_HAVE_PREHEADERS}: Forwarder blocks are created in such
a way that each loop has only one entry edge, and additionally, the
source block of this entry edge has only one successor.  This creates a
natural place where the code can be moved out of the loop, and ensures
that the entry edge of the loop leads from its immediate super-loop.
@item @code{LOOPS_HAVE_SIMPLE_LATCHES}: Forwarder blocks are created to
force the latch block of each loop to have only one successor.  This
ensures that the latch of the loop does not belong to any of its
sub-loops, and makes manipulation with the loops significantly easier.
Most of the loop manipulation functions assume that the loops are in
this shape.  Note that with this flag, the ``normal'' loop without any
control flow inside and with one exit consists of two basic blocks.
@item @code{LOOPS_HAVE_MARKED_IRREDUCIBLE_REGIONS}: Basic blocks and
edges in the strongly connected components that are not natural loops
(have more than one entry block) are marked with
@code{BB_IRREDUCIBLE_LOOP} and @code{EDGE_IRREDUCIBLE_LOOP} flags.  The
flag is not set for blocks and edges that belong to natural loops that
are in such an irreducible region (but it is set for the entry and exit
edges of such a loop, if they lead to/from this region).
@item @code{LOOPS_HAVE_RECORDED_EXITS}: The lists of exits are recorded
and updated for each loop.  This makes some functions (e.g.,
@code{get_loop_exit_edges}) more efficient.  Some functions (e.g.,
@code{single_exit}) can be used only if the lists of exits are
recorded.
@end itemize

These properties may also be computed/enforced later, using functions
@code{create_preheaders}, @code{force_single_succ_latches},
@code{mark_irreducible_loops} and @code{record_loop_exits}.

The memory occupied by the loops structures should be freed with
@code{loop_optimizer_finalize} function.

The CFG manipulation functions in general do not update loop structures.
Specialized versions that additionally do so are provided for the most
common tasks.  On GIMPLE, @code{cleanup_tree_cfg_loop} function can be
used to cleanup CFG while updating the loops structures if
@code{current_loops} is set.

@node Loop querying
@section Loop querying
@cindex Loop querying

The functions to query the information about loops are declared in
@file{cfgloop.h}.  Some of the information can be taken directly from
the structures.  @code{loop_father} field of each basic block contains
the innermost loop to that the block belongs.  The most useful fields of
loop structure (that are kept up-to-date at all times) are:

@itemize
@item @code{header}, @code{latch}: Header and latch basic blocks of the
loop.
@item @code{num_nodes}: Number of basic blocks in the loop (including
the basic blocks of the sub-loops).
@item @code{depth}: The depth of the loop in the loops tree, i.e., the
number of super-loops of the loop.
@item @code{outer}, @code{inner}, @code{next}: The super-loop, the first
sub-loop, and the sibling of the loop in the loops tree.
@end itemize

There are other fields in the loop structures, many of them used only by
some of the passes, or not updated during CFG changes; in general, they
should not be accessed directly.

The most important functions to query loop structures are:

@itemize
@item @code{flow_loops_dump}: Dumps the information about loops to a
file.
@item @code{verify_loop_structure}: Checks consistency of the loop
structures.
@item @code{loop_latch_edge}: Returns the latch edge of a loop.
@item @code{loop_preheader_edge}: If loops have preheaders, returns
the preheader edge of a loop.
@item @code{flow_loop_nested_p}: Tests whether loop is a sub-loop of
another loop.
@item @code{flow_bb_inside_loop_p}: Tests whether a basic block belongs
to a loop (including its sub-loops).
@item @code{find_common_loop}: Finds the common super-loop of two loops.
@item @code{superloop_at_depth}: Returns the super-loop of a loop with
the given depth.
@item @code{tree_num_loop_insns}, @code{num_loop_insns}: Estimates the
number of insns in the loop, on GIMPLE and on RTL.
@item @code{loop_exit_edge_p}: Tests whether edge is an exit from a
loop.
@item @code{mark_loop_exit_edges}: Marks all exit edges of all loops
with @code{EDGE_LOOP_EXIT} flag.
@item @code{get_loop_body}, @code{get_loop_body_in_dom_order},
@code{get_loop_body_in_bfs_order}: Enumerates the basic blocks in the
loop in depth-first search order in reversed CFG, ordered by dominance
relation, and breath-first search order, respectively.
@item @code{single_exit}: Returns the single exit edge of the loop, or
@code{NULL} if the loop has more than one exit.  You can only use this
function if LOOPS_HAVE_MARKED_SINGLE_EXITS property is used.
@item @code{get_loop_exit_edges}: Enumerates the exit edges of a loop.
@item @code{just_once_each_iteration_p}: Returns true if the basic block
is executed exactly once during each iteration of a loop (that is, it
does not belong to a sub-loop, and it dominates the latch of the loop).
@end itemize

@node Loop manipulation
@section Loop manipulation
@cindex Loop manipulation

The loops tree can be manipulated using the following functions:

@itemize
@item @code{flow_loop_tree_node_add}: Adds a node to the tree.
@item @code{flow_loop_tree_node_remove}: Removes a node from the tree.
@item @code{add_bb_to_loop}: Adds a basic block to a loop.
@item @code{remove_bb_from_loops}: Removes a basic block from loops.
@end itemize

Most low-level CFG functions update loops automatically.  The following
functions handle some more complicated cases of CFG manipulations:

@itemize
@item @code{remove_path}: Removes an edge and all blocks it dominates.
@item @code{split_loop_exit_edge}: Splits exit edge of the loop,
ensuring that PHI node arguments remain in the loop (this ensures that
loop-closed SSA form is preserved).  Only useful on GIMPLE.
@end itemize

Finally, there are some higher-level loop transformations implemented.
While some of them are written so that they should work on non-innermost
loops, they are mostly untested in that case, and at the moment, they
are only reliable for the innermost loops:

@itemize
@item @code{create_iv}: Creates a new induction variable.  Only works on
GIMPLE@.  @code{standard_iv_increment_position} can be used to find a
suitable place for the iv increment.
@item @code{duplicate_loop_to_header_edge},
@code{tree_duplicate_loop_to_header_edge}: These functions (on RTL and
on GIMPLE) duplicate the body of the loop prescribed number of times on
one of the edges entering loop header, thus performing either loop
unrolling or loop peeling.  @code{can_duplicate_loop_p}
(@code{can_unroll_loop_p} on GIMPLE) must be true for the duplicated
loop.
@item @code{loop_version}, @code{tree_ssa_loop_version}: These function
create a copy of a loop, and a branch before them that selects one of
them depending on the prescribed condition.  This is useful for
optimizations that need to verify some assumptions in runtime (one of
the copies of the loop is usually left unchanged, while the other one is
transformed in some way).
@item @code{tree_unroll_loop}: Unrolls the loop, including peeling the
extra iterations to make the number of iterations divisible by unroll
factor, updating the exit condition, and removing the exits that now
cannot be taken.  Works only on GIMPLE.
@end itemize

@node LCSSA
@section Loop-closed SSA form
@cindex LCSSA
@cindex Loop-closed SSA form

Throughout the loop optimizations on tree level, one extra condition is
enforced on the SSA form:  No SSA name is used outside of the loop in
that it is defined.  The SSA form satisfying this condition is called
``loop-closed SSA form'' -- LCSSA@.  To enforce LCSSA, PHI nodes must be
created at the exits of the loops for the SSA names that are used
outside of them.  Only the real operands (not virtual SSA names) are
held in LCSSA, in order to save memory.

There are various benefits of LCSSA:

@itemize
@item Many optimizations (value range analysis, final value
replacement) are interested in the values that are defined in the loop
and used outside of it, i.e., exactly those for that we create new PHI
nodes.
@item In induction variable analysis, it is not necessary to specify the
loop in that the analysis should be performed -- the scalar evolution
analysis always returns the results with respect to the loop in that the
SSA name is defined.
@item It makes updating of SSA form during loop transformations simpler.
Without LCSSA, operations like loop unrolling may force creation of PHI
nodes arbitrarily far from the loop, while in LCSSA, the SSA form can be
updated locally.  However, since we only keep real operands in LCSSA, we
cannot use this advantage (we could have local updating of real
operands, but it is not much more efficient than to use generic SSA form
updating for it as well; the amount of changes to SSA is the same).
@end itemize

However, it also means LCSSA must be updated.  This is usually
straightforward, unless you create a new value in loop and use it
outside, or unless you manipulate loop exit edges (functions are
provided to make these manipulations simple).
@code{rewrite_into_loop_closed_ssa} is used to rewrite SSA form to
LCSSA, and @code{verify_loop_closed_ssa} to check that the invariant of
LCSSA is preserved.

@node Scalar evolutions
@section Scalar evolutions
@cindex Scalar evolutions
@cindex IV analysis on GIMPLE

Scalar evolutions (SCEV) are used to represent results of induction
variable analysis on GIMPLE@.  They enable us to represent variables with
complicated behavior in a simple and consistent way (we only use it to
express values of polynomial induction variables, but it is possible to
extend it).  The interfaces to SCEV analysis are declared in
@file{tree-scalar-evolution.h}.  To use scalar evolutions analysis,
@code{scev_initialize} must be used.  To stop using SCEV,
@code{scev_finalize} should be used.  SCEV analysis caches results in
order to save time and memory.  This cache however is made invalid by
most of the loop transformations, including removal of code.  If such a
transformation is performed, @code{scev_reset} must be called to clean
the caches.

Given an SSA name, its behavior in loops can be analyzed using the
@code{analyze_scalar_evolution} function.  The returned SCEV however
does not have to be fully analyzed and it may contain references to
other SSA names defined in the loop.  To resolve these (potentially
recursive) references, @code{instantiate_parameters} or
@code{resolve_mixers} functions must be used.
@code{instantiate_parameters} is useful when you use the results of SCEV
only for some analysis, and when you work with whole nest of loops at
once.  It will try replacing all SSA names by their SCEV in all loops,
including the super-loops of the current loop, thus providing a complete
information about the behavior of the variable in the loop nest.
@code{resolve_mixers} is useful if you work with only one loop at a
time, and if you possibly need to create code based on the value of the
induction variable.  It will only resolve the SSA names defined in the
current loop, leaving the SSA names defined outside unchanged, even if
their evolution in the outer loops is known.

The SCEV is a normal tree expression, except for the fact that it may
contain several special tree nodes.  One of them is
@code{SCEV_NOT_KNOWN}, used for SSA names whose value cannot be
expressed.  The other one is @code{POLYNOMIAL_CHREC}.  Polynomial chrec
has three arguments -- base, step and loop (both base and step may
contain further polynomial chrecs).  Type of the expression and of base
and step must be the same.  A variable has evolution
@code{POLYNOMIAL_CHREC(base, step, loop)} if it is (in the specified
loop) equivalent to @code{x_1} in the following example

@smallexample
while (@dots{})
  @{
    x_1 = phi (base, x_2);
    x_2 = x_1 + step;
  @}
@end smallexample

Note that this includes the language restrictions on the operations.
For example, if we compile C code and @code{x} has signed type, then the
overflow in addition would cause undefined behavior, and we may assume
that this does not happen.  Hence, the value with this SCEV cannot
overflow (which restricts the number of iterations of such a loop).

In many cases, one wants to restrict the attention just to affine
induction variables.  In this case, the extra expressive power of SCEV
is not useful, and may complicate the optimizations.  In this case,
@code{simple_iv} function may be used to analyze a value -- the result
is a loop-invariant base and step.

@node loop-iv
@section IV analysis on RTL
@cindex IV analysis on RTL

The induction variable on RTL is simple and only allows analysis of
affine induction variables, and only in one loop at once.  The interface
is declared in @file{cfgloop.h}.  Before analyzing induction variables
in a loop L, @code{iv_analysis_loop_init} function must be called on L.
After the analysis (possibly calling @code{iv_analysis_loop_init} for
several loops) is finished, @code{iv_analysis_done} should be called.
The following functions can be used to access the results of the
analysis:

@itemize
@item @code{iv_analyze}: Analyzes a single register used in the given
insn.  If no use of the register in this insn is found, the following
insns are scanned, so that this function can be called on the insn
returned by get_condition.
@item @code{iv_analyze_result}: Analyzes result of the assignment in the
given insn.
@item @code{iv_analyze_expr}: Analyzes a more complicated expression.
All its operands are analyzed by @code{iv_analyze}, and hence they must
be used in the specified insn or one of the following insns.
@end itemize

The description of the induction variable is provided in @code{struct
rtx_iv}.  In order to handle subregs, the representation is a bit
complicated; if the value of the @code{extend} field is not
@code{UNKNOWN}, the value of the induction variable in the i-th
iteration is

@smallexample
delta + mult * extend_@{extend_mode@} (subreg_@{mode@} (base + i * step)),
@end smallexample

with the following exception:  if @code{first_special} is true, then the
value in the first iteration (when @code{i} is zero) is @code{delta +
mult * base}.  However, if @code{extend} is equal to @code{UNKNOWN},
then @code{first_special} must be false, @code{delta} 0, @code{mult} 1
and the value in the i-th iteration is

@smallexample
subreg_@{mode@} (base + i * step)
@end smallexample

The function @code{get_iv_value} can be used to perform these
calculations.

@node Number of iterations
@section Number of iterations analysis
@cindex Number of iterations analysis

Both on GIMPLE and on RTL, there are functions available to determine
the number of iterations of a loop, with a similar interface.  The
number of iterations of a loop in GCC is defined as the number of
executions of the loop latch.  In many cases, it is not possible to
determine the number of iterations unconditionally -- the determined
number is correct only if some assumptions are satisfied.  The analysis
tries to verify these conditions using the information contained in the
program; if it fails, the conditions are returned together with the
result.  The following information and conditions are provided by the
analysis:

@itemize
@item @code{assumptions}: If this condition is false, the rest of
the information is invalid.
@item @code{noloop_assumptions} on RTL, @code{may_be_zero} on GIMPLE: If
this condition is true, the loop exits in the first iteration.
@item @code{infinite}: If this condition is true, the loop is infinite.
This condition is only available on RTL@.  On GIMPLE, conditions for
finiteness of the loop are included in @code{assumptions}.
@item @code{niter_expr} on RTL, @code{niter} on GIMPLE: The expression
that gives number of iterations.  The number of iterations is defined as
the number of executions of the loop latch.
@end itemize

Both on GIMPLE and on RTL, it necessary for the induction variable
analysis framework to be initialized (SCEV on GIMPLE, loop-iv on RTL).
On GIMPLE, the results are stored to @code{struct tree_niter_desc}
structure.  Number of iterations before the loop is exited through a
given exit can be determined using @code{number_of_iterations_exit}
function.  On RTL, the results are returned in @code{struct niter_desc}
structure.  The corresponding function is named
@code{check_simple_exit}.  There are also functions that pass through
all the exits of a loop and try to find one with easy to determine
number of iterations -- @code{find_loop_niter} on GIMPLE and
@code{find_simple_exit} on RTL@.  Finally, there are functions that
provide the same information, but additionally cache it, so that
repeated calls to number of iterations are not so costly --
@code{number_of_latch_executions} on GIMPLE and @code{get_simple_loop_desc}
on RTL.

Note that some of these functions may behave slightly differently than
others -- some of them return only the expression for the number of
iterations, and fail if there are some assumptions.  The function
@code{number_of_latch_executions} works only for single-exit loops.
The function @code{number_of_cond_exit_executions} can be used to
determine number of executions of the exit condition of a single-exit
loop (i.e., the @code{number_of_latch_executions} increased by one).

@node Dependency analysis
@section Data Dependency Analysis
@cindex Data Dependency Analysis

The code for the data dependence analysis can be found in
@file{tree-data-ref.c} and its interface and data structures are
described in @file{tree-data-ref.h}.  The function that computes the
data dependences for all the array and pointer references for a given
loop is @code{compute_data_dependences_for_loop}.  This function is
currently used by the linear loop transform and the vectorization
passes.  Before calling this function, one has to allocate two vectors:
a first vector will contain the set of data references that are
contained in the analyzed loop body, and the second vector will contain
the dependence relations between the data references.  Thus if the
vector of data references is of size @code{n}, the vector containing the
dependence relations will contain @code{n*n} elements.  However if the
analyzed loop contains side effects, such as calls that potentially can
interfere with the data references in the current analyzed loop, the
analysis stops while scanning the loop body for data references, and
inserts a single @code{chrec_dont_know} in the dependence relation
array.

The data references are discovered in a particular order during the
scanning of the loop body: the loop body is analyzed in execution order,
and the data references of each statement are pushed at the end of the
data reference array.  Two data references syntactically occur in the
program in the same order as in the array of data references.  This
syntactic order is important in some classical data dependence tests,
and mapping this order to the elements of this array avoids costly
queries to the loop body representation.

Three types of data references are currently handled: ARRAY_REF,
INDIRECT_REF and COMPONENT_REF@. The data structure for the data reference
is @code{data_reference}, where @code{data_reference_p} is a name of a
pointer to the data reference structure. The structure contains the
following elements:

@itemize
@item @code{base_object_info}: Provides information about the base object
of the data reference and its access functions. These access functions
represent the evolution of the data reference in the loop relative to
its base, in keeping with the classical meaning of the data reference
access function for the support of arrays. For example, for a reference
@code{a.b[i][j]}, the base object is @code{a.b} and the access functions,
one for each array subscript, are:
@code{@{i_init, + i_step@}_1, @{j_init, +, j_step@}_2}.

@item @code{first_location_in_loop}: Provides information about the first
location accessed by the data reference in the loop and about the access
function used to represent evolution relative to this location. This data
is used to support pointers, and is not used for arrays (for which we
have base objects). Pointer accesses are represented as a one-dimensional
access that starts from the first location accessed in the loop. For
example:

@smallexample
      for1 i
         for2 j
          *((int *)p + i + j) = a[i][j];
@end smallexample

The access function of the pointer access is @code{@{0, + 4B@}_for2}
relative to @code{p + i}. The access functions of the array are
@code{@{i_init, + i_step@}_for1} and @code{@{j_init, +, j_step@}_for2}
relative to @code{a}.

Usually, the object the pointer refers to is either unknown, or we can't
prove that the access is confined to the boundaries of a certain object.

Two data references can be compared only if at least one of these two
representations has all its fields filled for both data references.

The current strategy for data dependence tests is as follows:
If both @code{a} and @code{b} are represented as arrays, compare
@code{a.base_object} and @code{b.base_object};
if they are equal, apply dependence tests (use access functions based on
base_objects).
Else if both @code{a} and @code{b} are represented as pointers, compare
@code{a.first_location} and @code{b.first_location};
if they are equal, apply dependence tests (use access functions based on
first location).
However, if @code{a} and @code{b} are represented differently, only try
to prove that the bases are definitely different.

@item Aliasing information.
@item Alignment information.
@end itemize

The structure describing the relation between two data references is
@code{data_dependence_relation} and the shorter name for a pointer to
such a structure is @code{ddr_p}.  This structure contains:

@itemize
@item a pointer to each data reference,
@item a tree node @code{are_dependent} that is set to @code{chrec_known}
if the analysis has proved that there is no dependence between these two
data references, @code{chrec_dont_know} if the analysis was not able to
determine any useful result and potentially there could exist a
dependence between these data references, and @code{are_dependent} is
set to @code{NULL_TREE} if there exist a dependence relation between the
data references, and the description of this dependence relation is
given in the @code{subscripts}, @code{dir_vects}, and @code{dist_vects}
arrays,
@item a boolean that determines whether the dependence relation can be
represented by a classical distance vector,
@item an array @code{subscripts} that contains a description of each
subscript of the data references.  Given two array accesses a
subscript is the tuple composed of the access functions for a given
dimension.  For example, given @code{A[f1][f2][f3]} and
@code{B[g1][g2][g3]}, there are three subscripts: @code{(f1, g1), (f2,
g2), (f3, g3)}.
@item two arrays @code{dir_vects} and @code{dist_vects} that contain
classical representations of the data dependences under the form of
direction and distance dependence vectors,
@item an array of loops @code{loop_nest} that contains the loops to
which the distance and direction vectors refer to.
@end itemize

Several functions for pretty printing the information extracted by the
data dependence analysis are available: @code{dump_ddrs} prints with a
maximum verbosity the details of a data dependence relations array,
@code{dump_dist_dir_vectors} prints only the classical distance and
direction vectors for a data dependence relations array, and
@code{dump_data_references} prints the details of the data references
contained in a data reference array.

@node Lambda
@section Linear loop transformations framework
@cindex Linear loop transformations framework

Lambda is a framework that allows transformations of loops using
non-singular matrix based transformations of the iteration space and
loop bounds. This allows compositions of skewing, scaling, interchange,
and reversal transformations.  These transformations are often used to
improve cache behavior or remove inner loop dependencies to allow
parallelization and vectorization to take place.

To perform these transformations, Lambda requires that the loopnest be
converted into an internal form that can be matrix transformed easily.
To do this conversion, the function
@code{gcc_loopnest_to_lambda_loopnest} is provided.  If the loop cannot
be transformed using lambda, this function will return NULL.

Once a @code{lambda_loopnest} is obtained from the conversion function,
it can be transformed by using @code{lambda_loopnest_transform}, which
takes a transformation matrix to apply.  Note that it is up to the
caller to verify that the transformation matrix is legal to apply to the
loop (dependence respecting, etc).  Lambda simply applies whatever
matrix it is told to provide.  It can be extended to make legal matrices
out of any non-singular matrix, but this is not currently implemented.
Legality of a matrix for a given loopnest can be verified using
@code{lambda_transform_legal_p}.

Given a transformed loopnest, conversion back into gcc IR is done by
@code{lambda_loopnest_to_gcc_loopnest}.  This function will modify the
loops so that they match the transformed loopnest.


@node Omega
@section Omega a solver for linear programming problems
@cindex Omega a solver for linear programming problems

The data dependence analysis contains several solvers triggered
sequentially from the less complex ones to the more sophisticated.
For ensuring the consistency of the results of these solvers, a data
dependence check pass has been implemented based on two different
solvers.  The second method that has been integrated to GCC is based
on the Omega dependence solver, written in the 1990's by William Pugh
and David Wonnacott.  Data dependence tests can be formulated using a
subset of the Presburger arithmetics that can be translated to linear
constraint systems.  These linear constraint systems can then be
solved using the Omega solver.

The Omega solver is using Fourier-Motzkin's algorithm for variable
elimination: a linear constraint system containing @code{n} variables
is reduced to a linear constraint system with @code{n-1} variables.
The Omega solver can also be used for solving other problems that can
be expressed under the form of a system of linear equalities and
inequalities.  The Omega solver is known to have an exponential worst
case, also known under the name of ``omega nightmare'' in the
literature, but in practice, the omega test is known to be efficient
for the common data dependence tests.

The interface used by the Omega solver for describing the linear
programming problems is described in @file{omega.h}, and the solver is
@code{omega_solve_problem}.