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author	Kai Tietz <kai.tietz@onevision.com>	2011-01-01 11:05:41 +0000
committer	Kai Tietz <ktietz@gcc.gnu.org>	2011-01-01 12:05:41 +0100
commit	220e83ca20ee9254033fd5a6adec8a18d5ef6eb9 (patch)
tree	965dc756aaa48aec86a2a583feae2d461252558b /gcc/tree-ssa-structalias.c
parent	49e38883586f5bcca7dc763550b541c4c83f135e (diff)
download	gcc-220e83ca20ee9254033fd5a6adec8a18d5ef6eb9.zip gcc-220e83ca20ee9254033fd5a6adec8a18d5ef6eb9.tar.gz gcc-220e83ca20ee9254033fd5a6adec8a18d5ef6eb9.tar.bz2

ChangeLog gcc/

2011-01-01 Kai Tietz <kai.tietz@onevision.com> PR target/38662 * tree.c (type_hash_eq): Call language hook for METHOD_TYPEs, too. ChangeLog gcc/cp 2011-01-01 Kai Tietz <kai.tietz@onevision.com> PR target/38662 * tree.c (cxx_type_hash_eq): Allow METHOD_TYPE, too. ChangeLog gcc/testsuite 2011-01-01 Kai Tietz <kai.tietz@onevision.com> PR target/38662 * g++.dg/eh/pr38662.C: New testcase. From-SVN: r168389

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/*  Loop transformation code generation
    Copyright (C) 2003, 2004 Free Software Foundation, Inc.
    Contributed by Daniel Berlin <dberlin@dberlin.org>

    This file is part of GCC.
    
    GCC is free software; you can redistribute it and/or modify it under
    the terms of the GNU General Public License as published by the Free
    Software Foundation; either version 2, or (at your option) any later
    version.
    
    GCC is distributed in the hope that it will be useful, but WITHOUT ANY
    WARRANTY; without even the implied warranty of MERCHANTABILITY or
    FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public License
    for more details.
    
    You should have received a copy of the GNU General Public License
    along with GCC; see the file COPYING.  If not, write to the Free
    Software Foundation, 59 Temple Place - Suite 330, Boston, MA
    02111-1307, USA.  */

#include "config.h"
#include "system.h"
#include "coretypes.h"
#include "tm.h"
#include "errors.h"
#include "ggc.h"
#include "tree.h"
#include "target.h"
#include "rtl.h"
#include "basic-block.h"
#include "diagnostic.h"
#include "tree-flow.h"
#include "tree-dump.h"
#include "timevar.h"
#include "cfgloop.h"
#include "expr.h"
#include "optabs.h"
#include "tree-chrec.h"
#include "tree-data-ref.h"
#include "tree-pass.h"
#include "tree-scalar-evolution.h"
#include "vec.h"
#include "lambda.h"

/* This loop nest code generation is based on non-singular matrix
   math.
 
 A little terminology and a general sketch of the algorithm.  See "A singular
 loop transformation framework based on non-singular matrices" by Wei Li and
 Keshav Pingali for formal proofs that the various statements below are
 correct. 

 A loop iteration space are the points traversed by the loop.  A point in the
 iteration space can be represented by a vector of size <loop depth>.  You can
 therefore represent the iteration space as a integral combinations of a set
 of basis vectors. 

 A loop iteration space is dense if every integer point between the loop
 bounds is a point in the iteration space.  Every loop with a step of 1
 therefore has a dense iteration space.

 for i = 1 to 3, step 1 is a dense iteration space.
   
 A loop iteration space is sparse if it is not dense.  That is, the iteration
 space skips integer points that are within the loop bounds.  

 for i = 1 to 3, step 2 is a sparse iteration space, because the integer point
 2 is skipped.

 Dense source spaces are easy to transform, because they don't skip any
 points to begin with.  Thus we can compute the exact bounds of the target
 space using min/max and floor/ceil.

 For a dense source space, we take the transformation matrix, decompose it
 into a lower triangular part (H) and a unimodular part (U). 
 We then compute the auxiliary space from the unimodular part (source loop
 nest . U = auxiliary space) , which has two important properties:
  1. It traverses the iterations in the same lexicographic order as the source
  space.
  2. It is a dense space when the source is a dense space (even if the target
  space is going to be sparse).
 
 Given the auxiliary space, we use the lower triangular part to compute the
 bounds in the target space by simple matrix multiplication.
 The gaps in the target space (IE the new loop step sizes) will be the
 diagonals of the H matrix.

 Sparse source spaces require another step, because you can't directly compute
 the exact bounds of the auxiliary and target space from the sparse space.
 Rather than try to come up with a separate algorithm to handle sparse source
 spaces directly, we just find a legal transformation matrix that gives you
 the sparse source space, from a dense space, and then transform the dense
 space.

 For a regular sparse space, you can represent the source space as an integer
 lattice, and the base space of that lattice will always be dense.  Thus, we
 effectively use the lattice to figure out the transformation from the lattice
 base space, to the sparse iteration space (IE what transform was applied to
 the dense space to make it sparse).  We then compose this transform with the
 transformation matrix specified by the user (since our matrix transformations
 are closed under composition, this is okay).  We can then use the base space
 (which is dense) plus the composed transformation matrix, to compute the rest
 of the transform using the dense space algorithm above.
 
 In other words, our sparse source space (B) is decomposed into a dense base
 space (A), and a matrix (L) that transforms A into B, such that A.L = B.
 We then compute the composition of L and the user transformation matrix (T),
 so that T is now a transform from A to the result, instead of from B to the
 result. 
 IE A.(LT) = result instead of B.T = result
 Since A is now a dense source space, we can use the dense source space
 algorithm above to compute the result of applying transform (LT) to A.

 Fourier-Motzkin elimination is used to compute the bounds of the base space
 of the lattice.  */

/* Lattice stuff that is internal to the code generation algorithm.  */

typedef struct
{
  /* Lattice base matrix.  */
  lambda_matrix base;
  /* Lattice dimension.  */
  int dimension;
  /* Origin vector for the coefficients.  */
  lambda_vector origin;
  /* Origin matrix for the invariants.  */
  lambda_matrix origin_invariants;
  /* Number of invariants.  */
  int invariants;
} *lambda_lattice;

#define LATTICE_BASE(T) ((T)->base)
#define LATTICE_DIMENSION(T) ((T)->dimension)
#define LATTICE_ORIGIN(T) ((T)->origin)
#define LATTICE_ORIGIN_INVARIANTS(T) ((T)->origin_invariants)
#define LATTICE_INVARIANTS(T) ((T)->invariants)

static bool lle_equal (lambda_linear_expression, lambda_linear_expression,
		       int, int);
static lambda_lattice lambda_lattice_new (int, int);
static lambda_lattice lambda_lattice_compute_base (lambda_loopnest);

static tree find_induction_var_from_exit_cond (struct loop *);

/* Create a new lambda body vector.  */

lambda_body_vector
lambda_body_vector_new (int size)
{
  lambda_body_vector ret;

  ret = ggc_alloc (sizeof (*ret));
  LBV_COEFFICIENTS (ret) = lambda_vector_new (size);
  LBV_SIZE (ret) = size;
  LBV_DENOMINATOR (ret) = 1;
  return ret;
}

/* Compute the new coefficients for the vector based on the
  *inverse* of the transformation matrix.  */

lambda_body_vector
lambda_body_vector_compute_new (lambda_trans_matrix transform,
				lambda_body_vector vect)
{
  lambda_body_vector temp;
  int depth;

  /* Make sure the matrix is square.  */
  gcc_assert (LTM_ROWSIZE (transform) == LTM_COLSIZE (transform));

  depth = LTM_ROWSIZE (transform);

  temp = lambda_body_vector_new (depth);
  LBV_DENOMINATOR (temp) =
    LBV_DENOMINATOR (vect) * LTM_DENOMINATOR (transform);
  lambda_vector_matrix_mult (LBV_COEFFICIENTS (vect), depth,
			     LTM_MATRIX (transform), depth,
			     LBV_COEFFICIENTS (temp));
  LBV_SIZE (temp) = LBV_SIZE (vect);
  return temp;
}

/* Print out a lambda body vector.  */

void
print_lambda_body_vector (FILE * outfile, lambda_body_vector body)
{
  print_lambda_vector (outfile, LBV_COEFFICIENTS (body), LBV_SIZE (body));
}

/* Return TRUE if two linear expressions are equal.  */

static bool
lle_equal (lambda_linear_expression lle1, lambda_linear_expression lle2,
	   int depth, int invariants)
{
  int i;

  if (lle1 == NULL || lle2 == NULL)
    return false;
  if (LLE_CONSTANT (lle1) != LLE_CONSTANT (lle2))
    return false;
  if (LLE_DENOMINATOR (lle1) != LLE_DENOMINATOR (lle2))
    return false;
  for (i = 0; i < depth; i++)
    if (LLE_COEFFICIENTS (lle1)[i] != LLE_COEFFICIENTS (lle2)[i])
      return false;
  for (i = 0; i < invariants; i++)
    if (LLE_INVARIANT_COEFFICIENTS (lle1)[i] !=
	LLE_INVARIANT_COEFFICIENTS (lle2)[i])
      return false;
  return true;
}

/* Create a new linear expression with dimension DIM, and total number
   of invariants INVARIANTS.  */

lambda_linear_expression
lambda_linear_expression_new (int dim, int invariants)
{
  lambda_linear_expression ret;

  ret = ggc_alloc_cleared (sizeof (*ret));

  LLE_COEFFICIENTS (ret) = lambda_vector_new (dim);
  LLE_CONSTANT (ret) = 0;
  LLE_INVARIANT_COEFFICIENTS (ret) = lambda_vector_new (invariants);
  LLE_DENOMINATOR (ret) = 1;
  LLE_NEXT (ret) = NULL;

  return ret;
}

/* Print out a linear expression EXPR, with SIZE coefficients, to OUTFILE.
   The starting letter used for variable names is START.  */

static void
print_linear_expression (FILE * outfile, lambda_vector expr, int size,
			 char start)
{
  int i;
  bool first = true;
  for (i = 0; i < size; i++)
    {
      if (expr[i] != 0)
	{
	  if (first)
	    {
	      if (expr[i] < 0)
		fprintf (outfile, "-");
	      first = false;
	    }
	  else if (expr[i] > 0)
	    fprintf (outfile, " + ");
	  else
	    fprintf (outfile, " - ");
	  if (abs (expr[i]) == 1)
	    fprintf (outfile, "%c", start + i);
	  else
	    fprintf (outfile, "%d%c", abs (expr[i]), start + i);
	}
    }
}

/* Print out a lambda linear expression structure, EXPR, to OUTFILE. The
   depth/number of coefficients is given by DEPTH, the number of invariants is
   given by INVARIANTS, and the character to start variable names with is given
   by START.  */

void
print_lambda_linear_expression (FILE * outfile,
				lambda_linear_expression expr,
				int depth, int invariants, char start)
{
  fprintf (outfile, "\tLinear expression: ");
  print_linear_expression (outfile, LLE_COEFFICIENTS (expr), depth, start);
  fprintf (outfile, " constant: %d ", LLE_CONSTANT (expr));
  fprintf (outfile, "  invariants: ");
  print_linear_expression (outfile, LLE_INVARIANT_COEFFICIENTS (expr),
			   invariants, 'A');
  fprintf (outfile, "  denominator: %d\n", LLE_DENOMINATOR (expr));
}

/* Print a lambda loop structure LOOP to OUTFILE.  The depth/number of
   coefficients is given by DEPTH, the number of invariants is 
   given by INVARIANTS, and the character to start variable names with is given
   by START.  */

void
print_lambda_loop (FILE * outfile, lambda_loop loop, int depth,
		   int invariants, char start)
{
  int step;
  lambda_linear_expression expr;

  gcc_assert (loop);

  expr = LL_LINEAR_OFFSET (loop);
  step = LL_STEP (loop);
  fprintf (outfile, "  step size = %d \n", step);

  if (expr)
    {
      fprintf (outfile, "  linear offset: \n");
      print_lambda_linear_expression (outfile, expr, depth, invariants,
				      start);
    }

  fprintf (outfile, "  lower bound: \n");
  for (expr = LL_LOWER_BOUND (loop); expr != NULL; expr = LLE_NEXT (expr))
    print_lambda_linear_expression (outfile, expr, depth, invariants, start);
  fprintf (outfile, "  upper bound: \n");
  for (expr = LL_UPPER_BOUND (loop); expr != NULL; expr = LLE_NEXT (expr))
    print_lambda_linear_expression (outfile, expr, depth, invariants, start);
}

/* Create a new loop nest structure with DEPTH loops, and INVARIANTS as the
   number of invariants.  */

lambda_loopnest
lambda_loopnest_new (int depth, int invariants)
{
  lambda_loopnest ret;
  ret = ggc_alloc (sizeof (*ret));

  LN_LOOPS (ret) = ggc_alloc_cleared (depth * sizeof (lambda_loop));
  LN_DEPTH (ret) = depth;
  LN_INVARIANTS (ret) = invariants;

  return ret;
}

/* Print a lambda loopnest structure, NEST, to OUTFILE.  The starting
   character to use for loop names is given by START.  */

void
print_lambda_loopnest (FILE * outfile, lambda_loopnest nest, char start)
{
  int i;
  for (i = 0; i < LN_DEPTH (nest); i++)
    {
      fprintf (outfile, "Loop %c\n", start + i);
      print_lambda_loop (outfile, LN_LOOPS (nest)[i], LN_DEPTH (nest),
			 LN_INVARIANTS (nest), 'i');
      fprintf (outfile, "\n");
    }
}

/* Allocate a new lattice structure of DEPTH x DEPTH, with INVARIANTS number
   of invariants.    */

static lambda_lattice
lambda_lattice_new (int depth, int invariants)
{
  lambda_lattice ret;
  ret = ggc_alloc (sizeof (*ret));
  LATTICE_BASE (ret) = lambda_matrix_new (depth, depth);
  LATTICE_ORIGIN (ret) = lambda_vector_new (depth);
  LATTICE_ORIGIN_INVARIANTS (ret) = lambda_matrix_new (depth, invariants);
  LATTICE_DIMENSION (ret) = depth;
  LATTICE_INVARIANTS (ret) = invariants;
  return ret;
}

/* Compute the lattice base for NEST.  The lattice base is essentially a
   non-singular transform from a dense base space to a sparse iteration space.
   We use it so that we don't have to specially handle the case of a sparse
   iteration space in other parts of the algorithm.  As a result, this routine
   only does something interesting (IE produce a matrix that isn't the
   identity matrix) if NEST is a sparse space.  */

static lambda_lattice
lambda_lattice_compute_base (lambda_loopnest nest)
{
  lambda_lattice ret;
  int depth, invariants;
  lambda_matrix base;

  int i, j, step;
  lambda_loop loop;
  lambda_linear_expression expression;

  depth = LN_DEPTH (nest);
  invariants = LN_INVARIANTS (nest);

  ret = lambda_lattice_new (depth, invariants);
  base = LATTICE_BASE (ret);
  for (i = 0; i < depth; i++)
    {
      loop = LN_LOOPS (nest)[i];
      gcc_assert (loop);
      step = LL_STEP (loop);
      /* If we have a step of 1, then the base is one, and the
         origin and invariant coefficients are 0.  */
      if (step == 1)
	{
	  for (j = 0; j < depth; j++)
	    base[i][j] = 0;
	  base[i][i] = 1;
	  LATTICE_ORIGIN (ret)[i] = 0;
	  for (j = 0; j < invariants; j++)
	    LATTICE_ORIGIN_INVARIANTS (ret)[i][j] = 0;
	}
      else
	{
	  /* Otherwise, we need the lower bound expression (which must
	     be an affine function)  to determine the base.  */
	  expression = LL_LOWER_BOUND (loop);
	  gcc_assert (expression && LLE_NEXT (expression) 
		      && LLE_DENOMINATOR (expression) == 1);

	  /* The lower triangular portion of the base is going to be the
	     coefficient times the step */
	  for (j = 0; j < i; j++)
	    base[i][j] = LLE_COEFFICIENTS (expression)[j]
	      * LL_STEP (LN_LOOPS (nest)[j]);
	  base[i][i] = step;
	  for (j = i + 1; j < depth; j++)
	    base[i][j] = 0;

	  /* Origin for this loop is the constant of the lower bound
	     expression.  */
	  LATTICE_ORIGIN (ret)[i] = LLE_CONSTANT (expression);

	  /* Coefficient for the invariants are equal to the invariant
	     coefficients in the expression.  */
	  for (j = 0; j < invariants; j++)
	    LATTICE_ORIGIN_INVARIANTS (ret)[i][j] =
	      LLE_INVARIANT_COEFFICIENTS (expression)[j];
	}
    }
  return ret;
}

/* Compute the greatest common denominator of two numbers (A and B) using
   Euclid's algorithm.  */

static int
gcd (int a, int b)
{

  int x, y, z;

  x = abs (a);
  y = abs (b);

  while (x > 0)
    {
      z = y % x;
      y = x;
      x = z;
    }

  return (y);
}

/* Compute the greatest common denominator of a VECTOR of SIZE numbers.  */

static int
gcd_vector (lambda_vector vector, int size)
{
  int i;
  int gcd1 = 0;

  if (size > 0)
    {
      gcd1 = vector[0];
      for (i = 1; i < size; i++)
	gcd1 = gcd (gcd1, vector[i]);
    }
  return gcd1;
}

/* Compute the least common multiple of two numbers A and B .  */

static int
lcm (int a, int b)
{
  return (abs (a) * abs (b) / gcd (a, b));
}

/* Compute the loop bounds for the auxiliary space NEST.
   Input system used is Ax <= b.  TRANS is the unimodular transformation.  */

static lambda_loopnest
lambda_compute_auxillary_space (lambda_loopnest nest,
				lambda_trans_matrix trans)
{
  lambda_matrix A, B, A1, B1, temp0;
  lambda_vector a, a1, temp1;
  lambda_matrix invertedtrans;
  int determinant, depth, invariants, size, newsize;
  int i, j, k;
  lambda_loopnest auxillary_nest;
  lambda_loop loop;
  lambda_linear_expression expression;
  lambda_lattice lattice;

  int multiple, f1, f2;

  depth = LN_DEPTH (nest);
  invariants = LN_INVARIANTS (nest);

  /* Unfortunately, we can't know the number of constraints we'll have
     ahead of time, but this should be enough even in ridiculous loop nest
     cases. We abort if we go over this limit.  */
  A = lambda_matrix_new (128, depth);
  B = lambda_matrix_new (128, invariants);
  a = lambda_vector_new (128);

  A1 = lambda_matrix_new (128, depth);
  B1 = lambda_matrix_new (128, invariants);
  a1 = lambda_vector_new (128);

  /* Store the bounds in the equation matrix A, constant vector a, and
     invariant matrix B, so that we have Ax <= a + B.
     This requires a little equation rearranging so that everything is on the
     correct side of the inequality.  */
  size = 0;
  for (i = 0; i < depth; i++)
    {
      loop = LN_LOOPS (nest)[i];

      /* First we do the lower bound.  */
      if (LL_STEP (loop) > 0)
	expression = LL_LOWER_BOUND (loop);
      else
	expression = LL_UPPER_BOUND (loop);

      for (; expression != NULL; expression = LLE_NEXT (expression))
	{
	  /* Fill in the coefficient.  */
	  for (j = 0; j < i; j++)
	    A[size][j] = LLE_COEFFICIENTS (expression)[j];

	  /* And the invariant coefficient.  */
	  for (j = 0; j < invariants; j++)
	    B[size][j] = LLE_INVARIANT_COEFFICIENTS (expression)[j];

	  /* And the constant.  */
	  a[size] = LLE_CONSTANT (expression);

	  /* Convert (2x+3y+2+b)/4 <= z to 2x+3y-4z <= -2-b.  IE put all
	     constants and single variables on   */
	  A[size][i] = -1 * LLE_DENOMINATOR (expression);
	  a[size] *= -1;
	  for (j = 0; j < invariants; j++)
	    B[size][j] *= -1;

	  size++;
	  /* Need to increase matrix sizes above.  */
	  gcc_assert (size <= 127);
	  
	}

      /* Then do the exact same thing for the upper bounds.  */
      if (LL_STEP (loop) > 0)
	expression = LL_UPPER_BOUND (loop);
      else
	expression = LL_LOWER_BOUND (loop);

      for (; expression != NULL; expression = LLE_NEXT (expression))
	{
	  /* Fill in the coefficient.  */
	  for (j = 0; j < i; j++)
	    A[size][j] = LLE_COEFFICIENTS (expression)[j];

	  /* And the invariant coefficient.  */
	  for (j = 0; j < invariants; j++)
	    B[size][j] = LLE_INVARIANT_COEFFICIENTS (expression)[j];

	  /* And the constant.  */
	  a[size] = LLE_CONSTANT (expression);

	  /* Convert z <= (2x+3y+2+b)/4 to -2x-3y+4z <= 2+b.  */
	  for (j = 0; j < i; j++)
	    A[size][j] *= -1;
	  A[size][i] = LLE_DENOMINATOR (expression);
	  size++;
	  /* Need to increase matrix sizes above.  */
	  gcc_assert (size <= 127);

	}
    }

  /* Compute the lattice base x = base * y + origin, where y is the
     base space.  */
  lattice = lambda_lattice_compute_base (nest);

  /* Ax <= a + B then becomes ALy <= a+B - A*origin.  L is the lattice base  */

  /* A1 = A * L */
  lambda_matrix_mult (A, LATTICE_BASE (lattice), A1, size, depth, depth);

  /* a1 = a - A * origin constant.  */
  lambda_matrix_vector_mult (A, size, depth, LATTICE_ORIGIN (lattice), a1);
  lambda_vector_add_mc (a, 1, a1, -1, a1, size);

  /* B1 = B - A * origin invariant.  */
  lambda_matrix_mult (A, LATTICE_ORIGIN_INVARIANTS (lattice), B1, size, depth,
		      invariants);
  lambda_matrix_add_mc (B, 1, B1, -1, B1, size, invariants);

  /* Now compute the auxiliary space bounds by first inverting U, multiplying
     it by A1, then performing fourier motzkin.  */

  invertedtrans = lambda_matrix_new (depth, depth);

  /* Compute the inverse of U.  */
  determinant = lambda_matrix_inverse (LTM_MATRIX (trans),
				       invertedtrans, depth);

  /* A = A1 inv(U).  */
  lambda_matrix_mult (A1, invertedtrans, A, size, depth, depth);

  /* Perform Fourier-Motzkin elimination to calculate the bounds of the
     auxillary nest.
     Fourier-Motzkin is a way of reducing systems of linear inequality so that
     it is easy to calculate the answer and bounds.
     A sketch of how it works:
     Given a system of linear inequalities, ai * xj >= bk, you can always
     rewrite the constraints so they are all of the form
     a <= x, or x <= b, or x >= constant for some x in x1 ... xj (and some b
     in b1 ... bk, and some a in a1...ai)
     You can then eliminate this x from the non-constant inequalities by
     rewriting these as a <= b, x >= constant, and delete the x variable.
     You can then repeat this for any remaining x variables, and then we have
     an easy to use variable <= constant (or no variables at all) form that we
     can construct our bounds from. 

     In our case, each time we eliminate, we construct part of the bound from
     the ith variable, then delete the ith variable. 

     Remember the constant are in our vector a, our coefficient matrix is A,
     and our invariant coefficient matrix is B  */

  /* Swap B and B1, and a1 and a */
  temp0 = B1;
  B1 = B;
  B = temp0;

  temp1 = a1;
  a1 = a;
  a = temp1;

  auxillary_nest = lambda_loopnest_new (depth, invariants);

  for (i = depth - 1; i >= 0; i--)
    {
      loop = lambda_loop_new ();
      LN_LOOPS (auxillary_nest)[i] = loop;
      LL_STEP (loop) = 1;

      for (j = 0; j < size; j++)
	{
	  if (A[j][i] < 0)
	    {
	      /* Lower bound.  */
	      expression = lambda_linear_expression_new (depth, invariants);

	      for (k = 0; k < i; k++)
		LLE_COEFFICIENTS (expression)[k] = A[j][k];
	      for (k = 0; k < invariants; k++)
		LLE_INVARIANT_COEFFICIENTS (expression)[k] = -1 * B[j][k];
	      LLE_DENOMINATOR (expression) = -1 * A[j][i];
	      LLE_CONSTANT (expression) = -1 * a[j];
	      /* Ignore if identical to the existing lower bound.  */
	      if (!lle_equal (LL_LOWER_BOUND (loop),
			      expression, depth, invariants))
		{
		  LLE_NEXT (expression) = LL_LOWER_BOUND (loop);
		  LL_LOWER_BOUND (loop) = expression;
		}

	    }
	  else if (A[j][i] > 0)
	    {
	      /* Upper bound.  */
	      expression = lambda_linear_expression_new (depth, invariants);
	      for (k = 0; k < i; k++)
		LLE_COEFFICIENTS (expression)[k] = -1 * A[j][k];
	      LLE_CONSTANT (expression) = a[j];

	      for (k = 0; k < invariants; k++)
		LLE_INVARIANT_COEFFICIENTS (expression)[k] = B[j][k];

	      LLE_DENOMINATOR (expression) = A[j][i];
	      /* Ignore if identical to the existing upper bound.  */
	      if (!lle_equal (LL_UPPER_BOUND (loop),
			      expression, depth, invariants))
		{
		  LLE_NEXT (expression) = LL_UPPER_BOUND (loop);
		  LL_UPPER_BOUND (loop) = expression;
		}

	    }
	}
      /* creates a new system by deleting the i'th variable.  */
      newsize = 0;
      for (j = 0; j < size; j++)
	{
	  if (A[j][i] == 0)
	    {
	      lambda_vector_copy (A[j], A1[newsize], depth);
	      lambda_vector_copy (B[j], B1[newsize], invariants);
	      a1[newsize] = a[j];
	      newsize++;
	    }
	  else if (A[j][i] > 0)
	    {
	      for (k = 0; k < size; k++)
		{
		  if (A[k][i] < 0)
		    {
		      multiple = lcm (A[j][i], A[k][i]);
		      f1 = multiple / A[j][i];
		      f2 = -1 * multiple / A[k][i];

		      lambda_vector_add_mc (A[j], f1, A[k], f2,
					    A1[newsize], depth);
		      lambda_vector_add_mc (B[j], f1, B[k], f2,
					    B1[newsize], invariants);
		      a1[newsize] = f1 * a[j] + f2 * a[k];
		      newsize++;
		    }
		}
	    }
	}

      temp0 = A;
      A = A1;
      A1 = temp0;

      temp0 = B;
      B = B1;
      B1 = temp0;

      temp1 = a;
      a = a1;
      a1 = temp1;

      size = newsize;
    }

  return auxillary_nest;
}

/* Compute the loop bounds for the target space, using the bounds of
   the auxiliary nest AUXILLARY_NEST, and the triangular matrix H.  This is
   done by matrix multiplication and then transformation of the new matrix
   back into linear expression form.
   Return the target loopnest.  */

static lambda_loopnest
lambda_compute_target_space (lambda_loopnest auxillary_nest,
			     lambda_trans_matrix H, lambda_vector stepsigns)
{
  lambda_matrix inverse, H1;
  int determinant, i, j;
  int gcd1, gcd2;
  int factor;

  lambda_loopnest target_nest;
  int depth, invariants;
  lambda_matrix target;

  lambda_loop auxillary_loop, target_loop;
  lambda_linear_expression expression, auxillary_expr, target_expr, tmp_expr;

  depth = LN_DEPTH (auxillary_nest);
  invariants = LN_INVARIANTS (auxillary_nest);

  inverse = lambda_matrix_new (depth, depth);
  determinant = lambda_matrix_inverse (LTM_MATRIX (H), inverse, depth);

  /* H1 is H excluding its diagonal.  */
  H1 = lambda_matrix_new (depth, depth);
  lambda_matrix_copy (LTM_MATRIX (H), H1, depth, depth);

  for (i = 0; i < depth; i++)
    H1[i][i] = 0;

  /* Computes the linear offsets of the loop bounds.  */
  target = lambda_matrix_new (depth, depth);
  lambda_matrix_mult (H1, inverse, target, depth, depth, depth);

  target_nest = lambda_loopnest_new (depth, invariants);

  for (i = 0; i < depth; i++)
    {

      /* Get a new loop structure.  */
      target_loop = lambda_loop_new ();
      LN_LOOPS (target_nest)[i] = target_loop;

      /* Computes the gcd of the coefficients of the linear part.  */
      gcd1 = gcd_vector (target[i], i);

      /* Include the denominator in the GCD  */
      gcd1 = gcd (gcd1, determinant);

      /* Now divide through by the gcd  */
      for (j = 0; j < i; j++)
	target[i][j] = target[i][j] / gcd1;

      expression = lambda_linear_expression_new (depth, invariants);
      lambda_vector_copy (target[i], LLE_COEFFICIENTS (expression), depth);
      LLE_DENOMINATOR (expression) = determinant / gcd1;
      LLE_CONSTANT (expression) = 0;
      lambda_vector_clear (LLE_INVARIANT_COEFFICIENTS (expression),
			   invariants);
      LL_LINEAR_OFFSET (target_loop) = expression;
    }

  /* For each loop, compute the new bounds from H */
  for (i = 0; i < depth; i++)
    {
      auxillary_loop = LN_LOOPS (auxillary_nest)[i];
      target_loop = LN_LOOPS (target_nest)[i];
      LL_STEP (target_loop) = LTM_MATRIX (H)[i][i];
      factor = LTM_MATRIX (H)[i][i];

      /* First we do the lower bound.  */
      auxillary_expr = LL_LOWER_BOUND (auxillary_loop);

      for (; auxillary_expr != NULL;
	   auxillary_expr = LLE_NEXT (auxillary_expr))
	{
	  target_expr = lambda_linear_expression_new (depth, invariants);
	  lambda_vector_matrix_mult (LLE_COEFFICIENTS (auxillary_expr),
				     depth, inverse, depth,
				     LLE_COEFFICIENTS (target_expr));
	  lambda_vector_mult_const (LLE_COEFFICIENTS (target_expr),
				    LLE_COEFFICIENTS (target_expr), depth,
				    factor);

	  LLE_CONSTANT (target_expr) = LLE_CONSTANT (auxillary_expr) * factor;
	  lambda_vector_copy (LLE_INVARIANT_COEFFICIENTS (auxillary_expr),
			      LLE_INVARIANT_COEFFICIENTS (target_expr),
			      invariants);
	  lambda_vector_mult_const (LLE_INVARIANT_COEFFICIENTS (target_expr),
				    LLE_INVARIANT_COEFFICIENTS (target_expr),
				    invariants, factor);
	  LLE_DENOMINATOR (target_expr) = LLE_DENOMINATOR (auxillary_expr);

	  if (!lambda_vector_zerop (LLE_COEFFICIENTS (target_expr), depth))
	    {
	      LLE_CONSTANT (target_expr) = LLE_CONSTANT (target_expr)
		* determinant;
	      lambda_vector_mult_const (LLE_INVARIANT_COEFFICIENTS
					(target_expr),
					LLE_INVARIANT_COEFFICIENTS
					(target_expr), invariants,
					determinant);
	      LLE_DENOMINATOR (target_expr) =
		LLE_DENOMINATOR (target_expr) * determinant;
	    }
	  /* Find the gcd and divide by it here, rather than doing it
	     at the tree level.  */
	  gcd1 = gcd_vector (LLE_COEFFICIENTS (target_expr), depth);
	  gcd2 = gcd_vector (LLE_INVARIANT_COEFFICIENTS (target_expr),
			     invariants);
	  gcd1 = gcd (gcd1, gcd2);
	  gcd1 = gcd (gcd1, LLE_CONSTANT (target_expr));
	  gcd1 = gcd (gcd1, LLE_DENOMINATOR (target_expr));
	  for (j = 0; j < depth; j++)
	    LLE_COEFFICIENTS (target_expr)[j] /= gcd1;
	  for (j = 0; j < invariants; j++)
	    LLE_INVARIANT_COEFFICIENTS (target_expr)[j] /= gcd1;
	  LLE_CONSTANT (target_expr) /= gcd1;
	  LLE_DENOMINATOR (target_expr) /= gcd1;
	  /* Ignore if identical to existing bound.  */
	  if (!lle_equal (LL_LOWER_BOUND (target_loop), target_expr, depth,
			  invariants))
	    {
	      LLE_NEXT (target_expr) = LL_LOWER_BOUND (target_loop);
	      LL_LOWER_BOUND (target_loop) = target_expr;
	    }
	}
      /* Now do the upper bound.  */
      auxillary_expr = LL_UPPER_BOUND (auxillary_loop);

      for (; auxillary_expr != NULL;
	   auxillary_expr = LLE_NEXT (auxillary_expr))
	{
	  target_expr = lambda_linear_expression_new (depth, invariants);
	  lambda_vector_matrix_mult (LLE_COEFFICIENTS (auxillary_expr),
				     depth, inverse, depth,
				     LLE_COEFFICIENTS (target_expr));
	  lambda_vector_mult_const (LLE_COEFFICIENTS (target_expr),
				    LLE_COEFFICIENTS (target_expr), depth,
				    factor);
	  LLE_CONSTANT (target_expr) = LLE_CONSTANT (auxillary_expr) * factor;
	  lambda_vector_copy (LLE_INVARIANT_COEFFICIENTS (auxillary_expr),
			      LLE_INVARIANT_COEFFICIENTS (target_expr),
			      invariants);
	  lambda_vector_mult_const (LLE_INVARIANT_COEFFICIENTS (target_expr),
				    LLE_INVARIANT_COEFFICIENTS (target_expr),
				    invariants, factor);
	  LLE_DENOMINATOR (target_expr) = LLE_DENOMINATOR (auxillary_expr);

	  if (!lambda_vector_zerop (LLE_COEFFICIENTS (target_expr), depth))
	    {
	      LLE_CONSTANT (target_expr) = LLE_CONSTANT (target_expr)
		* determinant;
	      lambda_vector_mult_const (LLE_INVARIANT_COEFFICIENTS
					(target_expr),
					LLE_INVARIANT_COEFFICIENTS
					(target_expr), invariants,
					determinant);
	      LLE_DENOMINATOR (target_expr) =
		LLE_DENOMINATOR (target_expr) * determinant;
	    }
	  /* Find the gcd and divide by it here, instead of at the
	     tree level.  */
	  gcd1 = gcd_vector (LLE_COEFFICIENTS (target_expr), depth);
	  gcd2 = gcd_vector (LLE_INVARIANT_COEFFICIENTS (target_expr),
			     invariants);
	  gcd1 = gcd (gcd1, gcd2);
	  gcd1 = gcd (gcd1, LLE_CONSTANT (target_expr));
	  gcd1 = gcd (gcd1, LLE_DENOMINATOR (target_expr));
	  for (j = 0; j < depth; j++)
	    LLE_COEFFICIENTS (target_expr)[j] /= gcd1;
	  for (j = 0; j < invariants; j++)
	    LLE_INVARIANT_COEFFICIENTS (target_expr)[j] /= gcd1;
	  LLE_CONSTANT (target_expr) /= gcd1;
	  LLE_DENOMINATOR (target_expr) /= gcd1;
	  /* Ignore if equal to existing bound.  */
	  if (!lle_equal (LL_UPPER_BOUND (target_loop), target_expr, depth,
			  invariants))
	    {
	      LLE_NEXT (target_expr) = LL_UPPER_BOUND (target_loop);
	      LL_UPPER_BOUND (target_loop) = target_expr;
	    }
	}
    }
  for (i = 0; i < depth; i++)
    {
      target_loop = LN_LOOPS (target_nest)[i];
      /* If necessary, exchange the upper and lower bounds and negate
         the step size.  */
      if (stepsigns[i] < 0)
	{
	  LL_STEP (target_loop) *= -1;
	  tmp_expr = LL_LOWER_BOUND (target_loop);
	  LL_LOWER_BOUND (target_loop) = LL_UPPER_BOUND (target_loop);
	  LL_UPPER_BOUND (target_loop) = tmp_expr;
	}
    }
  return target_nest;
}

/* Compute the step signs of TRANS, using TRANS and stepsigns.  Return the new
   result.  */

static lambda_vector
lambda_compute_step_signs (lambda_trans_matrix trans, lambda_vector stepsigns)
{
  lambda_matrix matrix, H;
  int size;
  lambda_vector newsteps;
  int i, j, factor, minimum_column;
  int temp;

  matrix = LTM_MATRIX (trans);
  size = LTM_ROWSIZE (trans);
  H = lambda_matrix_new (size, size);

  newsteps = lambda_vector_new (size);
  lambda_vector_copy (stepsigns, newsteps, size);

  lambda_matrix_copy (matrix, H, size, size);

  for (j = 0; j < size; j++)
    {
      lambda_vector row;
      row = H[j];
      for (i = j; i < size; i++)
	if (row[i] < 0)
	  lambda_matrix_col_negate (H, size, i);
      while (lambda_vector_first_nz (row, size, j + 1) < size)
	{
	  minimum_column = lambda_vector_min_nz (row, size, j);
	  lambda_matrix_col_exchange (H, size, j, minimum_column);

	  temp = newsteps[j];
	  newsteps[j] = newsteps[minimum_column];
	  newsteps[minimum_column] = temp;

	  for (i = j + 1; i < size; i++)
	    {
	      factor = row[i] / row[j];
	      lambda_matrix_col_add (H, size, j, i, -1 * factor);
	    }
	}
    }
  return newsteps;
}

/* Transform NEST according to TRANS, and return the new loopnest.
   This involves
   1. Computing a lattice base for the transformation
   2. Composing the dense base with the specified transformation (TRANS)
   3. Decomposing the combined transformation into a lower triangular portion,
   and a unimodular portion. 
   4. Computing the auxillary nest using the unimodular portion.
   5. Computing the target nest using the auxillary nest and the lower
   triangular portion.  */ 

lambda_loopnest
lambda_loopnest_transform (lambda_loopnest nest, lambda_trans_matrix trans)
{
  lambda_loopnest auxillary_nest, target_nest;

  int depth, invariants;
  int i, j;
  lambda_lattice lattice;
  lambda_trans_matrix trans1, H, U;
  lambda_loop loop;
  lambda_linear_expression expression;
  lambda_vector origin;
  lambda_matrix origin_invariants;
  lambda_vector stepsigns;
  int f;

  depth = LN_DEPTH (nest);
  invariants = LN_INVARIANTS (nest);

  /* Keep track of the signs of the loop steps.  */
  stepsigns = lambda_vector_new (depth);
  for (i = 0; i < depth; i++)
    {
      if (LL_STEP (LN_LOOPS (nest)[i]) > 0)
	stepsigns[i] = 1;
      else
	stepsigns[i] = -1;
    }

  /* Compute the lattice base.  */
  lattice = lambda_lattice_compute_base (nest);
  trans1 = lambda_trans_matrix_new (depth, depth);

  /* Multiply the transformation matrix by the lattice base.  */

  lambda_matrix_mult (LTM_MATRIX (trans), LATTICE_BASE (lattice),
		      LTM_MATRIX (trans1), depth, depth, depth);

  /* Compute the Hermite normal form for the new transformation matrix.  */
  H = lambda_trans_matrix_new (depth, depth);
  U = lambda_trans_matrix_new (depth, depth);
  lambda_matrix_hermite (LTM_MATRIX (trans1), depth, LTM_MATRIX (H),
			 LTM_MATRIX (U));

  /* Compute the auxiliary loop nest's space from the unimodular
     portion.  */
  auxillary_nest = lambda_compute_auxillary_space (nest, U);

  /* Compute the loop step signs from the old step signs and the
     transformation matrix.  */
  stepsigns = lambda_compute_step_signs (trans1, stepsigns);

  /* Compute the target loop nest space from the auxiliary nest and
     the lower triangular matrix H.  */
  target_nest = lambda_compute_target_space (auxillary_nest, H, stepsigns);
  origin = lambda_vector_new (depth);
  origin_invariants = lambda_matrix_new (depth, invariants);
  lambda_matrix_vector_mult (LTM_MATRIX (trans), depth, depth,
			     LATTICE_ORIGIN (lattice), origin);
  lambda_matrix_mult (LTM_MATRIX (trans), LATTICE_ORIGIN_INVARIANTS (lattice),
		      origin_invariants, depth, depth, invariants);

  for (i = 0; i < depth; i++)
    {
      loop = LN_LOOPS (target_nest)[i];
      expression = LL_LINEAR_OFFSET (loop);
      if (lambda_vector_zerop (LLE_COEFFICIENTS (expression), depth))
	f = 1;
      else
	f = LLE_DENOMINATOR (expression);

      LLE_CONSTANT (expression) += f * origin[i];

      for (j = 0; j < invariants; j++)
	LLE_INVARIANT_COEFFICIENTS (expression)[j] +=
	  f * origin_invariants[i][j];
    }

  return target_nest;

}

/* Convert a gcc tree expression EXPR to a lambda linear expression, and
   return the new expression.  DEPTH is the depth of the loopnest.
   OUTERINDUCTIONVARS is an array of the induction variables for outer loops
   in this nest.  INVARIANTS is the array of invariants for the loop.  EXTRA
   is the amount we have to add/subtract from the expression because of the
   type of comparison it is used in.  */

static lambda_linear_expression
gcc_tree_to_linear_expression (int depth, tree expr,
			       VEC(tree) *outerinductionvars,
			       VEC(tree) *invariants, int extra)
{
  lambda_linear_expression lle = NULL;
  switch (TREE_CODE (expr))
    {
    case INTEGER_CST:
      {
	lle = lambda_linear_expression_new (depth, 2 * depth);
	LLE_CONSTANT (lle) = TREE_INT_CST_LOW (expr);
	if (extra != 0)
	  LLE_CONSTANT (lle) = extra;

	LLE_DENOMINATOR (lle) = 1;
      }
      break;
    case SSA_NAME:
      {
	tree iv, invar;
	size_t i;
	for (i = 0; VEC_iterate (tree, outerinductionvars, i, iv); i++)
	  if (iv != NULL)
	    {
	      if (SSA_NAME_VAR (iv) == SSA_NAME_VAR (expr))
		{
		  lle = lambda_linear_expression_new (depth, 2 * depth);
		  LLE_COEFFICIENTS (lle)[i] = 1;
		  if (extra != 0)
		    LLE_CONSTANT (lle) = extra;

		  LLE_DENOMINATOR (lle) = 1;
		}
	    }
	for (i = 0; VEC_iterate (tree, invariants, i, invar); i++)
	  if (invar != NULL)
	    {
	      if (SSA_NAME_VAR (invar) == SSA_NAME_VAR (expr))
		{
		  lle = lambda_linear_expression_new (depth, 2 * depth);
		  LLE_INVARIANT_COEFFICIENTS (lle)[i] = 1;
		  if (extra != 0)
		    LLE_CONSTANT (lle) = extra;
		  LLE_DENOMINATOR (lle) = 1;
		}
	    }
      }
      break;
    default:
      return NULL;
    }

  return lle;
}

/* Return true if OP is invariant in LOOP and all outer loops.  */

static bool
invariant_in_loop (struct loop *loop, tree op)
{
  if (loop->depth == 0)
    return true;
  if (TREE_CODE (op) == SSA_NAME)
    {
      if (TREE_CODE (SSA_NAME_VAR (op)) == PARM_DECL
	  && IS_EMPTY_STMT (SSA_NAME_DEF_STMT (op)))
	return true;
      if (IS_EMPTY_STMT (SSA_NAME_DEF_STMT (op)))
	return false;
      if (loop->outer)
	if (!invariant_in_loop (loop->outer, op))
	  return false;
      return !flow_bb_inside_loop_p (loop,
				     bb_for_stmt (SSA_NAME_DEF_STMT (op)));
    }
  return false;
}

/* Generate a lambda loop from a gcc loop LOOP.  Return the new lambda loop,
   or NULL if it could not be converted.
   DEPTH is the depth of the loop.
   INVARIANTS is a pointer to the array of loop invariants.
   The induction variable for this loop should be stored in the parameter
   OURINDUCTIONVAR.
   OUTERINDUCTIONVARS is an array of induction variables for outer loops.  */

static lambda_loop
gcc_loop_to_lambda_loop (struct loop *loop, int depth,
			 VEC (tree) ** invariants,
			 tree * ourinductionvar,
			 VEC (tree) * outerinductionvars)
{
  tree phi;
  tree exit_cond;
  tree access_fn, inductionvar;
  tree step;
  lambda_loop lloop = NULL;
  lambda_linear_expression lbound, ubound;
  tree test;
  int stepint;
  int extra = 0;
  tree uboundvar;
  use_optype uses;

  /* Find out induction var and set the pointer so that the caller can
     append it to the outerinductionvars array later.  */

  inductionvar = find_induction_var_from_exit_cond (loop);
  *ourinductionvar = inductionvar;

  exit_cond = get_loop_exit_condition (loop);

  if (inductionvar == NULL || exit_cond == NULL)
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
	fprintf (dump_file,
		 "Unable to convert loop: Cannot determine exit condition or induction variable for loop.\n");
      return NULL;
    }

  test = TREE_OPERAND (exit_cond, 0);

  if (SSA_NAME_DEF_STMT (inductionvar) == NULL_TREE)
    {

      if (dump_file && (dump_flags & TDF_DETAILS))
	fprintf (dump_file,
		 "Unable to convert loop: Cannot find PHI node for induction variable\n");

      return NULL;
    }

  phi = SSA_NAME_DEF_STMT (inductionvar);
  if (TREE_CODE (phi) != PHI_NODE)
    {
      get_stmt_operands (phi);
      uses = STMT_USE_OPS (phi);

      if (!uses)
	{

	  if (dump_file && (dump_flags & TDF_DETAILS))
	    fprintf (dump_file,
		     "Unable to convert loop: Cannot find PHI node for induction variable\n");

	  return NULL;
	}

      phi = USE_OP (uses, 0);
      phi = SSA_NAME_DEF_STMT (phi);
      if (TREE_CODE (phi) != PHI_NODE)
	{

	  if (dump_file && (dump_flags & TDF_DETAILS))
	    fprintf (dump_file,
		     "Unable to convert loop: Cannot find PHI node for induction variable\n");
	  return NULL;
	}

    }

  access_fn = instantiate_parameters
    (loop, analyze_scalar_evolution (loop, PHI_RESULT (phi)));
  if (!access_fn)
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
	fprintf (dump_file,
		 "Unable to convert loop: Access function for induction variable phi is NULL\n");

      return NULL;
    }

  step = evolution_part_in_loop_num (access_fn, loop->num);
  if (!step || step == chrec_dont_know)
    {
      if (dump_file && (dump_flags & TDF_DETAILS))
	fprintf (dump_file,
		 "Unable to convert loop: Cannot determine step of loop.\n");

      return NULL;
    }
  if (TREE_CODE (step) != INTEGER_CST)
    {

      if (dump_file && (dump_flags & TDF_DETAILS))
	fprintf (dump_file,
		 "Unable to convert loop: Step of loop is not integer.\n");


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