[Mlir-commits] [mlir] [MLIR][Presburger][WIP] Implement vertex enumeration and chamber decomposition for polytope generating function computation. (PR #78987)

[Arjun P]

================
@@ -147,6 +149,307 @@ GeneratingFunction mlir::presburger::detail::unimodularConeGeneratingFunction(
                             std::vector({denominator}));
 }
 
+/// We use Gaussian elimination to find the solution to a set of d equations
+/// of the form
+/// a_1 x_1 + ... + a_d x_d + b_1 m_1 + ... + b_p m_p + c = 0
+/// where x_i are variables,
+/// m_i are parameters and
+/// a_i, b_i, c are rational coefficients.
+/// The solution expresses each x_i as an affine function of the m_i, and is
+/// therefore represented as a matrix of size d x (p+1).
+/// If there is no solution, we return null.
+std::optional<ParamPoint>
+mlir::presburger::detail::solveParametricEquations(FracMatrix equations) {
+  // equations is a d x (d + p + 1) matrix.
+  // Each row represents an equation.
+  unsigned d = equations.getNumRows();
+  unsigned numCols = equations.getNumColumns();
+
+  // If the determinant is zero, there is no unique solution.
+  // Thus we return null.
+  if (FracMatrix(equations.getSubMatrix(/*fromRow=*/0, /*toRow=*/d - 1,
+                                        /*fromColumn=*/0,
+                                        /*toColumn=*/d - 1))
+          .determinant() == 0)
+    return std::nullopt;
+
+  // Perform row operations to make each column all zeros except for the
+  // diagonal element, which is made to be one.
+  for (unsigned i = 0; i < d; ++i) {
+    // First ensure that the diagonal element is nonzero, by swapping
+    // it with a row that is non-zero at column i.
+    if (equations(i, i) != 0)
+      continue;
+    for (unsigned j = i + 1; j < d; ++j) {
+      if (equations(j, i) == 0)
+        continue;
+      equations.swapRows(j, i);
+      break;
+    }
+
+    Fraction diagElement = equations(i, i);
+
+    // Apply row operations to make all elements except the diagonal to zero.
+    for (unsigned j = 0; j < d; ++j) {
+      if (i == j)
+        continue;
+      if (equations(j, i) == 0)
+        continue;
+      // Apply row operations to make element (j, i) zero by subtracting the
+      // ith row, appropriately scaled.
+      Fraction currentElement = equations(j, i);
+      equations.addToRow(/*sourceRow=*/i, /*targetRow=*/j,
+                         /*scale=*/-currentElement / diagElement);
+    }
+  }
+
+  // Rescale diagonal elements to 1.
+  for (unsigned i = 0; i < d; ++i) {
+    Fraction diagElement = equations(i, i);
+    for (unsigned j = 0; j < numCols; ++j)
+      equations(i, j) = equations(i, j) / diagElement;
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Superty wrote:

implement a scaleRow function

https://github.com/llvm/llvm-project/pull/78987