Fix dense Cholesky refinement residual storage

DenseCholesky accepts a column-major lower triangle. Map that storage
explicitly and use its lower self-adjoint view for refinement residuals.

Poison unused triangles in the existing backend and size tests. Compute
the right-hand side before poisoning the input. Direct refiner tests
retain a full-matrix reference and factorization. Preserve tolerances
and iteration limits.

Issue: https://github.com/ceres-solver/ceres-solver/issues/863

Codex used.

Change-Id: I06d272ac382de5aecd6805a02d5f9c996aa88a4f

internal/ceres/dense_cholesky_test.cc

diff --git a/internal/ceres/dense_cholesky_test.cc b/internal/ceres/dense_cholesky_test.cc
index ac5075a..b5fa59b 100644
--- a/internal/ceres/dense_cholesky_test.cc
+++ b/internal/ceres/dense_cholesky_test.cc
@@ -99,6 +99,7 @@
       lhs += VectorType::Ones(num_cols).asDiagonal();
       Vector x = VectorType::Random(num_cols);
       Vector rhs = lhs * x;
+      lhs.triangularView<Eigen::StrictlyUpper>().setZero();
       Vector actual = Vector::Random(num_cols);
 
       LinearSolver::Summary summary;

internal/ceres/iterative_refiner.cc

diff --git a/internal/ceres/iterative_refiner.cc b/internal/ceres/iterative_refiner.cc
index 54d48f3..da3cc15 100644
--- a/internal/ceres/iterative_refiner.cc
+++ b/internal/ceres/iterative_refiner.cc
@@ -86,12 +86,12 @@
                                    DenseCholesky* cholesky,
                                    double* solution_ptr) {
   Allocate(num_cols);
-  ConstMatrixRef lhs(lhs_ptr, num_cols, num_cols);
+  Eigen::Map<const Eigen::MatrixXd> lhs(lhs_ptr, num_cols, num_cols);
   ConstVectorRef rhs(rhs_ptr, num_cols);
   VectorRef solution(solution_ptr, num_cols);
   std::string ignored_message;
   for (int i = 0; i < max_num_iterations_; ++i) {
-    residual_ = rhs - lhs * solution;
+    residual_ = rhs - lhs.selfadjointView<Eigen::Lower>() * solution;
     // solution += lhs^-1 residual
     cholesky->Solve(residual_.data(), correction_.data(), &ignored_message);
     solution += correction_;

internal/ceres/iterative_refiner_test.cc

diff --git a/internal/ceres/iterative_refiner_test.cc b/internal/ceres/iterative_refiner_test.cc
index a2557f2..ffa25dc 100644
--- a/internal/ceres/iterative_refiner_test.cc
+++ b/internal/ceres/iterative_refiner_test.cc
@@ -45,8 +45,10 @@
 
 // Macros to help us define virtual methods which we do not expect to
 // use/call in this test.
-#define DO_NOT_CALL \
-  { LOG(FATAL) << "DO NOT CALL"; }
+#define DO_NOT_CALL              \
+  {                              \
+    LOG(FATAL) << "DO NOT CALL"; \
+  }
 #define DO_NOT_CALL_WITH_RETURN(x) \
   {                                \
     LOG(FATAL) << "DO NOT CALL";   \
@@ -219,6 +221,7 @@
        RandomSolutionWithExactFactorizationConverges) {
   Matrix lhs = lhs_;
   FakeDenseCholesky<double> dense_cholesky(lhs);
+  lhs.triangularView<Eigen::StrictlyLower>().setZero();
   DenseIterativeRefiner refiner(max_num_iterations_);
   Vector refined_solution(num_cols_);
   refined_solution.setRandom();
@@ -238,6 +241,7 @@
   // Use a single precision Cholesky factorization of the double
   // precision matrix. This will give us an approximate factorization.
   FakeDenseCholesky<float> dense_cholesky(lhs_);
+  lhs.triangularView<Eigen::StrictlyLower>().setZero();
   DenseIterativeRefiner refiner(max_num_iterations_);
   Vector refined_solution(num_cols_);
   refined_solution.setRandom();