feat(Jacobian): Jacobian generation is now stateless.

Previously Jacobians were stored by engines and accessed through engine
accessors (e.g getJacobianMatrixEntry); however, this resulted in
desynced jacobian states. We have changed to a pattern of Engine creates
a jacobian and returns it to the caller. The caller can then do what
they will with it. Because of this the getJacobianMatrixEntry method has
been removed.

BREAKING CHANGE:
    - There is no longer any getJacobianMatrixEntry method on
DynamicEngine classes
    - the generateJacobian method signature has changed to return a
NetworkJacobian object. Internally this uses an Eigen Sparse Matrix to
store its data.

src/include/gridfire/engine/engine_graph.h

@@ -73,6 +73,7 @@ namespace gridfire {
 
 
 
+
     /**
      * @class GraphEngine
      * @brief A reaction network engine that uses a graph-based representation.
@@ -236,7 +237,7 @@ namespace gridfire {
          *
          * @see getJacobianMatrixEntry()
          */
-        void generateJacobianMatrix(
+        [[nodiscard]] NetworkJacobian generateJacobianMatrix(
             const fourdst::composition::CompositionAbstract &comp,
             double T9,
             double rho
@@ -254,7 +255,7 @@ namespace gridfire {
          * @see getJacobianMatrixEntry()
          * @see generateJacobianMatrix()
          */
-        void generateJacobianMatrix(
+        [[nodiscard]] NetworkJacobian generateJacobianMatrix(
             const fourdst::composition::CompositionAbstract &comp,
             double T9,
             double rho,
@@ -276,7 +277,7 @@ namespace gridfire {
          *
          * @see getJacobianMatrixEntry()
          */
-        void generateJacobianMatrix(
+        [[nodiscard]] NetworkJacobian generateJacobianMatrix(
             const fourdst::composition::CompositionAbstract &comp,
             double T9,
             double rho,
@@ -334,22 +335,6 @@ namespace gridfire {
          */
         void setNetworkReactions(const reaction::ReactionSet& reactions) override;
 
-        /**
-         * @brief Gets an entry from the previously generated Jacobian matrix.
-         *
-         * @param rowSpecies Species corresponding to the row index (i).
-         * @param colSpecies Species corresponding to the column index (j).
-         * @return Value of the Jacobian matrix at (i, j).
-         *
-         * The Jacobian must have been generated by `generateJacobianMatrix()` before calling this.
-         *
-         * @see generateJacobianMatrix()
-         */
-        [[nodiscard]] double getJacobianMatrixEntry(
-            const fourdst::atomic::Species& rowSpecies,
-            const fourdst::atomic::Species& colSpecies
-        ) const override;
-
         /**
          * @brief Gets the net stoichiometry for a given reaction.
          *
@@ -754,6 +739,8 @@ namespace gridfire {
          */
         fourdst::composition::Composition collectComposition(const fourdst::composition::CompositionAbstract &comp, double T9, double rho) const override;
 
+        SpeciesStatus getSpeciesStatus(const fourdst::atomic::Species &species) const override;
+
 
     private:
         struct PrecomputedReaction {
@@ -864,9 +851,6 @@ namespace gridfire {
 
         boost::numeric::ublas::compressed_matrix<int> m_stoichiometryMatrix; ///< Stoichiometry matrix (species x reactions).
 
-        mutable boost::numeric::ublas::compressed_matrix<double> m_jacobianMatrix; ///< Jacobian matrix (species x species).
-        mutable JacobianMatrixState m_jacobianMatrixState = JacobianMatrixState::UNINITIALIZED;
-
         mutable CppAD::ADFun<double> m_rhsADFun; ///< CppAD function for the right-hand side of the ODE.
         mutable CppAD::ADFun<double> m_epsADFun; ///< CppAD function for the energy generation rate.
         mutable CppAD::sparse_jac_work m_jac_work; ///< Work object for sparse Jacobian calculations.
@@ -921,14 +905,6 @@ namespace gridfire {
          */
         void populateSpeciesToIndexMap();
 
-        /**
-         * @brief Reserves space for the Jacobian matrix.
-         *
-         * This method reserves space for the Jacobian matrix, which is used
-         * to store the partial derivatives of the right-hand side of the ODE
-         * with respect to the species abundances.
-         */
-        void reserveJacobianMatrix() const;
 
         /**
          * @brief Records the AD tape for the right-hand side of the ODE.
@@ -1254,7 +1230,7 @@ namespace gridfire {
         std::unordered_map<fourdst::atomic::Species, int> reactant_counts;
         reactant_counts.reserve(reaction.reactants().size());
         for (const auto& reactant : reaction.reactants()) {
-            reactant_counts[reactant]++;
+            reactant_counts[reactant] = reaction.countReactantOccurrences(reactant);
         }
         const int totalReactants = static_cast<int>(reaction.reactants().size());
 
@@ -1285,7 +1261,6 @@ namespace gridfire {
         //       the tape more expensive to record, but it will also mean that we only need to record it once for
         //       the entire network.
         const T densityTerm = CppAD::pow(rho, totalReactants > 1 ? static_cast<T>(totalReactants - 1) : zero); // Density raised to the power of (N-1) for N reactants
-
         return molar_concentration_product * k_reaction * densityTerm;
     }