Added 2nd partial derivative test. Detects saddles, mins and maxes.

MLPP/LinAlg/LinAlg.cpp

@@ -704,6 +704,34 @@ namespace MLPP{
         return true;
     }
 
+    bool LinAlg::negativeDefiniteChecker(std::vector<std::vector<double>> A){
+        auto [eigenvectors, eigenvals] = eig(A);
+        std::vector<double> eigenvals_vec;
+        for(int i = 0; i < eigenvals.size(); i++){
+            eigenvals_vec.push_back(eigenvals[i][i]);
+        }
+        for(int i = 0; i < eigenvals_vec.size(); i++){
+            if(eigenvals_vec[i] >= 0){ // Simply check to ensure all eigenvalues are negative.
+                return false;
+            }
+        }
+        return true;
+    }
+
+    bool LinAlg::zeroEigenvalue(std::vector<std::vector<double>> A){
+        auto [eigenvectors, eigenvals] = eig(A);
+        std::vector<double> eigenvals_vec;
+        for(int i = 0; i < eigenvals.size(); i++){
+            eigenvals_vec.push_back(eigenvals[i][i]);
+        }
+        for(int i = 0; i < eigenvals_vec.size(); i++){
+            if(eigenvals_vec[i] == 0){ 
+                return true;
+            }
+        }
+        return false;
+    }
+
     void LinAlg::printMatrix(std::vector<std::vector<double>> A){
         for(int i = 0; i < A.size(); i++){
             for(int j = 0; j < A[i].size(); j++){

MLPP/LinAlg/LinAlg.hpp

@@ -106,6 +106,10 @@ namespace MLPP{
 
         bool positiveDefiniteChecker(std::vector<std::vector<double>> A);
 
+        bool negativeDefiniteChecker(std::vector<std::vector<double>> A);
+
+        bool zeroEigenvalue(std::vector<std::vector<double>> A);
+        
         void printMatrix(std::vector<std::vector<double>> A);
         
         // VECTOR FUNCTIONS

MLPP/NumericalAnalysis/NumericalAnalysis.cpp

@@ -260,4 +260,43 @@ namespace MLPP{
         }
         return laplacian;
     }
+
+    std::string NumericalAnalysis::secondPartialDerivativeTest(double(*function)(std::vector<double>), std::vector<double> x){
+        LinAlg alg;
+        std::vector<std::vector<double>> hessianMatrix = hessian(function, x);
+        /* 
+        The reason we do this is because the 2nd partial derivative test is less conclusive for functions of variables greater than
+        2, and the calculations specific to the bivariate case are less computationally intensive. 
+        */
+        if(x.size() == 2){ 
+            double det = alg.det(hessianMatrix, hessianMatrix.size());
+            double secondDerivative = numDiff_2(function, x, 0, 0);
+            if(secondDerivative > 0 && det > 0){
+                return "min";
+            }
+            else if(secondDerivative < 0 && det > 0){
+                return "max";
+            }
+            else if(det < 0){
+                return "saddle";
+            }
+            else{
+                return "test was inconclusive";
+            }
+        }
+        else {
+            if(alg.positiveDefiniteChecker(hessianMatrix)){
+                return "min";
+            }
+            else if(alg.negativeDefiniteChecker(hessianMatrix)){
+                return "max";
+            }
+            else if(!alg.zeroEigenvalue(hessianMatrix)){
+                return "saddle";
+            }
+            else{
+                return "test was inconclusive";
+            }
+        }
+    }
 }

MLPP/NumericalAnalysis/NumericalAnalysis.hpp

@@ -48,6 +48,8 @@ namespace MLPP{
 
             double laplacian(double(*function)(std::vector<double>), std::vector<double> x); // laplacian
 
+            std::string secondPartialDerivativeTest(double(*function)(std::vector<double>), std::vector<double> x);
+
     };
 }