Newton Forward method added

Author: yahya-mohammed07

numerical_methods/newton_forward.cpp

@@ -0,0 +1,56 @@
+#include <cstdint>
+#include <iostream>
+#include <vector>
+//
+[[nodiscard]] constexpr auto factorial(std::size_t val) -> std::size_t {
+    if (val <= 1) {
+        return 1;
+    }
+    return val * factorial(val - 1);
+}
+//
+/// @brief performs Newton's Forward Difference Formula on two dynamic
+/// containers
+/// @param xi container
+/// @param yi container
+/// @param x [by default it's `1`]
+/// @return Ty
+template <class Ty, class Cont>
+[[nodiscard]] auto newton_forward(const Cont &xi, Cont &yi, const Ty x) -> Ty {
+    Cont forward{yi[0]}; // holds forward difference of `yi` init = yi[0]
+    std::size_t size{yi.size()};
+    //
+    for (std::size_t i{}; i < size;
+         ++i) { // calculates forward difference of `yi`
+        Cont temp{};
+        for (std::size_t j{1}; j < size; ++j) {
+            temp.push_back(yi[j - 1] = (yi[j] - yi[j - 1]));
+            if (j == yi.size() - 1) {
+                yi.erase(yi.begin() + static_cast<int64_t>(size - 1));
+            }
+        }
+        size = temp.size();
+        forward.push_back(temp[0]);
+    }
+    forward.push_back(yi[0] = yi[1] - yi[0]);
+    const auto cal_u = [](const Ty u, const std::size_t times) -> Ty {
+        Ty result{u};
+        for (std::size_t i{1}; i < times; ++i) {
+            result *= u - static_cast<Ty>(i);
+        }
+        return result;
+    };
+    const Ty u{(x - xi[0]) / (xi[1] - xi[0])};
+    Ty y{(forward[0])};
+    //
+    for (std::size_t i{1}; i < xi.size(); ++i) {
+        y = y + (cal_u(u, i) * (forward[i])) / static_cast<Ty>(factorial(i));
+    }
+    return y;
+}
+//
+auto main() -> int {
+    std::vector<float> xi{1891, 1901, 1911, 1921, 1931};
+    std::vector<float> yi{46, 66, 81, 93, 101};
+    std::cout << newton_forward(xi, yi, 1.) << '\n';
+}