Added quadratic_equations_complex_numbers.cpp

Author: Renjian-buchai

math/quadratic_equations_complex_numbers.cpp

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+/**
+ * @file
+ * @brief Calculate quadratic equation with complex roots, i.e. b^2 - 4ac < 0.
+ * @see https://en.wikipedia.org/wiki/Quadratic_equation,
+ * https://en.wikipedia.org/wiki/Quadratic_equation#discriminant
+ *
+ * @author [Renjian-buchai](https://github.com/Renjian-buchai)
+ */
+
+#include <array>
+#include <cmath>
+#include <complex>
+#include <exception>
+
+/// @brief Quadratic equation calculator.
+/// @param a quadratic coefficient.
+/// @param b linear coefficient.
+/// @param c constant
+/// @return Array containing the roots of quadratic equation, incl. complex
+/// root.
+std::array<std::complex<long double>, 2> quadraticEquation(long double a,
+                                                           long double b,
+                                                           long double c) {
+    /**
+     * Calculates any quadratic equation in form ax^2 + bx + c.
+     *
+     * Quadratic equation:
+     * x = (-b +/- sqrt(b^2 - 4ac)) / 2a
+     *
+     * e.g.
+     * using namespace std;
+     * int main() {
+     *   array<complex<long double, 2> solutions = quadraticEquation(1, 2, 1);
+     *   cout << solutions[0] << " " << solutions[1] << "\n";
+     *
+     *   solutions = quadraticEquation(1, 1, 1);  // Reusing solutions.
+     *   cout << solutions[0] << " " << solutions[1] << "\n";
+     *   return 0;
+     * }
+     *
+     * Output:
+     * (-1, 0) (-1, 0)
+     * (-0.5,0.866025) (-0.5,0.866025)
+     */
+
+    if (a == 0)
+        throw std::invalid_argument("quadratic coefficient cannot be 0");
+
+    long double discriminant = b * b - 4 * a * c;
+    std::array<std::complex<long double>, 2> solutions{0, 0};
+
+    // Complex root (discriminant < 0)
+    // Note that the left term (-b / 2a) is always real. The imaginary part
+    // appears when b^2 - 4ac < 0, so sqrt(b^2 - 4ac) has no real roots. So, the
+    // imaginary component is i * (+/-)sqrt(abs(b^2 - 4ac)) / 2a.
+    if (discriminant < 0) {
+        // Since b^2 - 4ac is < 0, for faster computation, -discriminant is
+        // enough to make it positive.
+        solutions[0] = std::complex<long double>{
+            -b * 0.5 / a, -std::sqrt(-discriminant) * 0.5 / a};
+        solutions[1] = std::complex<long double>{
+            -b * 0.5 / a, std::sqrt(-discriminant) * 0.5 / a};
+    } else {
+        // Since discriminant > 0, there are only real roots. Therefore,
+        // imaginary component = 0.
+        solutions[0] = std::complex<long double>{
+            (-b - std::sqrt(discriminant)) * 0.5 / a, 0};
+        solutions[1] = std::complex<long double>{
+            (-b + std::sqrt(discriminant)) * 0.5 / a, 0};
+    }
+
+    return solutions;
+}