Quadratic equations complex numbers

math/quadratic_equations_complex_numbers.cpp

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+/**
+ * @file
+ * @brief Calculate quadratic equation with complex roots, i.e. b^2 - 4ac < 0.
+ *
+ * @author [Renjian-buchai](https://github.com/Renjian-buchai)
+ *
+ * @description Calculates any quadratic equation in form ax^2 + bx + c.
+ *
+ * Quadratic equation:
+ * x = (-b +/- sqrt(b^2 - 4ac)) / 2a
+ *
+ * @example
+ * int main() {
+ *   using std::array;
+ *   using std::complex;
+ *   using std::cout;
+ *
+ *   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)
+ */
+
+#include <array>      /// std::array
+#include <cassert>    /// assert
+#include <cmath>      /// std::sqrt, std::trunc, std::pow
+#include <complex>    /// std::complex
+#include <exception>  /// std::invalid_argument
+#include <iomanip>    /// std::setprecision
+#include <iostream>   /// std::cout
+
+/**
+ * @namespace
+ * @brief Mathematical algorithms
+ */
+namespace math {
+
+/**
+ * @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) {
+    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;
+}
+
+}  // namespace math
+
+/**
+ * @brief Asserts an array of complex numbers.
+ * @param input Input array of complex numbers. .
+ * @param expected Expected array of complex numbers.
+ * @param precision Precision to be asserted. Default=10
+ */
+void assertArray(std::array<std::complex<long double>, 2> input,
+                 std::array<std::complex<long double>, 2> expected,
+                 size_t precision = 10) {
+    long double exponent = std::pow(10, precision);
+    input[0].real(std::round(input[0].real() * exponent));
+    input[1].real(std::round(input[1].real() * exponent));
+    input[0].imag(std::round(input[0].imag() * exponent));
+    input[1].imag(std::round(input[1].imag() * exponent));
+
+    expected[0].real(std::round(expected[0].real() * exponent));
+    expected[1].real(std::round(expected[1].real() * exponent));
+    expected[0].imag(std::round(expected[0].imag() * exponent));
+    expected[1].imag(std::round(expected[1].imag() * exponent));
+
+    assert(input == expected);
+}
+
+/**
+ * @brief Self-test implementations to test quadraticEquation function.
+ * @note There are 4 different types of solutions: Real and equal, real,
+ * complex, complex and equal.
+ */
+static void test() {
+    // Values are equal and real.
+    std::cout << "Input: \n"
+                 "a=1 \n"
+                 "b=-2 \n"
+                 "c=1 \n"
+                 "Expected output: \n"
+                 "(1, 0), (1, 0)\n\n";
+    std::array<std::complex<long double>, 2> equalCase{
+        std::complex<long double>{1, 0}, std::complex<long double>{1, 0}};
+    assert(math::quadraticEquation(1, -2, 1) == equalCase);
+
+    // Values are equal and complex.
+    std::cout << "Input: \n"
+                 "a=1 \n"
+                 "b=4 \n"
+                 "c=5 \n"
+                 "Expected output: \n"
+                 "(-2, -1), (-2, 1)\n\n";
+    std::array<std::complex<long double>, 2> complexCase{
+        std::complex<long double>{-2, -1}, std::complex<long double>{-2, 1}};
+    assert(math::quadraticEquation(1, 4, 5) == complexCase);
+
+    // Values are real.
+    std::cout << "Input: \n"
+                 "a=1 \n"
+                 "b=5 \n"
+                 "c=1 \n"
+                 "Expected output: \n"
+                 "(-4.7912878475, 0), (-0.2087121525, 0)\n\n";
+    std::array<std::complex<long double>, 2> floatCase{
+        std::complex<long double>{-4.7912878475, 0},
+        std::complex<long double>{-0.2087121525, 0}};
+    assertArray(math::quadraticEquation(1, 5, 1), floatCase);
+
+    // Values are complex.
+    std::cout << "Input: \n"
+                 "a=1 \n"
+                 "b=1 \n"
+                 "c=1 \n"
+                 "Expected output: \n"
+                 "(-0.5, -0.8660254038), (-0.5, 0.8660254038)\n\n";
+    std::array<std::complex<long double>, 2> ifloatCase{
+        std::complex<long double>{-0.5, -0.8660254038},
+        std::complex<long double>{-0.5, 0.8660254038}};
+    assertArray(math::quadraticEquation(1, 1, 1), ifloatCase);
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    test();  // Run self-test implementation.
+    return 0;
+}