mode math algorithms to math folder

math/fibonacci_fast.cpp

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+/**
+ * @file
+ * @brief Faster computation of Fibonacci series
+ *
+ * An efficient way to calculate nth fibonacci number faster and simpler than
+ * \f$O(n\log n)\f$ method of matrix exponentiation This works by using both
+ * recursion and dynamic programming. as 93rd fibonacci exceeds 19 digits, which
+ * cannot be stored in a single long long variable, we can only use it till 92nd
+ * fibonacci we can use it for 10000th fibonacci etc, if we implement
+ * bigintegers. This algorithm works with the fact that nth fibonacci can easily
+ * found if we have already found n/2th or (n+1)/2th fibonacci It is a property
+ * of fibonacci similar to matrix exponentiation.
+ *
+ * @see fibonacci_large.cpp, fibonacci.cpp
+ */
+
+#include <cinttypes>
+#include <cstdio>
+#include <iostream>
+
+/** maximum number that can be computed - The result after 93 cannot be stored
+ * in a `uint64_t` data type. */
+const uint64_t MAX = 93;
+
+/** Array of computed fibonacci numbers */
+uint64_t f[MAX] = {0};
+
+/** Algorithm */
+uint64_t fib(uint64_t n) {
+    if (n == 0)
+        return 0;
+    if (n == 1 || n == 2)
+        return (f[n] = 1);
+
+    if (f[n])
+        return f[n];
+
+    uint64_t k = (n % 2 != 0) ? (n + 1) / 2 : n / 2;
+
+    f[n] = (n % 2 != 0) ? (fib(k) * fib(k) + fib(k - 1) * fib(k - 1))
+                        : (2 * fib(k - 1) + fib(k)) * fib(k);
+    return f[n];
+}
+
+/** Main function */
+int main() {
+    // Main Function
+    for (uint64_t i = 1; i < 93; i++) {
+        std::cout << i << " th fibonacci number is " << fib(i) << std::endl;
+    }
+    return 0;
+}