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50 lines
1.2 KiB
C++
50 lines
1.2 KiB
C++
/**
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* @file
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* @brief Calculate the square root of any positive number in \f$O(\log N)\f$
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* time, with precision fixed using [bisection
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* method](https://en.wikipedia.org/wiki/Bisection_method) of root-finding.
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*
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* @see Can be implemented using faster and better algorithms like
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* newton_raphson_method.cpp and false_position.cpp
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*/
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#include <cassert>
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#include <iostream>
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/** Bisection method implemented for the function \f$x^2-a=0\f$
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* whose roots are \f$\pm\sqrt{a}\f$ and only the positive root is returned.
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*/
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double Sqrt(double a) {
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if (a > 0 && a < 1) {
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return 1 / Sqrt(1 / a);
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}
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double l = 0, r = a;
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/* Epsilon is the precision.
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A great precision is
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between 1e-7 and 1e-12.
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double epsilon = 1e-12;
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*/
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double epsilon = 1e-12;
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while (l <= r) {
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double mid = (l + r) / 2;
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if (mid * mid > a) {
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r = mid;
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} else {
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if (a - mid * mid < epsilon) {
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return mid;
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}
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l = mid;
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}
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}
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return -1;
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}
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/** main function */
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int main() {
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double n{};
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std::cin >> n;
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assert(n >= 0);
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// Change this line for a better precision
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std::cout.precision(12);
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std::cout << std::fixed << Sqrt(n);
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}
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