模板之小试牛刀---我有点明白为什么这么多人骂c++了(终，算法改变世界)

上一个版本使用了2分法计算整数平方根，光光看代码也觉得有些麻烦吧，收敛也比较慢，vc6在80左右的时候就要说模板递归太深受不了了。

现在在来看一个用牛顿迭代法来加速收敛的方法。迭代次数大大减少，vc6可以支持到大约225了。

还是上代码。和上一份比，代码有些很小的变动。4个宏依然继续可以用。

上菜#include <iostream>using namespace std;//enum {stop_tag = (size_t)-1};//Some compiler will crash so be careful! Warning comes if size_t is not used!static const size_t stop_tag = (size_t)-1;//Try the one above if your compiler doesn't support static const./* Binary search between [M1,M2]*/template<size_t N, size_t M1 , size_t M2 >//Assert M1<=M2<=Nstruct sqrt_check{enum {middle = (M1 + M2)/2};enum {end = M1 * M1 == N || M2 * M2 == N || M1 == middle || middle * middle== N};enum {selector = middle * middle > N};enum { result = N == M1 * M1 ?M1 : N == M2 * M2 ?M2 : M1 == middle ?//|M1-M2|=1, which means M1*M1 <= N < =M2*M2M1 : N == middle * middle ?//Bsearch begin!middle : selector ? sqrt_check<end ? stop_tag : N, end ? stop_tag : M1, end ? stop_tag : middle>::result : sqrt_check<end ? stop_tag : N, end ? stop_tag : middle, end ? stop_tag : M2>::result};};template<>struct sqrt_check<stop_tag,stop_tag,stop_tag>{enum { result = -1 };//Stop here!};template <size_t N, size_t M1 , size_t M2 >struct sqrt_check_newton{enum {next = (M1 + N/M1)/2};//newton iterator methodenum {end = next == M1 || next == M2 || next * next== N};enum { result = N == next * next || M1 == next ?//Fixed Point不动点next : M2 == next ?//|M1-M2|=1, which means M1*M1 <= N < =M2*M2 来回摆动(M2 + M1)/2: sqrt_check_newton<(end ? stop_tag : N), (end ? stop_tag : next), (end ? stop_tag : M1)>::result};};template<>struct sqrt_check_newton<stop_tag,stop_tag,stop_tag>{enum { result = -1 };//Stop here!};/* Get the sqrt result of N*/template <size_t N>struct sqrt_nature{//enum {result = sqrt_check <N, 1, N/2+1>::result};//begin searching between [1, N/2+1]enum {result = sqrt_check_newton <N, N/2, N>::result};enum {square = result * result == N};};template <>struct sqrt_nature<1>{enum {result = 1};enum {square = true};};template <>struct sqrt_nature<0>{enum {result = 0};enum {square = true};};/* Check if N % D == 0*/template<size_t N, size_t D>struct prime_check{enum {mod = N % D == 0};enum {prime = D == 1 ? true : mod ? false : prime_check<mod ? stop_tag : N , mod ? stop_tag : D - 1>::prime };};template<>struct prime_check<stop_tag , stop_tag>{enum {prime = true };};/* Check if N is a prime*/template<size_t N>struct prime_nature{enum {prime = prime_check<N, sqrt_nature<N>::result>::prime};enum {composite = !prime};};template<>struct prime_nature<1>{enum {prime = false};enum {composite = false};};/* Get the count of prime smaller than N.*/ template<size_t N>struct prime_count_nature{enum {prime_count = prime_count_nature<N-1>::prime_count + (prime_nature<N>::prime ? 1 : 0)};enum {composite_count = N - prime_count - 1};};template<>struct prime_count_nature<1>{enum {prime_count = 0};enum {composite_count = 0};};/* The following 4 macros are used for test. You can uncomment any of 4.*/#define test(x)/do{/if (bool(prime_nature<x>::prime))/cout<<x<<endl;/}while(false)//#define test(x) cout<<prime_count_nature<x>::prime_count<<endl//#define test(x) cout<<x<<' '<<(bool(prime_nature<x>::prime) ? "Yes" : "No")<<endl//#define test(x) cout<<x<<' ' <<sqrt_nature<x>::result<<endl/*Test from 1 to N*/template<size_t N>struct test_nature:public test_nature<N-1>{test_nature():test_nature<N-1>(){test(N);}};template<>struct test_nature<1>{test_nature(){test(1);}};int main(){ test_nature<225> x;//vc6.0 will crash for nesting template over 225 //test(8);return 0;} 

 

要点：enum {next = (M1 + N/M1)/2};//newton iterator method

y=x²   =>   x=(x+y/x)/2 

而当y大于0时，(x+y/x)/2 的不动点只有一个：y^1/2

这个例子很直白的告诉我们，算法一点小的进步，coder就可以向前进一大步。

走路先走头，再走脚。你说这是什么科学家呢？