打印 pi 的值。 在您第一次獲得 3.14 之前，您必須使用這個系列的多少項？ 3.141？ 3.1415？ 3.14159？

Question

#include <stdio.h>
#include <math.h>

int main(void) {
    // Set the initial value of pi to 0
    double pi = 0.0;

    // Set the initial value of the term to 1
    double term = 1.0;

    // Set the initial value of the divisor to 1
    double divisor = 1.0;

    // Print the table header
    printf("%10s%25s\n", "Number of terms", "Approximation of pi");

    // Calculate and print the approximations of pi
    for (int i = 1; i <= 20; i++) {
        pi += term / divisor;
        printf("%10d%25.10f\n", i, pi*4.0);
        term *= -1.0;
        divisor += 2.0;
    }

    return 0;
}

我試圖更正代碼，但仍然無法接近我的老師在作業中要求的值......

問題是.. 從無限級數中計算 π 的值。 打印一張表格，顯示用該級數的一項、兩項、三項等近似的 π 值。 在您第一次獲得 3.14 之前，您必須使用這個系列的多少項？ 3.141？ 3.1415？ 3.14159？

Answer 1

在您第一次獲得 3.14 之前，您必須使用這個系列的多少項？ 3.141？ 3.1415？ 3.14159？

“first get 3.14”的細節有點不清楚。 下面嘗試類似於 OP 的目標，並說明了緩慢的收斂，因為計算時間與項數成正比。

大量的項，每一項都會在除法和加法中產生舍入誤差，最終導致此計算對於高項計數而言過於不准確。

int main(void) {
  double pi_true = 3.1415926535897932384626433832795;
  double threshold = 0.5;
  int dp = 0;

  // Set the initial value of pi to 0
  double pi = 0.0;

  // Set the initial value of the term to 1
  double term = 1.0;

  // Set the initial value of the divisor to 1
  double divisor = 1.0;

  // Print the table header
  printf("%7s %12s %-25.16f\n", "", "", pi_true);
  printf("%7s %12s %-25s\n", "", "# of terms", "Approximation of pi");

  // Calculate and print the approximations of pi
  for (long long i = 1; ; i++) {
    pi += term / divisor;
    double diff = fabs(4*pi - pi_true);
    if (diff <= threshold) {
      printf("%7.1e %12lld %-25.16f %-25.*f\n", diff, i, pi * 4.0, dp++, pi * 4.0);
      fflush(stdout);
      threshold /= 10;
      if (4*pi == pi_true) {
        break;
      }
    }
    term *= -1.0;
    divisor += 2.0;
  }
  puts("Done");
  return 0;
}

輸出

                     3.1415926535897931       
          # of terms Approximation of pi      
4.7e-01            2 2.6666666666666670        3                        
5.0e-02           20 3.0916238066678399        3.1                      
5.0e-03          200 3.1365926848388161        3.14                     
5.0e-04         2000 3.1410926536210413        3.141                    
5.0e-05        20000 3.1415426535898248        3.1415                   
5.0e-06       200001 3.1415976535647618        3.14160                  
5.0e-07      2000001 3.1415931535894743        3.141593                 
5.0e-08     19999992 3.1415926035897974        3.1415926                
5.0e-09    199984633 3.1415926585897931        3.14159266               
5.0e-10   1993125509 3.1415926540897927        3.141592654   
5.0e-11  19446391919 3.1415926536397927        3.1415926536 
...

Ref                  3.1415926535897931        

---

在第二次嘗試中，也許這更接近 OP 的目標

int main(void) {
  double pi_true = 3.1415926535897932384626433832795;

  double threshold_lo = 2.5;
  double threshold_hi = 3.5;
  double error_band = 0.5;
  int dp = 0;

  // Set the initial value of pi to 0
  double pi = 0.0;

  // Set the initial value of the term to 4
  double term = 4.0;

  // Set the initial value of the divisor to 1
  double divisor = 1.0;

  // Print the table header
  printf("%12s %-25.16f\n", "", pi_true);
  printf("%12s %-25s\n", "# of terms", "Approximation of pi");

  // Calculate and print the approximations of pi
  for (long long i = 1;; i++) {
    pi += term / divisor;
    if (pi > threshold_lo && pi < threshold_hi) {
      printf("%12lld %-25.16f %-25.*f\n", i, pi, dp++, pi);
      fflush(stdout);

      char buf[100] = "3.1415926535897932384626433832795";
      buf[dp + 2] = 0;
      error_band /= 10.0;
      double target = atof(buf);
      threshold_lo = target - error_band;
      threshold_hi = target + error_band;
    }
    term *= -1.0;
    divisor += 2.0;
  }
  puts("Done");
  return 0;
}

輸出

             3.1415926535897931       
  # of terms Approximation of pi      
           2 2.6666666666666670        3                        
          12 3.0584027659273332        3.1                      
         152 3.1350137774059244        3.14                     
         916 3.1405009508583017        3.141                    
        7010 3.1414500002381582        3.1415                   
      130658 3.1415850000208838        3.14159                  
      866860 3.1415915000009238        3.141592                 
     9653464 3.1415925500000141        3.1415926                
   116423306 3.1415926450000007        3.14159265               
   919102060 3.1415926525000004        3.141592653              
  7234029994 3.1415926534500005        3.1415926535