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Programme pour trouver la racine d'une équation en utilisant la méthode sécante

La méthode sécante est utilisée pour trouver la racine d'une équation f(x) = 0. Elle part de deux estimations distinctes x1 et x2 pour la racine. Il s'agit d'une procédure itérative impliquant une interpolation linéaire vers une racine. L'itération s'arrête si la différence entre deux valeurs intermédiaires est inférieure au facteur de convergence.

Exemples :  

Saisir : équation = x3+x-1 
             x1 = 0 x2 = 1 E = 0,0001
Sortir : Racine de l'équation donnée = 0,682326
               N° d'itération = 5




Algorithme  

Initialize: x1 x2 E n // E = convergence indicator calculate f(x1)f(x2) if(f(x1) * f(x2) = E); //repeat the loop until the convergence print 'x0' //value of the root print 'n' //number of iteration } else print 'can not found a root in the given interval'
C++
// C++ Program to find root of an  // equations using secant method #include    using namespace std; // function takes value of x and returns f(x) float f(float x) {  // we are taking equation as x^3+x-1  float f = pow(x 3) + x - 1;  return f; } void secant(float x1 float x2 float E) {  float n = 0 xm x0 c;  if (f(x1) * f(x2) < 0) {  do {  // calculate the intermediate value  x0 = (x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1));  // check if x0 is root of equation or not  c = f(x1) * f(x0);  // update the value of interval  x1 = x2;  x2 = x0;  // update number of iteration  n++;  // if x0 is the root of equation then break the loop  if (c == 0)  break;  xm = (x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1));  } while (fabs(xm - x0) >= E); // repeat the loop  // until the convergence  cout << 'Root of the given equation=' << x0 << endl;  cout << 'No. of iterations = ' << n << endl;  } else  cout << 'Can not find a root in the given interval'; } // Driver code int main() {  // initializing the values  float x1 = 0 x2 = 1 E = 0.0001;  secant(x1 x2 E);  return 0; } 
Java
// Java Program to find root of an  // equations using secant method class GFG {    // function takes value of x and   // returns f(x)  static float f(float x) {    // we are taking equation   // as x^3+x-1  float f = (float)Math.pow(x 3)   + x - 1;    return f;  }    static void secant(float x1 float x2  float E) {    float n = 0 xm x0 c;  if (f(x1) * f(x2) < 0)   {  do {    // calculate the intermediate  // value  x0 = (x1 * f(x2) - x2 * f(x1))  / (f(x2) - f(x1));    // check if x0 is root of  // equation or not  c = f(x1) * f(x0);    // update the value of interval  x1 = x2;  x2 = x0;    // update number of iteration  n++;    // if x0 is the root of equation   // then break the loop  if (c == 0)  break;  xm = (x1 * f(x2) - x2 * f(x1))   / (f(x2) - f(x1));    // repeat the loop until the   // convergence   } while (Math.abs(xm - x0) >= E);     System.out.println('Root of the' +  ' given equation=' + x0);    System.out.println('No. of '  + 'iterations = ' + n);  }     else  System.out.print('Can not find a'  + ' root in the given interval');  }    // Driver code  public static void main(String[] args) {    // initializing the values  float x1 = 0 x2 = 1 E = 0.0001f;  secant(x1 x2 E);  } } // This code is contributed by Anant Agarwal. 
Python3
# Python3 Program to find root of an  # equations using secant method  # function takes value of x  # and returns f(x)  def f(x): # we are taking equation  # as x^3+x-1  f = pow(x 3) + x - 1; return f; def secant(x1 x2 E): n = 0; xm = 0; x0 = 0; c = 0; if (f(x1) * f(x2) < 0): while True: # calculate the intermediate value  x0 = ((x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1))); # check if x0 is root of  # equation or not  c = f(x1) * f(x0); # update the value of interval  x1 = x2; x2 = x0; # update number of iteration  n += 1; # if x0 is the root of equation  # then break the loop  if (c == 0): break; xm = ((x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1))); if(abs(xm - x0) < E): break; print('Root of the given equation =' round(x0 6)); print('No. of iterations = ' n); else: print('Can not find a root in ' 'the given interval'); # Driver code  # initializing the values  x1 = 0; x2 = 1; E = 0.0001; secant(x1 x2 E); # This code is contributed by mits 
C#
// C# Program to find root of an  // equations using secant method using System; class GFG {    // function takes value of   // x and returns f(x)  static float f(float x)   {    // we are taking equation   // as x^3+x-1  float f = (float)Math.Pow(x 3)   + x - 1;  return f;  }    static void secant(float x1 float x2  float E)     {    float n = 0 xm x0 c;  if (f(x1) * f(x2) < 0)   {  do {    // calculate the intermediate  // value  x0 = (x1 * f(x2) - x2 * f(x1))  / (f(x2) - f(x1));    // check if x0 is root of  // equation or not  c = f(x1) * f(x0);    // update the value of interval  x1 = x2;  x2 = x0;    // update number of iteration  n++;    // if x0 is the root of equation   // then break the loop  if (c == 0)  break;  xm = (x1 * f(x2) - x2 * f(x1))   / (f(x2) - f(x1));    // repeat the loop until   // the convergence   } while (Math.Abs(xm - x0) >= E);     Console.WriteLine('Root of the' +  ' given equation=' + x0);    Console.WriteLine('No. of ' +   'iterations = ' + n);  }     else  Console.WriteLine('Can not find a' +   ' root in the given interval');  }    // Driver code  public static void Main(String []args)   {    // initializing the values  float x1 = 0 x2 = 1 E = 0.0001f;  secant(x1 x2 E);  } } // This code is contributed by vt_m. 
PHP
 // PHP Program to find root of an  // equations using secant method // function takes value of x  // and returns f(x) function f( $x) { // we are taking equation // as x^3+x-1 $f = pow($x 3) + $x - 1; return $f; } function secant($x1 $x2 $E) { $n = 0; $xm; $x0; $c; if (f($x1) * f($x2) < 0) { do { // calculate the intermediate value $x0 = ($x1 * f($x2) - $x2 * f($x1)) / (f($x2) - f($x1)); // check if x0 is root  // of equation or not $c = f($x1) * f($x0); // update the value of interval $x1 = $x2; $x2 = $x0; // update number of iteration $n++; // if x0 is the root of equation // then break the loop if ($c == 0) break; $xm = ($x1 * f($x2) - $x2 * f($x1)) / (f($x2) - f($x1)); // repeat the loop // until the convergence } while (abs($xm - $x0) >= $E); echo 'Root of the given equation='. $x0.'n' ; echo 'No. of iterations = '. $n ; } else echo 'Can not find a root in the given interval'; } // Driver code { // initializing the values $x1 = 0; $x2 = 1; $E = 0.0001; secant($x1 $x2 $E); return 0; } // This code is contributed by nitin mittal. ?> 
JavaScript
<script> // JavaScript Program to find root of an // equations using secant method // function takes value of x and returns f(x) function f(x) {  // we are taking equation as x^3+x-1  let f = Math.pow(x 3) + x - 1;  return f; } function secant(x1 x2 E) {  let n = 0 xm x0 c;  if (f(x1) * f(x2) < 0) {  do {  // calculate the intermediate value  x0 = (x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1));  // check if x0 is root of equation or not  c = f(x1) * f(x0);  // update the value of interval  x1 = x2;  x2 = x0;  // update number of iteration  n++;  // if x0 is the root of equation then break the loop  if (c == 0)  break;  xm = (x1 * f(x2) - x2 * f(x1)) / (f(x2) - f(x1));  } while (Math.abs(xm - x0) >= E); // repeat the loop  // until the convergence  document.write('Root of the given equation=' + x0.toFixed(6) + '  
'
); document.write('No. of iterations = ' + n + '
'
); } else document.write('Can not find a root in the given interval'); } // Driver code // initializing the values let x1 = 0 x2 = 1 E = 0.0001; secant(x1 x2 E); // This code is contributed by Surbhi Tyagi. </script>

Complexité temporelle : O(1)
Espace auxiliaire : O(1)

Référence  
https://en.wikipedia.org/wiki/Secant_method
 

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