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Secant Method/Description Page 1.PNG

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+"""
+Created on Mon Jan 30 12:53:04 2023
+
+@author: utkucaylan
+
+=============================================================================================
+Numberical Methods 
+    --> Roots of Equations (i. Bracketing ii. Open iii. Roots of Polynomials)
+        --> Open Methods: Only a single starting value of x is required
+                          i.   Simple Fixed-Point Iteration
+                          ii.  Newton Rapson Method
+                          iii. Secant Method
+                          iv.  Brent`s Method 
+                          v.   Multiple Roots
+                          vi.  Systems of Nonlinear Equations
+                          
+            --> Secant Method
+            
+A potential problem in implementing the Newton-Raphson method is the evaluation of the
+derivative. Although this is not inconvenient for polynomials and many other functions,
+there are certain functions whose derivatives may be extremely difficult or inconvenient to
+evaluate. For these cases, the derivative can be approximated by a backward finite divided
+difference.
+
+P.S. See the figures for more details.
+=============================================================================================
+"""
+
+"""
+
+Problem: Use secant method to locate the root of f(x) = e^{−x} − x. Start
+with initial estimates of x_{-1} = 0 and x_{0} = 1
+
+"""
+import math
+import numpy as np
+
+
+
+x_i = 0
+x_ii = 1
+counter = 0
+while counter < 100:
+  counter = counter + 1
+  #function value at x_i
+  f_i = math.exp(-x_i) - x_i
+  #function value at x_ii
+  f_ii = math.exp(-x_ii) - x_ii
+  
+  # compute the new root at x_iii
+  x_iii = x_ii - (f_ii*(x_i-x_ii))/(f_i-f_ii) 
+
+  # percentage error
+  delta_x = np.abs((x_iii - x_ii)/x_ii*100)
+  x_i = x_ii
+  x_ii = x_iii
+  
+  print("The percentage error %.8f" % delta_x)
+  if delta_x < 1e-6:
+    print("The total number of iterations is %.0f" % counter)
+    print("The root of the given function is %.4f" % x_iii)
+    break
+