## Newton's Method in Python We will work through a couple of examples of using Newton's method in Python. ### Example 1 Let's say we have a simple function given by the following equation: ```python def f(x): return x**2 - 2 ``` and we want to find the value of *x* for which *f(x) = 0*. Newton's method is a numerical method that allows us to approximate the solution to this equation. Here's the Python code that implements Newton's method to solve for *x*: ```python def f(x): return x**2 - 2 def df(x): return 2*x def f_solve(f, df, x0, delta): for i in range(n): print(i, x0) x1 = x0 - f(x0) / df(x0) if abs(x1 - x0) <= delta: return x1 assert False, "cannot find solution" x0 = x1 n = 1000 delta = 0.0001 x = f_solve(f, df, 1, n, delta) print("f(x) =", f(x)) print("x =", x) ``` In this code: * `f(x)` is our function, which takes a single input *x* and returns the value of *x� - 2*. * `df(x)` is the derivative of our function, which returns the value of *2x*. * `f_solve(f, df, x0, delta)` is the function that implements Newton's method. It takes the function `f`, its derivative `df`, an initial guess `x0`, and a tolerance `delta`. The function iterates until the difference between successive approximations is less than the tolerance. * `n` is the maximum number of iterations allowed. * `delta` is the tolerance value. **Running the code:** We set our initial guess to *1*, the maximum number of iterations to *1,000*, and the tolerance to *0.0001*. When we run this code, we get the following output: ```bash python newton_example.py ``` The output will show the iterations performed by the code until it finds a solution. The final value of *x* will be approximately *1.41421*, which is the square root of 2. ### Example 2 Now let's say we have a curve defined by the following equation: ```python def f(x, y, A, D): K = A * x * y / (D / 2)**2 return x * y + K - (D / 2)**2 - K * D**2 ``` We know the values of *x*, *A*, and *D*, and we want to find the value of *y* that satisfies this equation. To solve for *y*, we'll use Newton's method again. The Python code is similar to the previous example, but we need to define new functions for our curve and its derivative with respect to *y*. Here's the modified code: ```python def f(x, y, A, D): K = A * x * y / (D / 2)**2 return x * y + K - (D / 2)**2 - K * D**2 n = 100 delta = 0.0001 x = 120 y = 90 A = 10 D = 200 def f_D(d): return f(x, y, A, d) def df_d(d): return 4 * A * x * y * d / 2 - d / 2 d = f_solve(f_D, df_d, D, n, delta) print("d", d) print("f_D(d)", f_D(d)) def f_y(y): return f(x, y, A, d) def df_dy(y): return x + 4 * A * x * y * d / 2 - A * x * d * (y * d / 2) / d**2 y = f_solve(f_y, df_dy, y, n, delta) print("y", y) print("f_y(y)", f_y(y)) ``` Here, we've defined: * `f_y(y)` is the function for the curve, taking *y* as input and using the known values of *x*, *A*, and *D*. * `df_dy(y)` is the partial derivative of the curve's function with respect to *y*. **Running the code:** We'll execute this code and get the following output: ```bash python newton_example.py ``` The output shows that Newton's method finds a value for *y* that is close enough to satisfy the curve's equation. ### Conclusion Newton's method is a powerful tool for finding solutions to equations. We have seen how to use it in Python, and how to modify it to solve different problems.