Description of algorithms for solving equations using Newton Iteration Method and bipartite Method

1. Newton Iteration Method:

The method for finding a solid root near x0 by using the Newton Iteration Method f (x) = 0 is
(1) Select an approximate root X1 that is close to X;
(2) Use X1 to obtain F (X1 ). In ry, x = X1 is used, and f (x) is used in F (X1 );
(3) f (X1) is used as the tangent of f (x), and the X axis is X 2. You can use the formula to obtain X2. Therefore
(4) use X2 to obtain F (X2 );
(5) then f (X2) is used as the tangent of f (x), and the X axis is equal to X2;
(6) then use X3 to find F (X3 ),... Keep Seeking until it is close to the real root. When the root difference obtained twice | xn + N | ≤ε, we think xn + 1 is close enough to the real root.
The Newton iteration formula is:

Ii. Binary method

Take any two points x1 and X2 to judge whether there is any solid root in the (x1, x2) range. If the characters F (X1) and F (X2) are opposite, there is a solid root between them. Take the midpoint X of (x1, x2) and check whether f (x) and F (X1) are the same symbol. If different numbers are used, the real root is in the (x, X1) interval, in this way, the root search range has been reduced by half. And further narrow down the scope in the same way. Find the midpoint "X" of X1 and X2 (x2 = x) and discard the half interval. If f (x) is the same as F (X1), it means that the root is in the (x, X2) interval, then the midpoint of x and X2 is obtained, and half of the interval is discarded. Use this method to narrow down the range until the range is quite small.

From: http://hefuliang.cai.swufe.edu.cn/EXAMPLE/PROG0031.htm