Common algorithm design method (I)-Iterative Method

Iteration Method is a common algorithm design method used to obtain the approximate root of equations or equations. Set the equation to f (x) = 0, use a mathematical method to export the equivalent form X = g (x), and then follow these steps:
(1) Select the approximate root of an equation and assign it to variable x0;
(2) Save the value of x0 to the variable X1, calculate g (X1), and save the result to the variable x0;
(3) when the absolute value of the difference between x0 and X1 is smaller than the specified precision requirement, repeat the calculation in step (2.
If the equation has roots and the approximate root sequence obtained by the above method converges, The x0 obtained by the above method is considered as the root of the equation. The above algorithm is represented in the form of a C program:
[Algorithm] iterative method is used to find the root of the equation.
{X0 = initial approximate root;
Do {
X1 = x0;
X0 = g (X1);/* calculate a new approximate root based on a specific equation */
} While (FABS (x0-x1)> epsilon );
Printf ("the approximate root of the equation is % F/N", X0 );
}
Iterative algorithms are also often used to find the root of the equations.
X = (x0, X1 ,..., Xn-1)
Set the equations:
Xi = GI (x) (I = 0, 1 ,..., N-1)
The Iterative Algorithm for Finding the root of the equations can be described as follows:
[Algorithm] iterative method is used to obtain the root of the equations.
{For (I = 0; I <n; I ++)
X = Initial approximate root;
Do {
For (I = 0; I <n; I ++)
Y= X;
For (I = 0; I <n; I ++)
X= GI (X );
For (delta = 0.0, I = 0; I <n; I ++)
If (FABS (Y-X)> Delta) Delta = FABS (Y-X);
} While (delta> epsilon );
For (I = 0; I <n; I ++)
Printf ("the approximate root of variable X [% d] is % F", I, X);
Printf ("/N ");
}
Pay attention to the following two possible situations when using iterative methods to root out:
(1) If the equation has no solution, the approximate root sequence obtained by the algorithm will not converge, and the iteration process will become an endless loop. Therefore, before using the iterative algorithm, we should first check whether the equation has a solution, limit the number of iterations in the program;
(2) Although the equation has solutions, improper selection of iteration formulas or unreasonable selection of the initial approximate root of iteration may also lead to iteration failure.