Newton iterative formula

Let r be the root and choose it as the initial approximation of r, so that the tangent L of the curve passes through this point. The equation of L is, find the abscissa of the intersection of L and X axis, call x 1 as the first approximation of R, and find the abscissa of the intersection of tangent and X axis by making a point tangent to the curve, which is called the second approximation of R. Repeat the above process to get the approximate sequence of R, which is called the second approximation of R. The above formula is called Newton iterative formula.

Solving nonlinear equations by Newton iteration method is an approximate method to linearize nonlinear equations. Expand a Taylor series in a certain neighborhood of a certain point, and take its linear part (that is, the first two terms of Taylor series) to make it equal to 0, that is, take it as an approximate equation of a nonlinear equation. If it is, its solution is, and thus an iterative relationship of Newton's iteration method is obtained.

It has been proved that if it is continuous and the zero point of the solution is isolated, then there is a region around the zero point, and Newton's method will converge as long as the initial value is located in this adjacent region. And if it is not 0, Newton's method will have the performance of square convergence. Roughly speaking, this means that the effective number of Newton method results will double with each iteration.

Soldiers often use the method of alternating cover attack when attacking. If the points on the number axis represent the positions of A and B, and the number in front is greater than the number in the back, then A >；; b、B& gt； A appears alternately, but now suppose there is a coward in the army and everyone takes care of him at the same time. Every charge is to let him follow. Whenever the person in front occupies a new position, give him the position, and then others will come forward to occupy the new position. That is to say, A is always in front of B, A advances, B keeps up, A gives its position to B (that is, B = A), then A advances to occupy a new position, and B keeps up until it occupies all positions, and the advance is over. Such a method that two numbers gradually approach a certain position one by one is called iterative method.

Iterative method, also known as trial and error method, is a process of recursively deriving new values from the old values of variables. Corresponding to iterative method, it is direct method (or one-time solution), that is, one-time solution. Iterative algorithm is the basic method to solve problems with computers. It makes use of the characteristics of computer's fast operation speed and suitable for repeated operation, so that the computer can repeatedly execute a set of instructions (or some steps), and every time this set of instructions (or these steps) is executed, the new value of a variable is deduced from its original value.

Using iterative algorithm to solve the problem, we need to do the following three aspects:

First, the iterative variables are determined

Among the problems that can be solved by iterative algorithm, at least one variable can directly or indirectly derive new values from old values, and this variable is an iterative variable.

Second, establish an iterative relationship

The so-called iterative relationship refers to how to deduce the formula (or relationship) of the next value from the previous value of a variable. The establishment of iterative relationship is the key to solve iterative problems, which can usually be done by recursion or reverse derivation.

Third, control the iterative process.

When does the iterative process end? This is a problem that must be considered when writing iterative programs. We can't let the iterative process go on endlessly. The control of iterative process can usually be divided into two situations: one is that the required number of iterations is a certain value, which can be calculated; The other is that the number of iterations required cannot be determined. For the former case, a fixed number of loops can be constructed to control the iterative process; In the latter case, it is necessary to further analyze the conditions available for ending the iterative process.