Advantages and disadvantages of iterative algorithms, and what are the iterative algorithms?

There are no advantages or disadvantages, this algorithm is just a way to solve the problem of convergence. Advantages and disadvantages need to be compared, and there is no comparison object and the same comparison conditions, how to talk about advantages and disadvantages. There are many algorithms that can be solved for every problem, and it is not necessarily good or bad to iterate.

For a certain problem, the above different algorithms have advantages and disadvantages.

Iterative algorithms are a basic way to solve problems with computers. It takes advantage of the characteristics of the computer's fast computing speed and suitable for repetitive operations, allowing the computer to repeatedly execute a set of instructions (or a certain step), and each time the set of instructions (or these steps) is executed, a new value is pushed from the original value of the variable.

To solve problems with iterative algorithms, we need to do a good job in the following three aspects:

1. Determine the iteration variables. In the problem that can be solved by iterative algorithms, there is at least one variable that directly or indirectly continuously rolls out new values from old values, and this variable is the iterative variable.

2. Establish iterative relationships. An iterative relationship is a formula (or relation) on how to deduce the next value of a variable from its previous value. The establishment of iterative relationships is the key to solving iterative problems, and can usually be done using recursive or backward methods.

3. Control the iterative process. When does the iteration process end? This is something that must be considered when writing an iterative program.

You can't let the iteration process repeat itself endlessly. The control of the iterative process can usually be divided into two situations: one is that the number of iterations required is a definite value that can be calculated; The other is that the number of iterations required cannot be determined.

In the former case, a fixed number of loops can be built to control the iterative process; In the latter case, the conditions used to end the iterative process need to be further analyzed. When using the iterative method to find the root, you should pay attention to the following two possible situations:

1) If there is no solution to the equation, the approximate root sequence obtained by the algorithm will not converge, and the iterative process will become an endless loop, so whether the equation has a solution should be checked before using the iterative algorithm, and the number of iterations should be limited in the program.

2) Although the equation has a solution, the improper selection of the iterative formula or the unreasonable selection of the initial approximate root of the iteration will also lead to the failure of the iteration.

In computational mathematics, iteration solves a problem by finding a series of approximate solutions starting from an initial estimate (generallySolve equationsor a system of equations), which is the method used to achieve this process in cavity reform.

Iterative approach. The counterpart is the direct method (or one-time solution), which solves the problem at once. In general, direct solutions are always preferred if possible.

However, when we encounter complex problems, especially when there are many unknowns and the equation is nonlinear, we cannot find a direct solution (for example, there is no analytic solution for the algebraic equation of the fifth order and higher, see Abel's theorem, then we may be able to seek the approximate solution of the equation (system) through the large-code iterative method.

The most common iterative method is Newton's method.

Others include gradient descent, conjugate iteration, variable-scale iteration, and least squares.

Linear programming, nonlinear programming, simplex method, penalty function method, slope projection method, genetic algorithm.

Simulated annealing and much more.

Method. 1. Constant iterative method.

This method is easy to derive and easy to implement and analyze, but it can only ensure the convergence of some specific form matrix solutions. Examples of steady-iterative methods include the Jacobian method, the Gauss-Seidel iteration, and the successive superrelaxation iteration (SOR). The linear steady-iterative method is also known as the relaxation method.

2. Krylov subspace method.

The approximate solution is obtained by rounding the subspace which minimizes the margin. The prototype of Krylov's subspace method is the conjugate gradient method (CG), and other methods include the generalized minimum residual method (GMRES) and the double conjugate gradient method (BICG).

Fundamentals of the Iterative Approach:

The iterative method, also known as the tossing method, is a process of continuously using the old value of a variable to recursively deduce the new value, which is similar to the iterative method.

The corresponding is the direct method (or the one-time solution method), that is, the solution to the problem at one time.

Iterative algorithm is a basic method of solving problems with computers, which uses computers to calculate fast and suitable for repeatability.

The characteristics of the operation are that the computer repeats the execution of a set of instructions (or a certain step), and each time this set of instructions (or these steps) is executed, a new value is deduced from the original value of the variable and poor selling volume, and the iterative method is divided into exact iteration and approximate iteration. A typical iterative approach is the "dichotomy."

with"Newton's iterative method.

It is an approximate iterative method.

The convergence theorem of iterative methods can be divided into the following three categories:

1. Local convergence definite know-how theory: Assuming that the problem solution exists, it is concluded that the iterative method converges when the initial approximation is sufficiently close to the solution.

2. Semi-local convergence theorem: without assuming the existence of the solution, it is concluded that the iterative method converges to the solution of the problem according to the conditions satisfied by the iterative method at the initial approximation.

3. Large-scale convergence theorem: Under the condition that the approximation of the beginning of the destruction is not assumed to be fully close to the solution, it is concluded that the iterative method converges to the solution of the problem.

It is widely used in computing and other problems.

The iterative method, also known as the tossing and turning method, is a process of continuously using the old value of a variable to recursively deduce the new value, and the iterative method corresponds to the direct method, that is, solving the problem at one time. Iterative methods are further divided into exact iteration and approximate iteration.

Dichotomy. and "Newton's Iterative Method."

It is an approximate iterative method. Iterative algorithms are a basic way to solve problems with computers. It takes advantage of the characteristics of the computer's fast computing speed and suitable for repetitive operations, allowing the computer to repeatedly execute a set of finger cavity disturbance orders (or certain steps), and each time this set of instructions (or these steps) is executed, a new value is deduced from the original value of the variable.

Iterative is the process of numerical analysis to solve the Lichang problem (usually by solving an equation or a system of equations) by finding a series of approximate solutions from an initial estimate, and the methods used to achieve this process are collectively called iterative methods.

Tossing and dividing.

Also known as Euclidean's algorithm.

euclidean algorithm), is to find the greatest common divisor.

One of the methods. It is done by dividing the larger number by the smaller number, and then using the remainder that appears.

The first remainder) is removed from the divisor, and then the first remainder is removed with the resulting remainder (the second remainder), and so on until the last remainder is 0.

If you are finding the greatest common divisor of two numbers, then the final divisor is the greatest common divisor of the two numbers. Another way to find the greatest common divisor of two numbers is the method of more derogation.