regardless of the numerical algorithm, its design principle is
to transform the complexity into simple repetition , or, through simple repetitive generation complex, in the algorithm design and algorithm implementation process, the repetition is the power [1]. ----Preface
In general, a linear equation group can be expressed as
Ax = b
Where a is called the coefficient matrix, B is called the right-side item, and x is the unknown vector to be asked.
The application of the iterative method. If the number of unknowns is not many, the elimination of the method (including the choice of the main element of the way) can be solved perfectly, but in reality, such as the finite element equation, the coefficient matrix is often
large-scale sparse matrices(the so-called sparse, refers to the matrix only a small amount of element value is not 0), this time again using the elimination method (in fact, it can be transformed into matrix form of LU decomposition), it will destroy the sparsity of the coefficient matrix, resulting in a surge in computational capacity.
Look at the results in two simple cases first. If the coefficient matrix is diagonal, then the equation can be easily solved, slightly more complex, if the coefficient matrix is the lower triangular matrix, then you can first find out the first unknown, and then find the second unknown, has been sequentially continued until the last unknown.
Mentioned in [1] (P139),
To solve the iterative method of the linear equation Group, the essence is to gradually diagonalization or triangularThe solution process of the linear equations is processed into the repetition of the solution process of the diagonal equation group or the trigonometric equation Group. When the process of solving a diagonal equation group is repeated, it corresponds to the Jacobi iterative method, and corresponds to the Gauss-seidel iterative method when the process of solving the trigonometric equation Group is repeated. Further, if
using relaxation technique for Gauss-seidel iterative method(The method of weighting the two accompanying approximation to better approximate the target value, the key of which is the selection of the weighting coefficient i.e. the relaxation factor),
the iterative method of super-relaxation (SOR) is obtained.(the "super" here means that the weighted coefficients can not be used in the [0,1] range); Considering the coefficient matrix of symmetric positive definite, because its upper triangular matrix is the transpose of its lower triangular matrix, the Sor iterative method is used for the upper triangular form, and then the Sor iteration method can be used for the lower triangular form. Different from the former is only the sequence of unknown iterations, the latter is in reverse order. Well, since
two times using the SOR iteratorLaw, then naturally call it
Ssor Iterative Method[2]. Oh,no! In fact, it is not such a drop, ssor inside the first S is symmetrical (symmetric) The first letter of the word. Using the matrix in the Ssor iterative method as the preprocessing matrix in the conjugate gradient method, the corresponding SSOR-PCG method can be obtained [3].
whether the iterative method works, that is, whether the iterative sequence can approximate the true solution of the problem, it needs
Consider the problem of convergenceThe Rough analysis, from the difference between the previous and the two sequences, naturally requires the distance between them (can be given a different form of definition, the common 1-norm, 2-norm and infinity norm) is getting smaller, until the trend of 0. Notice that the sequence of iterations is a sequence of vectors that borrows the conclusion of a numerical sequence (considering that a geometric series a_n=k*a_{n-1} tends to be 0, then the absolute value of K must be less than 1), which requires that the norm of the iteration matrix (which can be regarded as the absolute value of the number) is less than 1.
PostScript: Strongly recommend Mr. Wang Nengsu's book.
Resources:
[1] "Computational Method-algorithm design and its implementation of MATLAB (2nd edition)" Chap5 of linear equations, Wang Nengsu, Huazhong University of Science and Technology Press, 2012
[2] Matrix calculation (3rd edition) 10.1.6 symmetric Sor method, Gene H. golub,charles F. Van Loan, Shing, people's post and Telecommunications press, 2011
[3] "parallel computation of finite element structure analysis" CHAP5 iterative method for solving large sparse finite element equations by parallel preprocessing, Chou wait, Science Press, 1994
Iterative method of linear equation group
