The linear equations are transformed into matrix form, that is, Ax=b, where a is the coefficient matrix, x is the unknown vector and b is the constant vector. The coefficient matrix A is decomposed into A=M-N, where m is a decomposition matrix of A and n is another decomposition matrix of A, and the linear equations are transformed into an iterative format, that is, x(k+1) = m (-1) NX (k)+m (-1) b.
Select the initial solution vector x(0) and perform iterative calculation until some convergence conditions are met. The basic idea of iterative method for linear equations is to gradually approximate the solution of linear equations and perform iterative calculation until certain convergence conditions are met. Its advantages are simple calculation and easy implementation, but it is necessary to choose appropriate decomposition matrix and convergence conditions, otherwise the iterative process may not converge or the convergence speed may be slow.
Brief introduction of linear equation;
Linear equations are a set of linear equations, in which each equation is a linear equation in the form of a 1 x 1+a2X2+…+anxn = b, where x1,x2, …, xn are unknowns, a1,a2, …, an are coefficients, and b. Linear equations are the basic concepts in mathematics, which are widely used in physics, engineering, economy and other fields.
There are many methods to solve linear equations, such as gauss elimination, LU decomposition and iteration. Among them, gauss elimination is a basic solution method, and its basic idea is to transform linear equations into simplified ladder matrix through elimination iteration, so as to solve unknown values.
LU decomposition method is a method to decompose the coefficient matrix into lower triangular matrix and upper triangular matrix, which can effectively reduce the amount of calculation. Iterative method is an iterative calculation method by gradually approaching the solution of linear equations until certain convergence conditions are met.