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However, for systems with a finite number of complex solutions, these polynomial equations have now been well understood, and there are effective methods to solve them. The method of moments is a method of discretizing continuous equations into algebraic equations, which is suitable for solving differential equations and integral equations. In the process of solving the method of moments, the generalized moment needs to be calculated, so it is named. The method of moments includes the following three basic processes: (1) discretization process: the main purpose is to transform operator equations into algebraic equations; (2) Sampling detection process: the main purpose is to convert the problem of solving algebraic equations into the problem of solving matrix equations; (3) Matrix inversion process: the work it does is to transform the integral equation into a difference equation, or to integrate the integral equation into a finite sum, so as to establish an algebraic equation group.

Many linear algebra books think that it is more suitable for computers to solve the inverse of a matrix, because it involves less algebraic operations, but it may not be effective for matrices in 3D. Cramer's rule is a method for solving linear equations. The solution of the unknown in the i-th column is the determinant of the replacement matrix generated by replacing the i-th column of the coefficient matrix with the constant column, divided by the number obtained by the determinant of the original matrix. In the operation of matrix data, STL or a third-party library are generally not used, but data pointers are directly used.

A variety of methods for discretizing various regions to generate grids have been developed, collectively known as grid generation technology. The production process of grid is the coordinate mapping process from computing plane to physical plane. Explain the neural network as a discrete format for solving differential equations? The field of numerical solution will pay attention to the numerical convergence of discrete schemes, but what is the connection between this and differential equations? How to map the input-output mapping of the network connection to the infinite dimensional mapping of differential equations? Using the knowledge of dynamic system to analyze the properties of neural network? As a numerical analysis method, the whole solution process of finite element method is completely different from those classical analytical methods in solving mathematical and physical equations. The following table lists the differences between the mechanical analytical solution and the finite element method. While the function represents the change of state, the differential equation studies the relationship between the change of state and its change field.

The time steps can be obtained very small, there is no convergence problem, and it is easy to write programs for parallel and distributed calculation. In eigen, there are many methods to solve sparse matrix equations (Systems). Due to the particularity and performance requirements of sparse matrices, special care should be taken in the solution process. This section introduces the sparse linear solver in eigen The main steps shared by all solvers are also introduced.

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