Gaussian elimination is the concrete realization of Gaussian algorithm, and its main steps include: selecting principal elements, exchanging lines, eliminating elements and substituting. Selecting the main element means selecting a non-zero element from each column as the main element, exchanging rows means exchanging the row where the main element of the current column is located with the row where the diagonal element is located, and eliminating elements means eliminating other elements of the current column into 0, substituting the solved unknowns into the original equation, and solving the remaining unknowns.
The advantage of Gaussian algorithm is that it can quickly solve linear equations, especially large linear equations, and its efficiency is much higher than that of direct method. In addition, Gaussian algorithm can also be used to find the inverse matrix and solve the linear least squares problem.
However, Gaussian algorithm also has some shortcomings. First of all, Gaussian algorithm needs to store a large number of intermediate results, which will occupy a lot of memory space. Secondly, the stability of Gaussian algorithm is poor, and when the coefficient matrix is close to singularity, it may lead to calculation errors. Finally, the calculation complexity of Gaussian algorithm is high, especially for large linear equations, its calculation time may be very long.
Generally speaking, Gaussian algorithm is a very practical numerical method, which is widely used in mathematics, physics, engineering and other fields. However, due to its own shortcomings, we need to pay attention to choosing appropriate methods and skills to improve the accuracy and efficiency of calculation.