Random Kaczmarz Algorithm And Function-valued Padé-type Approximate Method And Its Applications

The research of this paper is divided into two parts.In the first part of this paper we mainly study the related numerical algorithms of random Kaczmarz methods in solving large sparse linear equations and the improved randomized Kaczmarz reconstruction algorithm is applied to the calculation of signal reconstruction in compressed sensing.The related work we have done can be summarized as follows:1.Based on the research of the Kaczmarz algorithm,a new numerical iterative method of solving the large linear systems is proposed by means of random sampling method to calculate the part of the residual.In the method we randomly selects k indices of rows of the coefficient matrix of linear system and then compute the largest absolute relative residual and construct a randomized sampling Kaczmarz(RSK)algorithm.the numerical experiments show that the method is effective.In addition,we extend the RSK algorithm to signal reconstruction problems in compressed sensing and establish the corresponding randomized sparse sampling Kaczmarz(Ra SSK)algorithm,and show that Ra SSK performs very well for online compressed sensing.2.Random extended Kaczmarz(REK)method and greedy random Kaczmarz(GRK)method are random iterative algorithms for solving large linear equations.based on the study of the two Kaczmarz algorithms we put foward a multi-step greedy random extended Kaczmarz(MGREK)method.This algorithm mainly integrates the advantages of GRK and REK algorithms,and can make full use of the remaining information of REK method.This paper compares this new algorithm with the algorithm proposed by predecessors through numerical calculation of different types of cases,which shows that this new method can achieve faster calculation results under various circumstances.Integral equation is an important branch of modern mathematics,which is closely related to differential equation,computational mathematics and random analysis.In addition,it is an important tool for the application of mathematics in mechanics,mathematical physics and engineering.In the research fields of quantum mechanics,transportation,electromagnetic scattering and computer graphics manipulations,many of them involve solving Fredholm integral equations of the second kind.The second part of this paper is mainly about the recursive method of Pad?e-type approximation,and it is applied to the calculation of Fredholm integral equation of the second kind.The work we have done can be summarized as follows:1.In order to avoid the calculation of high order determinant of functionvalue Pad?e-type approximation,by constructing a projection operator and combining Sylvester-equality,a recursive algorithm shaped like an inverted triangle is formed with the help of Sylvester theorem for the generated polynomial of the function value Padé approximation.Finally,numerical examples are given to illustrate the effectiveness of the algorithm.2.With the help of formal orthogonal polynomials and the three recursive relations of polynomials on linear functional,a recursive algorithm is formed for the generation polynomials of function value padé-type approximation,and the calculation of high-order determinant is transformed into the calculation of low-order determinant.Finally,the corresponding numerical examples are given and analyzed.