A Unified Iterative Format For Solving Singular And Nonsingular Least-Squares Problems

Whether it is to solve partial differential equations or the existing data points for curve fitting, eventually we will be faced with solving a set of linear equations. With the development of science and technology, now the data is becoming more large then before. The coefficient matrix corresponding to Linear equations is large and sparse matrix. The traditional solving linear equations methods are direct method and iterative method. The direct method uses LU decomposition to make the matrix into upper or lower triangular matrix, and then back to the substitution of the solution.The iterative method is mainly setting the initial value and then iterate to approach the solution. In solving large sparse matrix and singular matrix, the direct method is not applicable. The Modified Richardson method has to solve the maximum and minimum eigenvalues when determine the optimal iteration factor in the method. The CG method is related to the construction of the conjugate vector group, LSQR relates to the Golub-Kahan bi-diagonalization, TSVD (Truncated Singular Value Decomposition) and BTSVD(Block Truncation Singular Value Decomposition) are related to the singular value decomposition, so these algorithms need to spend a long time to solve the large sparse matrix, especially when the matrix’s rank is deficiency.Aiming at the above problems, this paper proposes an iterative algorithm which can effectively avoid the singular value decomposition, also do not need to bi-diagonalization and construct the conjugate vector group. It can be used for solving large sparse linear equations, and the singularity of the coefficient matrix is not effect for the algorithm solution. We calculated the infinite norm of the matrix and sum each column. The diagonal elements of diagonal matrices which use to preprocess the coefficient matrix are the two product. Then use iteration method similar to the Modified Richardson for solving the equations. Through numerical experiments comparing the changes of the classical LSQR algorithm and Richardson iterative method for solving large sparse full rank matrix and rank deficiency matrix in the iterative process of residuals, and the selection of initial values is analyzed. Then the algorithm this paper shows is superior to the existing iterative algorithms in some aspects.