a b s t r a c tThe Moore-Penrose inverse of an arbitrary matrix (including singular and rectangular) has many applications in statistics, prediction theory, control system analysis, curve fitting and numerical analysis. In this paper, an algorithm based on the conjugate Gram-Schmidt process and the Moore-Penrose inverse of partitioned matrices is proposed for computing the pseudoinverse of an m × n real matrix A with m ≥ n and rank r ≤ n. Numerical experiments show that the resulting pseudoinverse matrix is reasonably accurate and its computation time is significantly less than that of pseudoinverses obtained by the other methods for large sparse matrices.

No abstract

No abstract

This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, the matrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus the approximate solutions are obtained. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. To illustrate the efficiency of our method, some numerical examples are given. ( ) ( ) + −1 ∑ =0 ( ) ( ) ( ) = ( ) , ≥ 1, = + , ∈ [ , ] , ∈ [ , ] ,with the initial conditions ( ) (0) = , = 0, 1, . . . , − 1.(2)

No abstract