A Class of Kung–Traub-Type Iterative Algorithms for Matrix Inversion

Abstract

Kung-Traub (J ACM 21:643-651, 1974) constructed two optimal general iterative methods without memory for finding solution of nonlinear equations. In this work, we are going to show that one of them can be applied for matrix inversion. It is observed that the convergence order 2 m can be attained using 2m matrix-matrix multiplications. Moreover, a method with the efficiency index 10 1/6 ≈ 1.4677 will be furnished. To justify that our procedure works efficiently, some numerical problems are included.

Constructing a Matrix Mid-Point Iterative Method for Matrix Square Roots and Applications

“…which shows a convergence rate of four for Equation (14). Note that the norm of sign(A) can be large arbitrarily, though its eigenvalues are ±1.…”

“…Noting that the iterative solver in Equation (10) does not satisfy the Kung-Traub conjecture regarding the construction of iterative schemes without memory to solve equations [14], but it has an important feature. As a matter of fact, if we pursue the optimality of the iterative solvers for nonlinear equations, then we lose the global convergence behavior in solving Equation ( 5), which clearly limits the matrix application of such solvers.…”

“…It is remarked that the convergence rate is not the only important point in an iterative scheme; the cost per iteration is also important for determining the total cost of the method. By comparing Equations ( 12) and (14) to Equations ( 17) and (18), it is clear that all the methods need four matrix-matrix multiplications and only one matrix inverse per cycle, which means that Equations ( 12)-( 14) are competitors to Padé variants of the same order with global convergence.…”

Calculating the Weighted Moore–Penrose Inverse by a High Order Iteration Scheme

“…Schulz-type methods for the calculation of the WMP inverse are sensitive to the choice of the initial value; that is, the initial choice of matrix must be close enough to the generalized inverse so as to guarantee the scheme to converge [20]. More precisely, convergence can only be observed if the starting matrix is chosen carefully.…”

An Improved Computationally Efficient Method for Finding the Drazin Inverse

“…Methods for calculating the generalized inverses are a topic of current investigation in computational mathematics (see, e.g., [1,2]). Enormous amount of work in the topic of generalized inverses has been done during the past 63 years.…”