On iterative methods for the quadratic matrix equation with M-matrix

Some predictor-corrector-type iterative schemes for solving nonsymmetric algebraic Riccati equations arising in transport theory

“…where k k 2 is the 2-norm for a vector and eps D 2 53 1.1 10 16 is the relative accuracy of the floating point operation of the computer. We compare ITER1 and ITER2 with the modified simple iteration (MSI) method (29), the NBJM (10), and the NBGSM (30) with different parameters˛and c. All these methods were tested in CPU time, number of iterations, and accuracy.…”

“…For the nonsymmetric algebraic Riccati equation (1), the solution of practical interest is the minimal positive solution. The corresponding numerical iterative methods for finding the minimal positive solution of (1) have been extensively studied [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].…”

A structure-preserving doubling algorithm for solving a class of quadratic matrix equation with $M$-matrix

“…Such qme arises from an overdamped vibrating system [24,25]. By left multiplying Ã−1 [29], without changing the M -matrix structure of it, qme (1.1) can be reduced to the following form…”

“…where B is a nonsingular M -matrix and C is an M -matrix such that B −1 C ≥ 0. It is known that (1.2) has a maximal nonpositive solvent Φ under the condition that [29] B − C − I is a nonsingular M -matrix. (1.3) This solvent Φ is the one of interest.…”

“…(1.3) This solvent Φ is the one of interest. Various iterative methods have been developed to obtain the maximal nonpositive solvent of qme (1.2) with assumption (1.3), including the Newton's method and Bernoulli-like methods (fixed-point iterative methods) [29], modified Bernoulli-like methods with diagonal update skill [18]. Newton's method is not competitive in terms of CPU time since there is a generalized Sylvester matrix equation to solve in each Newton's iterative step.…”