A New Iterative Criterion for H‐Matrices

Diagonal dominance, Schur complements and some classes of H-matrices and P-matrices

“…This calculation also shows that A (2) [2,3,4], the Schur complement of A [1], is not SDD 1 by rows.…”

“…The matrixà (1) associated with any symmetric diagonal dominant pivoting strategy is again a matrix SDD 1 by rows and is given byà (1) = P 1i AP T 1i . By Theorem 3.4 (i), its Schur complement A (2) [2, .…”

“…However, for large sparse matrices, it is more convenient the use of iterative methods for this purpose as well as for the construction of D. This is an important recent field of research and has been studied in the last years (cf. [1][2][3][10][11][12]). In these papers, the practical interest and applications of this subject is shown, as well as its difficulty.…”

Diagonal Matrix Scaling and H-Matrices

“…Since the coefficient matrix in system (2) is of order n × 2n, in solving the corresponding LP, it is needed O(n 2 ) computations in the worst case, at each iteration of the simplex algorithm. This complexity is the same as some iterative methods that proposed for characterization of H-matrices [2,12]. Those algorithms need to update a matrix of order n×n at each iteration and therefore the amount of computation that they need for one iteration is O(n 2 ).…”

“…Some iterative algorithms have been proposed based on this definition to find out if a matrix is an H-matrix or not, [2,3,12]. They all try to find a point diagonal matrix D to show that Definition 1.2 holds.…”