CKV-type matrices with applications

On Cvetković-Kostić-Varga type matrices

“…As reported in [8], [19], and [32], the relations of SDD, DSDD, DZ, DZ-type, CKV, and CKV-type matrices are…”

“…As shown in [8], by the nonsingularity of CKV-type matrices, a new eigenvalue localization set for matrices has been obtained (see [8,Theorem 16]). In the following, on the basis of Theorem 3.9, we give a new eigenvalue localization set equivalent to that of [8], but it involves the sparsity pattern and requires less computation. The proof of this result is similar to that of [8,Theorem 16] and hence omitted.…”

“…In the following, on the basis of Theorem 3.9, we give a new eigenvalue localization set equivalent to that of [8], but it involves the sparsity pattern and requires less computation. The proof of this result is similar to that of [8,Theorem 16] and hence omitted.…”

“…where r S i (A) := k∈S\{i} |a ik | and S := N \ S. In [9], Cvetković, Kostić, and Varga proved that S-SDD matrices are nonsingular H-matrices and applied this property to give a new eigenvalue localization set for matrices in the complex plane. Moreover, as in [7] and [8], the union of all S-SDD matrices is usually called Cvetković-Kostić-Varga (CKV) class or Σ-SDD class, that is, a matrix A belongs to the class of CKV matrices if there exists a nonempty proper subset S of N such that A is an S-SDD matrix. As reported in [8], the CKV-type class has potential applications in many fields of numerical linear algebra such as eigenvalue localization, infinity-norm bound for the inverse matrix, and pseudospectra localization, among many other problems.…”

“…Moreover, as in [7] and [8], the union of all S-SDD matrices is usually called Cvetković-Kostić-Varga (CKV) class or Σ-SDD class, that is, a matrix A belongs to the class of CKV matrices if there exists a nonempty proper subset S of N such that A is an S-SDD matrix. As reported in [8], the CKV-type class has potential applications in many fields of numerical linear algebra such as eigenvalue localization, infinity-norm bound for the inverse matrix, and pseudospectra localization, among many other problems. It is well-known (see [1,17,29]) that for many applications it is useful to know whether a given matrix is an Hmatrix, and up to now, many direct and iterative algorithms such as Noda iterations [17], Algorithm AH [1], and Algorithm YZ [29] have been developed for determining the Hmatrix characterization of the coefficient matrix in a linear system or a linear complementarity problem (LCP).…”