MatTriad 2015, Coimbra H-matrix theory and applications Maja Nedovi ć University of Novi Sad, Serbia joint work with Ljiljana Cvetkovi ć
Contents H-matrices and SDD-property � Benefits from H-subclasses • Breaking the SDD � Additive and multiplicative conditions • Partitioning the index set • Recursive row sums • Nonstrict conditions •
H-matrices and SDD-property A complex matrix A=[a ij ] nxn is an SDD- matrix if for each i from N it holds that ∑ ( ) = a ii > r i A a ij Deleted row sums j ∈ N , j ≠ i Lévy-Desplanques: Lev nonsingular
H-matrices and SDD-property A complex matrix A=[a ij ] nxn is an SDD- matrix if for each i from N it holds that ∑ ( ) = a ii > r i A a ij j ∈ N , j ≠ i A complex matrix A=[a ij ] nxn is an H-matrix if and only if there exists Lev a diagonal nonsingular matrix W such that AW is an SDD matrix.
H-matrices and SDD-property A complex matrix A=[a ij ] nxn is an SDD- matrix if for each i from N it holds that ∑ ( ) = a ii > r i A a ij H j ∈ N , j ≠ i HH SDD
H-matrices and SDD-property A complex matrix A=[a ij ] nxn is an SDD- matrix if for each i from N it holds that ∑ ( ) = a ii > r i A a ij H j ∈ N , j ≠ i HH SDD ?
Subclasses of H-matrices & diagonal scaling characterizations Benefits: 1. Nonsingularity result covering a wider matrix class 2. A tighter eigenvalue inclusion area (not just for the observed class) 3. A new bound for the max-norm of the inverse for a wider matrix class 4. A tighter bound for the max-norm of the inverse for some SDD matrices 5. Schur complement related results (closure and eigenvalues) 6. Convergence area for relaxation iterative methods 7. Sub-direct sums 8. Bounds for determinants
Breaking the SDD Recursive row sums SDD Additive and Non-strict multiplicative conditions conditions Partitioning the index set
Breaking the SDD Recursive row sums SDD Additive and Non-strict multiplicative conditions conditions Partitioning the index set
Breaking the SDD Recursive row sums SDD Additive and Non-strict multiplicative conditions conditions Partitioning the index set
Breaking the SDD Recursive row sums SDD Additive and Non-strict multiplicative conditions conditions Partitioning the index set
Ostrowski, A. M. (1937), Pupkov, V. A. (1983), Hoffman, A.J. (2000), Breaking the SDD Varga, R.S. : Ger š gorin and his circles (2004) Recursive row sums SDD Additive and Non-strict Ostrowski, multiplicative conditions Pupkovi conditions Partitioning the index set
Gao, Y.M., Xiao, H.W. (1992), Varga, R.S. (2004), Dashnic, L.S., Zusmanovich, M.S. (1970), Kolotilina, l. Yu.(2010), Breaking the SDD Cvetkovi ć , Lj., Nedovi ć , M. (2009), (2012), (2013). Recursive row sums SDD Additive and Non-strict Ostrowski, multiplicative conditions Pupkovi conditions Partitioning the S-SDD, index set PH
Mehmke, R., Nekrasov, P. (1892), Gudkov, V.V. (1965), Szulc, T. Breaking the SDD (1995), Li, W. (1998), Cvetkovi ć , Lj., Kosti ć , V., Nedovi ć , M. (2014). Recursive Nekrasov- row sums matrices SDD Additive and Non-strict Ostrowski, multiplicative conditions Pupkovi conditions Partitioning the S-SDD, index set PH
O. Taussky (1948), Beauwens (1976), Szulc,T. (1995), Li, W. (1998), Breaking the SDD Varga, R.S. (2004) Cvetkovi ć , Lj., Kosti ć , V. (2005) Recursive Nekrasov- row sums matrices SDD Additive and IDD, CDD Non-strict Ostrowski, multiplicative S-IDD, S-CDD conditions Pupkovi conditions Partitioning the S-SDD, index set PH
O. Taussky (1948), Beauwens (1976), Szulc,T. (1995), Li, W. (1998), Breaking the SDD Varga, R.S. (2004) Cvetkovi ć , Lj., Kosti ć , V. (2005) H Recursive Nekrasov- row sums matrices SDD Additive and IDD, CDD Non-strict Ostrowski, multiplicative S-IDD, S-CDD conditions Pupkovi conditions Partitioning the S-SDD, index set PH
I Additive and multiplicative conditions Ostrowski-matrices multiplicative condition: ( ) ( ) a a r A r A > ii jj i j Pupkov-matrices additive condition: a min{max { a }, r ( ) A } > ii j i ji i ≠ ( ) ( ) a a r A r A + > + ii jj i j Ostrowski, A. M. (1937), Pupkov, V. A. (1983), Hoffman, A.J. (2000), Varga, R.S. : Ger š gorin and his circles (2004)
II Partitioning the index set S-SDD -matrices Given any complex matrix A=[a ij ] nxn S and given any nonempty proper subset S of N, A is an S-SDD matrix if S ( ) a r A a , i S ∑ > = ∈ S ii i ij j S , j i ∈ ≠ ( ) ( ) S S S S ( ) ( ) ( ) ( ) a r A a r A r A r A , − − > ii i jj j i j i S , j S ∈ ∈ Gao, Y.M., Xiao, H.W. LAA (1992) Cvetkovi ć , Lj., Kosti ć , V., Varga, R. ETNA (2004)
II Partitioning the index set S-SDD - matrices A matrix A=[a ij ] nxn is an S-SDD matrix iff there exists a matrix W in W s � such that AW is an SDD matrix. ⎧ ⎫ W S ( ) W diag w , w ,..., w : w 0 for i S and w 1 for i S = = = γ > ∈ = ∈ ⎨ ⎬ 1 2 n i i ⎩ ⎭ γ . . S γ SDD 1 1 N\S . . 1
Diagonal scaling characterization & Scaling matrices γ . . S γ SDD 1 1 N\S . . 1 We choose parameter from the interval : ( ( ) ( ) ) , I A , A γ = γ γ 1 2 S S ( ) ( ) a r A r A − jj j ( ) 0 A max i ( ) , ( ) A min . ≤ γ = γ = 1 S 2 a r A S r ( ) A i S − j S ∈ ∈ ii i j
Diagonal scaling characterization & Scaling matrices S-SDD SDD
Diagonal scaling characterization & Scaling matrices γ S-SDD T-SDD γ γ γ SDD γ γ γ
Diagonal scaling characterization & Scaling matrices ∑ -SDD γ S-SDD T-SDD γ γ γ SDD γ γ γ
Diagonal scaling characterization & Scaling matrices ∑ -SDD γ γ γ γ SDD γ γ γ
Eigenvalue localization { } S ( ) S ( ) A z C : z a r A , i S , Γ = ∈ − ≤ ∈ i ii i { ( ) } ( ) S ( ) S ( ) S ( ) S ( ) ( ) S V A z C : z a r A z a r A r A r A , = ∈ − − − − ≤ ij ii i jj j i j i S , j S . ∈ ∈ ⎛ ⎞ ⎛ ⎞ ( ) S ( ) S ( ) S ( ) . A C A A V A ⎜ ⎟ ⎜ ⎟ σ ⊆ = Γ ⎜ ⎟ i ij ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ i S i S , j S ∈ ∈ ∈ Cvetkovi ć , L., Kosti ć , V., Varga R.S.: A new Ger š gorin-type eigenvalue inclusion set ETNA, 2004. Varga R.S.: Ger š gorin and his circles, Springer, Berlin, 2004.
Schur complement ( ) ( ) ( ) ( ) A A α , A A α , α α α α ( ) ( ) A α , A 0 Schur α α The Schur complement of a complex nxn matrix A, with respect to a proper subset α of index set N={1, 2, … , n}, is denoted by A/ α and defined to be: ( ) 1 A A A ( , ) ( ) ( A ) − ( , ) α − α α α α α
Schur complement ( ) ( ) ( ) ( ) A A α , A A α , α α α α SDD ( ) ( ) A α , A 0 Schur α α Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices. Czech. Math. J. 29 (104) (1979), 246-251.
Schur complement ( ) ( ) ( ) ( ) A A α , A A α , α α α α ( ) ( ) A α , A 0 Schur α α SDD Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices. Czech. Math. J. 29 (104) (1979), 246-251.
Schur complement ( ) ( ) ( ) ( ) A A α , A A α , α α α α Ostr ( ) ( ) A α , A 0 Schur α α Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices. Czech. Math. J. 29 (104) (1979) Li, B., Tsatsomeros, M.J. : Doubly diagonally dominant matrices. LAA 261 (1997)
Schur complement ( ) ( ) ( ) ( ) A A α , A A α , α α α α ( ) ( ) A α , A 0 Schur α α Ostr Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices. Czech. Math. J. 29 (104) (1979) Li, B., Tsatsomeros, M.J. : Doubly diagonally dominant matrices. LAA 261 (1997)
Schur complement ( ) ( ) ( ) ( ) A A α , A A α , α α α α $ ( ) ( ) A α , A 0 Schur α α Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices. Czech. Math. J. 29 (104) (1979) Li, B., Tsatsomeros, M.J. : Doubly diagonally dominant matrices. LAA 261 (1997) Zhang, F. : The Schur complement and its applications, Springer, NY, (2005).
Schur complement ( ) ( ) ( ) ( ) A A α , A A α , α α α α ( ) ( ) $ A α , A 0 Schur α α Carlson, D., Markham, T. : Schur complements of diagonally dominant matrices. Czech. Math. J. 29 (104) (1979) Li, B., Tsatsomeros, M.J. : Doubly diagonally dominant matrices. LAA 261 (1997) Zhang, F. : The Schur complement and its applications, Springer, NY, (2005).
Schur complements of S-SDD ( ) ( ) ( ) ( ) A A α , A A α , α α α α ∑ -SDD ( ) ( ) A α , A 0 Schur α α Cvetkovi ć , Lj., Kosti ć , V., Kova č evi ć , M., Szulc, T. : Further results on H-matrices and their Schur complements. AMC (2008) Liu, J., Huang, Y., Zhang, F. : The Schur complements of generalized doubly diagonally dominant matrices. LAA (2004)
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