Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
S-SDD class of matrices and its application Vladimir Kostić University of Novi Sad Faculty of Science Dept. of Mathematics and Informatics Introduction Equivalent definitions of S-SDD matrices Bounds for the determinants Convergence area of PDAOR Subdirect sums Equivalent definitions of S-SDD matrices _ S S ri S A aii for all i S ri S A rjS A aii ri S A a jj rjS A for all i S , j S S 1 A max iS a ii A ri S A ri S min jS rjS A 0 a jj rjS A r S j A S 2 L. Cvetkovic, V. Kostic and R. S. Varga, A new Gersgorin-type eigenvalue inclusion set, Electron. Trans. Numer. Anal., 18 (2004), 73–80 A Equivalent definitions of S-SDD matrices ri S A aii for all i S A S is SDD matrix A S is SDD matrix ri S A rjS A aii ri S A a jj rjS A for all i S , j S A is S-SDD matrix J A S 1S A , 2S A Equivalent definitions of S-SDD matrices S _ S AX is an SDD _ S S x 1 x J A S : 1S A , 2S A Equivalent definitions of S-SDD matrices H S-SDD 1 x x S 1 SDD S 2 Bounds for the determinants det A aii ri A Ostrowski 1937 det A aii ui A Price 1951 det A aii li A Ouder 1951 iN iN iN k 1 det A max aii ui A akk k 1..n max i 1..n i 1 li A ui A aii a n i k 1 ii li A , Ostrowski 1952 Bounds for the determinants 700 600 500 400 300 200 100 0 1 2 3 4 estimates improved estimates determinant Bounds for the determinants 700 600 500 400 300 200 100 0 1 2 3 4 estimates improved estimates determinant Bounds for the determinants 700 600 500 400 300 200 100 0 1 2 3 4 estimates improved estimates determinant Convergence of PDAOR A is block H-matrix n n1 n2 ... nN iff 1M nisi , an i M-matrix 1, 2,...N H-matrix A A Aij i , j 1,2.. N 1 ii n1 n2 n3 n4 n4 M mij i , j 1,2.. N mij 1 1 ij 1 ii A Aij nN m m n1 1 m12 m13 m14 n2 m21 1 m23 m24 n3 m31 m32 1 m34 41 m42 m43 1 N1 mN 2 mN 3 mN 4 nN m m m 1 m1N 2N 3N 4N Convergence of PDAOR N AL Lij i , j 1... AU U ij i , j 1... AD Dij i , j 1... AD diag D1 ,...Dn A AD AL AU AD X p1 AD AL 1 1, 2,..., N J1 J 2 ... J J i i 1 1 , i 1 2 ,..., i 1 AD AL AU AD X p AD AL 1 B Convergence of PDAOR Let A Aij i , j 1.. N , N 2 be block SDD matrix. Then LPDAOR , 1 if we chose parameters in the following way: 1 0 1, and 2 1 ri 1 ri 1 min i 1.. N 2l i 1.. N 2l i i min 1 li 1 2 min , i 1.. N 1 r 2l i i 3 1 2 min 1 li , 1 ri 2 li 4 1 li , 1 ri i 1.. N 1 2 min i 1.. N and 1 min and 2 1 ri 2li i 1.. N and or 2li or 0 1 or 1 ri 0. i 1.. N 2l i min Lj.Cvetkovic, J. Obrovski, Some convergence results of PD relaxation methods, AMC 107 (2000) 103-112 Convergence of PDAOR lm k 1 sJ k , j ) A1m m A m k 1 s m i 1 j 1 l7 A7,7 A7,1 1 A7,7 A7,5 1 A7,7 A7,9 1 A7,7 A7,2 1 A7,7 A7,6 1 A7,7 A7,10 Convergence of PDAOR l S m k 1 s k 1 sJ k , j ) S A1m m A m k 1 s m i 1 j 1 l4S A4,4 A4,1 1 A4,4 A4,3 1 l8S A8,8 A8,6 1 A8,8 A8,9 1 A4,4 A4,2 1 A4,4 A4,5 1 A8,8 A8,7 1 A8,8 A8,10 1 A8,8 A8,11 Convergence of PDAOR Let A Aij Then i , j 1.. N , N 2 be block S-SDD matrix for set S. LPDAOR , 1 2 3 4 where 1 if we chose parameters in the following way: 0 1 min S , S 1 min S , S , and 1 min S , S and 0 1, 1 min S , S and min S , S 0. or or 1 l 1 r l r 2 min , 1 r 2 l 1 r r 2 l r 1 l 1 r l r 1 l 1 r l r , 2 min 2 min 1 r 1 r r r 1 r 2 l 1 r r 2 l r 2 1 r 2l 1 r r 2l r . 1 min 2 l 1 r l r S i S i , jS S S j S i S j S S i S S i S i S i i S j S j S i S j S j S i S S j S i S j S S i i S j S S i i i , jS i , jS i S i i S 1 min S , S , and 1 min S , S 1 ri S 1 rjS ri S rjS 1 S min , 2 i , jS liS 1 rjS liS rjS S or S i S j S S j i , jS i S i S j S j S j Cvetković, Kostic, New subclasses of block H-matrices with applications to parallel decomposition-type relaxation methods, Numer. Alg. 42 (2006) S i S i S j S j , Convergence of PDAOR Subdirect sums A n1 n1 C A k B B n2 n2 Subdirect sums B11 B12 A11 A12 A22 , B11 B A B21 B22 n2 n2 A21 A22 n1n1 Same sign pattern on the diagonal A12 0 A11 C A21 A22 B11 B12 0 B21 B22 n n k n n k 1 2 1 2 k k Subdirect sums H NO A and B are is the matrix C YES SDD , too? Subdirect sums H NO A and B are is the matrix C , too? S-SDD YES SDD R. Bru, F. Pedroche, and D. B. Szyld, Subdirect sums of S-Strictly Diagonally Dominant matrices, Electron. J. Linear Algebra, 15 (2006), 201–209 Subdirect sums Subdirect sums S aii ri S A a jj rjS A ri S A rjS A for all i S , j S Subdirect sums Let S 1, 2,..., card S n1 k If A is S-SDD and B is SDD then C A k B is S-SDD. R. Bru, F. Pedroche, and D. B. Szyld, Subdirect sums of S-Strictly Diagonally Dominant matrices, Electron. J. Linear Algebra, 15 (2006), 201–209 Subdirect sums S Subdirect sums of S-SDD matrices SA Subdirect sums of S-SDD matrices SB Subdirect sums of S-SDD matrices Let If S be arbitrary A is S A - SDD, B is S B - SDD and J A S A J B SB then C A k B is S-SDD. Thank you for your attention