GMRES(m) , GMRES, , GMRES(m), Look-Back GMRES(m). Ax = b, A C n n, x, b C n (1) Krylov.

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1 , GMRES,.,., Look-Back.,, Ax = b, A C n n, x, b C n (1),., Krylov., GMRES [5],.,., Look-Back [3]., 2 Krylov,. 3, Look-Back,

2 Algorith 1 The GMRES ethod 1: Choose the initial guess x and copute r = b Ax 2: Set β = r 2, v 1 = r /β 3: For j = 1, 2,..., k, Do: 4: Copute w j = Av j 5: For i = 1, 2,..., j, Do: 6: h i,j = (w j, v i ) 7: w j = w j h i,j v i 8: End For 9: h j+1,j = w j 2. If h j+1,j = then Set k := j and go to 12 1: v j+1 = w j /h j+1,j 11: End For 12: Define the (k + 1) k Hessenberg atrix H k = {h i,j } 1 i k+1,1 j k 13: x k = x + V k s k, where s k = arg in s C k βe 1 H k s 2 (1) Krylov, A x C n r := b Ax Krylov.,, Krylov. [4, 6]. GMRES Arnoldi Krylov, Algorith 1. GMRES, Arnoldi, 1.,,, GMRES., 3. Part 1. 1 x (1). Part 2. l, Ax = b x (l) GMRES ( ), x (l). Part 3. x (l), (l + 1) x (l+1), x (l+1) := x (l). Part 1 3 Algorith 2. 2

3 Algorith 2 The ethod 1: Choose the restart frequency and the initial guess x (1) 2: For l = 1, 2,..., until convergence Do: 3: Solve (approxiately) Ax = b by iterations of GMRES with the initial guess x (l), and get the approxiate solution x (l) 4: Update the initial guess x (l+1) := x (l) 5: End For Look-Back, 1 GMRES,., Part 1 Part 2., Part 2, Part 1., Part 3 [2]., x (l+1) x (l+1) := x (l) x (l), y (l+1) C n, x (l+1) := x (l) + y (l+1). [2].,, Look-Back, Look-Back [3]., Look-Back y (l+1) y (l+1) := µ (l) x (l), x (l) := { x (l) x (l k 2 ) x (l) k 1 (l 2 ) (k : ) x (k : )., k (k 2, k N), µ (l) C µ (l) = arg in µ C r(l) µa x (l) 2 = (r(l), A x (l) ) (A x (l), A x (l) )., Look-Back. Look-Back Algorith 3., Look-Back 1 Table 1., Mat-Vec, AXPY Inner-Product, 3

4 Algorith 3 A Look-Back ethod 1: Choose the restart frequency, the paraeter k 2 and the initial guess x (1) 2: For l = 1, 2,..., until convergence Do: 3: Solve (approxiately) Ax = b by iterations of GMRES with the initial guess x (l), and get the approxiate solution x (l) 4: Copute the vector y (l+1) as follows: If l = 1 then y (l+1) = If l 2 then If (l = k = 2) or (k : even, l k k 1 2 ) or (k : odd, l 2 ) then x (l) := x (l) x (1) Else x (l) := End If x (l) x (l k 2 ) x (l) k 1 (l 2 ) (k : even) x (k : odd) y (l+1) = µ (l) x (l), µ (l) = arg in µ C r (l) µa x (l) 2 End If 5: Update the initial guess x (l+1) := x (l) + y (l+1) 6: End For Table 1 The nuber of operations per restart cycle and storage requireents of and Look-Back. Method Look-Back k : even Mat-Vec + 1 ( + 1) + 1 k : odd AXPY ( )/2 ( )/2 + 3 Inner-Product ( )/2 ( )/2 + 2 Storage ( + 2)n ( + 2)n + k k+1 2 n ( + 2)n + 2 n 1,., Strage.,. + 1 [5], Look-Back + 2.,. 4

5 , The University of Florida Sparse Matrix Collection [1] 6, Look-Back (Algorith 3),. AMD Pheno II X4 94 (3.GHz) FORTRAN 77. Table 2 Characteristics of the coefficient atrices of the test probles for and Look- Back. Matrix (Type) n Nnz Ave.Nnz Application area CAVITY1 (R) Coputational fluid dynaics KIM1 (C) D/3D proble LIGHT IN TISSUE (C) Electroagnetics proble RAJAT3 (R) Circuit siulation WAVEGUIDE3D (C) Electroagnetics proble XENON2 (R) Materials proble Table 2., (R) (C),,., n, Nnz, Ave.Nnz,, 1 (1 ). = 3, Look-Back k = 3., b = [1, 1,..., 1] T x (1) = [,,..., ] T, r k 2 / b [ ] Table 3, Fig. 1., Table (1 ).,, 1 ( ) 3.,. (Iter), Look-Back,,., CAVITY1 RAJAT3,, Look-Back (Table 3 TRR )., Look-Back., Fig. 1 Look-Back. KIM1 WAVEGUIDE3D Look-Back, CAVITY1, LIGHT IN TISSUE, RAJAT3 5

6 Table 3 Convergence results (Iter : nuber of iterations, t Total : total coputation tie, t Restart : coputation tie per restart cycle, TRR : log 1 of explicitly coputed relative residual 2-nor) of and Look-Back for = 3. Matrix Method Iter Tie[sec.] TRR t Total t Restart CAVITY LB KIM LB LIGHT IN TISSUE LB RAJAT LB WAVEGUIDE3D LB XENON LB XENON2, Look-Back, Look-Back., 1 ( ) (t Restart ). Look-Back, 1 1%., Look-Back 1 1 AXPY (Table 1 ). (t Total ). Look-Back,, (Iter),, 1 (t Restart ),, KIM1., Look-Back,., Look-Back 1, 6

7 log 1 of relative residual 2-nor (d) CAVITY1 Look-Back Nuber of Iterations log 1 of relative residual 2-nor (a) KIM1 Look-Back Nuber of Iterations log 1 of relative residual 2-nor (b) LIGHT IN TISSUE Look-Back Nuber of Iterations log 1 of relative residual 2-nor (d) RAJAT3-1 Look-Back Nuber of Iterations log 1 of relative residual 2-nor (d) WAVEGUIDE3D Look-Back Nuber of Iterations log 1 of relative residual 2-nor (f) XENON2 Look-Back Nuber of Iterations Fig.1 The relative residual 2-nor history of and Look-Back of = 3 without preconditioners for KIM1, LIGHT IN TISSUE, NS3DA, RAJAT3, RDB5 and XENON2.., Look-Back,,.,., AXPY,.. 7

8 [1] Davis, T. A., The University of Florida Sparse Matrix Collection, [2],,,,, 19(29), [3],,, Look-Back, 21, (21), [4] Saad, Y., Iterative ethods for sparse linear systes. 2nd edition, SIAM, Philadelphia, PA, 23. [5] Saad, Y. and Schultz, M. H., GMRES: A generalized inial residual algorith for solving nonsyetric linear systes, SIAM J. Sci. Stat. Coput., 7(1986), [6],,,,, 29. 8

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