Research Article Some Generalizations and Modifications of Iterative Methods for Solving Large Sparse Symmetric Indefinite Linear Systems

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1 Abstract and Applied Analysis Article ID pages Research Article Some Generalizations and Modifications of Iterative Methods for Solving Large Sparse Symmetric Indefinite Linear Systems Yu-Chien Li Jen-Yuan Chen and David R Kincaid 2 Department of Mathematics National Kaohsiung Normal University Kaohsiung 824 Taiwan 2 Institute for Computational Engineering and Sciences The University of Texas at Austin Austin TX 7872 USA Correspondence should be addressed to Jen-Yuan Chen; jchen@nknuedutw Received 27 November 203; Revised 0 January 204; Accepted 4 February 204; Published 3 April 204 Academic Editor: Chi-Ming Chen Copyright 204 Yu-Chien Li et al This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited We discuss a variety of iterative methods that are based on the Arnoldi process for solving large sparse symmetric indefinite linear systems We describe the SYMMLQ and SYMMQR methods as well as generalizations and modifications of them Then we cover the Lanczos/MSYMMLQ and Lanczos/MSYMMQR methods which arise from a double linear system We present pseudocodes for these algorithms The authors dedicate this paper to the memory of Professor David M Young Jr for his pioneering research inspirational teaching and exceptional life Introduction Frequently when computing numerical solutions of partial differential equations one needs to solve systems of very large sparse linear algebraic equations of the form Ax = b () A is an n nmatrix b is an n vector and one seeks a numericalsolutionvector x or a good approximation of it Particularly for large linear systems arising from partial differential equations in three dimensions well-known direct methods such as Gaussian elimination may become prohibitively expensive in terms of both computer storage and computer time On the other hand a variety of iterative methods may avoid these difficulties For linear systems involving symmetric positive definite (SPD) matrices the conjugate gradient (CG) method (and variations of it) may work well On the other hand when solving linear systems the coefficient matrix A is symmetric indefinite the choice of a suitable iterative method is not at all clear On the other hand the SYMMLQ and MINRES methods have been shown to be useful in certain situations (see Paige and Saunders ) For nonsymmetric systems Saad and Schultz 2 generalized the MINRES method to obtain the GMRES method In Section 2 wereviewthearnoldiprocessinsections 3 and 4 we describe the SYMMLQ and SYMMQR methods Then we can generalize them in Section 5andweoutlinethe modified SYMMLQ method in Section 6 Next in Section 7 we discuss applying the MSYMMLQ and MSYMMQR methods applied to a double linear system Finally we present pseudocodes in Sections 8 2 Arnoldi Process WebeginwithareviewoftheArnoldiprocess Theorem Suppose that A is an n nsymmetric matrix One can generate orthonormal vectors w (0) w () w (n 2) w (n ) using this short-term recurrence w (j+) Aw (j) α j w (j) β j w (j ) (0 j n 2) w (j+) =( σ j+ ) w (j+) σ j+ = w (j+) w (j+) (2)

2 2 Abstract and Applied Analysis Example 2 We illustrate Theorem for the case n=3 α j = Aw (j) w (j) β j = Aw (j) w (j ) (3) From (2)and(3) we have σ w () = w () = Aw (0) α 0 w (0) β 0 w ( ) Here one assumes that w ( ) =0and σ j =0foralljThenthe following properties hold for (0 i j n ): w (i) w (j) =δ ij β j =σ j (4) Proof If we let w (0) r (0) = b Au (0) then the subspace Span {w (0) w () w (n 2) w (n ) } (5) is equivalent to the Krylov subspace K n (r (0) A) = Span {r (0) Ar (0) A n 2 r (0) A n r (0) } (6) We obtain β j = Aw (j) w (j ) (by (3)) = w (j) Aw (j ) (A T = A) = w (j) (σ j w (j) +α j w (j ) +β j w (j 2) ) (by (2)) =σ j (7) since w (i) w (j) =δ ij From Theorem in matrix formit follows that σ 2 w (2) = w (2) = Aw () α w () β w (0) σ 3 w (3) = w (3) = Aw (2) α 2 w (2) β 2 w () Consequently we obtain since w ( ) =0 So we obtain AW 2 = A w (0) w () w (2) = Aw (0) Aw () Aw (2) = β 0 w ( ) +α 0 w (0) +σ w () β w (0) +α w () +σ 2 w (2) β 2 w () +α 2 w (2) +σ 3 w (3) = w (0) w () w (2) w (3) α 0 β 0 σ α β 2 0 σ 2 α σ T 3 = W 2 w (3) 0 0 σ 3 T (0) () AW 2 = W 2 T 3 +σ 3 w (3) e T 3 = W 3 T 4 (2) AW n = W n T n +σ n w (n) e T n = W n T n+ W n =w (0) w () w (n ) w (n 2) w (n ) n n α 0 β σ α β 2 T n e T n = n σ 2 α 2 β 3 σ n α n 3 β n 2 σ n 2 α n 2 β n σ n α n α 0 β σ α β 2 σ 2 α 2 β 3 T n+ σ n 3 α n 3 β n 2 σ n 2 α n 2 β n σ n α n σ n n n (n+) n (8) (9) 3 SYMMLQ Method We choose u (n) suchthatu (n) u (0) K n (r (0) A)Hencewe have u (n) = u (0) + W n y (n) (3) r (n) = r (0) AW n y (n) Imposing the Galerkin condition r (n) K n (r (0) A) we obtain We obtain because (r (n) ) T W n = 0 W T n r(n) = 0 W T n r(0) = W T n AW n y (n) (by (3)) (4) T n y (n) =σ 0 e (5) W T n AW n = T n (6) W T n r(0) =σ 0 e σ 0 = r(0)

3 Abstract and Applied Analysis 3 Instead of solving for y (n) directly from the triangular linear system (5) Paige and Saunders factorize the matrix T n into a lower triangular matrix with bandwidth three (resulting in the SYMMLQ method) Also we have γ 0 δ γ ε 2 δ 2 γ 2 T n Q n = L n = ε n 3 δ n 3 γ n 3 ε n 2 δ n 2 γ n 2 ε n δ n γ n n n (7) Q n Q 2 Q 23 Q n n is an orthogonal matrix and we have We let x (n) = x (0) + V n ẑ n (25) V n =v (0) v () v (n 2) v (n ) n n (26) z n =ζ 0 ζ ζ n 2 ζ n T n V n =v (0) v () v (n 2) v (n ) n n (27) L n z (n) =σ 0 e d c Q ii+ = i s i s i c i d c 2 i +s 2 i =SinceT n Q n = L n wehave n n (8) γ 0 δ γ ε 2 δ 2 γ 2 L n = ε n 3 δ n 3 γ n 3 ε n 2 δ n 2 γ n 2 ε n δ n γ n γ n = γ 2 n +β2 n c n = γ n γ n n n s n = β n γ n (28) From (2) and(28) we have ζ n =( ζ n /ζ n ) ζ n =c n ζn Since T n Q n Q n y(n) = L n Q n y(n) =σ 0 e (9) V n+ = v (0) v () v (n ) v (n) = W n Q n Letting = W n w (n) Q 2 Q 23 Q n n Q nn+ (29) then Next letting ẑ (n) = Q n y(n) (20) L n ẑ (n) =σ 0 e (2) we have = V n w (n) Q nn+ = v (0) v () v (n ) w (n) Q nn+ v (n ) =c n v (n ) +s n w (n) v (n) =s n v (n ) c n w (n) (30) we have Defining x (n) = u (0) + W n y (n) (22) x (n) = u (0) + W n Q n Q n y(n) = u (0) + W n Q n ẑ (n) V n = W n Q n (23) ẑ n =ζ 0 ζ ζ n 2 ζ n T n (24) If β n =0thenL n is nonsingular We can find z (n) by solving L n z (n) =σ 0 e u (n) = u (0) + V n z (n) = u (n ) +ζ n v (n ) 4 SYMMQR Method (3) We choose u (0) such that u (n) u (0) K n (r (0) A)Hencewe have u (n) = u (0) + W n y (n) r (n) = r (0) AW n y (n) (32)

4 4 Abstract and Applied Analysis Imposing the Galerkin condition r (n) K n (r (0) A) as before we obtain Since we have (r (n) ) T W n =0 W T n r(n) =0 W T n r(0) = W T n AW n y (n) (by (32)) (33) W T n AW n = T n (34) W T n r(0) =σ 0 e σ 0 = r(0) T n y (n) =σ 0 e (35) Instead of solving for y (n) directly from the triangular system (35) Paige and Saunders factorizedthematrixt n into a lower triangular matrix with bandwidth three We can use a different factorization of T n to obtain a slightly different method which is called the SYMMQR method We multiply the matrix T n by an orthogonal matrix ontheleft-handsideinsteadoftheright-handsidewehave Q n T n ŷ n = Q n σ 0 e (36) γ 0 δ ε 2 γ δ 2 ε 3 γ 2 δ 3 ε 4 Q n T n = R = γ n 3 δ n 2 ε n γ n 2 δ n γ n n n (37) We obtain the matrix Q n Q n n Q n 2n Q 2 d c Q ii+ = i s i s i c i d n n (38) with c 2 i +s 2 i =being the Givens rotation Letting ŷ n be the solution of Rŷ n = Q n σ 0 e (39) which satisfies the Galerkin condition r n K n (r (0) A) r n = b A x (n) Wenotethat γ n is not always nonzero and thus R n might be singular We assume that R n is nonsingular and then we define We have P n = W n R n (4) P n =p (0) p () p (n 2) p (n ) n n ẑ (n) = Q n σ 0 e =ζ 0 ζ ζ n 2 ζ n T (42) x (n) = u (0) + P n ẑ (n) (43) For the next iterate x (n+) we need to solve T n+ ŷ (n+) =σ 0 e (44) α 0 β β α β 2 β 2 α 2 β 3 T n+ = β n 2 α n 2 β n β n α n β n β n α n n+ n (45) Applying the Givens rotation Q n to both sides of (45) we have γ 0 δ ε 2 γ δ 2 ε 3 γ 2 δ 3 ε 4 Q n T n+ = γ n 3 δ n 2 ε n γ n 2 δ n ε n γ n ψ β n α n n+ n (46) 00000ε n ψα n T = Q n n β n α n T To eliminate β n we compute thenth Given rotation Q nn+ by then we have x (n) = u (0) + W n ŷ n = u (0) + W n R n Q n σ 0 e (40) γ n = γ 2 n +β2 n s n = β n γ n c n = γ n γ n (47)

5 Abstract and Applied Analysis 5 By multiplying Q nn+ times Q n T n and times Q n σ 0 e we have We have Q nn+ (Q n T n+ )= R n+ γ 0 δ ε 2 γ δ 2 ε 3 γ 2 δ 3 ε 4 = γ n 3 δ n 2 ε n γ n 2 δ n ε n γ n δ n γ n n+ n Q nn+ (Q n σ 0 e )=ẑ (n+) =ζ 0 ζ ζ n ζ n T (48) Q nn+q n ( T n+ y (n) σ 0 e ) ζ 0 ζ R ζ = n 0000 y (n) n+ n ζ n ζ n (53) Hence the solution y (n) from R n y (n) = z (n) minimizes T n+ y (n) σ 0 e and ζ n = T n+ y (n) σ 0 e Let 00000ε n δ n γ n T u (n) = u (0) + W n y (n) = u (0) + W n R n z(n) (54) = Q nn ε n ψα n T ζ n =c n ζn ζ n =s n ζn (49) P n R n = W n P n =p (0) p () p (n 2) p (n ) (55) Let We have γ 0 δ ε 2 γ δ 2 ε 3 γ 2 δ 3 ε 4 R n = γ n 3 δ n 2 ε n γ n 2 δ n γ n n n (50) Since p (n ) =( γ n )w (n ) ε n p (n 3) δ n p (n 2) u (n) = u (0) + P n z (n) = u (n ) +γ n p (n ) (56) We define z (n) = ζ 0 ζ ζ n 2 ζ n T Since β n = w (n) =0thenγ n =0 and R n is nonsingular We can solve for y (n) from we obtain x (n+) = u (0) + W n ŷ (n+) (57) R n y (n) = z (n) (5) We discuss the case β n = w (n) later Consider solving the least square problem involving y (n) minimizing T n+ y (n) σ 0 e x (n+) = u (0) + W n R n+ẑ(n+) = u (0) + P n ẑ (n+) = u (0) + P n z (n) + ζ n p (n) = u (n) + ζ n p (n) (58) α 0 β β α β 2 β 2 α 2 β 3 T n+ = β n 2 α n 2 β n β n α n β n n+ n (52) We note that x (n) is the estimated solution vector satisfying the Galerkin condition while u (n) = u (0) + W n y (n) (59) with y (n) minimizing T n+ y (n) σ 0 e

6 6 Abstract and Applied Analysis 5 Generalized SYMMLQ and SYMMQR Methods Now we generalize the SYMMLQ and SYMMQR methods Theorem 3 Suppose that E is an n nsymmetric positive definite (SPD) matrix and EA is an n n symmetric matrix One can generate orthonormal vectors w (0) w () w (n 2) w (n ) using this short-term recurrence w (j+) Aw (j) α j w (j) β j w (j ) (0 j n 2) w (j+) =( γ j+ ) w (j+) γ j+ = E w(j+) w (j+) α j = EAw (j) w (j) β j = EAw (j) w (j ) (60) (6) As before we let x (n) = x (0) + W n y (n) r (n) = r (0) AW n y (n) ImposingtheGalerkinconditionagainwehave (Er (n) ) T W n = 0 (EW n ) T r (n) = 0 (EW n ) T r (0) =(EW n ) T AW n y (n) (by (67)) We obtain T n y (n) =β e β = Er(0) r (0) because (EW n ) T AW n = T n (EW n ) T W n = I (67) (68) (69) (70) Then the following properties hold for (0 i j n ): Proof We obtain w (j) Ew (j) =δ ij β j =γ j (62) β j = EAw (j) w (j ) (by (6)) = w (j) EAw (j ) ((EA) T = EA) = w (j) E (γ j w (j) +α j w (j ) +β j w (j 2) ) = w (j) γ j Ew (j) =γ j Since w (i) Ew (j) =δ ij As before we let Moreover we have (by (60)) W n =w () w (2) w (n ) w (n) n n α β 2 σ 2 α 2 β 3 T n = e T n = n (63) (64) AW n = W n T n +γ n+ w (n+) e T n (65) σ 3 α 3 β 4 σ n 2 α n 2 β n σ n α n β n σ n α n n n (66) (EW n ) T r (0) =β e Since T n is symmetric we can apply the same techniques as in the SYMMLQ method Also if E = Ithemethodreduces to the SYMMQR method 6 Modified SYMMLQ Method Next we outline the modified SYMMLQ method Theorem 4 Suppose that E is an n nsymmetric (not necessary positive definite) matrix and EA is an n n symmetric matrix One can generate orthonormal vectors w (0) w () w (n 2) w (n ) using this short-term recurrence w (j+) Aw (j) α j w (j) β j w (j ) (0 j n 2) w (j+) =( γ j+ ) w (j+) + = E w(j+) w (j+) α j = EAw(j) w (j) Ew (j) w (j) β j = EAw(j) w (j ) Ew (j ) w (j ) Then the following properties hold for (0 j n 2): d j+ = Ew (j+) w (j+) and for (0 i j n ) = { if E w(j+) w (j+) >0 if E w (j+) w (j+) <0 (7) (72) (73) Ew (i) w (j) =δ ij (74)

7 Abstract and Applied Analysis 7 From Theorem 4in matrix formwe obtain Then we have AW n = W n T n +γ n+ w (n+) e T n (75) (EW n ) T AW n =(EW n ) T W n T n +(EW n ) T (γ n+ w (n+) e T n ) = D n T n + 0 (78) Herethesecondtermontheright-handsideisthezero matrix! In addition we have W n =w () w (2) w (n 2) w (n ) w (n) n n x (n) = x (0) + W n y (n) r (n) = r (0) AW n y (n) (79) α β 2 T n = e T n = n γ 2 α 2 β 3 γ 3 α 3 β 4 γ n 2 α n β n γ n α n β n γ n α n n n (76) Imposing the Galerkin condition r (n) K n (r (0) A) aswe did before we obtain (Er (n) ) T W n = 0 (EW n ) T r (n) = 0 (EW n ) T r (0) =(EW n ) T AW n y (n) (by (79)) (EW n ) T r (0) = D n T n y (n) (by (78)) In other words we use (80) Moreover from Theorem 4 we obtain (EW n ) T W n = d d2 d 3 d d n 2 dn D n d n n n (77) (EW n ) T AW n = D n T n (8) (EW n ) T W n = D n We obtain (EW n ) T r (0) =β e (82) because D n T n y (n) =β e β = Er(0) r (0) (83) Here d α d β 2 d 2 γ 2 d 2 α 2 d 2 β 3 d 3 γ 3 d 3 α 3 d 3 α 4 D n T n = d n 2 γ n 2 d n 2 α n 2 d n 2 β n d n γ n d n α n d n β n d n γ n d n α n n n (84) We note that D n T n is symmetric for ( i n ): d i β i+ =d i EAw (i+) w (i) Ew (i) w (i) = d i w (i+) EAw (i) Ew (i) w (i) (by (72)) ((EA) T = EA) =d i w (i+) E (+ w (i+) +α i w (i) +β i w (i ) Ew (i) w (i) =d i+ γ j+ (by (7)) (85)

8 8 Abstract and Applied Analysis 7 Lanczos/MSYMMLQ Method Next we consider this 2n 2n double linear system: A 0 0 A T x x =b b (86) We obtain the block symmetric matrices A E andea A = A 0 0 A T E = 0 I I 0 EA = 0 AT A 0 (87) For example the modified SYMMLQ method and the modified SYMMQR method can be applied to the double linear system (86) This leads us to the LAN/MSYMMLQ method and the LAN/MSYMMQR method The pseudocodes for these methods are given in the following sections For additional details see Li 3 See the books by Golub and Van Loan 4 and Saad 5 as well as the papers by Lanczos 6 and Kincaid et al 7 among others 8 MSYMMLQ Pseudocode x () = x (0) + ζ v () c = d α α 2 +β2 2 s = d β 2 α 2 +β2 2 γ =(d α )c +(d β 2 )s ζ =( β γ ) v () =c i v () +s w (2) x () = x (0) +ζ v () for i=23n α i =( d i ) EAw (i) w (i) β i =( d i ) EAw (i) w (i ) w (i+) = Aw (i) α i w (i) β i w (i ) β i+ = E w(i+) w (i+) w (i+) =( β i+ ) w (i+) (88) r (0) = b Ax (0) β = Er(0) r (0) w () =( ) r (0) β d = { if Er (0) Er (0) >0 { { if Er (0) Er (0) <0 ε =ε 2 =0 s 0 =0 c 0 = α =( ) EAw () w () d w (2) = Aw () α w () d i+ ={ if E w(i+) w (i+) >0 if E w (i+) w (i+) <0 ε i =(d i β i )s i 2 δ i = (d i β i )c i 2 hh = δ i δ i = (hh) c i +(d i α i )s i = (hh) s i (d i α i )c i ζ i =( )( ε i ζ i 2 δ i ζ i ) v (i) =s i v (i ) c i w (i) x (i) = x (i ) + ζ i v (i) = α 2 i +β 2 i+ β 2 = E w(2) w (2) w (2) =( β 2 ) w (2) c i =d i ( α i ) s i =d i ( β i+ ) ζ i =( γ )( ε i ζ i 2 δ i ζ i ) d 2 ={ if E w(2) w (2) >0 if E w (2) w (2) <0 v (i) =c v (i) +s i w (i+) ζ = β α v () = w () end for x (i) = x (i ) +ζ i v (i) (89)

9 Abstract and Applied Analysis 9 9 MSYMMQR Pseudocode r (0) = b Ax (0) β = Er(0) r (0) w () =( β ) r (0) d ={ if Er(0) Er (0) >0 if Er (0) Er (0) <0 ε =ε 2 =0 s 0 =0 c 0 = α =( d ) EAw () w () w (2) = Aw () α w () β 2 = E w(2) w (2) w (2) =( β 2 ) w (2) d 2 ={ if E w(2) w (2) >0 if E w (2) w (2) <0 ζ =β d p () =( α d ) w () x () = x (0) + ζ p () c = d α α 2 +β2 2 s = d β 2 α 2 +β2 2 γ =(d α )c (d β 2 )s ζ =c ζ p () =( γ ) w () end for d i+ ={ if E w(i+) w (i+) >0 if E w (i+) w i+ < 0 ε i = (d i β i )s i 2 δ i = (d i β i )c i 2 hh = δ i δ i = (hh) c i (d i α i )s i = (hh) s i +(d i α i )c i ζi =s i ζi p (i) =( )w (i) ε i p (i 2) δ i p (i ) x (i) = x (i ) + ζ i p (i) = ζ2 i +β 2 i+ c i = s i = ( )(d i β i+ ) ζ i =c i ζi p (i) =( )w (i) ε i p (i 2) δ i p (i ) x (i) = x (i ) +ζ i p (i) 0 LAN/MSYMMLQ Pseudocode r (0) = b Ax (0) r (0) = b A T x (0) β = 2 r(0) r (0) w () =( β ) r (0) (9) x () = x (0) +ζ p () for i=23n α i =( d i ) EAw (i) w (i) β i =( d i ) EAw (i) w (i ) w (i+) = Aw (i) α i w (i) β i w (i ) β i+ = E w(i+) w (i+) w (i+) =( β i+ ) w (i+) (90) ŵ () =( β ) r (0) d ={ if 2 r(0) r (0) >0 if 2 r (0) r (0) <0 ε =ε 2 =0 s 0 =0 c 0 = α =( 2 d ) ŵ () w () w (2) = Aw () α w () ŵ (2) = A T w () α ŵ () β 2 = 2 ŵ (2) w (2)

10 0 Abstract and Applied Analysis w 2 =( β 2 ) w (2) ŵ (2) =( β 2 ) ŵ (2) d 2 = { if 2 ŵ (2) w (2) >0 { { if 2 ŵ (2) w (2) <0 ζ = β α v () = w () x () = x (0) + ζ v () c = d α s = d β 2 α 2 +β2 2 α 2 +β2 2 end for v (i) =s i v (i ) c i w (i) x (i) = x (i ) + ζ i v (i) = α 2 i +β 2 i+ c i =d i ( α i ) ζ i =( )( ε i ζ i 2 δ i ζ i ) v (i) =c i v (i) +s i w (i+) x (i) = x (i ) +ζ i v (i) LAN/MSYMMQR Pseudocode s i =d i ( β i+ ) (93) γ =(d α )c +(d β 2 )s ζ = β γ v () =c v () +s w (2) x () = x (0) +ζ v () for i = 2 3 N α i =( 2 d i ) Aw (i) ŵ (i) (92) r (0) = b Ax (0) r (0) = b A T x (0) β = 2 r(0) r (0) w () =( ) r (0) β ŵ () =( ) r (0) β β i =( d i ) Aw (i) ŵ (i ) + ŵ (i) Aw (i ) w (i+) = Aw (i) α i w (i) β i w (i ) ŵ (i+) = A T ŵ (i) α i ŵ (i) β i ŵ (i ) β i+ = 2 w(i+) w (i+) w (i+) =( β i+ ) w (i+) ŵ (i+) =( β i+ ) ŵ (i+) d i+ ={ if 2 w(i+) w (i+) >0 if 2 w (i+) w i+ <0 ε i =d i β i s i 2 δ i = (d i β i )c i 2 hh = δ i δ i = (hh) c i +(d i α i )s i = (hh) s i (d i α i )c i ζ i =( )( ε i ζ i 2 δ i ζ i ) d ={ if 2 r(0) r (0) >0 if 2 r (0) r (0) <0 ε =ε 2 =0 s 0 =0 c 0 = α =( 2 d ) ŵ () w () w (2) = Aw () α w () ŵ (2) = A T w () α ŵ (2) β 2 = 2 ŵ (2) w (2) w (2) =( β 2 ) w (2) ŵ (2) =( β 2 ) ŵ (2) d 2 = { if 2 ŵ (2) w (2) >0 { if 2 ŵ (2) w { (2) <0 p () =( α i d i ) w ()

11 Abstract and Applied Analysis ζ = β α x () = x (0) + ζ p () c = d α α 2 +β2 2 s = d β 2 α 2 +β2 2 end for p (i) =( )w (i) ε i p (i 2) δ i p (i ) x (i) = x (i ) +ζ i p (i) (95) γ =(d α )c (d β 2 )s ζ =β γ p () =( γ ) w () x () = x (0) +ζ p () for i=23n α i =( 2 d i ) Aw (i) ŵ (i) β i =( d i ) Aw (i) ŵ (i ) + ŵ (i) Aw (i ) w (i+) = Aw (i) α i w (i) β i w (i ) ŵ (i+) = A T ŵ (i) α i ŵ (i) β i ŵ (i ) β i+ = 2 w(i+) w (i+) w (i+) =( β i+ ) w (i+) ŵ (i+) =( β i+ ) ŵ (i+) d i+ ={ if 2 w(i+) w (i+) >0 if 2 w (i+) w (i+) <0 (94) Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper References C C Paige and M A Saunders Solutions of sparse indefinite systems of linear equations SIAM Journal on Numerical Analysisvol2no4pp Y Saad and M H Schultz GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM Journal on Scientific and Statistical Computingvol7no 3pp Y Li The Modified MINRES methods for solving large sparse nonsymmetric linear systems MS thesis Department of Mathematics National Kaohsiung Normal University Kaohsiung Taiwan GHGolubandCFVanLoanMatrix Computations Johns Hopkins University Press Baltimore Md USA 3rd edition Y Saad IterativeMethodsforSparseLinearSystemsSIAM2nd edition C Lanczos An iteration method for the solution of the eigenvalue problem of linear differential and integral operators Research of the National Bureau of Standardsvol45 no 4 pp D R Kincaid D M Young and J-Y Chen An overview of MGMRES and LAN/MGMRES methods for solving nonsymmetric linear systems Taiwanese Mathematics vol 4 no 3 pp ε i =(d i β i )s i 2 δ i =(d i β i )c i 2 hh = δ i δ i = (hh) c i d i (α i s i ) = (hh) sc i +d i (α i c i ) ζ i = ζi s i p (i) =( )w (i) ε i p (i) δ i p (i ) x (i) = x (i ) + ζ i p (i) = γ 2 +β 2 i+ c i =( ) s i = d i ( β i+ ) ζ i =c i ζi

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