1.Start:Choose xo and compute ro =b-Axo and B=lroll,v1 ro/B. 2.Iterate:For j=1,2,...,m do: h,i=(（Avj,v),i=1,2,…,j +1=4妈-含4 合1 hj+1,=l⑦j+1ll: Uj+1=Di+1/h+1 3.Form the approximate solution: Em=o+Vmym;where ym minimizes lBe1-Hmyll,yE Rm.Here Hm is the (m+1)by m matrix whose only nonzero entries are the elements hij defined in step 2.Vm [v1,v2,...,Um]and the vector el is the first column of the (m +1)x(m +1)identity matrix. 4.Restart: Compute rm =b-Azm,if satisfied then stop else compute zo : Im,ro :=rm,B :=lroll,v1 :ro/B and go to 2. If A is not positive real,then ro span{Aro,A2ro,.,Amro}may happen. In this situation the restarted GMRES method is stationary.To avoid this disad- vantage,we introduce and analyze a new convergent restarted GMRES method. Conveniently,we use the term CGMRES(m)to denote the method. 2.CGMRES(m) The linear systems associated with (1.1)can be taken as the following form [-a[]=[ (2.1) where I E Rnxn is the identity matrix,while u*E Rn is a given vector and f u*+b,g =-ATu*E R".Since A is nonsingular,thus the system (2.1) u* has an unique solution z*= x* Let zo is the initial approximate solution 2B-[]%=] B2o.Solving the systems (2.1)with GMRES(m),where m >2,we have the following results: Proposition 2.1 Denoting by Bi,i=1,2,...,n,the eigenvalues of ATA and supposing 3≥32≥…≥3m21/4, (2.2) then we have that the eigenvalues of matrix B have positive real part. Proof We have .- (2.3) 21. Start: Choose x0 and compute r0 = b − Ax0 and β = kr0k, v1 = r0/β. 2. Iterate: For j = 1, 2, · · · , m do: hi,j = (Avj , vi), i = 1, 2, · · · , j, v¯j+1 = Avj − P j i=1 hi,jvi , hj+1,j = kv¯j+1k, vj+1 = ¯vj+1/hj+1,j . 3. Form the approximate solution: xm = x0 + Vmym, where ym minimizes kβe1 − Hmyk, y ∈ Rm. Here Hm is the (m+ 1) by m matrix whose only nonzero entries are the elements hi,j defined in step 2. Vm = [v1, v2, · · · , vm] and the vector e1 is the first column of the (m + 1) × (m + 1) identity matrix. 4. Restart: Compute rm = b − Axm, if satisfied then stop else compute x0 := xm, r0 := rm, β := kr0k, v1 := r0/β and go to 2. If A is not positive real, then r0 ⊥ span{Ar0, A2 r0, · · · , Amr0} may happen. In this situation the restarted GMRES method is stationary. To avoid this disad￾vantage, we introduce and analyze a new convergent restarted GMRES method. Conveniently , we use the term CGMRES(m) to denote the method. 2. CGMRES(m) The linear systems associated with (1.1) can be taken as the following form " I A −AT 0 # " u ∗ x # = " f g # , (2.1) where I ∈ Rn×n is the identity matrix , while u ∗ ∈ Rn is a given vector and f = u ∗ + b, g = −AT u ∗ ∈ Rn . Since A is nonsingular, thus the system (2.1) has an unique solution z ∗ = " u ∗ x ∗ # . Let z0 is the initial approximate solution of (2.1), B = " I A −AT 0 # , r¯0 = " f g # − Bz0. Solving the systems (2.1) with GMRES(m),where m ≥ 2 , we have the following results: Proposition 2.1 Denoting by βi , i = 1, 2, · · · , n, the eigenvalues of AT A and supposing β1 ≥ β2 ≥ · · · ≥ βn ≥ 1/4, (2.2) then we have that the eigenvalues of matrix B have positive real part. Proof We have |λI1 − B| = ¯ ¯ ¯ ¯ ¯ (λ − 1)I −A AT λI ¯ ¯ ¯ ¯ ¯ , (2.3) 2