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+function gmres{T1,T2}(A::SparseMatrixCSC{T1,Int},b::Array{T2,1},restrt::Int; kwargs...)
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+ Ax = zeros(promote_type(T1,T2),size(A,1))
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+ return gmres(x -> A_mul_B!(1.0,A,x,0.0,Ax),b,restrt;kwargs...)
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+end
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+
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+"""
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+x,flag,err,iter,resvec = gmres(A,b,restrt,tol=1e-2,maxIter=100,M=1,x=[],out=0)
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+
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+Generalized Minimal residual (GMRESm) method with restarts applied to A*x = b.
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+
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+Input:
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+
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+A - function computing A*xor
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+
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+M -preconditioner,function computing M\\x
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+"""
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+function gmres(A::Function,b::Vector,restrt::Int,tol::Real=1e-2,maxIter::Int=100,M::Function=identity,x::Vector=[],out::Int=0,storeInterm::Bool=false)
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+ n = length(b)
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+ if norm(b)==0;return zeros(eltype(b),n),-9,0.0,0,[0.0];end
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+ if isempty(x)
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+ x = zeros(n)
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+ r = M(b)
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+ else
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+ r = M(b-A(x))
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+ end
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+ if storeInterm
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+ x = zeros(n,maxIter)
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+ end
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+
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+ bnrm2 = norm(b)
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+ if bnrm2 == 0.0;bnrm2 = 1.0;end
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+
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+ err = norm( r ) / bnrm2
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+
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+ if err < tol; return x,err;end
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+
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+ #重启次数 m
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+ restrt = min(restrt,n-1)
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+ #Arnoldi向量 m+1
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+ V = zeros(n,restrt+1)
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+ #m+1,m
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+ H = zeros(restrt+1,restrt)
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+ #Gj的cs
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+ cs = zeros(restrt)
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+ #Gj的sn
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+ sn = zeros(restrt)
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+ #e1
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+ e1 = zeros(n)
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+ e1[1] = 1.0
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+
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+ #暂时不考虑complex
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+
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+ #残差数组
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+ resvec = zeros((1+restrt)*maxIter)
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+ if out==2
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+ println(@sprintf("=== gmres ===\n%4s\t%7s\n","iter","relres"))
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+ end
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+
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+ #初始化
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+ iter = 0
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+ flag = -1
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+ cnt = 1
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+
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+ for iter = 1:maxIter
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+ # v1 = r0/belta
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+ V[:,1] = r / norm(r)
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+ #theta = belta*e1
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+ s = norm(r)*e1;
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+ #显示迭代次数
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+ if out==2;;print(@sprintf("%3d\t",iter));end
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+ #开始迭代 带重启
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+ for i = 1:restrt
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+ #wi = Avi
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+ w = A(V[:,i])
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+ w = M(w)
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+
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+ #Arnoldi process
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+ for k = 1:i
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+ # hij = （wj,vi)
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+ H[k,i] = dot(w,V[:,k])
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+ # wj = wj - hijVi
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+ w -= H[k,i] * V[:,k]
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+ end
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+ #hj+1,j = ||wj||2
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+ H[i+1,i] = norm(w)
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+
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+
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+ #Apply Givens rotation
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+ for k = 1:i-1
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+ temp = cs[k]*H[k,i] + sn[k]*H[k+1,i]
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+ H[k+1,i] = -sn[k] *H[k,i] + cs[k]*H[k+1,i]
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+ H[k,i] = temp
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+ end
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+
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+ #跳出
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+ if H[i+1,i] == 0
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+
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+ end
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+
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+ #vj+1 = wj/h(j+1,j)
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+ V[:,i+1] = w/H[i+1,i]
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+
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+ #From the Givens rotation Gj
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+ cs[i],sn[i] = symOrtho(H[i,i],H[i+1,i])
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+
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+ #Apply Gj to last column of Hj+1,j
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+ H[i,i] = cs[i]*H[i,i]+sn[i,i]*H[i+1,i]
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+ H[i+1,i] = 0.0
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+
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+ #Apply Gj to right-hand side
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+ s[i+1] = -sn[i]*s[i]
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+ s[i] = cs[i]*s[i]
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+
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+ #check convergence
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+ err = abs(s[i+1])/bnrm2
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+
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+ if out == 2;print(@sprintf("%1.1e",err));end
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+
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+ resvec[cnt] = err
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+
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+ if err <= tol
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+ y = H[1:i,1:i] \ s[1:i]
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+ x += V[:,1:i]*y
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+ if out == 2;print("\n"); end
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+ flag = 0;break;
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+ end
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+ cnt = cnt + 1
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+ end
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+ if err <= tol
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+ flag = 0
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+ break
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+ end
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+ y = H[1:restrt,1:restrt]\s[1:restrt]
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+ x += V[:,1:restrt]*y
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+ if storeInterm; X[:,iter] = x; end
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+
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+ r = b - A(x)
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+ r = M(r)
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+
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+ s[restrt+1] = norm(r)
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+ resvec[cnt] = abs(s[restrt+1]) / bnrm2
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+
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+ if out==2; print(@sprintf("\t %1.1e\n", err)); end
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+
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+ end
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+ if out>=0
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+ if flag==-1
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+ println(@sprintf("gmres iterated maxIter (=%d) times without achieving the desired tolerance.",maxIter))
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+ elseif flag==0 && out>=1
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+ println(@sprintf("gmres achieved desired tolerance at iteration %d. Residual norm is %1.2e.",iter,resvec[cnt]))
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+ end
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+ end
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+ if storeInterm
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+ return X[:,1:iter],flag,resvec[cnt],iter,resvec[1:cnt]
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+ else
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+ return x,flag,resvec[cnt],iter,resvec[1:cnt]
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+ end
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+end
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+
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+
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+"""
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+c,s,r = SymOrtho(a,b)
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+
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+Computes a Givens rotation
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+
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+Implementation is based on Table 2.9 in
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+
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+Choi, S.-C. T. (2006).
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+Iterative Methods for Singular Linear Equations and Least-squares Problems.
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+Phd thesis, Stanford University.
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+"""
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+function symOrtho(a,b)
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+ c = 0.0; s = 0.0; r = 0.0
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+ if b==0
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+ s = 0.0
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+ r = abs(a)
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+ c = (a==0) ? c=1.0 : c = sign(a)
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+ elseif a == 0
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+ c = 0.0
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+ s = sign(b)
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+ r = abs(b)
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+ elseif abs(b) > abs(a)
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+ tau = a/b
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+ s = sign(b)/sqrt(1+tau^2)
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+ c = s*tau
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+ r = b/s
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+ elseif abs(a) > abs(b)
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+ tau = b/a
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+ c = sign(a)/sqrt(1+tau^2)
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+ s = c*tau
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+ r = a/c
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+ end
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+ return c,s,r
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+end
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