【数值算法】系数矩阵非对称时线性方程组求解方法：稳定双共轭梯度法(Bicgstab)

program bicgstab_main
    implicit none
    integer,parameter::n=8
    real(8)::a(n,n),b(n)
    real(8)::x(n),x0(n)
    integer::i,j
    integer,parameter::m=20
    real(8)::c(m,m),d(m)
    real(8)::y(m),y0(m)
    
    
    a=reshape((/6,5,1,2,0,0,2,1, &
                0,5,1,1,0,0,3,0,&
                1,1,623,1,2,0,1001,2, &
                2,1,1,7,1,2,1,1,&
                0,0,2,31,6,0,2,1,&
                0,0,0,2,0,4,1,0,&
                2,3,1,1,23,1,5,213,&
                123,0,12,1,1,0,1,3/),(/n,n/))
    b=(/1,1,1,1,1,1,1,1/)
    write(*,*)"矩阵A"
    write(*,"(8f10.4)")(a(i,:),i=1,n)
    write(*,*)"向量b"
    write(*,"(f10.4)")b
    x0=100.0
    x=0.0
    call bicgstab(a,b,x,x0,n)
    end program bicgstab_main
subroutine bicgstab(a,b,x,x0,n)
    implicit none
    integer,intent(in)::n
    real(kind=8),intent(in)::a(n,n),b(n),x0(n)
    real(kind=8),intent(out)::x(n)
    real(kind=8)::p0(n),r0(n)
    real(kind=8)::tol=1.0d-6
    integer::nmax=1000
    real(kind=8)::rj(n),pj(n)
    real(kind=8)::alphaj
    real(kind=8)::r0_bar(n)
    real(kind=8)::sj(n)
    real(kind=8)::xjp1(n),xj(n)
    real(kind=8)::wj
    real(kind=8)::rjp1(n)
    real(kind=8)::betaj
    integer::n_iter
    real(kind=8)::apj(n),asj(n)
 
    r0=b-matmul(a,x0)
    r0_bar=r0
    if(abs(dot_product(r0,r0_bar))<tol) then
        write(*,*)"unvalid r0_bar,select an other!"
        return    
    endif
    p0=r0
     rj=r0
     pj=p0
     xj=x0
     
     n_iter=0
    do
        if(n_iter>nmax) then
            write(*,*)"exceed max iter!"
            exit
        endif
        alphaj=dot_product(rj,r0_bar)/dot_product(matmul(a,pj),r0_bar)
        apj=matmul(a,pj)
        sj=rj-alphaj*apj
        if(norm2(sj)<tol) then
            xjp1=xj+alphaj*pj
            x=xjp1
            exit
        endif
        asj=matmul(a,sj)
    wj=dot_product(asj,sj)/(norm2(asj))**2
    xjp1=xj+alphaj*pj+wj*sj
    rjp1=sj-wj*asj
    
    if(norm2(rjp1)<tol) then
        x=xjp1
        exit
    endif
    betaj=alphaj*dot_product(rjp1,r0_bar)/(wj*dot_product(rj,r0_bar))
    pj=rjp1+betaj*(pj-wj*apj)
    xj=xjp1
    rj=rjp1
    n_iter=n_iter+1
    write(*,*)"the",n_iter,"iter!"
    enddo
    
    write(*,*)"the solution of equation:"
    write(*,"(es18.8)")x
    end subroutine bicgstab