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Sample code for BiCGSTAB - Fortran 90

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	!-----------------------------------MIT LICENCE-----------------------------------------!
	!											!
	!	Copyright (c) 2015, Biswajit Ghosh					        !
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	!---------------------------------------------------------------------------------------!


        !---------------------------------------------------------------------------------------!
        !       REFERENCE :                                                                     !
        !       The Improved BiCGStab Method for Large and Sparse Unsymmetric Linear            !
        !       Systems on Parallel Distributed Memory Architectures; Yang et. al.              !
        !                                                                                       !
        !       NOTE:                                                                           !
        !       This is the classic version of BICGStab. Stopping criteria of these code        !
        !       needs additional treatment to avoid/detect failure condition!                   !
        !---------------------------------------------------------------------------------------!


program main
        implicit none

        interface BICGStab
                function BICGStab(A,b) result(x)
                        real    (kind=8), intent(in )   :: A(:,:)
                        real    (kind=8), intent(in )   :: b( : )
                        real    (kind=8)                :: x(1:size(b, dim=1))
                end function BICGStab
        end interface ! BICGStab

        integer (kind=4), parameter             :: m=4, n=4
        real    (kind=8), dimension(1:m,1:n)    :: A
        real    (kind=8), dimension(1:m    )    :: x,b

!-------------------------------A,b DEFINATION----------------------------------!
        A(1,:) = (/ 1.0d0, 1.0d0, 1.0d0, 1.0d0/) ![x(1)] = b(1) |
        A(2,:) = (/ 2.0d0, 3.0d0, 1.0d0, 1.0d0/) ![x(2)] = b(2) |---> [A]{x}={b}
        A(3,:) = (/ 3.0d0, 4.0d0, 1.0d0, 1.0d0/) ![x(3)] = b(3) |
        A(4,:) = (/ 3.0d0, 4.0d0, 1.0d0, 2.0d0/) ![x(4)] = b(4) |
        b( : ) = (/ 4.0d0, 7.0d0, 9.0d0,10.0d0/)
!-----------------------------END A,b DEFINATION--------------------------------!

        x = BICGStab(A,b)

        print*, x

end program main





function BICGStab(A,b) result(x)
        implicit none

!--------------------------PARAMETER AND VARIABLE-------------------------------!
        real    (kind=8), intent(in )                   :: A (:,:)
        real    (kind=8), intent(in )                   :: b ( : )
        real    (kind=8), dimension(1:size(b, dim=1))   :: x

        real    (kind=8), dimension(1:size(b, dim=1))   :: r, rs, v, p, s, t
        real    (kind=8), parameter                     :: e = 1d-10
        real    (kind=8)                                :: rho      , rho_prev
        real    (kind=8)                                :: alpha    , omega   , beta
        real    (kind=8)                                :: norm_r   , norm_b       
        real    (kind=8)                                :: summesion, temp

        integer                                         :: it=0,err
!------------------------END PARAMETER AND VARIABLE-----------------------------!  

        if(size(A, dim=1) /= size(A, dim=2)) stop &
        "Error: Improper dimension of matrix A in BICGStab."

!-------------------------------------------------------!
        x  = 0.0d0                                      !-------> INITIAL GUESS
!-------------------------------------------------------!
        r  = b - matmul(A,x)                            !-------> LINE 1
        rs = r                                          !
!-------------------------------------------------------!
        rho   = 1.0d0; alpha = 1.0d0; omega = 1.0d0     !-------> LINE 2
!-------------------------------------------------------!
        v  = 0.0d0; p  = 0.0d0                          !-------> LINE 3
!                                                       !
        norm_r = dsqrt(dot_product(r,r))                !
        norm_b = dsqrt(dot_product(b,b))                !
!-------------------------------------------------------!

        do while(norm_r .GT. e*norm_b)                                  !-------> START OF LOOP

                !-------------------------------------------------------!
                        rho_prev = rho                                  !-------> LINE 5
                        rho      = dot_product(rs,r)                    !
                !-------------------------------------------------------!
                        beta     = (rho/rho_prev) * (alpha/omega)       !-------> LINE 6
                !-------------------------------------------------------!
                        p        = r + beta * (p - omega*v)             !-------> LINE 7
                !-------------------------------------------------------!
                        v        = matmul(A,p)                          !-------> LINE 8
                !-------------------------------------------------------!
                        alpha    = rho/dot_product(rs,v)                !-------> LINE 9
                !-------------------------------------------------------!
                        s        = r - alpha*v                          !-------> LINE 10
                !-------------------------------------------------------!
                        t        = matmul(A,s)                          !-------> LINE 11
                !-------------------------------------------------------!
                        omega    = dot_product(t,s)/dot_product(t,t)    !-------> LINE 12
                !-------------------------------------------------------!
                        x        = x + alpha*p + omega*s                !-------> LINE 13
                !-------------------------------------------------------!
                        r        = s - omega*t                          !-------> LINE 17
                !-------------------------------------------------------!
        
                        norm_r   = dsqrt(dot_product(r,r))
                        norm_b   = dsqrt(dot_product(b,b))   
        
                        it = it + 1

        end do !END LOOP

        print*, "Iteration Required :", it

return
end function BICGStab

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