用Fortran编写Inverse-free Krylov程序，给个例子

以下是使用Fortran编写的Inverse-free Krylov程序示例：

program inverse_free_krylov
    implicit none
    integer :: n, maxiter, i, j, iter
    real :: tol, alpha, beta, error
    real, dimension(:), allocatable :: x, v, w, q, h, y
    real, dimension(:,:), allocatable :: qh, r
    character(len=80) :: inputfile

    ! 读取输入文件中的矩阵和向量
    write(*,*) '请输入矩阵和向量的文件名：'
    read(*,'(a)') inputfile
    call read_input_file(inputfile, n, v)

    ! 初始化参数
    maxiter = 100
    tol = 1e-6
    iter = 0
    error = 0.0
    allocate(q(n,maxiter), h(maxiter,maxiter), qh(n,maxiter+1), r(maxiter,maxiter+1), y(maxiter))

    ! 开始迭代
    do while (iter < maxiter .and. error > tol)
        iter = iter + 1

        ! 计算新向量
        call matvec(n, v, q(:,iter))
        do j = 1, iter
            h(j,iter) = dot_product(q(:,iter), q(:,j))
            q(:,iter) = q(:,iter) - h(j,iter)*q(:,j)
        enddo
        h(iter+1,iter) = sqrt(dot_product(q(:,iter), q(:,iter)))
        q(:,iter) = q(:,iter)/h(iter+1,iter)

        ! 用QR分解求解Hessenberg矩阵的特征值和特征向量
        call qrfactorization(iter+1, iter, h, qh, r)

        ! 更新解向量
        y = solve_upper_triangular(iter, r, qh(:,1:iter), q(:,1:iter), h(iter+1,iter)*v)
        x = x + q(:,1:iter)*y

        ! 判断收敛
        error = norm2(q(:,iter+1)*h(iter+1,iter))

    enddo

    ! 输出解向量
    write(*,*) '解向量：'
    do i = 1, n
        write(*,'(f8.4)') x(i)
    enddo

    ! 释放内存
    deallocate(q, h, qh, r, y, v)

contains

    ! 矩阵向量乘法
    subroutine matvec(n, x, y)
        implicit none
        integer :: n, i
        real, dimension(n), intent(in) :: x
        real, dimension(n), intent(out) :: y

        y = 0.0
        do i = 1, n
            y = y + x(i)*a(:,i)
        enddo
    end subroutine matvec

    ! QR分解
    subroutine qrfactorization(m, n, a, q, r)
        implicit none
        integer :: m, n, i, j, k
        real, dimension(m,n), intent(in) :: a
        real, dimension(m,n), intent(out) :: q
        real, dimension(n,n), intent(out) :: r

        q = 0.0
        r = 0.0

        do k = 1, n
            r(k,k) = norm2(a(:,k))
            q(:,k) = a(:,k)/r(k,k)
            do j = k+1, n
                r(k,j) = dot_product(q(:,k), a(:,j))
                a(:,j) = a(:,j) - r(k,j)*q(:,k)
            enddo
        enddo
    end subroutine qrfactorization

    ! 解上三角线性方程组
    function solve_upper_triangular(n, a, b, c, d)
        implicit none
        integer :: n, i, j
        real, dimension(n,n), intent(in) :: a
        real, dimension(n), intent(in) :: d
        real, dimension(n), intent(in) :: b(:,1:n-1)
        real, dimension(n), intent(in) :: c(:,1:n-1)
        real, dimension(n) :: x

        x(n) = d(n)/a(n,n)
        do i = n-1, 1, -1
            x(i) = (d(i) - dot_product(a(i,i+1:n), x(i+1:n)) - &
                dot_product(b(i,1:i-1), x(1:i-1)) - dot_product(c(i,1:i-1), x(1:i-1))) / a(i,i)
        enddo

        solve_upper_triangular = x
    end function solve_upper_triangular

    ! 求向量的2范数
    function norm2(x)
        implicit none
        integer :: i, n
        real, dimension(:), intent(in) :: x
        real :: norm2

        n = size(x)
        norm2 = 0.0
        do i = 1, n
            norm2 = norm2 + x(i)**2
        enddo
        norm2 = sqrt(norm2)
    end function norm2

    ! 读取输入文件中的矩阵和向量
    subroutine read_input_file(filename, n, v)
        implicit none
        integer :: n, i
        real, dimension(:), allocatable :: v
        character(len=*), intent(in) :: filename
        integer :: status
        integer :: unit

        ! 打开输入文件
        open(newunit=unit, file=trim(filename), status='old', action='read', iostat=status)
        if (status /= 0) then
            write(*,*) '无法打开文件：', trim(filename)
            stop
        endif

        ! 读取矩阵和向量
        read(unit,*) n
        allocate(v(n))
        do i = 1, n
            read(unit,*) v(i)
        enddo

        ! 关闭文件并返回结果
        close(unit)
    end subroutine read_input_file
end program inverse_free_krylov

这个程序实现了一个简单的Inverse-free Krylov算法，其中包括矩阵向量乘法、QR分解、解上三角线性方程组等基本操作。在实际使用中，需要根据具体的问题进行修改和优化。