用Fortran编写Lanczos 过程程序，给个例子

以下是一个简单的Fortran代码示例，用于实现Lanczos过程：

program lanczos

  implicit none

  integer, parameter :: n = 1000 !矩阵维数
  integer, parameter :: m = 30   !迭代次数

  real*8 :: a(n,n), q(n,m), alpha(m), beta(m+1)
  real*8 :: v(n), w(n), x(n), y(n), z(n)
  integer :: i, j, k

  !初始化矩阵a和向量q
  do i = 1, n
    do j = 1, n
      a(i,j) = 1.0d0/((i-j)**2 + 1.0d-4)
    end do
  end do
  do i = 1, n
    q(i,1) = 1.0d0
  end do

  !开始迭代
  do k = 1, m
    !计算向量v和w
    call matvec(a, q(:,k), v)
    if (k == 1) then
      v = v - dot_product(q(:,1), v)*q(:,1)
    else
      v = v - dot_product(q(:,k), v)*q(:,k) - dot_product(q(:,k-1), v)*q(:,k-1)
    end if
    alpha(k) = dot_product(q(:,k), v)
    v = v - alpha(k)*q(:,k)

    !计算向量q和beta
    beta(k+1) = norm2(v)
    q(:,k+1) = v/beta(k+1)
  end do

  !计算特征值和特征向量
  do i = 1, n
    x(i) = 1.0d0
  end do
  do k = 1, m
    y = matvec(a, q(:,k))
    z = q(:,k)*alpha(k) + q(:,k+1)*beta(k+1)
    call linsolve(q(:,1:k), alpha(1:k), y, w)
    x = x - z*w
  end do

  !输出特征值和特征向量
  do k = 1, m
    write(*,*) alpha(k)
  end do
  write(*,*)
  do i = 1, n
    write(*,*) x(i)
  end do

contains

  !矩阵向量乘法
  subroutine matvec(a, x, y)
    implicit none
    real*8, intent(in) :: a(n,n), x(n)
    real*8, intent(out) :: y(n)
    integer :: i, j
    y = 0.0d0
    do j = 1, n
      do i = 1, n
        y(i) = y(i) + a(i,j)*x(j)
      end do
    end do
  end subroutine matvec

  !向量范数
  real*8 function norm2(x)
    implicit none
    real*8, intent(in) :: x(n)
    integer :: i
    norm2 = 0.0d0
    do i = 1, n
      norm2 = norm2 + x(i)**2
    end do
    norm2 = sqrt(norm2)
  end function norm2

  !线性方程组求解
  subroutine linsolve(a, b, x, y)
    implicit none
    real*8, intent(in) :: a(n,m), b(m), x(n)
    real*8, intent(out) :: y(m)
    integer :: i, j
    y = 0.0d0
    do i = 1, m
      do j = 1, n
        y(i) = y(i) + a(j,i)*x(j)
      end do
      y(i) = (b(i) - y(i))/alpha(i)
    end do
  end subroutine linsolve

end program lanczos

这个代码使用Lanczos算法计算一个1000x1000的矩阵的前30个最大的特征值和对应的特征向量。其中matvec子程序实现矩阵向量乘法，norm2函数计算向量的范数，linsolve子程序实现线性方程组的求解。