Fortran编写代码实现用QMR算法求解大型线性方程组

可以使用Fortran语言编写代码来实现用QMR算法求解大型线性方程组。下面是一份Fortran代码的示例：

PROGRAM QMR

IMPLICIT NONE

INTEGER :: N, ITER_MAX, I, J
REAL(KIND=8) :: TOL, RHO, ALPHA, BETA, GAMMA, ETA, EPSILON, TAU, D, D1, D2
REAL(KIND=8), DIMENSION(:), POINTER :: X, S, V, R, T, Q, P
REAL(KIND=8), DIMENSION(:,:), POINTER :: A

! Read in the problem size and tolerance
WRITE(*,*) 'Enter the problem size:'
READ(*,*) N
WRITE(*,*) 'Enter the maximum number of iterations:'
READ(*,*) ITER_MAX
WRITE(*,*) 'Enter the tolerance:'
READ(*,*) TOL

! Allocate memory for the arrays
ALLOCATE(A(N,N), X(N), S(N), V(N), R(N), T(N), Q(N), P(N))

! Read in the matrix A and the vector b
WRITE(*,*) 'Enter the matrix A:'
DO I = 1, N
  DO J = 1, N
    READ(*,*) A(I,J)
  END DO
END DO
WRITE(*,*) 'Enter the vector b:'
DO I = 1, N
  READ(*,*) R(I)
END DO

! Initialize the algorithm
X = 0.0d0
RHO = 1.0d0
ALPHA = 0.0d0
BETA = 0.0d0
GAMMA = 0.0d0
ETA = 0.0d0
EPSILON = TOL**2

! Main QMR loop
DO I = 1, ITER_MAX
  CALL MATRIX_VECTOR_PRODUCT(A, X, S)
  R = R - S
  CALL MATRIX_VECTOR_PRODUCT(A, R, V)
  TAU = DOT_PRODUCT(R, V)
  IF (TAU == 0.0d0) THEN
    WRITE(*,*) 'Error: solution not found'
    EXIT
  END IF
  IF (I == 1) THEN
    P = R
  ELSE
    BETA = (TAU/RHO)*ALPHA/GAMMA
    P = R + BETA*(P - GAMMA*V)
  END IF
  CALL MATRIX_VECTOR_PRODUCT(A, P, Q)
  ALPHA = TAU/DOT_PRODUCT(P, Q)
  S = S + ALPHA*Q
  T = T - ALPHA*V
  CALL MATRIX_VECTOR_PRODUCT(A, T, Q)
  GAMMA = DOT_PRODUCT(Q, T)/DOT_PRODUCT(Q, Q)
  X = X + ALPHA*P + GAMMA*T
  R = R - ALPHA*Q - GAMMA*V
  RHO = DOT_PRODUCT(R, R)
  IF (RHO < EPSILON) THEN
    WRITE(*,*) 'Solution found after', I, 'iterations'
    EXIT
  END IF
  IF (I == 1) THEN
    D = RHO
  ELSE
    D1 = D
    D2 = DOT_PRODUCT(R, R)/ALPHA**2
    ETA = D2 - 2.0d0*TAU*GAMMA/ALPHA + GAMMA**2*D1
    IF (ETA < 0.0d0) THEN
      D = RHO
    ELSE
      D = RHO/SQRT(ETA)
    END IF
  END IF
END DO

! Output the solution
WRITE(*,*) 'The solution is:'
DO I = 1, N
  WRITE(*,*) X(I)
END DO

! Deallocate memory
DEALLOCATE(A, X, S, V, R, T, Q, P)

END PROGRAM QMR

SUBROUTINE MATRIX_VECTOR_PRODUCT(A, X, Y)

IMPLICIT NONE

INTEGER :: N, I, J
REAL(KIND=8), DIMENSION(:,:), POINTER :: A
REAL(KIND=8), DIMENSION(:), POINTER :: X, Y

! Read in the problem size
N = SIZE(X)

! Compute the matrix-vector product
DO I = 1, N
  Y(I) = 0.0d0
  DO J = 1, N
    Y(I) = Y(I) + A(I,J)*X(J)
  END DO
END DO

END SUBROUTINE MATRIX_VECTOR_PRODUCT

注意，在上面的代码中，我们使用了 `DIMENSION(:), POINTER` 这样的语法来声明变量，这是Fortran中动态数组的用法。在程序的开始部分，我们读入了问题的规模和精度要求，并分配了内存以存储矩阵、向量和算法中的其他变量。然后，我们进入了QMR循环，其中我们计算了矩阵向量积、更新步长、更新解向量等。循环在满足精度要求或达到最大迭代次数时退出，并输出计算得到的解向量。最后，我们释放了分配的内存。

需要注意的是，上面的代码仅仅是一个QMR算法的示例，实际问题的求解可能需要更多的代码和复杂的算法。