如何制作 Makefile 以运行调用英特尔 MKL 的 gfortran 文件

问：

我正在尝试使用英特尔 MKL 中的 PARDISO 子程序求解大型稀疏矩阵系统。我用 gfortran 和 makefile 完成了代码的编写。但是当我执行makefile时，弹出错误消息说“未定义对'mkl_pardiso_'的引用”。我在 Ubuntu 22.04 LTS 上运行代码，gfortran 代码和 makefile 如下：

program large_sparse_solver
  use omp_lib  
  
  ! Include Intel MKL headers
  include 'mkl.fi' 


  ! Define the matrix dimensions
  integer, parameter :: eps = 5.0d-15
  integer, parameter :: p = 12 
  integer, parameter :: m = 2*(p+1) + 4*p
  integer, parameter :: nb = p*(m-p-1)         ! Block size
  integer, parameter :: n = (m-p-1)**2          ! Size of the system
  integer, parameter :: num_blocks = n/nb ! Number of blocks in the sparse matrix

  ! Define arrays
  real(8), dimension(n,n) :: A
  real(8), dimension(n) :: b, x

  ! Declarations and variables for timing and residual
  integer :: i, j, iTimes1, iTimes2, rate
  real(8) :: elapsed_time
  real(8) :: rsdl(n) ! Residual

  ! Initialize MKL parameters
  integer, parameter :: max_threads = 28 ! Number of OpenMP threads

  ! Load A, b
  OPEN(1, FILE='./data/stifk1n5', STATUS='old', ACCESS='SEQUENTIAL', FORM='FORMATTED', ACTION='READ')
  OPEN(2, FILE='./data/rhsk1n5', STATUS='old', ACCESS='SEQUENTIAL', FORM='FORMATTED', ACTION='READ')
  do i = 1, n
      read(1, *) (A(i,j), j=1,n)
      read(2, *) b(i)
  enddo
  close(1); close(2)

  ! Initialize x
  x(:) = 0.0d0

  call system_clock(count_rate=rate)

  ! Start timing
  call system_clock(iTimes1)

  ! Set the number of threads for OpenMP
  call omp_set_num_threads(max_threads)

  ! Initialize MKL
  call mkl_set_num_threads(max_threads)

  ! Use the PARDISO solver from MKL for solving
  call solve_sparse_system(A, b, x, n)

  ! Stop timing
  call system_clock(iTimes2)

  ! Calculate and print the execution time
  elapsed_time = real(iTimes2-iTimes1)/real(rate)
  print *, "Execution time (seconds): ", elapsed_time

  ! Clean up
  call mkl_free_buffers()

contains 

! PARDISO in Intel MKL for Fortran
! Calculates the solution of a set of sparse linear equations with single or multiple right-hand sides.
! https://www.intel.com/content/www/us/en/docs/onemkl/developer-reference-fortran/2023-0/pardiso.html#GUID-431916D5-B76D-48A1-ABB5-1A0613FDC0FA

  subroutine solve_sparse_system(A, b, x, n)
    integer, intent(in) :: n
    real(8), intent(inout) :: A(n,n), b(n)
    real(8), intent(out) :: x(n)

    ! Input parameters of PARDISO
    integer :: mtype, nrhs, maxfct, mnum, phase
    ! Output parameters of PARDISO  
    integer :: perm(n), iparm(64), error, msglvl

    integer :: i, j, k, nz, ia(n+1)  
    integer, allocatable :: ja(:)
    real(8), allocatable :: nza(:)

    TYPE(mkl_pardiso_handle) :: pt(64)

    ! Convert the matrix system into three array variation of CSR format
    call get_CSR(A, b, n, nza, ia, ja, nz)

    ! Initialize PARDISO solver parameters
    ! iparm(:) = 0
    ! pt(:) = 0
    mtype = 11
    maxfct = 1
    mnum = 1
    nrhs = 1
    phase = 23
    msglvl = 0
    call pardisoinit (pt, mtype, iparm)
   
    ! iparm(2) = 2  ! Parallel factorization
    ! iparm(3) = 3  ! Parallel solve
    ! iparm(4) = 0  ! No iterative refinement
    ! iparm(8) = 2  ! Use Intel MKL PARDISO

    ! Solve the matrix system using PARDISO
    call mkl_pardiso(pt, maxfct, mnum, mtype, phase, n, a, ia, ja, perm, nrhs, iparm, msglvl, b, x, error)

    if (error /= 0) then
      print *, "PARDISO solver returned error code: ", error
      stop  
    end if
 
    ! Deallocate memory
    ! deallocate(ia, ja, nza) 
 
  end subroutine solve_sparse_system

  ! Convert the matrix system into three array variation of CSR format
  ! Output : 
  ! nz = the number of non-zero elements in A(n,n)
  ! nza = the arrya of non-zero elements of A, (nz by 1) array
  ! ia(i) = the index of element in 'nza' corresponding to the index of column of the first non-zero element in the i-th row of A (rowIndex), (n+1 by 1) array, ia(n+1) = nz + 1
  ! ja(j) = the index of column in A corresponding to the index of element in 'nza', (nz by 1) array

  subroutine get_CSR(A, b, n, nza, ia, ja, nz)
    integer, intent(in) :: n
    real*8, intent(in) :: A(n,n), b(n)

    integer, intent(inout) :: nz, ia(n+1)
    integer, allocatable, intent(inout) :: ja(:)
    real*8, allocatable, intent(inout) :: nza(:)

    logical :: non_mask(n,n)

    integer :: i, j, k, ii, jj, kk

    ! Count the number of non-zero elements in matrix A
    non_mask = A(:,:) < -eps .or. A(:,:) > eps
    nz = count(non_mask)

    ! Allocate the size of ja, nza
    allocate(ja(nz), nza(nz))

    ! k = 0
    ! do i = 1, n
    !     do j = 1, n
    !         if (dabs(A(i,j))>eps) then 
    !             k = k + 1
    !         endif
    !     enddo
    ! enddo

    k = 0
    do i = 1, n
        kk = 0
        do j = 1, n
            if (dabs(A(i,j))>eps) then 
                k = k + 1
                nza(k) = A(i,j)
                ja(k) = j
                if (kk==0) then 
                    ia(i) = k
                    kk = 1
                endif
            endif
        enddo
    enddo
    ia(n+1) = nz + 1

  end subroutine get_CSR

end program large_sparse_solver

这是 makefile。

# Makefile for compiling and running the large_sparse_solver Fortran program

# Compiler settings
FC = gfortran
FLAGS = -O3 -fopenmp -ffixed-line-length-none -ffree-line-length-none


# Libraries (Intel MKL)
MKLROOT = /opt/intel/oneapi/mkl/2023.2.0
MKL_LIB = -L$(MKLROOT)/lib/intel64 -lmkl_intel_lp64 -lmkl_gnu_thread -lmkl_core -lgomp -lpthread -lm -ldl
# MKL_LIB = -L$(MKLROOT)/lib/intel64 -libmkl_intel_lp64 -libmkl_gnu_thread -libmkl_core -libgomp -lpthread -lm -ldl 
# MKL_PARDISO = -L$(MKLROOT)/include/mkl_pardiso.f90


# Include directory for MKL headers
MKL_INCLUDE = -I$(MKLROOT)/include

# Source files
SRC = large_sparse_solver.f90
# Executable name
EXEC = test_run

# Targets
all: $(EXEC)

$(EXEC): $(SRC)
    $(FC) $(FLAGS) $(MKL_INCLUDE) $^ -o $@ $(MKL_LIB)

clean:
    rm -f *.mod $(EXEC)

我需要一个 makefile 来执行调用英特尔 MKL 子例程的 gfortran 文件。