// 轉載自： http://fortranwiki.org/fortran/show/Matrix+inversion
// 這裏僅適用於 小型矩陣 2*2,3*3，4*4 的矩陣求逆。事實上，對於這種規模的矩陣，直接寫出求逆的結果速度更快。
// 這裏3*3 的矩陣我已經驗證過了，其余的未驗證。一般矩陣的求逆方法原鏈接中也有。



pure function matinv2(A) result(B)
    !! Performs a direct calculation of the inverse of a 2×2 matrix.
    complex(wp), intent(in) :: A(2,2)   !! Matrix
    complex(wp)             :: B(
 
2,2)   !! Inverse matrix
    complex(wp)             :: detinv

    ! Calculate the inverse determinant of the matrix
    detinv = 1/(A(1,1)*A(2,2) - A(1,2)*A(2,1))

    ! Calculate the inverse of the matrix
    B(1,1) = +detinv * A(2,2)
    B(2,1) = -detinv * A(2,1)
    B(1,2) = -detinv * A(1,2)
    B(
 
2,2) = +detinv * A(1,1)
  end function

  pure function matinv3(A) result(B)
    !! Performs a direct calculation of the inverse of a 3×3 matrix.
    complex(wp), intent(in) :: A(3,3)   !! Matrix
    complex(wp)             :: B(3,3)   !! Inverse matrix
    complex(wp)             :: detinv

    ! Calculate the inverse determinant of the matrix
    detinv 
 
= 1/(A(1,1)*A(2,2)*A(3,3) - A(1,1)*A(2,3)*A(3,2)&
              - A(1,2)*A(2,1)*A(3,3) + A(1,2)*A(2,3)*A(3,1)&
              + A(1,3)*A(2,1)*A(3,2) - A(1,3)*A(2,2)*A(3,1))

    ! Calculate the inverse of the matrix
    B(1,1) = +detinv * (A(2,2)*A(3,3) - A(2,3)*A(3,2))
    B(2,1) = -detinv * (A(2,1)*A(3,3) - A(2,3)*A(3,1))
    B(3,1) = +detinv * (A(2,1)*A(3,2) - A(2,2)*A(3,1))
    B(1,2) = -detinv * (A(1,2)*A(3,3) - A(1,3)*A(3,2))
    B(2,2) = +detinv * (A(1,1)*A(3,3) - A(1,3)*A(3,1))
    B(3,2) = -detinv * (A(1,1)*A(3,2) - A(1,2)*A(3,1))
    B(1,3) = +detinv * (A(1,2)*A(2,3) - A(1,3)*A(2,2))
    B(2,3) = -detinv * (A(1,1)*A(2,3) - A(1,3)*A(2,1))
    B(3,3) = +detinv * (A(1,1)*A(2,2) - A(1,2)*A(2,1))
  end function

  pure function matinv4(A) result(B)
    !! Performs a direct calculation of the inverse of a 4×4 matrix.
    complex(wp), intent(in) :: A(4,4)   !! Matrix
    complex(wp)             :: B(4,4)   !! Inverse matrix
    complex(wp)             :: detinv

    ! Calculate the inverse determinant of the matrix
    detinv = &
      1/(A(1,1)*(A(2,2)*(A(3,3)*A(4,4)-A(3,4)*A(4,3))+A(2,3)*(A(3,4)*A(4,2)-A(3,2)*A(4,4))+A(2,4)*(A(3,2)*A(4,3)-A(3,3)*A(4,2)))&
       - A(1,2)*(A(2,1)*(A(3,3)*A(4,4)-A(3,4)*A(4,3))+A(2,3)*(A(3,4)*A(4,1)-A(3,1)*A(4,4))+A(2,4)*(A(3,1)*A(4,3)-A(3,3)*A(4,1)))&
       + A(1,3)*(A(2,1)*(A(3,2)*A(4,4)-A(3,4)*A(4,2))+A(2,2)*(A(3,4)*A(4,1)-A(3,1)*A(4,4))+A(2,4)*(A(3,1)*A(4,2)-A(3,2)*A(4,1)))&
       - A(1,4)*(A(2,1)*(A(3,2)*A(4,3)-A(3,3)*A(4,2))+A(2,2)*(A(3,3)*A(4,1)-A(3,1)*A(4,3))+A(2,3)*(A(3,1)*A(4,2)-A(3,2)*A(4,1))))

    ! Calculate the inverse of the matrix
    B(1,1) = detinv*(A(2,2)*(A(3,3)*A(4,4)-A(3,4)*A(4,3))+A(2,3)*(A(3,4)*A(4,2)-A(3,2)*A(4,4))+A(2,4)*(A(3,2)*A(4,3)-A(3,3)*A(4,2)))
    B(2,1) = detinv*(A(2,1)*(A(3,4)*A(4,3)-A(3,3)*A(4,4))+A(2,3)*(A(3,1)*A(4,4)-A(3,4)*A(4,1))+A(2,4)*(A(3,3)*A(4,1)-A(3,1)*A(4,3)))
    B(3,1) = detinv*(A(2,1)*(A(3,2)*A(4,4)-A(3,4)*A(4,2))+A(2,2)*(A(3,4)*A(4,1)-A(3,1)*A(4,4))+A(2,4)*(A(3,1)*A(4,2)-A(3,2)*A(4,1)))
    B(4,1) = detinv*(A(2,1)*(A(3,3)*A(4,2)-A(3,2)*A(4,3))+A(2,2)*(A(3,1)*A(4,3)-A(3,3)*A(4,1))+A(2,3)*(A(3,2)*A(4,1)-A(3,1)*A(4,2)))
    B(1,2) = detinv*(A(1,2)*(A(3,4)*A(4,3)-A(3,3)*A(4,4))+A(1,3)*(A(3,2)*A(4,4)-A(3,4)*A(4,2))+A(1,4)*(A(3,3)*A(4,2)-A(3,2)*A(4,3)))
    B(2,2) = detinv*(A(1,1)*(A(3,3)*A(4,4)-A(3,4)*A(4,3))+A(1,3)*(A(3,4)*A(4,1)-A(3,1)*A(4,4))+A(1,4)*(A(3,1)*A(4,3)-A(3,3)*A(4,1)))
    B(3,2) = detinv*(A(1,1)*(A(3,4)*A(4,2)-A(3,2)*A(4,4))+A(1,2)*(A(3,1)*A(4,4)-A(3,4)*A(4,1))+A(1,4)*(A(3,2)*A(4,1)-A(3,1)*A(4,2)))
    B(4,2) = detinv*(A(1,1)*(A(3,2)*A(4,3)-A(3,3)*A(4,2))+A(1,2)*(A(3,3)*A(4,1)-A(3,1)*A(4,3))+A(1,3)*(A(3,1)*A(4,2)-A(3,2)*A(4,1)))
    B(1,3) = detinv*(A(1,2)*(A(2,3)*A(4,4)-A(2,4)*A(4,3))+A(1,3)*(A(2,4)*A(4,2)-A(2,2)*A(4,4))+A(1,4)*(A(2,2)*A(4,3)-A(2,3)*A(4,2)))
    B(2,3) = detinv*(A(1,1)*(A(2,4)*A(4,3)-A(2,3)*A(4,4))+A(1,3)*(A(2,1)*A(4,4)-A(2,4)*A(4,1))+A(1,4)*(A(2,3)*A(4,1)-A(2,1)*A(4,3)))
    B(3,3) = detinv*(A(1,1)*(A(2,2)*A(4,4)-A(2,4)*A(4,2))+A(1,2)*(A(2,4)*A(4,1)-A(2,1)*A(4,4))+A(1,4)*(A(2,1)*A(4,2)-A(2,2)*A(4,1)))
    B(4,3) = detinv*(A(1,1)*(A(2,3)*A(4,2)-A(2,2)*A(4,3))+A(1,2)*(A(2,1)*A(4,3)-A(2,3)*A(4,1))+A(1,3)*(A(2,2)*A(4,1)-A(2,1)*A(4,2)))
    B(1,4) = detinv*(A(1,2)*(A(2,4)*A(3,3)-A(2,3)*A(3,4))+A(1,3)*(A(2,2)*A(3,4)-A(2,4)*A(3,2))+A(1,4)*(A(2,3)*A(3,2)-A(2,2)*A(3,3)))
    B(2,4) = detinv*(A(1,1)*(A(2,3)*A(3,4)-A(2,4)*A(3,3))+A(1,3)*(A(2,4)*A(3,1)-A(2,1)*A(3,4))+A(1,4)*(A(2,1)*A(3,3)-A(2,3)*A(3,1)))
    B(3,4) = detinv*(A(1,1)*(A(2,4)*A(3,2)-A(2,2)*A(3,4))+A(1,2)*(A(2,1)*A(3,4)-A(2,4)*A(3,1))+A(1,4)*(A(2,2)*A(3,1)-A(2,1)*A(3,2)))
    B(4,4) = detinv*(A(1,1)*(A(2,2)*A(3,3)-A(2,3)*A(3,2))+A(1,2)*(A(2,3)*A(3,1)-A(2,1)*A(3,3))+A(1,3)*(A(2,1)*A(3,2)-A(2,2)*A(3,1)))
  end function

矩陣求逆