Wrong inverse matrix using ZGETRI in Fortran

问题

I am trying to compute the inverse of a complex matrix with ZGETRI, but even if it executes without error (info = 0), it does not give me the correct inverse matrix and I have absolutely no idea where the error comes from.

PROGRAM solvelinear
implicit none
INTEGER                        :: i,j,info,lwork
INTEGER,dimension(3)        :: ipiv
COMPLEX(16), dimension(3,3) :: C,Cinv,M,LU
COMPLEX(16),allocatable :: work(:)

info=0
lwork=100
allocate(work(lwork))
ipiv=0
work=(0.d0,0.d0)

C(1,1)=(0.d0,-1.d0)
C(1,2)=(1.d0,5.d0)
C(1,3)=(2.d0,-2.d0)
C(2,1)=(4.d0,-1.d0)
C(2,2)=(2.d0,-3.d0)
C(2,3)=(-1.d0,2.d0)
C(3,1)=(1.d0,0.d0)
C(3,2)=(3.d0,-2.d0)
C(3,3)=(0.d0,1.d0)

write(*,*)"C = "
do i=1,3
   write(*,10)(C(i,j),j=1,3)
end do

!-- LU factorisation
LU=C
CALL ZGETRF(3,3,LU,3,ipiv,info)
write(*,*)'info = ',info
write(*,*)"LU = "
do i=1,3
   write(*,10)(LU(i,j),j=1,3)
end do

!-- Inversion of matrix C using the LU

Cinv=LU
CALL ZGETRI(3,Cinv,3,ipiv,work,lwork,info)
write(*,*)'info = ',info
write(*,*)"Cinv = "
do i=1,3
   write(*,10)(Cinv(i,j),j=1,3)
end do

!-- computation of C^-1 * C to check the inverse
M = matmul(Cinv,C)
write(*,*)"M = "
do i=1,3
   write(*,10)(M(i,j),j=1,3)
end do
          10 FORMAT(3('(',F20.10,',',F20.10,') '))

END PROGRAM solvelinear

I compile with ifort (and my LAPACK librairies version 3.7.1 are also compiled with ifort). Makefile:

#$Id: Makefile $
.SUFFIXES: .f90 .f .c .o
FC = ifort
FFLAGS = -g -check all -zmuldefs -i8
LIBS = -L/path/to/lapack-3.7.1 -llapack -ltmglib -lrefblas
MAIN = prog.o
EXEC = xx
all:  ${MAIN} Makefile
    ${FC} ${FFLAGS} -o ${EXEC} ${MAIN} ${LIBS}
.f.o: ${MODS} Makefile
    ${FC} ${FFLAGS} -c $<
.f90.o: ${MODS} Makefile
    ${FC} ${FFLAGS} -c $<

I have no errors when compiling. Here is my output:

 C = 
(       0.00000,      -1.00000) (       1.00000,       5.00000) (       2.00000,      -2.00000) 
(       4.00000,      -1.00000) (       2.00000,      -3.00000) (      -1.00000,       2.00000) 
(       1.00000,       0.00000) (       3.00000,      -2.00000) (       0.00000,       1.00000) 
 info =                      0
 LU = 
(       4.00000,       0.00000) (       2.00000,  120470.58824) (       2.00000,      -2.00000) 
(       0.00000,       0.00000) (28003147.29412,      -3.00000) (      -1.00000,       2.00000) 
(       1.00000,       0.00000) (       3.00000,      -2.00000) (       0.00000,       1.00000) 
 info =                      0
 Cinv = 
(       0.00000,       0.00000) (      -0.00000,      -0.00000) (       2.00000,      -2.00000) 
(      -0.00000,       0.00000) (      -0.00000,      -3.00000) (      -1.00000,       2.00000) 
(      -0.00000,      -0.00000) (       3.00000,      -2.00000) (       0.00000,       1.00000) 
 M = 
(       2.00000,      -2.00000) (       2.00000,     -10.00000) (       2.00000,       2.00000) 
(      -4.00000,     -10.00000) (      -8.00000,       2.00000) (       4.00000,       2.00000) 
(      10.00000,     -10.00000) (       2.00000,     -10.00000) (      -0.00000,       8.00000)

And M should be the identity if I'm not wrong.

回答1:

I suggest you to NOT use the kind notation with literal numbers like REAL(4) or COMPLEX(16).

First, it is ugly and not portable.

Second, it can be confusing for complex variables.

Here you define your variables as COMPLEX(16), but ZGETRI, and all other LAPACK Z routines, expects COMPLEX*16. These are NOT the same.

COMPLEX*16 is a non-standard notation for complex numbers with REAL*8 components. REAL*8 is a nonstandard notation for 8 byte real numbers that are normally equivalent to DOUBLE PRECISION.

COMPLEX(16) is a complex number with two REAL(16) components, provided such a kind exists. In compilers which provide REAL(16) this real is a quadruple precision, not double precision.

So you are effectively passing 32-byte complex variables instead of 16-byte complex variables.

There are enough resources where to learn how to use Fortran kinds properly. You can start with