Computing the inverse of a matrix using lapack in C

Question

I would like to be able to compute the inverse of a general NxN matrix in C/C++ using lapack.

My understanding is that the way to do an inversion in la

Answer 1

First, M has to be a two-dimensional array, like double M[3][3]. Your array is, mathematically speaking, a 1x9 vector, which is not invertible.

• N is a pointer to an int for the order of the matrix - in this case, N=3.

• A is a pointer to the LU factorization of the matrix, which you can get by running the LAPACK routine dgetrf.

• LDA is an integer for the "leading element" of the matrix, which lets you pick out a subset of a bigger matrix if you want to just invert a little piece. If you want to invert the whole matrix, LDA should just be equal to N.

• IPIV is the pivot indices of the matrix, in other words, it's a list of instructions of what rows to swap in order to invert the matrix. IPIV should be generated by the LAPACK routine dgetrf.

• LWORK and WORK are the "workspaces" used by LAPACK. If you are inverting the whole matrix, LWORK should be an int equal to N^2, and WORK should be a double array with LWORK elements.

• INFO is just a status variable to tell you whether the operation completed successfully. Since not all matrices are invertible, I would recommend that you send this to some sort of error-checking system. INFO=0 for successful operation, INFO=-i if the i'th argument had an incorrect input value, and INFO > 0 if the matrix is not invertible.

So, for your code, I would do something like this:

int main(){

    double M[3][3] = { {1 , 2 , 3},
                       {4 , 5 , 6},
                       {7 , 8 , 9}}
    double pivotArray[3]; //since our matrix has three rows
    int errorHandler;
    double lapackWorkspace[9];

    // dgetrf(M,N,A,LDA,IPIV,INFO) means invert LDA columns of an M by N matrix 
    // called A, sending the pivot indices to IPIV, and spitting error 
    // information to INFO.
    // also don't forget (like I did) that when you pass a two-dimensional array
    // to a function you need to specify the number of "rows"
    dgetrf_(3,3,M[3][],3,pivotArray[3],&errorHandler);
    //some sort of error check

    dgetri_(3,M[3][],3,pivotArray[3],9,lapackWorkspace,&errorHandler);
    //another error check

    }

Answer 2

Here is a working version of the above using OpenBlas interface to LAPACKE. Link with openblas library (LAPACKE is already contained)

#include <stdio.h>
#include "cblas.h"
#include "lapacke.h"

// inplace inverse n x n matrix A.
// matrix A is Column Major (i.e. firts line, second line ... *not* C[][] order)
// returns:
//   ret = 0 on success
//   ret < 0 illegal argument value
//   ret > 0 singular matrix

lapack_int matInv(double *A, unsigned n)
{
    int ipiv[n+1];
    lapack_int ret;

    ret =  LAPACKE_dgetrf(LAPACK_COL_MAJOR,
                          n,
                          n,
                          A,
                          n,
                          ipiv);

    if (ret !=0)
        return ret;


    ret = LAPACKE_dgetri(LAPACK_COL_MAJOR,
                       n,
                       A,
                       n,
                       ipiv);
    return ret;
}

int main()
{
    double A[] = {
        0.378589,   0.971711,   0.016087,   0.037668,   0.312398,
        0.756377,   0.345708,   0.922947,   0.846671,   0.856103,
        0.732510,   0.108942,   0.476969,   0.398254,   0.507045,
        0.162608,   0.227770,   0.533074,   0.807075,   0.180335,
        0.517006,   0.315992,   0.914848,   0.460825,   0.731980
    };

    for (int i=0; i<25; i++) {
        if ((i%5) == 0) putchar('\n');
        printf("%+12.8f ",A[i]);
    }
    putchar('\n');

    matInv(A,5);

    for (int i=0; i<25; i++) {
        if ((i%5) == 0) putchar('\n');
        printf("%+12.8f ",A[i]);
    }
    putchar('\n');
}

Example:

% g++ -I [OpenBlas path]/include/ example.cpp [OpenBlas path]/lib/libopenblas.a
% a.out

+0.37858900  +0.97171100  +0.01608700  +0.03766800  +0.31239800 
+0.75637700  +0.34570800  +0.92294700  +0.84667100  +0.85610300 
+0.73251000  +0.10894200  +0.47696900  +0.39825400  +0.50704500 
+0.16260800  +0.22777000  +0.53307400  +0.80707500  +0.18033500 
+0.51700600  +0.31599200  +0.91484800  +0.46082500  +0.73198000 

+0.24335255  -2.67946180  +3.57538817  +0.83711880  +0.34704217 
+1.02790497  -1.05086895  -0.07468137  +0.71041070  +0.66708313 
-0.21087237  -4.47765165  +1.73958308  +1.73999641  +3.69324020 
-0.14100897  +2.34977565  -0.93725915  +0.47383541  -2.15554470 
-0.26329660  +6.46315378  -4.07721533  -3.37094863  -2.42580445 

Answer 3

Here is a working version of Spencer Nelson's example above. One mystery about it is that the input matrix is in row-major order, even though it appears to call the underlying fortran routine dgetri. I am led to believe that all the underlying fortran routines require column-major order, but I am no expert on LAPACK, in fact, I'm using this example to help me learn it. But, that one mystery aside:

The input matrix in the example is singular. LAPACK tries to tell you that by returning a 3 in the errorHandler. I changed the 9 in that matrix to a 19, getting an errorHandler of 0 signalling success, and compared the result to that from Mathematica. The comparison was also successful and confirmed that the matrix in the example should be in row-major order, as presented.

Here is the working code:

#include <stdio.h>
#include <stddef.h>
#include <lapacke.h>

int main() {
    int N = 3;
    int NN = 9;
    double M[3][3] = { {1 , 2 , 3},
                       {4 , 5 , 6},
                       {7 , 8 , 9} };
    int pivotArray[3]; //since our matrix has three rows
    int errorHandler;
    double lapackWorkspace[9];

    // dgetrf(M,N,A,LDA,IPIV,INFO) means invert LDA columns of an M by N matrix
    // called A, sending the pivot indices to IPIV, and spitting error information
    // to INFO. also don't forget (like I did) that when you pass a two-dimensional
    // array to a function you need to specify the number of "rows"
    dgetrf_(&N, &N, M[0], &N, pivotArray, &errorHandler);
    printf ("dgetrf eh, %d, should be zero\n", errorHandler);

    dgetri_(&N, M[0], &N, pivotArray, lapackWorkspace, &NN, &errorHandler);
    printf ("dgetri eh, %d, should be zero\n", errorHandler);

    for (size_t row = 0; row < N; ++row)
    {   for (size_t col = 0; col < N; ++col)
        {   printf ("%g", M[row][col]);
            if (N-1 != col)
            {   printf (", ");   }   }
        if (N-1 != row)
        {   printf ("\n");   }   }

    return 0;   }

I built and ran it as follows on a Mac:

gcc main.c -llapacke -llapack
./a.out

I did an nm on the LAPACKE library and found the following:

liblapacke.a(lapacke_dgetri.o):
                 U _LAPACKE_dge_nancheck
0000000000000000 T _LAPACKE_dgetri
                 U _LAPACKE_dgetri_work
                 U _LAPACKE_xerbla
                 U _free
                 U _malloc

liblapacke.a(lapacke_dgetri_work.o):
                 U _LAPACKE_dge_trans
0000000000000000 T _LAPACKE_dgetri_work
                 U _LAPACKE_xerbla
                 U _dgetri_
                 U _free
                 U _malloc

and it looks like there is a LAPACKE [sic] wrapper that would presumably relieve us of having to take addresses everywhere for fortran's convenience, but I am probably not going to get around to trying it because I have a way forward.

EDIT

Here is a working version that bypasses LAPACKE [sic], using LAPACK fortran routines directly. I do not understand why a row-major input produces correct results, but I confirmed it again in Mathematica.

#include <stdio.h>
#include <stddef.h>

int main() {
    int N = 3;
    int NN = 9;
    double M[3][3] = { {1 , 2 ,  3},
                       {4 , 5 ,  6},
                       {7 , 8 , 19} };
    int pivotArray[3]; //since our matrix has three rows
    int errorHandler;
    double lapackWorkspace[9];
    /* from http://www.netlib.no/netlib/lapack/double/dgetrf.f
      SUBROUTINE DGETRF( M, N, A, LDA, IPIV, INFO )
      *
      *  -- LAPACK routine (version 3.1) --
      *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
      *     November 2006
      *
      *     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
      *     ..
      *     .. Array Arguments ..
      INTEGER            IPIV( * )
      DOUBLE PRECISION   A( LDA, * )
    */

    extern void dgetrf_ (int * m, int * n, double * A, int * LDA, int * IPIV,
                         int * INFO);

    /* from http://www.netlib.no/netlib/lapack/double/dgetri.f
       SUBROUTINE DGETRI( N, A, LDA, IPIV, WORK, LWORK, INFO )
       *
       *  -- LAPACK routine (version 3.1) --
       *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
       *     November 2006
       *
       *     .. Scalar Arguments ..
       INTEGER            INFO, LDA, LWORK, N
       *     ..
       *     .. Array Arguments ..
       INTEGER            IPIV( * )
       DOUBLE PRECISION   A( LDA, * ), WORK( * )
    */

    extern void dgetri_ (int * n, double * A, int * LDA, int * IPIV,
                         double * WORK, int * LWORK, int * INFO);

    // dgetrf(M,N,A,LDA,IPIV,INFO) means invert LDA columns of an M by N matrix
    // called A, sending the pivot indices to IPIV, and spitting error information
    // to INFO. also don't forget (like I did) that when you pass a two-dimensional
    // array to a function you need to specify the number of "rows"
    dgetrf_(&N, &N, M[0], &N, pivotArray, &errorHandler);
    printf ("dgetrf eh, %d, should be zero\n", errorHandler);

    dgetri_(&N, M[0], &N, pivotArray, lapackWorkspace, &NN, &errorHandler);
    printf ("dgetri eh, %d, should be zero\n", errorHandler);

    for (size_t row = 0; row < N; ++row)
    {   for (size_t col = 0; col < N; ++col)
        {   printf ("%g", M[row][col]);
            if (N-1 != col)
            {   printf (", ");   }   }
        if (N-1 != row)
        {   printf ("\n");   }   }
    return 0;   }

built and run like this:

$ gcc foo.c -llapack
$ ./a.out
dgetrf eh, 0, should be zero
dgetri eh, 0, should be zero
-1.56667, 0.466667, 0.1
1.13333, 0.0666667, -0.2
0.1, -0.2, 0.1

EDIT

The mystery no longer appears to be a mystery. I think the computations are being done in column-major order, as they must, but I am both inputting and printing the matrices as if they were in row-major order. I have two bugs that cancel each other out so things look row-ish even though they're column-ish.

Answer 4

Here is the working code for computing the inverse of a matrix using lapack in C/C++:

#include <cstdio>

extern "C" {
    // LU decomoposition of a general matrix
    void dgetrf_(int* M, int *N, double* A, int* lda, int* IPIV, int* INFO);

    // generate inverse of a matrix given its LU decomposition
    void dgetri_(int* N, double* A, int* lda, int* IPIV, double* WORK, int* lwork, int* INFO);
}

void inverse(double* A, int N)
{
    int *IPIV = new int[N];
    int LWORK = N*N;
    double *WORK = new double[LWORK];
    int INFO;

    dgetrf_(&N,&N,A,&N,IPIV,&INFO);
    dgetri_(&N,A,&N,IPIV,WORK,&LWORK,&INFO);

    delete[] IPIV;
    delete[] WORK;
}

int main(){

    double A [2*2] = {
        1,2,
        3,4
    };

    inverse(A, 2);

    printf("%f %f\n", A[0], A[1]);
    printf("%f %f\n", A[2], A[3]);

    return 0;
}