Exemplo n.º 1
0
/**
    Purpose
    -------
    Solves a system of linear equations
       A * X = B
    where A is a general N-by-N matrix and X and B are N-by-NRHS matrices.
    The LU decomposition with partial pivoting and row interchanges is
    used to factor A as
       A = P * L * U,
    where P is a permutation matrix, L is unit lower triangular, and U is
    upper triangular.  The factored form of A is then used to solve the
    system of equations A * X = B.

    Arguments
    ---------
    @param[in]
    n       INTEGER
            The order of the matrix A.  N >= 0.

    @param[in]
    nrhs    INTEGER
            The number of right hand sides, i.e., the number of columns
            of the matrix B.  NRHS >= 0.

    @param[in,out]
    dA      DOUBLE_PRECISION array on the GPU, dimension (LDDA,N).
            On entry, the M-by-N matrix to be factored.
            On exit, the factors L and U from the factorization
            A = P*L*U; the unit diagonal elements of L are not stored.

    @param[in]
    ldda    INTEGER
            The leading dimension of the array A.  LDA >= max(1,N).

    @param[out]
    ipiv    INTEGER array, dimension (min(M,N))
            The pivot indices; for 1 <= i <= min(M,N), row i of the
            matrix was interchanged with row IPIV(i).

    @param[in,out]
    dB      DOUBLE_PRECISION array on the GPU, dimension (LDB,NRHS)
            On entry, the right hand side matrix B.
            On exit, the solution matrix X.

    @param[in]
    lddb    INTEGER
            The leading dimension of the array B.  LDB >= max(1,N).

    @param[out]
    info    INTEGER
      -     = 0:  successful exit
      -     < 0:  if INFO = -i, the i-th argument had an illegal value

    @ingroup magma_dgesv_driver
    ********************************************************************/
extern "C" magma_int_t
magma_dgesv_gpu(
    magma_int_t n, magma_int_t nrhs,
    magmaDouble_ptr dA, magma_int_t ldda, magma_int_t *ipiv,
    magmaDouble_ptr dB, magma_int_t lddb,
    magma_int_t *info)
{
    *info = 0;
    if (n < 0) {
        *info = -1;
    } else if (nrhs < 0) {
        *info = -2;
    } else if (ldda < max(1,n)) {
        *info = -4;
    } else if (lddb < max(1,n)) {
        *info = -7;
    }
    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }

    /* Quick return if possible */
    if (n == 0 || nrhs == 0) {
        return *info;
    }

    magma_dgetrf_gpu( n, n, dA, ldda, ipiv, info );
    if ( *info == 0 ) {
        magma_dgetrs_gpu( MagmaNoTrans, n, nrhs, dA, ldda, ipiv, dB, lddb, info );
    }
    
    return *info;
}
Exemplo n.º 2
0
void magmaf_dgetrf_gpu(
    magma_int_t *m, magma_int_t *n,
    devptr_t *dA, magma_int_t *ldda,
    magma_int_t *ipiv,
    magma_int_t *info )
{
    magma_dgetrf_gpu(
        *m, *n,
        magma_ddevptr(dA), *ldda,
        ipiv,
        info );
}
Exemplo n.º 3
0
SEXP magLU(SEXP a)
{
   SEXP gpu = GET_SLOT(a, install("gpu")),
        b = PROTECT(NEW_OBJECT(MAKE_CLASS("magmaLU")));
   int *DIMA = INTEGER(GET_DIM(a)), M = DIMA[0], N = DIMA[1], LDA = M,
       MIN_MN = M < N ? M : N, *ipiv, info;
   double *A = REAL(PROTECT(AS_NUMERIC(a)));

   b = SET_SLOT(b, install(".Data"), AS_NUMERIC(a));
   SET_SLOT(b, install("pivot"), NEW_INTEGER(MIN_MN));
   ipiv = INTEGER(GET_SLOT(b, install("pivot")));
   SET_SLOT(b, install("gpu"), duplicate(gpu));

   if(LOGICAL_VALUE(gpu)) {
      double *dA;

      magma_malloc((void**)&dA, (M*N)*sizeof(double));

      magma_dsetmatrix(M, N, A, LDA, dA, LDA);
      magma_dgetrf_gpu(M, N, dA, LDA, ipiv, &info);
      magma_dgetmatrix(M, N, dA, LDA, REAL(b), LDA);

      magma_free(dA);
   } else {
      double *hA;

      magma_malloc_pinned((void**)&hA, (M*N)*sizeof(double));

      lapackf77_dlacpy(MagmaUpperLowerStr, &M, &N, A, &LDA, hA, &LDA);
      magma_dgetrf(M, N, hA, LDA, ipiv, &info);
      lapackf77_dlacpy(MagmaUpperLowerStr, &M, &N, hA, &LDA, REAL(b), &LDA);

      magma_free_pinned(hA);
   }

   if(info < 0) error("illegal argument %d in 'magLU'", -1 * info);
   else if(info > 0) error("factor U is singular");

   UNPROTECT(2);

   return b;
}
Exemplo n.º 4
0
extern "C" magma_int_t
magma_dsgesv_gpu(char trans, magma_int_t n, magma_int_t nrhs,
                 double *dA, magma_int_t ldda,
                 magma_int_t *ipiv,  magma_int_t *dipiv,
                 double *dB, magma_int_t lddb,
                 double *dX, magma_int_t lddx,
                 double *dworkd, float *dworks,
                 magma_int_t *iter, magma_int_t *info)
{
/*  -- MAGMA (version 1.4.0) --
       Univ. of Tennessee, Knoxville
       Univ. of California, Berkeley
       Univ. of Colorado, Denver
       August 2013

    Purpose
    =======
    DSGESV computes the solution to a real system of linear equations
       A * X = B or A' * X = B
    where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

    DSGESV first attempts to factorize the matrix in real SINGLE PRECISION
    and use this factorization within an iterative refinement procedure
    to produce a solution with real DOUBLE PRECISION norm-wise backward error
    quality (see below). If the approach fails the method switches to a
    real DOUBLE PRECISION factorization and solve.

    The iterative refinement is not going to be a winning strategy if
    the ratio real SINGLE PRECISION performance over real DOUBLE PRECISION
    performance is too small. A reasonable strategy should take the
    number of right-hand sides and the size of the matrix into account.
    This might be done with a call to ILAENV in the future. Up to now, we
    always try iterative refinement.
    
    The iterative refinement process is stopped if
        ITER > ITERMAX
    or for all the RHS we have:
        RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
    where
        o ITER is the number of the current iteration in the iterative
          refinement process
        o RNRM is the infinity-norm of the residual
        o XNRM is the infinity-norm of the solution
        o ANRM is the infinity-operator-norm of the matrix A
        o EPS is the machine epsilon returned by DLAMCH('Epsilon')
    The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.

    Arguments
    =========
    TRANS   (input) CHARACTER*1
            Specifies the form of the system of equations:
            = 'N':  A * X = B  (No transpose)
            = 'T':  A'* X = B  (Transpose)
            = 'C':  A'* X = B  (Conjugate transpose = Transpose)

    N       (input) INTEGER
            The number of linear equations, i.e., the order of the
            matrix A.  N >= 0.

    NRHS    (input) INTEGER
            The number of right hand sides, i.e., the number of columns
            of the matrix B.  NRHS >= 0.

    dA      (input or input/output) DOUBLE PRECISION array on the GPU, dimension (ldda,N)
            On entry, the N-by-N coefficient matrix A.
            On exit, if iterative refinement has been successfully used
            (info.EQ.0 and ITER.GE.0, see description below), A is
            unchanged. If double precision factorization has been used
            (info.EQ.0 and ITER.LT.0, see description below), then the
            array dA contains the factors L and U from the factorization
            A = P*L*U; the unit diagonal elements of L are not stored.

    ldda    (input) INTEGER
            The leading dimension of the array dA.  ldda >= max(1,N).

    IPIV    (output) INTEGER array, dimension (N)
            The pivot indices that define the permutation matrix P;
            row i of the matrix was interchanged with row IPIV(i).
            Corresponds either to the single precision factorization
            (if info.EQ.0 and ITER.GE.0) or the double precision
            factorization (if info.EQ.0 and ITER.LT.0).

    dIPIV   (output) INTEGER array on the GPU, dimension (min(M,N))
            The pivot indices; for 1 <= i <= min(M,N), row i of the
            matrix was moved to row IPIV(i).

    dB      (input) DOUBLE PRECISION array on the GPU, dimension (lddb,NRHS)
            The N-by-NRHS right hand side matrix B.

    lddb    (input) INTEGER
            The leading dimension of the array dB.  lddb >= max(1,N).

    dX      (output) DOUBLE PRECISION array on the GPU, dimension (lddx,NRHS)
            If info = 0, the N-by-NRHS solution matrix X.

    lddx    (input) INTEGER
            The leading dimension of the array dX.  lddx >= max(1,N).

    dworkd  (workspace) DOUBLE PRECISION array on the GPU, dimension (N*NRHS)
            This array is used to hold the residual vectors.

    dworks  (workspace) SINGLE PRECISION array on the GPU, dimension (N*(N+NRHS))
            This array is used to store the real single precision matrix
            and the right-hand sides or solutions in single precision.

    iter    (output) INTEGER
            < 0: iterative refinement has failed, double precision
                 factorization has been performed
                 -1 : the routine fell back to full precision for
                      implementation- or machine-specific reasons
                 -2 : narrowing the precision induced an overflow,
                      the routine fell back to full precision
                 -3 : failure of SGETRF
                 -31: stop the iterative refinement after the 30th iteration
            > 0: iterative refinement has been successfully used.
                 Returns the number of iterations
 
    info   (output) INTEGER
            = 0:  successful exit
            < 0:  if info = -i, the i-th argument had an illegal value
            > 0:  if info = i, U(i,i) computed in DOUBLE PRECISION is
                  exactly zero.  The factorization has been completed,
                  but the factor U is exactly singular, so the solution
                  could not be computed.
    =====================================================================    */

    #define dB(i,j)     (dB + (i) + (j)*lddb)
    #define dX(i,j)     (dX + (i) + (j)*lddx)
    #define dR(i,j)     (dR + (i) + (j)*lddr)
    
    double c_neg_one = MAGMA_D_NEG_ONE;
    double c_one     = MAGMA_D_ONE;
    magma_int_t     ione  = 1;
    double *dR;
    float  *dSA, *dSX;
    double Xnrmv, Rnrmv;
    double          Anrm, Xnrm, Rnrm, cte, eps;
    magma_int_t     i, j, iiter, lddsa, lddr;
    
    /* Check arguments */
    *iter = 0;
    *info = 0;
    if ( n < 0 )
        *info = -1;
    else if ( nrhs < 0 )
        *info = -2;
    else if ( ldda < max(1,n))
        *info = -4;
    else if ( lddb < max(1,n))
        *info = -8;
    else if ( lddx < max(1,n))
        *info = -10;
    
    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }
    
    if ( n == 0 || nrhs == 0 )
        return *info;

    lddsa = n;
    lddr  = n;
    
    dSA = dworks;
    dSX = dSA + lddsa*n;
    dR  = dworkd;
    
    eps  = lapackf77_dlamch("Epsilon");
    Anrm = magmablas_dlange('I', n, n, dA, ldda, (double*)dworkd );
    cte  = Anrm * eps * pow((double)n, 0.5) * BWDMAX;
    
    /*
     * Convert to single precision
     */
    //magmablas_dlag2s( n, nrhs, dB, lddb, dSX, lddsx, info );  // done inside dsgetrs with pivots
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }
    
    magmablas_dlag2s( n, n, dA, ldda, dSA, lddsa, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }
    
    // factor dSA in single precision
    magma_sgetrf_gpu( n, n, dSA, lddsa, ipiv, info );
    if (*info != 0) {
        *iter = -3;
        goto FALLBACK;
    }
    
    // Generate parallel pivots
    {
        magma_int_t *newipiv;
        magma_imalloc_cpu( &newipiv, n );
        if ( newipiv == NULL ) {
            *iter = -3;
            goto FALLBACK;
        }
        swp2pswp( trans, n, ipiv, newipiv );
        magma_setvector( n, sizeof(magma_int_t), newipiv, 1, dipiv, 1 );
        magma_free_cpu( newipiv );
    }
    
    // solve dSA*dSX = dB in single precision
    // converts dB to dSX and applies pivots, solves, then converts result back to dX
    magma_dsgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dB, lddb, dX, lddx, dSX, info );
    
    // residual dR = dB - dA*dX in double precision
    magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dR, lddr );
    if ( nrhs == 1 ) {
        magma_dgemv( trans, n, n,
                     c_neg_one, dA, ldda,
                                dX, 1,
                     c_one,     dR, 1 );
    }
    else {
        magma_dgemm( trans, MagmaNoTrans, n, nrhs, n,
                     c_neg_one, dA, ldda,
                                dX, lddx,
                     c_one,     dR, lddr );
    }
    
    // TODO: use MAGMA_D_ABS( dX(i,j) ) instead of dlange?
    for( j=0; j < nrhs; j++ ) {
        i = magma_idamax( n, dX(0,j), 1) - 1;
        magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
        Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
        
        i = magma_idamax ( n, dR(0,j), 1 ) - 1;
        magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
        Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
        
        if ( Rnrm >  Xnrm*cte ) {
            goto REFINEMENT;
        }
    }
    
    *iter = 0;
    return *info;

REFINEMENT:
    for( iiter=1; iiter < ITERMAX; ) {
        *info = 0;
        // convert residual dR to single precision dSX
        // solve dSA*dSX = R in single precision
        // convert result back to double precision dR
        // it's okay that dR is used for both dB input and dX output.
        magma_dsgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dR, lddr, dR, lddr, dSX, info );
        if (*info != 0) {
            *iter = -3;
            goto FALLBACK;
        }
        
        // Add correction and setup residual
        // dX += dR  --and--
        // dR = dB
        // This saves going through dR a second time (if done with one more kernel).
        // -- not really: first time is read, second time is write.
        for( j=0; j < nrhs; j++ ) {
            magmablas_daxpycp( n, dR(0,j), dX(0,j), dB(0,j) );
        }
        
        // residual dR = dB - dA*dX in double precision
        if ( nrhs == 1 ) {
            magma_dgemv( trans, n, n,
                         c_neg_one, dA, ldda,
                                    dX, 1,
                         c_one,     dR, 1 );
        }
        else {
            magma_dgemm( trans, MagmaNoTrans, n, nrhs, n,
                         c_neg_one, dA, ldda,
                                    dX, lddx,
                         c_one,     dR, lddr );
        }
        
        /*  Check whether the nrhs normwise backward errors satisfy the
         *  stopping criterion. If yes, set ITER=IITER>0 and return. */
        for( j=0; j < nrhs; j++ ) {
            i = magma_idamax( n, dX(0,j), 1) - 1;
            magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
            Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
            
            i = magma_idamax ( n, dR(0,j), 1 ) - 1;
            magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
            Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
            
            if ( Rnrm >  Xnrm*cte ) {
                goto L20;
            }
        }
        
        /*  If we are here, the nrhs normwise backward errors satisfy
         *  the stopping criterion, we are good to exit. */
        *iter = iiter;
        return *info;
        
      L20:
        iiter++;
    }
    
    /* If we are at this place of the code, this is because we have
     * performed ITER=ITERMAX iterations and never satisified the
     * stopping criterion. Set up the ITER flag accordingly and follow
     * up on double precision routine. */
    *iter = -ITERMAX - 1;
    
FALLBACK:
    /* Single-precision iterative refinement failed to converge to a
     * satisfactory solution, so we resort to double precision. */
    magma_dgetrf_gpu( n, n, dA, ldda, ipiv, info );
    if (*info == 0) {
        magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
        magma_dgetrs_gpu( trans, n, nrhs, dA, ldda, ipiv, dX, lddx, info );
    }
    
    return *info;
}
Exemplo n.º 5
0
/* ////////////////////////////////////////////////////////////////////////////
   -- Testing dgetrf
*/
int main( int argc, char** argv)
{
    TESTING_INIT();

    real_Double_t   gflops, gpu_perf, gpu_time, cpu_perf=0, cpu_time=0;
    double          error;
    double *h_A;
    magmaDouble_ptr d_A;
    magma_int_t     *ipiv;
    magma_int_t M, N, n2, lda, ldda, info, min_mn;
    magma_int_t status   = 0;

    magma_opts opts;
    parse_opts( argc, argv, &opts );

    double tol = opts.tolerance * lapackf77_dlamch("E");
    
    if ( opts.check == 2 ) {
        printf("    M     N   CPU GFlop/s (sec)   GPU GFlop/s (sec)   |Ax-b|/(N*|A|*|x|)\n");
    }
    else {
        printf("    M     N   CPU GFlop/s (sec)   GPU GFlop/s (sec)   |PA-LU|/(N*|A|)\n");
    }
    printf("=========================================================================\n");
    for( int itest = 0; itest < opts.ntest; ++itest ) {
        for( int iter = 0; iter < opts.niter; ++iter ) {
            M = opts.msize[itest];
            N = opts.nsize[itest];
            min_mn = min(M, N);
            lda    = M;
            n2     = lda*N;
            ldda   = ((M+31)/32)*32;
            gflops = FLOPS_DGETRF( M, N ) / 1e9;
            
            TESTING_MALLOC_CPU( ipiv, magma_int_t,        min_mn );
            TESTING_MALLOC_CPU( h_A,  double, n2     );
            TESTING_MALLOC_DEV( d_A,  double, ldda*N );
            
            /* =====================================================================
               Performs operation using LAPACK
               =================================================================== */
            if ( opts.lapack ) {
                init_matrix( M, N, h_A, lda );
                
                cpu_time = magma_wtime();
                lapackf77_dgetrf(&M, &N, h_A, &lda, ipiv, &info);
                cpu_time = magma_wtime() - cpu_time;
                cpu_perf = gflops / cpu_time;
                if (info != 0)
                    printf("lapackf77_dgetrf returned error %d: %s.\n",
                           (int) info, magma_strerror( info ));
            }
            
            /* ====================================================================
               Performs operation using MAGMA
               =================================================================== */
            init_matrix( M, N, h_A, lda );
            magma_dsetmatrix( M, N, h_A, lda, d_A, ldda );
            
            gpu_time = magma_wtime();
            magma_dgetrf_gpu( M, N, d_A, ldda, ipiv, &info);
            gpu_time = magma_wtime() - gpu_time;
            gpu_perf = gflops / gpu_time;
            if (info != 0)
                printf("magma_dgetrf_gpu returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            /* =====================================================================
               Check the factorization
               =================================================================== */
            if ( opts.lapack ) {
                printf("%5d %5d   %7.2f (%7.2f)   %7.2f (%7.2f)",
                       (int) M, (int) N, cpu_perf, cpu_time, gpu_perf, gpu_time );
            }
            else {
                printf("%5d %5d     ---   (  ---  )   %7.2f (%7.2f)",
                       (int) M, (int) N, gpu_perf, gpu_time );
            }
            if ( opts.check == 2 ) {
                magma_dgetmatrix( M, N, d_A, ldda, h_A, lda );
                error = get_residual( M, N, h_A, lda, ipiv );
                printf("   %8.2e   %s\n", error, (error < tol ? "ok" : "failed"));
                status += ! (error < tol);
            }
            else if ( opts.check ) {
                magma_dgetmatrix( M, N, d_A, ldda, h_A, lda );
                error = get_LU_error( M, N, h_A, lda, ipiv );
                printf("   %8.2e   %s\n", error, (error < tol ? "ok" : "failed"));
                status += ! (error < tol);
            }
            else {
                printf("     ---  \n");
            }
            
            TESTING_FREE_CPU( ipiv );
            TESTING_FREE_CPU( h_A );
            TESTING_FREE_DEV( d_A );
            fflush( stdout );
        }
        if ( opts.niter > 1 ) {
            printf( "\n" );
        }
    }

    TESTING_FINALIZE();
    return status;
}
extern "C" magma_err_t
magma_dgesv_gpu( magma_int_t n, magma_int_t nrhs, 
                 magmaDouble_ptr dA, size_t dA_offset, magma_int_t ldda,
                 magma_int_t *ipiv, 
                 magmaDouble_ptr dB, size_t dB_offset, magma_int_t lddb, 
                 magma_err_t *info, magma_queue_t queue )
{
/*  -- clMagma (version 0.1) --
       Univ. of Tennessee, Knoxville
       Univ. of California, Berkeley
       Univ. of Colorado, Denver
       April 2012

    Purpose
    =======

    Solves a system of linear equations
       A * X = B
    where A is a general N-by-N matrix and X and B are N-by-NRHS matrices.
    The LU decomposition with partial pivoting and row interchanges is
    used to factor A as
       A = P * L * U,
    where P is a permutation matrix, L is unit lower triangular, and U is
    upper triangular.  The factored form of A is then used to solve the
    system of equations A * X = B.

    Arguments
    =========

    N       (input) INTEGER
            The order of the matrix A.  N >= 0.

    NRHS    (input) INTEGER
            The number of right hand sides, i.e., the number of columns
            of the matrix B.  NRHS >= 0.

    A       (input/output) DOUBLE_PRECISION array on the GPU, dimension (LDDA,N).
            On entry, the M-by-N matrix to be factored.
            On exit, the factors L and U from the factorization
            A = P*L*U; the unit diagonal elements of L are not stored.

    LDA     (input) INTEGER
            The leading dimension of the array A.  LDA >= max(1,N).

    IPIV    (output) INTEGER array, dimension (min(M,N))
            The pivot indices; for 1 <= i <= min(M,N), row i of the
            matrix was interchanged with row IPIV(i).

    B       (input/output) DOUBLE_PRECISION array on the GPU, dimension (LDB,NRHS)
            On entry, the right hand side matrix B.
            On exit, the solution matrix X.

    LDB     (input) INTEGER
            The leading dimension of the array B.  LDB >= max(1,N).

    INFO    (output) INTEGER
            = 0:  successful exit
            < 0:  if INFO = -i, the i-th argument had an illegal value
    =====================================================================    */

    magma_err_t ret;

    *info = 0;
    if (n < 0) {
        *info = -1;
    } else if (nrhs < 0) {
        *info = -2;
    } else if (ldda < max(1,n)) {
        *info = -4;
    } else if (lddb < max(1,n)) {
        *info = -7;
    }
    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }

    /* Quick return if possible */
    if (n == 0 || nrhs == 0) {
        return *info;
    }

    ret = magma_dgetrf_gpu( n, n, dA, dA_offset, ldda, ipiv, info, queue);
    if ( (ret != MAGMA_SUCCESS) || (*info != 0) ) {
        return ret;
    }

    ret = magma_dgetrs_gpu( MagmaNoTrans, n, nrhs, dA, dA_offset, ldda, ipiv, dB, dB_offset, lddb, info, queue );
    
    return ret;
}
Exemplo n.º 7
0
/* ////////////////////////////////////////////////////////////////////////////
   -- Testing dgetrf
*/
int main( int argc, char** argv )
{
    TESTING_INIT();

    real_Double_t   gflops, gpu_perf, gpu_time, cpu_perf, cpu_time;
    double *h_A, *h_R, *work;
    magmaDouble_ptr d_A, dwork;
    double c_neg_one = MAGMA_D_NEG_ONE;
    magma_int_t N, n2, lda, ldda, info, lwork, ldwork;
    magma_int_t ione     = 1;
    magma_int_t ISEED[4] = {0,0,0,1};
    double tmp;
    double error, rwork[1];
    magma_int_t *ipiv;
    magma_int_t status = 0;
    
    magma_opts opts;
    parse_opts( argc, argv, &opts );
    opts.lapack |= opts.check;  // check (-c) implies lapack (-l)
    
    // need looser bound (3000*eps instead of 30*eps) for tests
    // TODO: should compute ||I - A*A^{-1}|| / (n*||A||*||A^{-1}||)
    opts.tolerance = max( 3000., opts.tolerance );
    double tol = opts.tolerance * lapackf77_dlamch("E");
    
    printf("    N   CPU GFlop/s (sec)   GPU GFlop/s (sec)   ||R||_F / (N*||A||_F)\n");
    printf("=================================================================\n");
    for( int itest = 0; itest < opts.ntest; ++itest ) {
        for( int iter = 0; iter < opts.niter; ++iter ) {
            N = opts.nsize[itest];
            lda    = N;
            n2     = lda*N;
            ldda   = ((N+31)/32)*32;
            ldwork = N * magma_get_dgetri_nb( N );
            gflops = FLOPS_DGETRI( N ) / 1e9;
            
            // query for workspace size
            lwork = -1;
            lapackf77_dgetri( &N, NULL, &lda, NULL, &tmp, &lwork, &info );
            if (info != 0)
                printf("lapackf77_dgetri returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            lwork = int( MAGMA_D_REAL( tmp ));
            
            TESTING_MALLOC_CPU( ipiv,  magma_int_t,        N      );
            TESTING_MALLOC_CPU( work,  double, lwork  );
            TESTING_MALLOC_CPU( h_A,   double, n2     );
            
            TESTING_MALLOC_PIN( h_R,   double, n2     );
            
            TESTING_MALLOC_DEV( d_A,   double, ldda*N );
            TESTING_MALLOC_DEV( dwork, double, ldwork );
            
            /* Initialize the matrix */
            lapackf77_dlarnv( &ione, ISEED, &n2, h_A );
            error = lapackf77_dlange( "f", &N, &N, h_A, &lda, rwork );  // norm(A)
            
            /* Factor the matrix. Both MAGMA and LAPACK will use this factor. */
            magma_dsetmatrix( N, N, h_A, lda, d_A, 0, ldda, opts.queue );
            magma_dgetrf_gpu( N, N, d_A, 0, ldda, ipiv, opts.queue, &info );
            magma_dgetmatrix( N, N, d_A, 0, ldda, h_A, lda, opts.queue );
            if ( info != 0 )
                printf("magma_dgetrf_gpu returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            // check for exact singularity
            //h_A[ 10 + 10*lda ] = MAGMA_D_MAKE( 0.0, 0.0 );
            //magma_dsetmatrix( N, N, h_A, lda, d_A, 0, ldda, opts.queue );
            
            /* ====================================================================
               Performs operation using MAGMA
               =================================================================== */
            gpu_time = magma_wtime();
            magma_dgetri_gpu( N, d_A, 0, ldda, ipiv, dwork, 0, ldwork, opts.queues2, &info );
            gpu_time = magma_wtime() - gpu_time;
            gpu_perf = gflops / gpu_time;
            if (info != 0)
                printf("magma_dgetri_gpu returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            magma_dgetmatrix( N, N, d_A, 0, ldda, h_R, lda, opts.queue );
            
            /* =====================================================================
               Performs operation using LAPACK
               =================================================================== */
            if ( opts.lapack ) {
                cpu_time = magma_wtime();
                lapackf77_dgetri( &N, h_A, &lda, ipiv, work, &lwork, &info );
                cpu_time = magma_wtime() - cpu_time;
                cpu_perf = gflops / cpu_time;
                if (info != 0)
                    printf("lapackf77_dgetri returned error %d: %s.\n",
                           (int) info, magma_strerror( info ));
                
                /* =====================================================================
                   Check the result compared to LAPACK
                   =================================================================== */
                blasf77_daxpy( &n2, &c_neg_one, h_A, &ione, h_R, &ione );
                error = lapackf77_dlange( "f", &N, &N, h_R, &lda, rwork ) / (N*error);
                
                printf( "%5d   %7.2f (%7.2f)   %7.2f (%7.2f)   %8.2e   %s\n",
                        (int) N, cpu_perf, cpu_time, gpu_perf, gpu_time,
                        error, (error < tol ? "ok" : "failed"));
                status += ! (error < tol);
            }
            else {
                printf( "%5d     ---   (  ---  )   %7.2f (%7.2f)     ---\n",
                        (int) N, gpu_perf, gpu_time );
            }
            
            TESTING_FREE_CPU( ipiv  );
            TESTING_FREE_CPU( work  );
            TESTING_FREE_CPU( h_A   );
            
            TESTING_FREE_PIN( h_R   );
            
            TESTING_FREE_DEV( d_A   );
            TESTING_FREE_DEV( dwork );
            fflush( stdout );
        }
        if ( opts.niter > 1 ) {
            printf( "\n" );
        }
    }

    TESTING_FINALIZE();
    return status;
}
Exemplo n.º 8
0
/* ////////////////////////////////////////////////////////////////////////////
   -- Testing dgetri
*/
int main( int argc, char** argv )
{
    TESTING_INIT();

    // constants
    const double c_zero    = MAGMA_D_ZERO;
    const double c_one     = MAGMA_D_ONE;
    const double c_neg_one = MAGMA_D_NEG_ONE;
    
    real_Double_t   gflops, gpu_perf, gpu_time, cpu_perf, cpu_time;
    double *h_A, *h_Ainv, *h_R, *work;
    magmaDouble_ptr d_A, dwork;
    magma_int_t N, n2, lda, ldda, info, lwork, ldwork;
    magma_int_t ione     = 1;
    magma_int_t ISEED[4] = {0,0,0,1};
    double tmp;
    double error, rwork[1];
    magma_int_t *ipiv;
    magma_int_t status = 0;
    
    magma_opts opts;
    opts.parse_opts( argc, argv );
    
    double tol = opts.tolerance * lapackf77_dlamch("E");
    
    printf("%%   N   CPU Gflop/s (sec)   GPU Gflop/s (sec)   ||I - A*A^{-1}||_1 / (N*cond(A))\n");
    printf("%%===============================================================================\n");
    for( int itest = 0; itest < opts.ntest; ++itest ) {
        for( int iter = 0; iter < opts.niter; ++iter ) {
            N = opts.nsize[itest];
            lda    = N;
            n2     = lda*N;
            ldda   = magma_roundup( N, opts.align );  // multiple of 32 by default
            ldwork = N * magma_get_dgetri_nb( N );
            gflops = FLOPS_DGETRI( N ) / 1e9;
            
            // query for workspace size
            lwork = -1;
            lapackf77_dgetri( &N, NULL, &lda, NULL, &tmp, &lwork, &info );
            if (info != 0) {
                printf("lapackf77_dgetri returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            }
            lwork = magma_int_t( MAGMA_D_REAL( tmp ));
            
            TESTING_MALLOC_CPU( ipiv,   magma_int_t,        N      );
            TESTING_MALLOC_CPU( work,   double, lwork  );
            TESTING_MALLOC_CPU( h_A,    double, n2     );
            TESTING_MALLOC_CPU( h_Ainv, double, n2     );
            TESTING_MALLOC_CPU( h_R,    double, n2     );
            
            TESTING_MALLOC_DEV( d_A,    double, ldda*N );
            TESTING_MALLOC_DEV( dwork,  double, ldwork );
            
            /* Initialize the matrix */
            lapackf77_dlarnv( &ione, ISEED, &n2, h_A );
            
            /* Factor the matrix. Both MAGMA and LAPACK will use this factor. */
            magma_dsetmatrix( N, N, h_A, lda, d_A, ldda, opts.queue );
            magma_dgetrf_gpu( N, N, d_A, ldda, ipiv, &info );
            magma_dgetmatrix( N, N, d_A, ldda, h_Ainv, lda, opts.queue );
            if (info != 0) {
                printf("magma_dgetrf_gpu returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            }
            
            // check for exact singularity
            //h_Ainv[ 10 + 10*lda ] = MAGMA_D_MAKE( 0.0, 0.0 );
            //magma_dsetmatrix( N, N, h_Ainv, lda, d_A, ldda, opts.queue );
            
            /* ====================================================================
               Performs operation using MAGMA
               =================================================================== */
            gpu_time = magma_wtime();
            magma_dgetri_gpu( N, d_A, ldda, ipiv, dwork, ldwork, &info );
            gpu_time = magma_wtime() - gpu_time;
            gpu_perf = gflops / gpu_time;
            if (info != 0) {
                printf("magma_dgetri_gpu returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            }
            
            /* =====================================================================
               Performs operation using LAPACK
               =================================================================== */
            if ( opts.lapack ) {
                cpu_time = magma_wtime();
                lapackf77_dgetri( &N, h_Ainv, &lda, ipiv, work, &lwork, &info );
                cpu_time = magma_wtime() - cpu_time;
                cpu_perf = gflops / cpu_time;
                if (info != 0) {
                    printf("lapackf77_dgetri returned error %d: %s.\n",
                           (int) info, magma_strerror( info ));
                }
                printf( "%5d   %7.2f (%7.2f)   %7.2f (%7.2f)",
                        (int) N, cpu_perf, cpu_time, gpu_perf, gpu_time );
            }
            else {
                printf( "%5d     ---   (  ---  )   %7.2f (%7.2f)",
                        (int) N, gpu_perf, gpu_time );
            }
            
            /* =====================================================================
               Check the result
               =================================================================== */
            if ( opts.check ) {
                magma_dgetmatrix( N, N, d_A, ldda, h_Ainv, lda, opts.queue );
                
                // compute 1-norm condition number estimate, following LAPACK's zget03
                double normA, normAinv, rcond;
                normA    = lapackf77_dlange( "1", &N, &N, h_A,    &lda, rwork );
                normAinv = lapackf77_dlange( "1", &N, &N, h_Ainv, &lda, rwork );
                if ( normA <= 0 || normAinv <= 0 ) {
                    rcond = 0;
                    error = 1 / (tol/opts.tolerance);  // == 1/eps
                }
                else {
                    rcond = (1 / normA) / normAinv;
                    // R = I
                    // R -= A*A^{-1}
                    // err = ||I - A*A^{-1}|| / ( N ||A||*||A^{-1}|| ) = ||R|| * rcond / N, using 1-norm
                    lapackf77_dlaset( "full", &N, &N, &c_zero, &c_one, h_R, &lda );
                    blasf77_dgemm( "no", "no", &N, &N, &N,
                                   &c_neg_one, h_A,    &lda,
                                               h_Ainv, &lda,
                                   &c_one,     h_R,    &lda );
                    error = lapackf77_dlange( "1", &N, &N, h_R, &lda, rwork );
                    error = error * rcond / N;
                }
                
                bool okay = (error < tol);
                status += ! okay;
                printf( "   %8.2e   %s\n",
                        error, (okay ? "ok" : "failed"));
            }
            else {
                printf( "\n" );
            }
            
            TESTING_FREE_CPU( ipiv   );
            TESTING_FREE_CPU( work   );
            TESTING_FREE_CPU( h_A    );
            TESTING_FREE_CPU( h_Ainv );
            TESTING_FREE_CPU( h_R    );
            
            TESTING_FREE_DEV( d_A    );
            TESTING_FREE_DEV( dwork  );
            fflush( stdout );
        }
        if ( opts.niter > 1 ) {
            printf( "\n" );
        }
    }

    opts.cleanup();
    TESTING_FINALIZE();
    return status;
}
Exemplo n.º 9
0
SEXP magma_dgeMatrix_LU_(SEXP x, Rboolean warn_sing)
{
#ifdef HIPLAR_WITH_MAGMA
	SEXP val = get_factors(x, "LU");
	int *dims, npiv, info;

	if (val != R_NilValue) {
//		R_ShowMessage("already in slot");	/* nothing to do if it's there in 'factors' slot */
		return val;
	}

	dims = INTEGER(GET_SLOT(x, Matrix_DimSym));
	if (dims[0] < 1 || dims[1] < 1)
		error(_("Cannot factor a matrix with zero extents"));
	npiv = (dims[0] < dims[1]) ? dims[0] : dims[1];
	val = PROTECT(NEW_OBJECT(MAKE_CLASS("denseLU")));
	slot_dup(val, x, Matrix_xSym);
	slot_dup(val, x, Matrix_DimSym);
	double *h_R = REAL(GET_SLOT(val, Matrix_xSym));
	int *ipiv = INTEGER(ALLOC_SLOT(val, Matrix_permSym, INTSXP, npiv));
	
	if(GPUFlag == 0){
#ifdef HIPLAR_DBG
		R_ShowMessage("DBG: LU decomposition using dgetrf;");
#endif
		F77_CALL(dgetrf)(dims, dims + 1, h_R,
				dims,
				ipiv,
				&info);
	}
	else if(GPUFlag == 1 && Interface == 0){
#ifdef HIPLAR_DBG
		R_ShowMessage("DBG: LU decomposition using magma_dgetrf;");
#endif
			magma_dgetrf(dims[0], dims[1], h_R, dims[0], ipiv, &info);
		}
		else if(GPUFlag == 1 && Interface == 1) {
			
#ifdef HIPLAR_DBG
			R_ShowMessage("DBG: LU decomposition using magma_dgetrf_gpu;");	
#endif
			double *d_A;
			int N2 = dims[0] * dims[1];
			cublasStatus retStatus;

			cublasAlloc( N2 , sizeof(double), (void**)&d_A);
			
			/* Error Checking */
			retStatus = cublasGetError ();
			if (retStatus != CUBLAS_STATUS_SUCCESS) 
				error(_("CUBLAS: Error in Memory Allocation"));
			/********************************************/

			cublasSetVector(N2, sizeof(double), h_R, 1, d_A, 1);
			
			/* Error Checking */
			retStatus = cublasGetError ();
				if (retStatus != CUBLAS_STATUS_SUCCESS) 
					error(_("CUBLAS: Error in Date Transfer to Device"));
			/********************************************/

			magma_dgetrf_gpu(dims[0],dims[1], d_A, dims[0], ipiv,  &info);
			
			cublasGetVector( N2, sizeof(double), d_A, 1, h_R, 1);		
			
			/* Error Checking */
			retStatus = cublasGetError ();
				if (retStatus != CUBLAS_STATUS_SUCCESS) 
					error(_("CUBLAS: Error in Date Transfer from Device"));
			/********************************************/
			
				cublasFree(d_A);
		
			/* Error Checking */
			retStatus = cublasGetError ();
				if (retStatus != CUBLAS_STATUS_SUCCESS) 
					error(_("CUBLAS: Error freeing data"));
			/********************************************/
		}
		else
			error(_("MAGMA/LAPACK/Interface Flag not defined correctly"));
		
	if (info < 0)
		error(_("Lapack routine %s returned error code %d"), "dgetrf", info);
	else if (info > 0 && warn_sing)
		warning(_("Exact singularity detected during LU decomposition: %s, i=%d."),
				"U[i,i]=0", info);
	UNPROTECT(1);

	return set_factors(x, val, "LU");
#endif

	    return R_NilValue;
}
Exemplo n.º 10
0
extern "C" magma_int_t
magma_dgesv(     magma_int_t n, magma_int_t nrhs, 
                 double *A, magma_int_t lda,
                 magma_int_t *ipiv, 
                 double *B, magma_int_t ldb, 
                 magma_int_t *info)
{
/*  -- MAGMA (version 1.3.0) --
       Univ. of Tennessee, Knoxville
       Univ. of California, Berkeley
       Univ. of Colorado, Denver
       November 2012

    Purpose
    =======

    Solves a system of linear equations
       A * X = B
    where A is a general N-by-N matrix and X and B are N-by-NRHS matrices.
    The LU decomposition with partial pivoting and row interchanges is
    used to factor A as
       A = P * L * U,
    where P is a permutation matrix, L is unit lower triangular, and U is
    upper triangular.  The factored form of A is then used to solve the
    system of equations A * X = B.

    Arguments
    =========

    N       (input) INTEGER
            The order of the matrix A.  N >= 0.

    NRHS    (input) INTEGER
            The number of right hand sides, i.e., the number of columns
            of the matrix B.  NRHS >= 0.

    A       (input/output) DOUBLE_PRECISION array, dimension (LDA,N).
            On entry, the M-by-N matrix to be factored.
            On exit, the factors L and U from the factorization
            A = P*L*U; the unit diagonal elements of L are not stored.

    LDA     (input) INTEGER
            The leading dimension of the array A.  LDA >= max(1,N).

    IPIV    (output) INTEGER array, dimension (min(M,N))
            The pivot indices; for 1 <= i <= min(M,N), row i of the
            matrix was interchanged with row IPIV(i).

    B       (input/output) DOUBLE_PRECISION array, dimension (LDB,NRHS)
            On entry, the right hand side matrix B.
            On exit, the solution matrix X.

    LDB     (input) INTEGER
            The leading dimension of the array B.  LDB >= max(1,N).

    INFO    (output) INTEGER
            = 0:  successful exit
            < 0:  if INFO = -i, the i-th argument had an illegal value
    =====================================================================    */

    magma_int_t num_gpus, ldda, lddb;
    
    *info = 0;
    if (n < 0) {
        *info = -1;
    } else if (nrhs < 0) {
        *info = -2;
    } else if (lda < max(1,n)) {
        *info = -4;
    } else if (ldb < max(1,n)) {
        *info = -7;
    }
    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }

    /* Quick return if possible */
    if (n == 0 || nrhs == 0) {
        return *info;
    }
    
    /* If single-GPU and allocation suceeds, use GPU interface. */
    num_gpus = magma_num_gpus();
    double *dA, *dB;
    if ( num_gpus > 1 ) {
        goto CPU_INTERFACE;
    }
    ldda = ((n+31)/32)*32;
    lddb = ldda;
    if ( MAGMA_SUCCESS != magma_dmalloc( &dA, ldda*n )) {
        goto CPU_INTERFACE;
    }
    if ( MAGMA_SUCCESS != magma_dmalloc( &dB, lddb*nrhs )) {
        magma_free( dA );
        dA = NULL;
        goto CPU_INTERFACE;
    }
    assert( num_gpus == 1 && dA != NULL && dB != NULL );
    magma_dsetmatrix( n, n, A, lda, dA, ldda );
    magma_dgetrf_gpu( n, n, dA, ldda, ipiv, info );
    magma_dgetmatrix( n, n, dA, ldda, A, lda );
    if ( *info == 0 ) {
        magma_dsetmatrix( n, nrhs, B, ldb, dB, lddb );
        magma_dgetrs_gpu( MagmaNoTrans, n, nrhs, dA, ldda, ipiv, dB, lddb, info );
        magma_dgetmatrix( n, nrhs, dB, lddb, B, ldb );
    }
    magma_free( dA );
    magma_free( dB );
    return *info;

CPU_INTERFACE:
    /* If multi-GPU or allocation failed, use CPU interface and LAPACK.
     * Faster to use LAPACK for getrs than to copy A to GPU. */
    magma_dgetrf( n, n, A, lda, ipiv, info );
    if ( *info == 0 ) {
        lapackf77_dgetrs( MagmaNoTransStr, &n, &nrhs, A, &lda, ipiv, B, &ldb, info );
    }
    return *info;
}
Exemplo n.º 11
0
/**
    Purpose
    -------
    DSGESV computes the solution to a real system of linear equations
       A * X = B,  A**T * X = B,  or  A**H * X = B,
    where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

    DSGESV first attempts to factorize the matrix in real SINGLE PRECISION
    and use this factorization within an iterative refinement procedure
    to produce a solution with real DOUBLE PRECISION norm-wise backward error
    quality (see below). If the approach fails the method switches to a
    real DOUBLE PRECISION factorization and solve.

    The iterative refinement is not going to be a winning strategy if
    the ratio real SINGLE PRECISION performance over real DOUBLE PRECISION
    performance is too small. A reasonable strategy should take the
    number of right-hand sides and the size of the matrix into account.
    This might be done with a call to ILAENV in the future. Up to now, we
    always try iterative refinement.
    
    The iterative refinement process is stopped if
        ITER > ITERMAX
    or for all the RHS we have:
        RNRM < SQRT(N)*XNRM*ANRM*EPS*BWDMAX
    where
        o ITER is the number of the current iteration in the iterative
          refinement process
        o RNRM is the infinity-norm of the residual
        o XNRM is the infinity-norm of the solution
        o ANRM is the infinity-operator-norm of the matrix A
        o EPS is the machine epsilon returned by DLAMCH('Epsilon')
    The value ITERMAX and BWDMAX are fixed to 30 and 1.0D+00 respectively.

    Arguments
    ---------
    @param[in]
    trans   magma_trans_t
            Specifies the form of the system of equations:
      -     = MagmaNoTrans:    A    * X = B  (No transpose)
      -     = MagmaTrans:      A**T * X = B  (Transpose)
      -     = MagmaConjTrans:  A**H * X = B  (Conjugate transpose)

    @param[in]
    n       INTEGER
            The number of linear equations, i.e., the order of the
            matrix A.  N >= 0.

    @param[in]
    nrhs    INTEGER
            The number of right hand sides, i.e., the number of columns
            of the matrix B.  NRHS >= 0.

    @param[in,out]
    dA      DOUBLE PRECISION array on the GPU, dimension (ldda,N)
            On entry, the N-by-N coefficient matrix A.
            On exit, if iterative refinement has been successfully used
            (info.EQ.0 and ITER.GE.0, see description below), A is
            unchanged. If double precision factorization has been used
            (info.EQ.0 and ITER.LT.0, see description below), then the
            array dA contains the factors L and U from the factorization
            A = P*L*U; the unit diagonal elements of L are not stored.

    @param[in]
    ldda    INTEGER
            The leading dimension of the array dA.  ldda >= max(1,N).

    @param[out]
    ipiv    INTEGER array, dimension (N)
            The pivot indices that define the permutation matrix P;
            row i of the matrix was interchanged with row IPIV(i).
            Corresponds either to the single precision factorization
            (if info.EQ.0 and ITER.GE.0) or the double precision
            factorization (if info.EQ.0 and ITER.LT.0).

    @param[out]
    dipiv   INTEGER array on the GPU, dimension (N)
            The pivot indices; for 1 <= i <= N, after permuting, row i of the
            matrix was moved to row dIPIV(i).
            Note this is different than IPIV, where interchanges
            are applied one-after-another.

    @param[in]
    dB      DOUBLE PRECISION array on the GPU, dimension (lddb,NRHS)
            The N-by-NRHS right hand side matrix B.

    @param[in]
    lddb    INTEGER
            The leading dimension of the array dB.  lddb >= max(1,N).

    @param[out]
    dX      DOUBLE PRECISION array on the GPU, dimension (lddx,NRHS)
            If info = 0, the N-by-NRHS solution matrix X.

    @param[in]
    lddx    INTEGER
            The leading dimension of the array dX.  lddx >= max(1,N).

    @param
    dworkd  (workspace) DOUBLE PRECISION array on the GPU, dimension (N*NRHS)
            This array is used to hold the residual vectors.

    @param
    dworks  (workspace) SINGLE PRECISION array on the GPU, dimension (N*(N+NRHS))
            This array is used to store the real single precision matrix
            and the right-hand sides or solutions in single precision.

    @param[out]
    iter    INTEGER
      -     < 0: iterative refinement has failed, double precision
                 factorization has been performed
        +        -1 : the routine fell back to full precision for
                      implementation- or machine-specific reasons
        +        -2 : narrowing the precision induced an overflow,
                      the routine fell back to full precision
        +        -3 : failure of SGETRF
        +        -31: stop the iterative refinement after the 30th iteration
      -     > 0: iterative refinement has been successfully used.
                 Returns the number of iterations
 
    @param[out]
    info   INTEGER
      -     = 0:  successful exit
      -     < 0:  if info = -i, the i-th argument had an illegal value
      -     > 0:  if info = i, U(i,i) computed in DOUBLE PRECISION is
                  exactly zero.  The factorization has been completed,
                  but the factor U is exactly singular, so the solution
                  could not be computed.

    @ingroup magma_dgesv_driver
    ********************************************************************/
extern "C" magma_int_t
magma_dsgesv_gpu(magma_trans_t trans, magma_int_t n, magma_int_t nrhs,
                 double *dA, magma_int_t ldda,
                 magma_int_t *ipiv,  magma_int_t *dipiv,
                 double *dB, magma_int_t lddb,
                 double *dX, magma_int_t lddx,
                 double *dworkd, float *dworks,
                 magma_int_t *iter, magma_int_t *info)
{
    #define dB(i,j)     (dB + (i) + (j)*lddb)
    #define dX(i,j)     (dX + (i) + (j)*lddx)
    #define dR(i,j)     (dR + (i) + (j)*lddr)
    
    double c_neg_one = MAGMA_D_NEG_ONE;
    double c_one     = MAGMA_D_ONE;
    magma_int_t     ione  = 1;
    double *dR;
    float  *dSA, *dSX;
    double Xnrmv, Rnrmv;
    double          Anrm, Xnrm, Rnrm, cte, eps;
    magma_int_t     i, j, iiter, lddsa, lddr;
    
    /* Check arguments */
    *iter = 0;
    *info = 0;
    if ( n < 0 )
        *info = -1;
    else if ( nrhs < 0 )
        *info = -2;
    else if ( ldda < max(1,n))
        *info = -4;
    else if ( lddb < max(1,n))
        *info = -8;
    else if ( lddx < max(1,n))
        *info = -10;
    
    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }
    
    if ( n == 0 || nrhs == 0 )
        return *info;

    lddsa = n;
    lddr  = n;
    
    dSA = dworks;
    dSX = dSA + lddsa*n;
    dR  = dworkd;
    
    eps  = lapackf77_dlamch("Epsilon");
    Anrm = magmablas_dlange(MagmaInfNorm, n, n, dA, ldda, (double*)dworkd );
    cte  = Anrm * eps * pow((double)n, 0.5) * BWDMAX;
    
    /*
     * Convert to single precision
     */
    //magmablas_dlag2s( n, nrhs, dB, lddb, dSX, lddsx, info );  // done inside dsgetrs with pivots
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }
    
    magmablas_dlag2s( n, n, dA, ldda, dSA, lddsa, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }
    
    // factor dSA in single precision
    magma_sgetrf_gpu( n, n, dSA, lddsa, ipiv, info );
    if (*info != 0) {
        *iter = -3;
        goto FALLBACK;
    }
    
    // Generate parallel pivots
    {
        magma_int_t *newipiv;
        magma_imalloc_cpu( &newipiv, n );
        if ( newipiv == NULL ) {
            *iter = -3;
            goto FALLBACK;
        }
        swp2pswp( trans, n, ipiv, newipiv );
        magma_setvector( n, sizeof(magma_int_t), newipiv, 1, dipiv, 1 );
        magma_free_cpu( newipiv );
    }
    
    // solve dSA*dSX = dB in single precision
    // converts dB to dSX and applies pivots, solves, then converts result back to dX
    magma_dsgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dB, lddb, dX, lddx, dSX, info );
    
    // residual dR = dB - dA*dX in double precision
    magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dR, lddr );
    if ( nrhs == 1 ) {
        magma_dgemv( trans, n, n,
                     c_neg_one, dA, ldda,
                                dX, 1,
                     c_one,     dR, 1 );
    }
    else {
        magma_dgemm( trans, MagmaNoTrans, n, nrhs, n,
                     c_neg_one, dA, ldda,
                                dX, lddx,
                     c_one,     dR, lddr );
    }
    
    // TODO: use MAGMA_D_ABS( dX(i,j) ) instead of dlange?
    for( j=0; j < nrhs; j++ ) {
        i = magma_idamax( n, dX(0,j), 1) - 1;
        magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
        Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
        
        i = magma_idamax ( n, dR(0,j), 1 ) - 1;
        magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
        Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
        
        if ( Rnrm >  Xnrm*cte ) {
            goto REFINEMENT;
        }
    }
    
    *iter = 0;
    return *info;

REFINEMENT:
    for( iiter=1; iiter < ITERMAX; ) {
        *info = 0;
        // convert residual dR to single precision dSX
        // solve dSA*dSX = R in single precision
        // convert result back to double precision dR
        // it's okay that dR is used for both dB input and dX output.
        magma_dsgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dR, lddr, dR, lddr, dSX, info );
        if (*info != 0) {
            *iter = -3;
            goto FALLBACK;
        }
        
        // Add correction and setup residual
        // dX += dR  --and--
        // dR = dB
        // This saves going through dR a second time (if done with one more kernel).
        // -- not really: first time is read, second time is write.
        for( j=0; j < nrhs; j++ ) {
            magmablas_daxpycp( n, dR(0,j), dX(0,j), dB(0,j) );
        }
        
        // residual dR = dB - dA*dX in double precision
        if ( nrhs == 1 ) {
            magma_dgemv( trans, n, n,
                         c_neg_one, dA, ldda,
                                    dX, 1,
                         c_one,     dR, 1 );
        }
        else {
            magma_dgemm( trans, MagmaNoTrans, n, nrhs, n,
                         c_neg_one, dA, ldda,
                                    dX, lddx,
                         c_one,     dR, lddr );
        }
        
        /*  Check whether the nrhs normwise backward errors satisfy the
         *  stopping criterion. If yes, set ITER=IITER > 0 and return. */
        for( j=0; j < nrhs; j++ ) {
            i = magma_idamax( n, dX(0,j), 1) - 1;
            magma_dgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
            Xnrm = lapackf77_dlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
            
            i = magma_idamax ( n, dR(0,j), 1 ) - 1;
            magma_dgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
            Rnrm = lapackf77_dlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
            
            if ( Rnrm >  Xnrm*cte ) {
                goto L20;
            }
        }
        
        /*  If we are here, the nrhs normwise backward errors satisfy
         *  the stopping criterion, we are good to exit. */
        *iter = iiter;
        return *info;
        
      L20:
        iiter++;
    }
    
    /* If we are at this place of the code, this is because we have
     * performed ITER=ITERMAX iterations and never satisified the
     * stopping criterion. Set up the ITER flag accordingly and follow
     * up on double precision routine. */
    *iter = -ITERMAX - 1;
    
FALLBACK:
    /* Single-precision iterative refinement failed to converge to a
     * satisfactory solution, so we resort to double precision. */
    magma_dgetrf_gpu( n, n, dA, ldda, ipiv, info );
    if (*info == 0) {
        magmablas_dlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
        magma_dgetrs_gpu( trans, n, nrhs, dA, ldda, ipiv, dX, lddx, info );
    }
    
    return *info;
}