Example #1
0
extern "C" magma_int_t
magma_zcposv_gpu(char uplo, magma_int_t n, magma_int_t nrhs,
                 magmaDoubleComplex *dA, magma_int_t ldda,
                 magmaDoubleComplex *dB, magma_int_t lddb,
                 magmaDoubleComplex *dX, magma_int_t lddx,
                 magmaDoubleComplex *dworkd, magmaFloatComplex *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
    =======
    ZCPOSV computes the solution to a complex system of linear equations
       A * X = B,
    where A is an N-by-N Hermitian positive definite matrix and X and B
    are N-by-NRHS matrices.

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

    The iterative refinement is not going to be a winning strategy if
    the ratio complex SINGLE PRECISION performance over complex 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
    =========
    UPLO    (input) CHARACTER
            = 'U':  Upper triangle of A is stored;
            = 'L':  Lower triangle of A is stored.

    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) COMPLEX_16 array on the GPU, dimension (LDDA,N)
            On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
            N-by-N upper triangular part of A contains the upper
            triangular part of the matrix A, and the strictly lower
            triangular part of A is not referenced.  If UPLO = 'L', the
            leading N-by-N lower triangular part of A contains the lower
            triangular part of the matrix A, and the strictly upper
            triangular part of A is not referenced.
            On exit, if iterative refinement has been successfully used
            (INFO.EQ.0 and ITER.GE.0, see description below), then A is
            unchanged, if double factorization has been used
            (INFO.EQ.0 and ITER.LT.0, see description below), then the
            array dA contains the factor U or L from the Cholesky
            factorization A = U**T*U or A = L*L**T.

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

    dB      (input) COMPLEX_16 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) COMPLEX_16 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) COMPLEX_16 array on the GPU, dimension (N*NRHS)
            This array is used to hold the residual vectors.

    dworks  (workspace) COMPLEX array on the GPU, dimension (N*(N+NRHS))
            This array is used to store the complex 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 SPOTRF
                 -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, the leading minor of order i of (DOUBLE
                  PRECISION) A is not positive definite, so the
                  factorization could not be completed, and the solution
                  has not been 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)
    #define dSX(i,j)    (dSX + (i) + (j)*lddsx)

    magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE;
    magmaDoubleComplex c_one     = MAGMA_Z_ONE;
    magma_int_t     ione  = 1;
    magmaDoubleComplex *dR;
    magmaFloatComplex  *dSA, *dSX;
    magmaDoubleComplex Xnrmv, Rnrmv;
    double          Anrm, Xnrm, Rnrm, cte, eps;
    magma_int_t     i, j, iiter, lddsa, lddsx, 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 = -7;
    else if ( lddx < max(1,n))
        *info = -9;

    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }

    if ( n == 0 || nrhs == 0 )
        return *info;

    lddsa = n;
    lddsx = n;
    lddr  = n;
    
    dSA = dworks;
    dSX = dSA + lddsa*n;
    dR  = dworkd;

    eps  = lapackf77_dlamch("Epsilon");
    Anrm = magmablas_zlanhe('I', uplo, n, dA, ldda, (double*)dworkd );
    cte  = Anrm * eps * pow((double)n, 0.5) * BWDMAX;

    /*
     * Convert to single precision
     */
    magmablas_zlag2c( n, nrhs, dB, lddb, dSX, lddsx, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }

    magmablas_zlat2c( uplo, n, dA, ldda, dSA, lddsa, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }
    
    // factor dSA in single precision
    magma_cpotrf_gpu( uplo, n, dSA, lddsa, info );
    if (*info != 0) {
        *iter = -3;
        goto FALLBACK;
    }
    
    // solve dSA*dSX = dB in single precision
    magma_cpotrs_gpu( uplo, n, nrhs, dSA, lddsa, dSX, lddsx, info );

    // residual dR = dB - dA*dX in double precision
    magmablas_clag2z( n, nrhs, dSX, lddsx, dX, lddx, info );
    magmablas_zlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dR, lddr );
    if ( nrhs == 1 ) {
        magma_zhemv( uplo, n,
                     c_neg_one, dA, ldda,
                                dX, 1,
                     c_one,     dR, 1 );
    }
    else {
        magma_zhemm( MagmaLeft, uplo, n, nrhs,
                     c_neg_one, dA, ldda,
                                dX, lddx,
                     c_one,     dR, lddr );
    }

    // TODO: use MAGMA_Z_ABS( dX(i,j) ) instead of zlange?
    for( j=0; j < nrhs; j++ ) {
        i = magma_izamax( n, dX(0,j), 1) - 1;
        magma_zgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
        Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );

        i = magma_izamax ( n, dR(0,j), 1 ) - 1;
        magma_zgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
        Rnrm = lapackf77_zlange( "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
        magmablas_zlag2c( n, nrhs, dR, lddr, dSX, lddsx, info );
        if (*info != 0) {
            *iter = -2;
            goto FALLBACK;
        }
        // solve dSA*dSX = R in single precision
        magma_cpotrs_gpu( uplo, n, nrhs, dSA, lddsa, dSX, lddsx, info );

        // Add correction and setup residual
        // dX += dSX [including conversion]  --and--
        // dR = dB
        for( j=0; j < nrhs; j++ ) {
            magmablas_zcaxpycp( n, dSX(0,j), dX(0,j), dB(0,j), dR(0,j) );
        }

        // residual dR = dB - dA*dX in double precision
        if ( nrhs == 1 ) {
            magma_zhemv( uplo, n,
                         c_neg_one, dA, ldda,
                                    dX, 1,
                         c_one,     dR, 1 );
        }
        else {
            magma_zhemm( MagmaLeft, uplo, n, nrhs,
                         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_izamax( n, dX(0,j), 1) - 1;
            magma_zgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
            Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );

            i = magma_izamax ( n, dR(0,j), 1 ) - 1;
            magma_zgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
            Rnrm = lapackf77_zlange( "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_zpotrf_gpu( uplo, n, dA, ldda, info );
    if (*info == 0) {
        magmablas_zlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
        magma_zpotrs_gpu( uplo, n, nrhs, dA, ldda, dX, lddx, info );
    }
    
    return *info;
}
/* ////////////////////////////////////////////////////////////////////////////
   -- Testing zcgeqrsv
*/
int main( int argc, char** argv)
{
    TESTING_INIT();

    real_Double_t   gflops, gpu_perf, gpu_time, cpu_perf, cpu_time, gpu_perfd, gpu_perfs;
    double          error, gpu_error, cpu_error, Anorm, work[1];
    magmaDoubleComplex c_one     = MAGMA_Z_ONE;
    magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE;
    magmaDoubleComplex *h_A, *h_A2, *h_B, *h_X, *h_R;
    magmaDoubleComplex_ptr d_A, d_B, d_X, d_T;
    magmaFloatComplex  *d_SA, *d_SB;
    magmaDoubleComplex *h_workd, *tau, tmp[1];
    magmaFloatComplex  *h_works;
    magma_int_t lda,  ldb, lhwork, lworkgpu;
    magma_int_t ldda, lddb, lddx;
    magma_int_t M, N, nrhs, qrsv_iters, info, size, min_mn, max_mn, nb;
    magma_int_t ione     = 1;
    magma_int_t ISEED[4] = {0,0,0,1};

    printf("Epsilon(double): %8.6e\n"
           "Epsilon(single): %8.6e\n\n",
           lapackf77_dlamch("Epsilon"), lapackf77_slamch("Epsilon") );
    magma_int_t status = 0;

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

    double tol = opts.tolerance * lapackf77_dlamch("E");
    
    nrhs = opts.nrhs;
    
    printf("                    CPU Gflop/s   GPU  Gflop/s                         |b-Ax|| / (N||A||)   ||dx-x||/(N||A||)\n");
    printf("    M     N  NRHS    double        double    single     mixed   Iter   CPU        GPU                        \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];
            if ( M < N ) {
                printf( "%5d %5d %5d   skipping because M < N is not yet supported.\n", (int) M, (int) N, (int) nrhs );
                continue;
            }
            min_mn = min(M, N);
            max_mn = max(M, N);
            lda    = M;
            ldb    = max_mn;
            ldda   = ((M+31)/32) * 32;
            lddb   = ((max_mn+31)/32)*32;
            lddx   = ((N+31)/32) * 32;
            nb     = max( magma_get_zgeqrf_nb( M ), magma_get_cgeqrf_nb( M ) );
            gflops = (FLOPS_ZGEQRF( M, N ) + FLOPS_ZGEQRS( M, N, nrhs )) / 1e9;
            
            lworkgpu = (M - N + nb)*(nrhs + nb) + nrhs*nb;
            
            // query for workspace size
            lhwork = -1;
            lapackf77_zgels( MagmaNoTransStr, &M, &N, &nrhs,
                             NULL, &lda, NULL, &ldb, tmp, &lhwork, &info );
            lhwork = (magma_int_t) MAGMA_Z_REAL( tmp[0] );
            lhwork = max( lhwork, lworkgpu );
            
            TESTING_MALLOC_CPU( tau,     magmaDoubleComplex, min_mn   );
            TESTING_MALLOC_CPU( h_A,     magmaDoubleComplex, lda*N    );
            TESTING_MALLOC_CPU( h_A2,    magmaDoubleComplex, lda*N    );
            TESTING_MALLOC_CPU( h_B,     magmaDoubleComplex, ldb*nrhs );
            TESTING_MALLOC_CPU( h_X,     magmaDoubleComplex, ldb*nrhs );
            TESTING_MALLOC_CPU( h_R,     magmaDoubleComplex, ldb*nrhs );
            TESTING_MALLOC_CPU( h_workd, magmaDoubleComplex, lhwork   );
            h_works = (magmaFloatComplex*)h_workd;
            
            TESTING_MALLOC_DEV( d_A, magmaDoubleComplex, ldda*N      );
            TESTING_MALLOC_DEV( d_B, magmaDoubleComplex, lddb*nrhs   );
            TESTING_MALLOC_DEV( d_X, magmaDoubleComplex, lddx*nrhs   );
            TESTING_MALLOC_DEV( d_T, magmaDoubleComplex, ( 2*min_mn + (N+31)/32*32 )*nb );
            
            /* Initialize the matrices */
            size = lda*N;
            lapackf77_zlarnv( &ione, ISEED, &size, h_A );
            lapackf77_zlacpy( MagmaUpperLowerStr, &M, &N, h_A, &lda, h_A2, &lda );
            
            // make random RHS
            size = ldb*nrhs;
            lapackf77_zlarnv( &ione, ISEED, &size, h_B );
            lapackf77_zlacpy( MagmaUpperLowerStr, &M, &nrhs, h_B, &ldb, h_R, &ldb );
            
            magma_zsetmatrix( M, N,    h_A, lda, d_A, ldda );
            magma_zsetmatrix( M, nrhs, h_B, ldb, d_B, lddb );
            
            //=====================================================================
            //              Mixed Precision Iterative Refinement - GPU
            //=====================================================================
            gpu_time = magma_wtime();
            magma_zcgeqrsv_gpu( M, N, nrhs,
                                d_A, ldda, d_B, lddb,
                                d_X, lddx, &qrsv_iters, &info );
            gpu_time = magma_wtime() - gpu_time;
            gpu_perf = gflops / gpu_time;
            if (info != 0)
                printf("magma_zcgeqrsv returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            // compute the residual
            magma_zgetmatrix( N, nrhs, d_X, lddx, h_X, ldb );
            blasf77_zgemm( MagmaNoTransStr, MagmaNoTransStr, &M, &nrhs, &N,
                           &c_neg_one, h_A, &lda,
                                       h_X, &ldb,
                           &c_one,     h_R, &ldb);
            Anorm = lapackf77_zlange("f", &M, &N,    h_A, &lda, work);
            
            //=====================================================================
            //                 Double Precision Solve
            //=====================================================================
            magma_zsetmatrix( M, N,    h_A, lda, d_A, ldda );
            magma_zsetmatrix( M, nrhs, h_B, ldb, d_B, lddb );
            
            gpu_time = magma_wtime();
            magma_zgels_gpu( MagmaNoTrans, M, N, nrhs, d_A, ldda,
                             d_B, lddb, h_workd, lworkgpu, &info);
            gpu_time = magma_wtime() - gpu_time;
            gpu_perfd = gflops / gpu_time;
            
            //=====================================================================
            //                 Single Precision Solve
            //=====================================================================
            magma_zsetmatrix( M, N,    h_A, lda, d_A, ldda );
            magma_zsetmatrix( M, nrhs, h_B, ldb, d_B, lddb );
            
            /* The allocation of d_SA and d_SB is done here to avoid
             * to double the memory used on GPU with zcgeqrsv */
            TESTING_MALLOC_DEV( d_SA, magmaFloatComplex, ldda*N    );
            TESTING_MALLOC_DEV( d_SB, magmaFloatComplex, lddb*nrhs );
            magmablas_zlag2c( M, N,    d_A, ldda, d_SA, ldda, &info );
            magmablas_zlag2c( N, nrhs, d_B, lddb, d_SB, lddb, &info );
            
            gpu_time = magma_wtime();
            magma_cgels_gpu( MagmaNoTrans, M, N, nrhs, d_SA, ldda,
                             d_SB, lddb, h_works, lhwork, &info);
            gpu_time = magma_wtime() - gpu_time;
            gpu_perfs = gflops / gpu_time;
            TESTING_FREE_DEV( d_SA );
            TESTING_FREE_DEV( d_SB );
            
            /* =====================================================================
               Performs operation using LAPACK
               =================================================================== */
            lapackf77_zlacpy( MagmaUpperLowerStr, &M, &nrhs, h_B, &ldb, h_X, &ldb );
            
            cpu_time = magma_wtime();
            lapackf77_zgels( MagmaNoTransStr, &M, &N, &nrhs,
                             h_A, &lda, h_X, &ldb, h_workd, &lhwork, &info );
            cpu_time = magma_wtime() - cpu_time;
            cpu_perf = gflops / cpu_time;
            if (info != 0)
                printf("lapackf77_zgels returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            blasf77_zgemm( MagmaNoTransStr, MagmaNoTransStr, &M, &nrhs, &N,
                           &c_neg_one, h_A2, &lda,
                                       h_X,  &ldb,
                           &c_one,     h_B,  &ldb );
            
            cpu_error = lapackf77_zlange("f", &M, &nrhs, h_B, &ldb, work) / (min_mn*Anorm);
            gpu_error = lapackf77_zlange("f", &M, &nrhs, h_R, &ldb, work) / (min_mn*Anorm);
            
            // error relative to LAPACK
            size = M*nrhs;
            blasf77_zaxpy( &size, &c_neg_one, h_B, &ione, h_R, &ione );
            error = lapackf77_zlange("f", &M, &nrhs, h_R, &ldb, work) / (min_mn*Anorm);
            
            printf("%5d %5d %5d   %7.2f       %7.2f   %7.2f   %7.2f   %4d   %8.2e   %8.2e   %8.2e   %s\n",
                   (int) M, (int) N, (int) nrhs,
                   cpu_perf, gpu_perfd, gpu_perfs, gpu_perf,
                   (int) qrsv_iters,
                   cpu_error, gpu_error, error, (error < tol ? "ok" : "failed"));
            status += ! (error < tol);
            
            TESTING_FREE_CPU( tau  );
            TESTING_FREE_CPU( h_A  );
            TESTING_FREE_CPU( h_A2 );
            TESTING_FREE_CPU( h_B  );
            TESTING_FREE_CPU( h_X  );
            TESTING_FREE_CPU( h_R  );
            TESTING_FREE_CPU( h_workd );
            
            TESTING_FREE_DEV( d_A );
            TESTING_FREE_DEV( d_B );
            TESTING_FREE_DEV( d_X );
            TESTING_FREE_DEV( d_T );
            fflush( stdout );
        }
        if ( opts.niter > 1 ) {
            printf( "\n" );
        }
    }
    
    TESTING_FINALIZE();
    return status;
}
Example #3
0
/* ////////////////////////////////////////////////////////////////////////////
   -- Testing zlag2c and clag2z
*/
int main( int argc, char** argv )
{
    TESTING_INIT();
    
    real_Double_t   gbytes, gpu_perf, gpu_time, cpu_perf, cpu_time;
    double error, work[1];
    float serror, swork[1];
    magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE;
    magmaFloatComplex  s_neg_one = MAGMA_C_NEG_ONE;
    magma_int_t ione = 1;
    magma_int_t m, n, lda, ldda, size, info;
    magma_int_t ISEED[4] = {0,0,0,1};
    magma_int_t status = 0;
    magmaFloatComplex   *SA, *SR;
    magmaDoubleComplex   *A,  *R;
    magmaFloatComplex  *dSA;
    magmaDoubleComplex_ptr dA;
    
    magma_opts opts;
    parse_opts( argc, argv, &opts );
    
    printf("func       M     N     CPU GB/s (ms)       GPU GB/s (ms)     ||R||_F\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];
            lda  = m;
            ldda = ((m+31)/32)*32;
            // m*n double-complex loads and m*n single-complex stores (and vice-versa for clag2z)
            gbytes = (real_Double_t) m*n * (sizeof(magmaDoubleComplex) + sizeof(magmaFloatComplex)) / 1e9;
            size = ldda*n;  // ldda >= lda
            
            TESTING_MALLOC_CPU(  SA, magmaFloatComplex,  size );
            TESTING_MALLOC_CPU(   A, magmaDoubleComplex, size );
            TESTING_MALLOC_CPU(  SR, magmaFloatComplex,  size );
            TESTING_MALLOC_CPU(   R, magmaDoubleComplex, size );
            
            TESTING_MALLOC_DEV( dSA, magmaFloatComplex,  size );
            TESTING_MALLOC_DEV(  dA, magmaDoubleComplex, size );
            
            lapackf77_zlarnv( &ione, ISEED, &size,  A );
            lapackf77_clarnv( &ione, ISEED, &size, SA );
            
            magma_zsetmatrix( m, n, A,  lda, dA,  ldda );
            magma_csetmatrix( m, n, SA, lda, dSA, ldda );
            
            /* =====================================================================
               Performs operation using LAPACK zlag2c
               =================================================================== */
            cpu_time = magma_wtime();
            lapackf77_zlag2c( &m, &n, A, &lda, SA, &lda, &info );
            cpu_time = magma_wtime() - cpu_time;
            cpu_perf = gbytes / cpu_time;
            if (info != 0)
                printf("lapackf77_zlag2c returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            /* ====================================================================
               Performs operation using MAGMA zlag2c
               =================================================================== */            
            gpu_time = magma_sync_wtime(0);
            magmablas_zlag2c( m, n, dA, ldda, dSA, ldda, &info );
            gpu_time = magma_sync_wtime(0) - gpu_time;
            gpu_perf = gbytes / gpu_time;
            if (info != 0)
                printf("magmablas_zlag2c returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            magma_cgetmatrix( m, n, dSA, ldda, SR, lda );
            
            /* =====================================================================
               compute error |SA_magma - SA_lapack|
               should be zero if both are IEEE compliant
               =================================================================== */
            blasf77_caxpy( &size, &s_neg_one, SA, &ione, SR, &ione );
            serror = lapackf77_clange( "Fro", &m, &n, SR, &lda, swork );
            
            printf( "zlag2c %5d %5d   %7.2f (%7.2f)   %7.2f (%7.2f)   %8.2e   %s\n",
                    (int) m, (int) n,
                    cpu_perf, cpu_time*1000., gpu_perf, gpu_time*1000.,
                    serror, (serror == 0 ? "ok" : "failed") );
            status += ! (serror == 0);
            
            
            /* =====================================================================
               Reset matrices
               =================================================================== */
            lapackf77_zlarnv( &ione, ISEED, &size,  A );
            lapackf77_clarnv( &ione, ISEED, &size, SA );
            
            magma_zsetmatrix( m, n, A,  lda, dA,  ldda );
            magma_csetmatrix( m, n, SA, lda, dSA, ldda );
            
            /* =====================================================================
               Performs operation using LAPACK clag2z
               =================================================================== */
            cpu_time = magma_wtime();
            lapackf77_clag2z( &m, &n, SA, &lda, A, &lda, &info );
            cpu_time = magma_wtime() - cpu_time;
            cpu_perf = gbytes / cpu_time;
            if (info != 0)
                printf("lapackf77_clag2z returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            /* ====================================================================
               Performs operation using MAGMA clag2z
               =================================================================== */
            magma_csetmatrix( m, n, SA, lda, dSA, ldda );
            
            gpu_time = magma_sync_wtime(0);
            magmablas_clag2z( m, n, dSA, ldda, dA, ldda, &info );
            gpu_time = magma_sync_wtime(0) - gpu_time;
            gpu_perf = gbytes / gpu_time;
            if (info != 0)
                printf("magmablas_clag2z returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            magma_zgetmatrix( m, n, dA, ldda, R, lda );
            
            /* =====================================================================
               compute error |A_magma - A_lapack|
               should be zero if both are IEEE compliant
               =================================================================== */
            blasf77_zaxpy( &size, &c_neg_one, A, &ione, R, &ione );
            error = lapackf77_zlange( "Fro", &m, &n, R, &lda, work );
            
            printf( "clag2z %5d %5d   %7.2f (%7.2f)   %7.2f (%7.2f)   %8.2e   %s\n",
                    (int) m, (int) n,
                    cpu_perf, cpu_time*1000., gpu_perf, gpu_time*1000.,
                    error, (error == 0 ? "ok" : "failed") );
            status += ! (error == 0);
            
            TESTING_FREE_CPU(  SA );
            TESTING_FREE_CPU(   A );
            TESTING_FREE_CPU(  SR );
            TESTING_FREE_CPU(   R );
                                 
            TESTING_FREE_DEV( dSA );
            TESTING_FREE_DEV(  dA );
            printf( "\n" );
            fflush( stdout );
        }
        if ( opts.niter > 1 ) {
            printf( "\n" );
        }
    }
    
    TESTING_FINALIZE();
    return status;
}
Example #4
0
/**
    Purpose
    -------
    ZCGESV computes the solution to a complex 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.

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

    The iterative refinement is not going to be a winning strategy if
    the ratio complex SINGLE PRECISION performance over complex 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      COMPLEX_16 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      COMPLEX_16 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      COMPLEX_16 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) COMPLEX_16 array on the GPU, dimension (N*NRHS)
            This array is used to hold the residual vectors.

    @param
    dworks  (workspace) COMPLEX array on the GPU, dimension (N*(N+NRHS))
            This array is used to store the complex 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_zgesv_driver
    ********************************************************************/
extern "C" magma_int_t
magma_zcgesv_gpu(magma_trans_t trans, magma_int_t n, magma_int_t nrhs,
                 magmaDoubleComplex *dA, magma_int_t ldda,
                 magma_int_t *ipiv,  magma_int_t *dipiv,
                 magmaDoubleComplex *dB, magma_int_t lddb,
                 magmaDoubleComplex *dX, magma_int_t lddx,
                 magmaDoubleComplex *dworkd, magmaFloatComplex *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)

    magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE;
    magmaDoubleComplex c_one     = MAGMA_Z_ONE;
    magma_int_t     ione  = 1;
    magmaDoubleComplex *dR;
    magmaFloatComplex  *dSA, *dSX;
    magmaDoubleComplex 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_zlange(MagmaInfNorm, n, n, dA, ldda, (double*)dworkd );
    cte  = Anrm * eps * pow((double)n, 0.5) * BWDMAX;

    /*
     * Convert to single precision
     */
    //magmablas_zlag2c( n, nrhs, dB, lddb, dSX, lddsx, info );  // done inside zcgetrs with pivots
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }

    magmablas_zlag2c( n, n, dA, ldda, dSA, lddsa, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }

    // factor dSA in single precision
    magma_cgetrf_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_zcgetrs_gpu( trans, n, nrhs, dSA, lddsa, dipiv, dB, lddb, dX, lddx, dSX, info );

    // residual dR = dB - dA*dX in double precision
    magmablas_zlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dR, lddr );
    if ( nrhs == 1 ) {
        magma_zgemv( trans, n, n,
                     c_neg_one, dA, ldda,
                     dX, 1,
                     c_one,     dR, 1 );
    }
    else {
        magma_zgemm( trans, MagmaNoTrans, n, nrhs, n,
                     c_neg_one, dA, ldda,
                     dX, lddx,
                     c_one,     dR, lddr );
    }

    // TODO: use MAGMA_Z_ABS( dX(i,j) ) instead of zlange?
    for( j=0; j < nrhs; j++ ) {
        i = magma_izamax( n, dX(0,j), 1) - 1;
        magma_zgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
        Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );

        i = magma_izamax ( n, dR(0,j), 1 ) - 1;
        magma_zgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
        Rnrm = lapackf77_zlange( "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_zcgetrs_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_zaxpycp( n, dR(0,j), dX(0,j), dB(0,j) );
        }

        // residual dR = dB - dA*dX in double precision
        if ( nrhs == 1 ) {
            magma_zgemv( trans, n, n,
                         c_neg_one, dA, ldda,
                         dX, 1,
                         c_one,     dR, 1 );
        }
        else {
            magma_zgemm( 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_izamax( n, dX(0,j), 1) - 1;
            magma_zgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
            Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );

            i = magma_izamax ( n, dR(0,j), 1 ) - 1;
            magma_zgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
            Rnrm = lapackf77_zlange( "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_zgetrf_gpu( n, n, dA, ldda, ipiv, info );
    if (*info == 0) {
        magmablas_zlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
        magma_zgetrs_gpu( trans, n, nrhs, dA, ldda, ipiv, dX, lddx, info );
    }

    return *info;
}
Example #5
0
int main(int argc, char **argv)
{
    TESTING_INIT();

    real_Double_t   gflopsF, gflopsS, gpu_perf, gpu_time /*cpu_perf, cpu_time*/;
    real_Double_t   gpu_perfdf, gpu_perfds;
    real_Double_t   gpu_perfsf, gpu_perfss;
    double          error, Rnorm, Anorm;
    magmaDoubleComplex c_one     = MAGMA_Z_ONE;
    magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE;
    magmaDoubleComplex *h_A, *h_B, *h_X;
    magmaDoubleComplex *d_A, *d_B, *d_X, *d_workd;
    magmaFloatComplex  *d_As, *d_Bs, *d_works;
    double          *h_workd;
    magma_int_t lda, ldb, ldx;
    magma_int_t N, nrhs, posv_iter, info, size;
    magma_int_t ione     = 1;
    magma_int_t ISEED[4] = {0,0,0,1};
    
    printf("Epsilon(double): %8.6e\n"
           "Epsilon(single): %8.6e\n\n",
           lapackf77_dlamch("Epsilon"), lapackf77_slamch("Epsilon") );
    magma_int_t status = 0;
    
    magma_opts opts;
    parse_opts( argc, argv, &opts );

    double tol = opts.tolerance * lapackf77_dlamch("E");
    
    nrhs = opts.nrhs;
    
    printf("using: uplo = %s\n",
           lapack_uplo_const(opts.uplo));

    printf("    N NRHS   DP-Factor  DP-Solve  SP-Factor  SP-Solve  MP-Solve  Iter   |b-Ax|/|A|\n");
    printf("=====================================================================================\n");
    for( int itest = 0; itest < opts.ntest; ++itest ) {
        for( int iter = 0; iter < opts.niter; ++iter ) {
            N = opts.nsize[itest];
            ldb = ldx = lda = N;
            gflopsF = FLOPS_ZPOTRF( N ) / 1e9;
            gflopsS = gflopsF + FLOPS_ZPOTRS( N, nrhs ) / 1e9;
            
            TESTING_MALLOC_CPU( h_A,     magmaDoubleComplex, lda*N    );
            TESTING_MALLOC_CPU( h_B,     magmaDoubleComplex, ldb*nrhs );
            TESTING_MALLOC_CPU( h_X,     magmaDoubleComplex, ldx*nrhs );
            TESTING_MALLOC_CPU( h_workd, double,             N        );
            
            TESTING_MALLOC_DEV( d_A,     magmaDoubleComplex, lda*N        );
            TESTING_MALLOC_DEV( d_B,     magmaDoubleComplex, ldb*nrhs     );
            TESTING_MALLOC_DEV( d_X,     magmaDoubleComplex, ldx*nrhs     );
            TESTING_MALLOC_DEV( d_works, magmaFloatComplex,  lda*(N+nrhs) );
            TESTING_MALLOC_DEV( d_workd, magmaDoubleComplex, N*nrhs       );
            
            /* Initialize the matrix */
            size = lda * N ;
            lapackf77_zlarnv( &ione, ISEED, &size, h_A );
            magma_zmake_hpd( N, h_A, lda );
            
            size = ldb * nrhs ;
            lapackf77_zlarnv( &ione, ISEED, &size, h_B );
            
            magma_zsetmatrix( N, N,    h_A, lda, d_A, lda );
            magma_zsetmatrix( N, nrhs, h_B, ldb, d_B, ldb );
            
            //=====================================================================
            //              Mixed Precision Iterative Refinement - GPU
            //=====================================================================
            gpu_time = magma_wtime();
            magma_zcposv_gpu(opts.uplo, N, nrhs, d_A, lda, d_B, ldb, d_X, ldx,
                             d_workd, d_works, &posv_iter, &info);
            gpu_time = magma_wtime() - gpu_time;
            gpu_perf = gflopsS / gpu_time;
            if (info != 0)
                printf("magma_zcposv returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            //=====================================================================
            //                 Error Computation
            //=====================================================================
            magma_zgetmatrix( N, nrhs, d_X, ldx, h_X, ldx ) ;
            
            Anorm = lapackf77_zlanhe( "I", lapack_uplo_const(opts.uplo), &N, h_A, &N, h_workd);
            blasf77_zhemm( "L", lapack_uplo_const(opts.uplo), &N, &nrhs,
                           &c_one,     h_A, &lda,
                                       h_X, &ldx,
                           &c_neg_one, h_B, &ldb);
            Rnorm = lapackf77_zlange( "I", &N, &nrhs, h_B, &ldb, h_workd);
            error = Rnorm / Anorm;
            
            //=====================================================================
            //                 Double Precision Factor
            //=====================================================================
            magma_zsetmatrix( N, N, h_A, lda, d_A, lda );
            
            gpu_time = magma_wtime();
            magma_zpotrf_gpu(opts.uplo, N, d_A, lda, &info);
            gpu_time = magma_wtime() - gpu_time;
            gpu_perfdf = gflopsF / gpu_time;
            if (info != 0)
                printf("magma_zpotrf returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            //=====================================================================
            //                 Double Precision Solve
            //=====================================================================
            magma_zsetmatrix( N, N,    h_A, lda, d_A, lda );
            magma_zsetmatrix( N, nrhs, h_B, ldb, d_B, ldb );
            
            gpu_time = magma_wtime();
            magma_zpotrf_gpu(opts.uplo, N, d_A, lda, &info);
            magma_zpotrs_gpu(opts.uplo, N, nrhs, d_A, lda, d_B, ldb, &info);
            gpu_time = magma_wtime() - gpu_time;
            gpu_perfds = gflopsS / gpu_time;
            if (info != 0)
                printf("magma_zpotrs returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            //=====================================================================
            //                 Single Precision Factor
            //=====================================================================
            d_As = d_works;
            d_Bs = d_works + lda*N;
            magma_zsetmatrix( N, N,    h_A, lda, d_A, lda );
            magma_zsetmatrix( N, nrhs, h_B, ldb, d_B, ldb );
            magmablas_zlag2c( N, N,    d_A, lda, d_As, N, &info );
            magmablas_zlag2c( N, nrhs, d_B, ldb, d_Bs, N, &info );
            
            gpu_time = magma_wtime();
            magma_cpotrf_gpu(opts.uplo, N, d_As, N, &info);
            gpu_time = magma_wtime() - gpu_time;
            gpu_perfsf = gflopsF / gpu_time;
            if (info != 0)
                printf("magma_cpotrf returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            //=====================================================================
            //                 Single Precision Solve
            //=====================================================================
            magmablas_zlag2c(N, N,    d_A, lda, d_As, N, &info );
            magmablas_zlag2c(N, nrhs, d_B, ldb, d_Bs, N, &info );
            
            gpu_time = magma_wtime();
            magma_cpotrf_gpu(opts.uplo, N, d_As, lda, &info);
            magma_cpotrs_gpu(opts.uplo, N, nrhs, d_As, N, d_Bs, N, &info);
            gpu_time = magma_wtime() - gpu_time;
            gpu_perfss = gflopsS / gpu_time;
            if (info != 0)
                printf("magma_cpotrs returned error %d: %s.\n",
                       (int) info, magma_strerror( info ));
            
            printf("%5d %5d   %7.2f   %7.2f   %7.2f   %7.2f   %7.2f    %4d   %8.2e   %s\n",
                   (int) N, (int) nrhs,
                   gpu_perfdf, gpu_perfds, gpu_perfsf, gpu_perfss, gpu_perf,
                   (int) posv_iter, error, (error < tol ? "ok" : "failed"));
            status += ! (error < tol);
            
            TESTING_FREE_CPU( h_A );
            TESTING_FREE_CPU( h_B );
            TESTING_FREE_CPU( h_X );
            TESTING_FREE_CPU( h_workd );
            
            TESTING_FREE_DEV( d_A );
            TESTING_FREE_DEV( d_B );
            TESTING_FREE_DEV( d_X );
            TESTING_FREE_DEV( d_works );
            TESTING_FREE_DEV( d_workd );
            fflush( stdout );
        }
        if ( opts.niter > 1 ) {
            printf( "\n" );
        }
    }

    TESTING_FINALIZE();
    return status;
}
Example #6
0
/**
    Purpose
    -------
    ZCGEQRSV solves the least squares problem
       min || A*X - B ||,
    where A is an M-by-N matrix and X and B are M-by-NRHS matrices.

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

    The iterative refinement is not going to be a winning strategy if
    the ratio complex SINGLE PRECISION performance over complex 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]
    m       INTEGER
            The number of rows of the matrix A. M >= 0.

    @param[in]
    n       INTEGER
            The number of columns of the matrix A. M >= 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      COMPLEX_16 array on the GPU, dimension (LDDA,N)
            On entry, the M-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 QR factorization of A as returned by
            function DGEQRF_GPU.

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

    @param[in,out]
    dB      COMPLEX_16 array on the GPU, dimension (LDDB,NRHS)
            The M-by-NRHS right hand side matrix B.
            May be overwritten (e.g., if refinement fails).

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

    @param[out]
    dX      COMPLEX_16 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[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 SGEQRF
        +        -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

    @ingroup magma_zgels_driver
    ********************************************************************/
extern "C" magma_int_t
magma_zcgeqrsv_gpu(
    magma_int_t m, magma_int_t n, magma_int_t nrhs,
    magmaDoubleComplex_ptr dA,  magma_int_t ldda,
    magmaDoubleComplex_ptr dB,  magma_int_t lddb,
    magmaDoubleComplex_ptr dX,  magma_int_t lddx,
    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)
    #define dSX(i,j)    (dSX + (i) + (j)*lddsx)
    
    magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE;
    magmaDoubleComplex c_one     = MAGMA_Z_ONE;
    magma_int_t     ione  = 1;
    magmaDoubleComplex *hworkd;
    magmaFloatComplex  *hworks;
    magmaDoubleComplex *tau;
    magmaFloatComplex  *stau;
    magmaDoubleComplex_ptr dworkd;
    magmaFloatComplex_ptr  dworks;
    magmaDoubleComplex_ptr dR, dT;
    magmaFloatComplex_ptr  dSA, dSX, dST;
    magmaDoubleComplex Xnrmv, Rnrmv;
    double          Anrm, Xnrm, Rnrm, cte, eps;
    magma_int_t     i, j, iiter, lddsa, lddsx, lddr, nb, lhwork, minmn, size, ldworkd;

    /* Check arguments */
    *iter = 0;
    *info = 0;
    if ( m < 0 )
        *info = -1;
    else if ( n < 0 || n > m )
        *info = -2;
    else if ( nrhs < 0 )
        *info = -3;
    else if ( ldda < max(1,m))
        *info = -5;
    else if ( lddb < max(1,m))
        *info = -7;
    else if ( lddx < max(1,n))
        *info = -9;

    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }

    if ( m == 0 || n == 0 || nrhs == 0 )
        return *info;

    nb   = magma_get_cgeqrf_nb(m);
    minmn= min(m, n);
    
    /* dSX contains both B and X, so must be max(m or lddb,n). */
    lddsa = ldda;
    lddsx = max(lddb,n);
    lddr  = lddb;
    
    /*
     * Allocate temporary buffers
     */
    /* dworks(dSA + dSX + dST) */
    size = lddsa*n + lddsx*nrhs + ( 2*minmn + ((n+31)/32)*32 )*nb;
    if (MAGMA_SUCCESS != magma_cmalloc( &dworks, size )) {
        fprintf(stderr, "Allocation of dworks failed (%d)\n", (int) size);
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
    dSA = dworks;
    dSX = dSA + lddsa*n;
    dST = dSX + lddsx*nrhs;

    /* dworkd(dR) = lddr*nrhs */
    ldworkd = lddr*nrhs;
    if (MAGMA_SUCCESS != magma_zmalloc( &dworkd, ldworkd )) {
        magma_free( dworks );
        fprintf(stderr, "Allocation of dworkd failed\n");
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
    dR = dworkd;

    /* hworks(workspace for cgeqrs + stau) = min(m,n) + lhworks */
    lhwork = (m - n + nb)*(nrhs + nb) + nrhs*nb;
    size = lhwork + minmn;
    magma_cmalloc_cpu( &hworks, size );
    if ( hworks == NULL ) {
        magma_free( dworks );
        magma_free( dworkd );
        fprintf(stderr, "Allocation of hworks failed\n");
        *info = MAGMA_ERR_HOST_ALLOC;
        return *info;
    }
    stau = hworks + lhwork;

    eps  = lapackf77_dlamch("Epsilon");
    Anrm = magmablas_zlange(MagmaInfNorm, m, n, dA, ldda, (double*)dworkd );
    cte  = Anrm * eps * pow((double)n, 0.5) * BWDMAX;

    /*
     * Convert to single precision
     */
    magmablas_zlag2c( m, nrhs, dB, lddb, dSX, lddsx, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }

    magmablas_zlag2c( m, n, dA, ldda, dSA, lddsa, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }

    // factor dSA in single precision
    magma_cgeqrf_gpu( m, n, dSA, lddsa, stau, dST, info );
    if (*info != 0) {
        *iter = -3;
        goto FALLBACK;
    }

    // solve dSA*dSX = dB in single precision
    magma_cgeqrs_gpu( m, n, nrhs, dSA, lddsa, stau, dST, dSX, lddsx, hworks, lhwork, info );
    if (*info != 0) {
        *iter = -3;
        goto FALLBACK;
    }

    // residual dR = dB - dA*dX in double precision
    magmablas_clag2z( n, nrhs, dSX, lddsx, dX, lddx, info );
    magmablas_zlacpy( MagmaUpperLower, m, nrhs, dB, lddb, dR, lddr );
    if ( nrhs == 1 ) {
        magma_zgemv( MagmaNoTrans, m, n,
                     c_neg_one, dA, ldda,
                                dX, 1,
                     c_one,     dR, 1 );
    }
    else {
        magma_zgemm( MagmaNoTrans, MagmaNoTrans, m, nrhs, n,
                     c_neg_one, dA, ldda,
                                dX, lddx,
                     c_one,     dR, lddr );
    }

    // TODO: use MAGMA_Z_ABS( dX(i,j) ) instead of zlange?
    for( j=0; j < nrhs; j++ ) {
        i = magma_izamax( n, dX(0,j), 1) - 1;
        magma_zgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
        Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );

        i = magma_izamax ( m, dR(0,j), 1 ) - 1;
        magma_zgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
        Rnrm = lapackf77_zlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );

        if ( Rnrm >  Xnrm*cte ) {
            goto REFINEMENT;
        }
    }

    *iter = 0;

    /* Free workspaces */
    magma_free( dworks );
    magma_free( dworkd );
    magma_free_cpu( hworks );
    return *info;

REFINEMENT:
    /* TODO: this iterative refinement algorithm works only for compatibile
     * systems (B in colspan of A).
     * See Matrix Computations (3rd ed) p. 267 for correct algorithm. */
    for( iiter=1; iiter < ITERMAX; ) {
        *info = 0;
        // convert residual dR to single precision dSX
        magmablas_zlag2c( m, nrhs, dR, lddr, dSX, lddsx, info );
        if (*info != 0) {
            *iter = -2;
            goto FALLBACK;
        }
        // solve dSA*dSX = R in single precision
        magma_cgeqrs_gpu( m, n, nrhs, dSA, lddsa, stau, dST, dSX, lddsx, hworks, lhwork, info );
        if (*info != 0) {
            *iter = -3;
            goto FALLBACK;
        }

        // Add correction and setup residual
        // dX += dSX [including conversion]  --and--
        // dR[1:n] = dB[1:n]   (only n rows, not whole m rows! -- useless if m > n)
        for( j=0; j < nrhs; j++ ) {
            magmablas_zcaxpycp( n, dSX(0,j), dX(0,j), dB(0,j), dR(0,j) );
        }
        // dR = dB  (whole m rows)
        magmablas_zlacpy( MagmaUpperLower, m, nrhs, dB, lddb, dR, lddr );
        
        // residual dR = dB - dA*dX in double precision
        if ( nrhs == 1 ) {
            magma_zgemv( MagmaNoTrans, m, n,
                         c_neg_one, dA, ldda,
                                    dX, 1,
                         c_one,     dR, 1 );
        }
        else {
            magma_zgemm( MagmaNoTrans, MagmaNoTrans, m, 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_izamax( n, dX(0,j), 1) - 1;
            magma_zgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
            Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );

            i = magma_izamax ( m, dR(0,j), 1 ) - 1;
            magma_zgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
            Rnrm = lapackf77_zlange( "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;

        /* Free workspaces */
        magma_free( dworks );
        magma_free( dworkd );
        magma_free_cpu( hworks );
        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_free( dworks );
    magma_free_cpu( hworks );

    /*
     * Allocate temporary buffers
     */
    /* dworkd = dT for zgeqrf */
    nb   = magma_get_zgeqrf_nb( m );
    size = (2*min(m, n) + (n+31)/32*32 )*nb;
    if ( size > ldworkd ) {
        magma_free( dworkd );
        if (MAGMA_SUCCESS != magma_zmalloc( &dworkd, size )) {
            fprintf(stderr, "Allocation of dworkd2 failed\n");
            *info = MAGMA_ERR_DEVICE_ALLOC;
            return *info;
        }
    }
    dT = dworkd;

    /* hworkd(dtau + workspace for zgeqrs) = min(m,n) + lhwork */
    size = lhwork + minmn;
    magma_zmalloc_cpu( &hworkd, size );
    if ( hworkd == NULL ) {
        magma_free( dworkd );
        fprintf(stderr, "Allocation of hworkd2 failed\n");
        *info = MAGMA_ERR_HOST_ALLOC;
        return *info;
    }
    tau = hworkd + lhwork;

    magma_zgeqrf_gpu( m, n, dA, ldda, tau, dT, info );
    if (*info == 0) {
        // if m > n, then dB won't fit in dX, so solve with dB and copy n rows to dX
        magma_zgeqrs_gpu( m, n, nrhs, dA, ldda, tau, dT, dB, lddb, hworkd, lhwork, info );
        magmablas_zlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
    }

    magma_free( dworkd );
    magma_free_cpu( hworkd );
    return *info;
}
Example #7
0
extern "C" magma_int_t
magma_zcgeqrsv_gpu(magma_int_t m, magma_int_t n, magma_int_t nrhs,
                   magmaDoubleComplex *dA,  magma_int_t ldda,
                   magmaDoubleComplex *dB,  magma_int_t lddb,
                   magmaDoubleComplex *dX,  magma_int_t lddx,
                   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
    =======
    ZCGEQRSV solves the least squares problem
       min || A*X - B ||,
    where A is an M-by-N matrix and X and B are M-by-NRHS matrices.

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

    The iterative refinement is not going to be a winning strategy if
    the ratio complex SINGLE PRECISION performance over complex 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
    =========
    M       (input) INTEGER
            The number of rows of the matrix A. M >= 0.

    N       (input) INTEGER
            The number of columns of the matrix A. M >= 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) COMPLEX_16 array on the GPU, dimension (LDDA,N)
            On entry, the M-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 QR factorization of A as returned by
            function DGEQRF_GPU.

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

    dB      (input or input/output) COMPLEX_16 array on the GPU, dimension (LDDB,NRHS)
            The M-by-NRHS right hand side matrix B.
            May be overwritten (e.g., if refinement fails).

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

    dX      (output) COMPLEX_16 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).

    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 SGEQRF
                 -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

    =====================================================================    */

    #define dB(i,j)     (dB + (i) + (j)*lddb)
    #define dX(i,j)     (dX + (i) + (j)*lddx)
    #define dR(i,j)     (dR + (i) + (j)*lddr)
    #define dSX(i,j)    (dSX + (i) + (j)*lddsx)
    
    magmaDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE;
    magmaDoubleComplex c_one     = MAGMA_Z_ONE;
    magma_int_t     ione  = 1;
    magmaDoubleComplex *dworkd, *hworkd;
    magmaFloatComplex  *dworks, *hworks;
    magmaDoubleComplex *dR, *tau, *dT;
    magmaFloatComplex  *dSA, *dSX, *dST, *stau;
    magmaDoubleComplex Xnrmv, Rnrmv;
    double          Anrm, Xnrm, Rnrm, cte, eps;
    magma_int_t     i, j, iiter, lddsa, lddsx, lddr, nb, lhwork, minmn, size, ldworkd;

    /* Check arguments */
    *iter = 0;
    *info = 0;
    if ( m < 0 )
        *info = -1;
    else if ( n < 0 || n > m )
        *info = -2;
    else if ( nrhs < 0 )
        *info = -3;
    else if ( ldda < max(1,m))
        *info = -5;
    else if ( lddb < max(1,m))
        *info = -7;
    else if ( lddx < max(1,n))
        *info = -9;

    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }

    if ( m == 0 || n == 0 || nrhs == 0 )
        return *info;

    nb   = magma_get_cgeqrf_nb(m);
    minmn= min(m, n);
    
    /* dSX contains both B and X, so must be max(m or lddb,n). */
    lddsa = ldda;
    lddsx = max(lddb,n);
    lddr  = lddb;
    
    /*
     * Allocate temporary buffers
     */
    /* dworks(dSA + dSX + dST) */
    size = lddsa*n + lddsx*nrhs + ( 2*minmn + ((n+31)/32)*32 )*nb;
    if (MAGMA_SUCCESS != magma_cmalloc( &dworks, size )) {
        fprintf(stderr, "Allocation of dworks failed (%d)\n", (int) size);
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
    dSA = dworks;
    dSX = dSA + lddsa*n;
    dST = dSX + lddsx*nrhs;

    /* dworkd(dR) = lddr*nrhs */
    ldworkd = lddr*nrhs;
    if (MAGMA_SUCCESS != magma_zmalloc( &dworkd, ldworkd )) {
        magma_free( dworks );
        fprintf(stderr, "Allocation of dworkd failed\n");
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
    dR = dworkd;

    /* hworks(workspace for cgeqrs + stau) = min(m,n) + lhworks */
    lhwork = (m - n + nb)*(nrhs + nb) + nrhs*nb;
    size = lhwork + minmn;
    magma_cmalloc_cpu( &hworks, size );
    if ( hworks == NULL ) {
        magma_free( dworks );
        magma_free( dworkd );
        fprintf(stderr, "Allocation of hworks failed\n");
        *info = MAGMA_ERR_HOST_ALLOC;
        return *info;
    }
    stau = hworks + lhwork;

    eps  = lapackf77_dlamch("Epsilon");
    Anrm = magmablas_zlange('I', m, n, dA, ldda, (double*)dworkd );
    cte  = Anrm * eps * pow((double)n, 0.5) * BWDMAX;

    /*
     * Convert to single precision
     */
    magmablas_zlag2c( m, nrhs, dB, lddb, dSX, lddsx, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }

    magmablas_zlag2c( m, n, dA, ldda, dSA, lddsa, info );
    if (*info != 0) {
        *iter = -2;
        goto FALLBACK;
    }

    // factor dSA in single precision
    magma_cgeqrf_gpu( m, n, dSA, lddsa, stau, dST, info );
    if (*info != 0) {
        *iter = -3;
        goto FALLBACK;
    }

    // solve dSA*dSX = dB in single precision
    magma_cgeqrs_gpu( m, n, nrhs, dSA, lddsa, stau, dST, dSX, lddsx, hworks, lhwork, info );
    if (*info != 0) {
        *iter = -3;
        goto FALLBACK;
    }

    // residual dR = dB - dA*dX in double precision
    magmablas_clag2z( n, nrhs, dSX, lddsx, dX, lddx, info );
    magmablas_zlacpy( MagmaUpperLower, m, nrhs, dB, lddb, dR, lddr );
    if ( nrhs == 1 ) {
        magma_zgemv( MagmaNoTrans, m, n,
                     c_neg_one, dA, ldda,
                                dX, 1,
                     c_one,     dR, 1 );
    }
    else {
        magma_zgemm( MagmaNoTrans, MagmaNoTrans, m, nrhs, n,
                     c_neg_one, dA, ldda,
                                dX, lddx,
                     c_one,     dR, lddr );
    }

    // TODO: use MAGMA_Z_ABS( dX(i,j) ) instead of zlange?
    for( j=0; j < nrhs; j++ ) {
        i = magma_izamax( n, dX(0,j), 1) - 1;
        magma_zgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
        Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );

        i = magma_izamax ( m, dR(0,j), 1 ) - 1;
        magma_zgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
        Rnrm = lapackf77_zlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );

        if ( Rnrm >  Xnrm*cte ) {
            goto REFINEMENT;
        }
    }

    *iter = 0;

    /* Free workspaces */
    magma_free( dworks );
    magma_free( dworkd );
    magma_free_cpu( hworks );
    return *info;

REFINEMENT:
    /* TODO: this iterative refinement algorithm works only for compatibile
     * systems (B in colspan of A).
     * See Matrix Computations (3rd ed) p. 267 for correct algorithm. */
    for( iiter=1; iiter < ITERMAX; ) {
        *info = 0;
        // convert residual dR to single precision dSX
        magmablas_zlag2c( m, nrhs, dR, lddr, dSX, lddsx, info );
        if (*info != 0) {
            *iter = -2;
            goto FALLBACK;
        }
        // solve dSA*dSX = R in single precision
        magma_cgeqrs_gpu( m, n, nrhs, dSA, lddsa, stau, dST, dSX, lddsx, hworks, lhwork, info );
        if (*info != 0) {
            *iter = -3;
            goto FALLBACK;
        }

        // Add correction and setup residual
        // dX += dSX [including conversion]  --and--
        // dR[1:n] = dB[1:n]   (only n rows, not whole m rows! -- useless if m > n)
        for( j=0; j < nrhs; j++ ) {
            magmablas_zcaxpycp( n, dSX(0,j), dX(0,j), dB(0,j), dR(0,j) );
        }
        // dR = dB  (whole m rows)
        magmablas_zlacpy( MagmaUpperLower, m, nrhs, dB, lddb, dR, lddr );
        
        // residual dR = dB - dA*dX in double precision
        if ( nrhs == 1 ) {
            magma_zgemv( MagmaNoTrans, m, n,
                         c_neg_one, dA, ldda,
                                    dX, 1,
                         c_one,     dR, 1 );
        }
        else {
            magma_zgemm( MagmaNoTrans, MagmaNoTrans, m, 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_izamax( n, dX(0,j), 1) - 1;
            magma_zgetmatrix( 1, 1, dX(i,j), 1, &Xnrmv, 1 );
            Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );

            i = magma_izamax ( m, dR(0,j), 1 ) - 1;
            magma_zgetmatrix( 1, 1, dR(i,j), 1, &Rnrmv, 1 );
            Rnrm = lapackf77_zlange( "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;

        /* Free workspaces */
        magma_free( dworks );
        magma_free( dworkd );
        magma_free_cpu( hworks );
        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_free( dworks );
    magma_free_cpu( hworks );

    /*
     * Allocate temporary buffers
     */
    /* dworkd = dT for zgeqrf */
    nb   = magma_get_zgeqrf_nb( m );
    size = (2*min(m, n) + (n+31)/32*32 )*nb;
    if ( size > ldworkd ) {
        magma_free( dworkd );
        if (MAGMA_SUCCESS != magma_zmalloc( &dworkd, size )) {
            fprintf(stderr, "Allocation of dworkd2 failed\n");
            *info = MAGMA_ERR_DEVICE_ALLOC;
            return *info;
        }
    }
    dT = dworkd;

    /* hworkd(dtau + workspace for zgeqrs) = min(m,n) + lhwork */
    size = lhwork + minmn;
    magma_zmalloc_cpu( &hworkd, size );
    if ( hworkd == NULL ) {
        magma_free( dworkd );
        fprintf(stderr, "Allocation of hworkd2 failed\n");
        *info = MAGMA_ERR_HOST_ALLOC;
        return *info;
    }
    tau = hworkd + lhwork;

    magma_zgeqrf_gpu( m, n, dA, ldda, tau, dT, info );
    if (*info == 0) {
        // if m > n, then dB won't fit in dX, so solve with dB and copy n rows to dX
        magma_zgeqrs_gpu( m, n, nrhs, dA, ldda, tau, dT, dB, lddb, hworkd, lhwork, info );
        magmablas_zlacpy( MagmaUpperLower, n, nrhs, dB, lddb, dX, lddx );
    }

    magma_free( dworkd );
    magma_free_cpu( hworkd );
    return *info;
}
Example #8
0
/**
    Purpose
    -------
    ZCGETRS solves a system of linear equations
       A * X = B,  A**T * X = B,  or  A**H * X = B
    with a general N-by-N matrix A using the LU factorization computed
    by MAGMA_CGETRF_GPU. B and X are in COMPLEX_16, and A is in COMPLEX.
    This routine is used in the mixed precision iterative solver
    magma_zcgesv.

    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 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]
    dA      COMPLEX array on the GPU, dimension (LDDA,N)
            The factors L and U from the factorization A = P*L*U
            as computed by CGETRF_GPU.

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

    @param[in]
    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 from ZGETRF, where interchanges
            are applied one-after-another.

    @param[in]
    dB      COMPLEX_16 array on the GPU, dimension (LDDB,NRHS)
            On entry, the right hand side matrix B.

    @param[in]
    lddb    INTEGER
            The leading dimension of the arrays X and B.  LDDB >= max(1,N).

    @param[out]
    dX      COMPLEX_16 array on the GPU, dimension (LDDX, NRHS)
            On exit, the solution matrix dX.

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

    @param
    dSX     (workspace) COMPLEX array on the GPU used as workspace,
            dimension (N, NRHS)

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

    @ingroup magma_zgesv_comp
    ********************************************************************/
extern "C" magma_int_t
magma_zcgetrs_gpu(
    magma_trans_t trans, magma_int_t n, magma_int_t nrhs,
    magmaFloatComplex_ptr  dA, magma_int_t ldda,
    magmaInt_ptr        dipiv,
    magmaDoubleComplex_ptr dB, magma_int_t lddb,
    magmaDoubleComplex_ptr dX, magma_int_t lddx,
    magmaFloatComplex_ptr dSX,
    magma_int_t *info)
{
    /* Constants */
    magmaFloatComplex c_one = MAGMA_C_ONE;
    
    /* Local variables */
    bool notran = (trans == MagmaNoTrans);
    magma_int_t inc;
    magma_int_t lddsx = n;

    *info = 0;
    if ( (! notran) &&
         (trans != MagmaTrans) &&
         (trans != MagmaConjTrans) ) {
        *info = -1;
    } else if (n < 0) {
        *info = -2;
    } else if (nrhs < 0) {
        *info = -3;
    } else if (ldda < n) {
        *info = -5;
    } else if (lddb < n) {
        *info = -8;
    } else if (lddx < n) {
        *info = -10;
    }
    // I think this is resolved, but it is unclear what the issue ever was.
    //else if (lddx != lddb) { /* TODO: remove it when zclaswp will have the correct interface */
    //    *info = -10;
    //}
    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }

    /* Quick return if possible */
    if (n == 0 || nrhs == 0) {
        return *info;
    }
    
    magma_queue_t queue;
    magma_device_t cdev;
    magma_getdevice( &cdev );
    magma_queue_create( cdev, &queue );
    
    if (notran) {
        inc = 1;
        
        /* Get X by row applying interchanges to B and cast to single */
        /*
         * TODO: clean zclaswp interface to have interface closer to zlaswp
         */
        magmablas_zclaswp( nrhs, dB, lddb, dSX, lddsx,
                           n, dipiv, inc, queue );
        
        /* Solve L*X = B, overwriting B with SX. */
        magma_ctrsm( MagmaLeft, MagmaLower, MagmaNoTrans, MagmaUnit,
                     n, nrhs, c_one, dA, ldda, dSX, lddsx, queue );
        
        /* Solve U*X = B, overwriting B with X. */
        magma_ctrsm( MagmaLeft, MagmaUpper, MagmaNoTrans, MagmaNonUnit,
                     n, nrhs, c_one, dA, ldda, dSX, lddsx, queue );
        
        magmablas_clag2z( n, nrhs, dSX, lddsx, dX, lddx, queue, info );
    }
    else {
        inc = -1;
        
        /* Cast the COMPLEX_16 RHS to COMPLEX */
        magmablas_zlag2c( n, nrhs, dB, lddb, dSX, lddsx, queue, info );
        
        /* Solve A**T * X = B, or A**H * X = B */
        magma_ctrsm( MagmaLeft, MagmaUpper, trans, MagmaNonUnit,
                     n, nrhs, c_one, dA, ldda, dSX, lddsx, queue );
        
        magma_ctrsm( MagmaLeft, MagmaLower, trans, MagmaUnit,
                     n, nrhs, c_one, dA, ldda, dSX, lddsx, queue );
        
        magmablas_zclaswp( nrhs, dX, lddx, dSX, lddsx,
                           n, dipiv, inc, queue );
    }
    
    magma_queue_destroy( queue );

    return *info;
} /* magma_zcgetrs */
Example #9
0
extern "C" magma_int_t
magma_zcgeqrsv_gpu(magma_int_t M, magma_int_t N, magma_int_t NRHS, 
                   cuDoubleComplex *dA,  magma_int_t ldda, 
                   cuDoubleComplex *dB,  magma_int_t lddb, 
                   cuDoubleComplex *dX,  magma_int_t lddx, 
                   magma_int_t *iter, magma_int_t *info)
{
/*  -- MAGMA (version 1.3.0) --
       Univ. of Tennessee, Knoxville
       Univ. of California, Berkeley
       Univ. of Colorado, Denver
       November 2012

    Purpose
    =======

    ZCGEQRSV solves the least squares problem 
       min || A*X - B ||,
    where A is an M-by-N matrix and X and B are M-by-NRHS matrices.

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

    The iterative refinement is not going to be a winning strategy if
    the ratio SINGLE PRECISION performance over 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
    =========

    M       (input) INTEGER   
            The number of rows of the matrix A. M >= 0.

    N       (input) INTEGER
            The number of columns of the matrix A. M >= 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 or input/output) DOUBLE PRECISION array, dimension (ldda,N)
            On entry, the M-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 A contains the QR factorization of A as returned by
            function DGEQRF_GPU.

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

    B       (input) DOUBLE PRECISION array, dimension (LDB,NRHS)
            The M-by-NRHS right hand side matrix B.

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

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

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

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

    SWORK   (workspace) REAL array, dimension (M*(N+NRHS))
            This array is used to store the 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
                      iterations
            > 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

    TAU     (output) REAL array, dimension (N)
            On exit, TAU(i) contains the scalar factor of the elementary
            reflector H(i), as returned by magma_cgeqrf_gpu.

    LWORK   (input) INTEGER   
            The dimension of the array H_WORK.  LWORK >= (M+N+NB)*NB,   
            where NB can be obtained through magma_get_sgeqrf_nb(M).

    H_WORK  (workspace/output) REAL array, dimension (MAX(1,LWORK))   
            Higher performance is achieved if H_WORK is in pinned memory, e.g.
            allocated using magma_malloc_pinned.

    D_WORK  (workspace/output)  REAL array on the GPU, dimension 2*N*NB,
            where NB can be obtained through magma_get_sgeqrf_nb(M).
            It starts with NB*NB blocks that store the triangular T 
            matrices, followed by the NB*NB blocks of the diagonal 
            inverses for the R matrix.

    TAU_D   (output) DOUBLE REAL array, dimension (N)
            On exit, if the matrix had to be factored in double precision,
            TAU(i) contains the scalar factor of the elementary
            reflector H(i), as returned by magma_zgeqrf_gpu.

    LWORK_D (input) INTEGER   
            The dimension of the array H_WORK_D. LWORK_D >= (M+N+NB)*NB,   
            where NB can be obtained through magma_get_dgeqrf_nb(M).

    H_WORK_D (workspace/output) DOUBLE REAL array, dimension (MAX(1,LWORK_D))
            This memory is unattached if the iterative refinement worked, 
            otherwise it is used as workspace to factor the matrix in
            double precision. Higher performance is achieved if H_WORK_D is 
            in pinned memory, e.g. allocated using magma_malloc_pinned. 

    D_WORK_D (workspace/output) DOUBLE REAL array on the GPU, dimension 2*N*NB,
            where NB can be obtained through magma_get_dgeqrf_nb(M).
            This memory is unattached if the iterative refinement worked, 
            otherwise it is used as workspace to factor the matrix in
            double precision. It starts with NB*NB blocks that store the 
            triangular T matrices, followed by the NB*NB blocks of the 
            diagonal inverses for the R matrix.

    =====================================================================    */

    cuDoubleComplex c_neg_one = MAGMA_Z_NEG_ONE;
    cuDoubleComplex c_one     = MAGMA_Z_ONE;
    magma_int_t     ione  = 1;
    cuDoubleComplex *dworkd, *hworkd;
    cuFloatComplex  *dworks, *hworks;
    cuDoubleComplex *dR, *tau, *dT;
    cuFloatComplex  *dSA, *dSX, *dST, *stau;
    cuDoubleComplex Xnrmv, Rnrmv; 
    double          Anrm, Xnrm, Rnrm, cte, eps; 
    magma_int_t     i, j, iiter, nb, lhwork, minmn, size;
    
    /*
      Check The Parameters. 
    */
    *iter = 0 ;
    *info = 0 ;
    if ( N < 0 )
        *info = -1;
    else if(NRHS<0)
        *info = -3;
    else if( ldda < max(1,N))
        *info = -5;
    else if( lddb < max(1,N))
        *info = -7;
    else if( lddx < max(1,N))
        *info = -9;

    if (*info != 0) {
        magma_xerbla( __func__, -(*info) );
        return *info;
    }

    if( N == 0 || NRHS == 0 )
        return *info;

    nb   = magma_get_cgeqrf_nb(M);
    minmn= min(M, N);

    /*
     * Allocate temporary buffers
     */
    /* dworks(dSA + dSX + dST) */
    size = ldda*N + N*NRHS + ( 2*minmn + ((N+31)/32)*32 )*nb;
    if (MAGMA_SUCCESS != magma_cmalloc( &dworks, size )) {
        fprintf(stderr, "Allocation of dworks failed (%d)\n", (int) size);
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
    dSA = dworks;
    dSX = dSA + ldda*N;
    dST = dSX + N*NRHS;
    
    /* dworkd(dR) = N*NRHS */
    size = N*NRHS;
    if (MAGMA_SUCCESS != magma_zmalloc( &dworkd, size )) {
        magma_free( dworks );
        fprintf(stderr, "Allocation of dworkd failed\n");
        *info = MAGMA_ERR_DEVICE_ALLOC;
        return *info;
    }
    dR = dworkd;

    /* hworks(stau + workspace for cgeqrs) = min(M,N) + lhworks */
    lhwork = nb*max((M-N+nb+2*(NRHS)), 1);
    lhwork = max(lhwork, N*nb); /* We hope that magma nb is bigger than lapack nb to have enough memory in workspace */
    size = minmn + lhwork;
    magma_cmalloc_cpu( &hworks, size );
    if ( hworks == NULL ) {
        magma_free( dworks );
        magma_free( dworkd );
        fprintf(stderr, "Allocation of hworks failed\n");
        *info = MAGMA_ERR_HOST_ALLOC;
        return *info;
    }
    stau = hworks + lhwork;

    eps  = lapackf77_dlamch("Epsilon");
    Anrm = magmablas_zlange('I', M, N, dA, ldda, (double*)dworkd );
    cte  = Anrm * eps *  pow((double)N, 0.5) * BWDMAX ;

    /*
     * Convert to single precision
     */
    magmablas_zlag2c(N, NRHS, dB, lddb, dSX, N, info );
    if( *info != 0 ) {
        *iter = -2; goto L40;
    }

    magmablas_zlag2c(N, N, dA, ldda, dSA, ldda, info );
    if(*info !=0){
        *iter = -2; goto L40;
    }

    // In an ideal version these variables should come from user.
    magma_cgeqrf_gpu(M, N, dSA, ldda, stau, dST, info);
    if( *info != 0 ) {
        *iter = -3; goto L40;
    }

    magma_cgeqrs_gpu(M, N, NRHS, dSA, ldda, stau, dST, dSX, N, hworks, lhwork, info);

    // dX = dSX
    magmablas_clag2z(N, NRHS, dSX, N, dX, lddx, info);

    // dR = dB
    magmablas_zlacpy(MagmaUpperLower, N, NRHS, dB, lddb, dR, N);

    // dR = dB - dA * dX
    if( NRHS == 1 )
        magma_zgemv( MagmaNoTrans, N, N, 
                     c_neg_one, dA, ldda, 
                                dX, 1, 
                     c_one,     dR, 1);
    else
        magma_zgemm( MagmaNoTrans, MagmaNoTrans, N, NRHS, N, 
                     c_neg_one, dA, ldda, 
                                dX, lddx, 
                     c_one,     dR, N );

    for(i=0; i<NRHS; i++){
        j = magma_izamax( N, dX+i*N, 1);
        magma_zgetmatrix( 1, 1, dX+i*N+j-1, 1, &Xnrmv, 1 );
        Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
      
        j = magma_izamax ( N, dR+i*N, 1 );
        magma_zgetmatrix( 1, 1, dR+i*N+j-1, 1, &Rnrmv, 1 );
        Rnrm = lapackf77_zlange( "F", &ione, &ione, &Rnrmv, &ione, NULL );
      
        if( Rnrm >  Xnrm *cte ) goto L10;
    }

    *iter = 0;

    /* Free workspaces */
    magma_free( dworks );
    magma_free( dworkd );
    magma_free_cpu( hworks );
    return *info;

  L10:
    for(iiter=1; iiter<ITERMAX; ) {
        *info = 0 ;
        /*  Convert R from double precision to single precision
            and store the result in SX.
            Solve the system SA*SX = SR.
            -- These two Tasks are merged here. */
        // make SWORK = WORK ... residuals... 
        magmablas_zlag2c( N, NRHS, dR, N, dSX, N, info );
        magma_cgeqrs_gpu( M, N, NRHS, dSA, ldda, stau, dST, dSX, N, hworks, lhwork, info);

        if( *info != 0 ){
            *iter = -3; goto L40;
        }

        for(i=0; i<NRHS; i++) {
            magmablas_zcaxpycp( dSX+i*N, dX+i*lddx, N, dB+i*lddb, dR+i*N );
        }

        /* unnecessary may be */
        magmablas_zlacpy(MagmaUpperLower, N, NRHS, dB, lddb, dR, N);
        if( NRHS == 1 )
            magma_zgemv( MagmaNoTrans, N, N, 
                         c_neg_one, dA, ldda,
                                    dX, 1,
                         c_one,     dR, 1);
        else
            magma_zgemm( MagmaNoTrans, MagmaNoTrans, N, NRHS, N, 
                         c_neg_one, dA, ldda,
                                    dX, lddx,
                         c_one,     dR, N);

        /*  Check whether the NRHS normwise backward errors satisfy the
            stopping criterion. If yes, set ITER=IITER>0 and return.     */
        for(i=0;i<NRHS;i++)
        {
            j = magma_izamax( N, dX+i*N, 1);
            magma_zgetmatrix( 1, 1, dX+i*N+j-1, 1, &Xnrmv, 1 );
            Xnrm = lapackf77_zlange( "F", &ione, &ione, &Xnrmv, &ione, NULL );
            
            j = magma_izamax ( N, dR+i*N, 1 );
            magma_zgetmatrix( 1, 1, dR+i*N+j-1, 1, &Rnrmv, 1 );
            Rnrm = lapackf77_zlange( "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 ;

        /* Free workspaces */
        magma_free( dworks );
        magma_free( dworkd );
        magma_free_cpu( hworks );
        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 ;

  L40:
    magma_free( dworks );

    /*
     * Allocate temporary buffers
     */
    /* dworkd(dT + tau) = min_mn + min_mn*nb*3 */
    nb   = magma_get_zgeqrf_nb(M);
    size = minmn * (3 * nb + 1);
    if ( size > (N*NRHS) ) {
        magma_free( dworkd );
        if (MAGMA_SUCCESS != magma_zmalloc( &dworkd, size )) {
            fprintf(stderr, "Allocation of dworkd2 failed\n");
            *info = MAGMA_ERR_DEVICE_ALLOC;
            return *info;
        }
    }
    tau = dworkd;
    dT  = tau + minmn;

    /* hworks(stau + workspace for cgeqrs) = min(M,N) + lhworks */
    /* re-use hworks memory for hworkd if possible, else re-allocate. */
    if ( (2*lhwork) <= (minmn+lhwork) ) {
        hworkd = (cuDoubleComplex*) hworks;
    }
    else {
        magma_free_cpu( hworks );
        magma_zmalloc_cpu( &hworkd, lhwork );
        if ( hworkd == NULL ) {
            magma_free( dworkd );
            fprintf(stderr, "Allocation of hworkd2 failed\n");
            *info = MAGMA_ERR_HOST_ALLOC;
            return *info;
        }
    }

    /* Single-precision iterative refinement failed to converge to a
       satisfactory solution, so we resort to double precision.           */
    magma_zgeqrf_gpu(M, N, dA, ldda, tau, dT, info);
    if ( *info == 0 ) {
        magmablas_zlacpy(MagmaUpperLower, N, NRHS, dB, lddb, dX, lddx);
        magma_zgeqrs_gpu(M, N, NRHS, dA, ldda, tau, dT, dX, lddx, hworkd, lhwork, info);
    }
    
    magma_free( dworkd );
    magma_free_cpu( hworkd );
    return *info;
}