FLA_Error FLA_Svd_uv_unb_var1( dim_t n_iter_max, FLA_Obj A, FLA_Obj s, FLA_Obj U, FLA_Obj V, dim_t k_accum, dim_t b_alg )
{
FLA_Error r_val = FLA_SUCCESS;
FLA_Datatype dt;
FLA_Datatype dt_real;
FLA_Datatype dt_comp;
FLA_Obj scale, T, S, rL, rR, d, e, G, H;
dim_t m_A, n_A;
dim_t min_m_n;
dim_t n_GH;
double crossover_ratio = 17.0 / 9.0;
n_GH = k_accum;
m_A = FLA_Obj_length( A );
n_A = FLA_Obj_width( A );
min_m_n = FLA_Obj_min_dim( A );
dt = FLA_Obj_datatype( A );
dt_real = FLA_Obj_datatype_proj_to_real( A );
dt_comp = FLA_Obj_datatype_proj_to_complex( A );
// Create matrices to hold block Householder transformations.
FLA_Bidiag_UT_create_T( A, &T, &S );
// Create vectors to hold the realifying scalars.
FLA_Obj_create( dt, min_m_n, 1, 0, 0, &rL );
FLA_Obj_create( dt, min_m_n, 1, 0, 0, &rR );
// Create vectors to hold the diagonal and sub-diagonal.
FLA_Obj_create( dt_real, min_m_n, 1, 0, 0, &d );
FLA_Obj_create( dt_real, min_m_n-1, 1, 0, 0, &e );
// Create matrices to hold the left and right Givens scalars.
FLA_Obj_create( dt_comp, min_m_n-1, n_GH, 0, 0, &G );
FLA_Obj_create( dt_comp, min_m_n-1, n_GH, 0, 0, &H );
// Create a real scaling factor.
FLA_Obj_create( dt_real, 1, 1, 0, 0, &scale );
// Compute a scaling factor; If none is needed, sigma will be set to one.
FLA_Svd_compute_scaling( A, scale );
// Scale the matrix if scale is non-unit.
if ( !FLA_Obj_equals( scale, FLA_ONE ) )
FLA_Scal( scale, A );
if ( m_A < crossover_ratio * n_A )
{
// Reduce the matrix to bidiagonal form.
// Apply scalars to rotate elements on the superdiagonal to the real domain.
// Extract the diagonal and superdiagonal from A.
FLA_Bidiag_UT( A, T, S );
FLA_Bidiag_UT_realify( A, rL, rR );
FLA_Bidiag_UT_extract_real_diagonals( A, d, e );
// Form U and V.
FLA_Bidiag_UT_form_U( A, T, U );
FLA_Bidiag_UT_form_V( A, S, V );
// Apply the realifying scalars in rL and rR to U and V, respectively.
{
FLA_Obj UL, UR;
FLA_Obj VL, VR;
FLA_Part_1x2( U, &UL, &UR, min_m_n, FLA_LEFT );
FLA_Part_1x2( V, &VL, &VR, min_m_n, FLA_LEFT );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_CONJUGATE, rL, UL );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_NO_CONJUGATE, rR, VL );
}
// Perform a singular value decomposition on the bidiagonal matrix.
r_val = FLA_Bsvd_v_opt_var1( n_iter_max, d, e, G, H, U, V, b_alg );
}
else // if ( crossover_ratio * n_A <= m_A )
{
FLA_Obj TQ, R;
FLA_Obj AT,
AB;
FLA_Obj UL, UR;
// Perform a QR factorization on A and form Q in U.
FLA_QR_UT_create_T( A, &TQ );
FLA_QR_UT( A, TQ );
FLA_QR_UT_form_Q( A, TQ, U );
FLA_Obj_free( &TQ );
// Set the lower triangle of R to zero and then copy the upper
// triangle of A to R.
FLA_Part_2x1( A, &AT,
&AB, n_A, FLA_TOP );
FLA_Obj_create( dt, n_A, n_A, 0, 0, &R );
FLA_Setr( FLA_LOWER_TRIANGULAR, FLA_ZERO, R );
FLA_Copyr( FLA_UPPER_TRIANGULAR, AT, R );
// Reduce the matrix to bidiagonal form.
// Apply scalars to rotate elements on the superdiagonal to the real domain.
// Extract the diagonal and superdiagonal from A.
FLA_Bidiag_UT( R, T, S );
FLA_Bidiag_UT_realify( R, rL, rR );
FLA_Bidiag_UT_extract_real_diagonals( R, d, e );
// Form V from right Householder vectors in upper triangle of R.
FLA_Bidiag_UT_form_V( R, S, V );
// Form U in R.
FLA_Bidiag_UT_form_U( R, T, R );
// Apply the realifying scalars in rL and rR to U and V, respectively.
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_CONJUGATE, rL, R );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_NO_CONJUGATE, rR, V );
// Perform a singular value decomposition on the bidiagonal matrix.
r_val = FLA_Bsvd_v_opt_var1( n_iter_max, d, e, G, H, R, V, b_alg );
// Multiply R into U, storing the result in A and then copying back
// to U.
FLA_Part_1x2( U, &UL, &UR, n_A, FLA_LEFT );
FLA_Gemm( FLA_NO_TRANSPOSE, FLA_NO_TRANSPOSE,
FLA_ONE, UL, R, FLA_ZERO, A );
FLA_Copy( A, UL );
FLA_Obj_free( &R );
}
// Copy the converged eigenvalues to the output vector.
FLA_Copy( d, s );
// Sort the singular values and singular vectors in descending order.
FLA_Sort_svd( FLA_BACKWARD, s, U, V );
// If the matrix was scaled, rescale the singular values.
if ( !FLA_Obj_equals( scale, FLA_ONE ) )
FLA_Inv_scal( scale, s );
FLA_Obj_free( &scale );
FLA_Obj_free( &T );
FLA_Obj_free( &S );
FLA_Obj_free( &rL );
FLA_Obj_free( &rR );
FLA_Obj_free( &d );
FLA_Obj_free( &e );
FLA_Obj_free( &G );
FLA_Obj_free( &H );
return r_val;
}
FLA_Error FLA_Svd_uv_var2_components( dim_t n_iter_max, dim_t k_accum, dim_t b_alg,
FLA_Obj A, FLA_Obj s, FLA_Obj U, FLA_Obj V,
double* dtime_bred, double* dtime_bsvd, double* dtime_appq,
double* dtime_qrfa, double* dtime_gemm )
{
FLA_Error r_val = FLA_SUCCESS;
FLA_Datatype dt;
FLA_Datatype dt_real;
FLA_Datatype dt_comp;
FLA_Obj T, S, rL, rR, d, e, G, H, RG, RH, W;
dim_t m_A, n_A;
dim_t min_m_n;
dim_t n_GH;
double crossover_ratio = 17.0 / 9.0;
double dtime_temp;
n_GH = k_accum;
m_A = FLA_Obj_length( A );
n_A = FLA_Obj_width( A );
min_m_n = FLA_Obj_min_dim( A );
dt = FLA_Obj_datatype( A );
dt_real = FLA_Obj_datatype_proj_to_real( A );
dt_comp = FLA_Obj_datatype_proj_to_complex( A );
// If the matrix is a scalar, then the SVD is easy.
if ( min_m_n == 1 )
{
FLA_Copy( A, s );
FLA_Set_to_identity( U );
FLA_Set_to_identity( V );
return FLA_SUCCESS;
}
// Create matrices to hold block Householder transformations.
FLA_Bidiag_UT_create_T( A, &T, &S );
// Create vectors to hold the realifying scalars.
FLA_Obj_create( dt, min_m_n, 1, 0, 0, &rL );
FLA_Obj_create( dt, min_m_n, 1, 0, 0, &rR );
// Create vectors to hold the diagonal and sub-diagonal.
FLA_Obj_create( dt_real, min_m_n, 1, 0, 0, &d );
FLA_Obj_create( dt_real, min_m_n-1, 1, 0, 0, &e );
// Create matrices to hold the left and right Givens scalars.
FLA_Obj_create( dt_comp, min_m_n-1, n_GH, 0, 0, &G );
FLA_Obj_create( dt_comp, min_m_n-1, n_GH, 0, 0, &H );
// Create matrices to hold the left and right Givens matrices.
FLA_Obj_create( dt_real, min_m_n, min_m_n, 0, 0, &RG );
FLA_Obj_create( dt_real, min_m_n, min_m_n, 0, 0, &RH );
FLA_Obj_create( dt, m_A, n_A, 0, 0, &W );
if ( m_A >= n_A )
{
if ( m_A < crossover_ratio * n_A )
{
dtime_temp = FLA_Clock();
{
// Reduce the matrix to bidiagonal form.
// Apply scalars to rotate elements on the sub-diagonal to the real domain.
// Extract the diagonal and sub-diagonal from A.
FLA_Bidiag_UT( A, T, S );
FLA_Bidiag_UT_realify( A, rL, rR );
FLA_Bidiag_UT_extract_diagonals( A, d, e );
}
*dtime_bred = FLA_Clock() - dtime_temp;
dtime_temp = FLA_Clock();
{
// Form U and V.
FLA_Bidiag_UT_form_U( A, T, U );
FLA_Bidiag_UT_form_V( A, S, V );
}
*dtime_appq = FLA_Clock() - dtime_temp;
// Apply the realifying scalars in rL and rR to U and V, respectively.
{
FLA_Obj UL, UR;
FLA_Obj VL, VR;
FLA_Part_1x2( U, &UL, &UR, min_m_n, FLA_LEFT );
FLA_Part_1x2( V, &VL, &VR, min_m_n, FLA_LEFT );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_CONJUGATE, rL, UL );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_NO_CONJUGATE, rR, VL );
}
dtime_temp = FLA_Clock();
{
// Perform a singular value decomposition on the bidiagonal matrix.
r_val = FLA_Bsvd_v_opt_var2( n_iter_max, d, e, G, H, RG, RH, W, U, V, b_alg );
}
*dtime_bsvd = FLA_Clock() - dtime_temp;
}
else // if ( crossover_ratio * n_A <= m_A )
{
FLA_Obj TQ, R;
FLA_Obj AT,
AB;
FLA_Obj UL, UR;
//FLA_QR_UT_create_T( A, &TQ );
FLA_Obj_create( dt, 32, n_A, 0, 0, &TQ );
dtime_temp = FLA_Clock();
{
// Perform a QR factorization on A and form Q in U.
FLA_QR_UT( A, TQ );
}
*dtime_qrfa = FLA_Clock() - dtime_temp;
dtime_temp = FLA_Clock();
{
FLA_QR_UT_form_Q( A, TQ, U );
}
*dtime_appq = FLA_Clock() - dtime_temp;
FLA_Obj_free( &TQ );
// Set the lower triangle of R to zero and then copy the upper
// triangle of A to R.
FLA_Part_2x1( A, &AT,
&AB, n_A, FLA_TOP );
FLA_Obj_create( dt, n_A, n_A, 0, 0, &R );
FLA_Setr( FLA_LOWER_TRIANGULAR, FLA_ZERO, R );
FLA_Copyr( FLA_UPPER_TRIANGULAR, AT, R );
dtime_temp = FLA_Clock();
{
// Reduce the matrix to bidiagonal form.
// Apply scalars to rotate elements on the superdiagonal to the real domain.
// Extract the diagonal and superdiagonal from A.
FLA_Bidiag_UT( R, T, S );
FLA_Bidiag_UT_realify( R, rL, rR );
FLA_Bidiag_UT_extract_diagonals( R, d, e );
}
*dtime_bred = FLA_Clock() - dtime_temp;
dtime_temp = FLA_Clock();
{
// Form V from right Householder vectors in upper triangle of R.
FLA_Bidiag_UT_form_V( R, S, V );
// Form U in R.
FLA_Bidiag_UT_form_U( R, T, R );
}
*dtime_appq += FLA_Clock() - dtime_temp;
// Apply the realifying scalars in rL and rR to U and V, respectively.
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_CONJUGATE, rL, R );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_NO_CONJUGATE, rR, V );
dtime_temp = FLA_Clock();
{
// Perform a singular value decomposition on the bidiagonal matrix.
r_val = FLA_Bsvd_v_opt_var2( n_iter_max, d, e, G, H, RG, RH, W, R, V, b_alg );
}
*dtime_bsvd = FLA_Clock() - dtime_temp;
dtime_temp = FLA_Clock();
{
// Multiply R into U, storing the result in A and then copying back
// to U.
FLA_Part_1x2( U, &UL, &UR, n_A, FLA_LEFT );
FLA_Gemm( FLA_NO_TRANSPOSE, FLA_NO_TRANSPOSE,
FLA_ONE, UL, R, FLA_ZERO, A );
FLA_Copy( A, UL );
}
*dtime_gemm = FLA_Clock() - dtime_temp;
FLA_Obj_free( &R );
}
}
else // if ( m_A < n_A )
{
FLA_Check_error_code( FLA_NOT_YET_IMPLEMENTED );
}
// Copy the converged eigenvalues to the output vector.
FLA_Copy( d, s );
// Sort the singular values and singular vectors in descending order.
FLA_Sort_svd( FLA_BACKWARD, s, U, V );
FLA_Obj_free( &T );
FLA_Obj_free( &S );
FLA_Obj_free( &rL );
FLA_Obj_free( &rR );
FLA_Obj_free( &d );
FLA_Obj_free( &e );
FLA_Obj_free( &G );
FLA_Obj_free( &H );
FLA_Obj_free( &RG );
FLA_Obj_free( &RH );
FLA_Obj_free( &W );
return r_val;
}
int main( int argc, char** argv ) {
FLA_Datatype datatype = TESTTYPE;
FLA_Obj A, A_flame, A_lapack, C;
int m;
FLA_Error init_result;
FLA_Obj TU, TV, U_flame, V_flame, d_flame, e_flame, B_flame;
FLA_Obj tauq, taup, d_lapack, e_lapack, U_lapack, V_lapack, W, B_lapack;
testtype *buff_tauq, *buff_taup, *buff_d_lapack, *buff_e_lapack,
*buff_W, *buff_A_lapack, *buff_U_lapack, *buff_V_lapack;
int lwork, info, is_flame;
if ( argc == 3 ) {
m = atoi(argv[1]);
is_flame = atoi(argv[2]);
} else {
fprintf(stderr, " \n");
fprintf(stderr, "Usage: %s m is_flame\n", argv[0]);
fprintf(stderr, " m : matrix length\n");
fprintf(stderr, " is_flame : 1 yes, 0 no\n");
fprintf(stderr, " \n");
return -1;
}
if ( m == 0 )
return 0;
FLA_Init_safe( &init_result );
fprintf( stdout, "lapack2flame: %d x %d: \n", m, m);
FLA_Obj_create( datatype, m, m, 0, 0, &A );
FLA_Random_matrix( A );
FLA_Obj_create_copy_of( FLA_NO_TRANSPOSE, A, &A_flame );
FLA_Obj_create_copy_of( FLA_NO_TRANSPOSE, A, &A_lapack );
FLA_Obj_create( datatype, m, m, 0, 0, &C );
FLA_Random_matrix( C );
if ( is_flame ) {
fprintf( stdout, " flame executed\n");
FLA_Bidiag_UT_create_T( A_flame, &TU, &TV );
FLA_Bidiag_UT( A_flame, TU, TV );
FLA_Obj_create_copy_of( FLA_NO_TRANSPOSE, A_flame, &U_flame );
FLA_Obj_create_copy_of( FLA_NO_TRANSPOSE, A_flame, &V_flame );
FLA_Bidiag_UT_form_U( U_flame, TU, U_flame );
FLA_Bidiag_UT_form_V( V_flame, TV, V_flame );
FLA_Obj_create( datatype, m, 1, 0, 0, &d_flame );
FLA_Obj_create( datatype, m - 1, 1, 0, 0, &e_flame );
FLA_Bidiag_UT_extract_diagonals( A_flame, d_flame, e_flame );
FLA_Obj_create( datatype, m, m, 0, 0, &B_flame ); FLA_Set( FLA_ZERO, B_flame );
{
FLA_Obj BTL, BTR, BBL, BBR;
FLA_Part_2x2( B_flame, &BTL, &BTR, &BBL, &BBR, 1,1, FLA_BL );
FLA_Set_diagonal_matrix( d_flame, B_flame );
FLA_Set_diagonal_matrix( e_flame, BTR );
}
if (1) {
fprintf( stdout, " - FLAME ----------\n");
FLA_Obj_fshow( stdout, " - Given A - ", A, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - A - ", A_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - U - ", U_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - V - ", V_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - d - ", d_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - e - ", e_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - B - ", B_flame, "% 6.4e", "------");
}
} else {
fprintf( stdout, " lapack executed\n");
FLA_Obj_create( datatype, m, 1, 0, 0, &tauq );
FLA_Obj_create( datatype, m, 1, 0, 0, &taup );
FLA_Obj_create( datatype, m, 1, 0, 0, &d_lapack );
FLA_Obj_create( datatype, m - 1, 1, 0, 0, &e_lapack );
buff_A_lapack = (testtype*)FLA_Obj_buffer_at_view( A_lapack );
buff_tauq = (testtype*)FLA_Obj_buffer_at_view( tauq );
buff_taup = (testtype*)FLA_Obj_buffer_at_view( taup );
buff_d_lapack = (testtype*)FLA_Obj_buffer_at_view( d_lapack );
buff_e_lapack = (testtype*)FLA_Obj_buffer_at_view( e_lapack );
lwork = 32*m;
FLA_Obj_create( datatype, lwork, 1, 0, 0, &W );
buff_W = (testtype*)FLA_Obj_buffer_at_view( W );
sgebrd_( &m, &m,
buff_A_lapack, &m,
buff_d_lapack,
buff_e_lapack,
buff_tauq,
buff_taup,
buff_W,
&lwork,
&info );
FLA_Obj_create( datatype, m, m, 0, 0, &U_lapack );
FLA_Obj_create( datatype, m, m, 0, 0, &V_lapack );
FLA_Copy( A_lapack, U_lapack );
FLA_Copy( A_lapack, V_lapack );
buff_U_lapack = (testtype*)FLA_Obj_buffer_at_view( U_lapack );
buff_V_lapack = (testtype*)FLA_Obj_buffer_at_view( V_lapack );
sorgbr_( "Q", &m, &m, &m,
buff_U_lapack, &m,
buff_tauq,
buff_W,
&lwork,
&info );
sorgbr_( "P", &m, &m, &m,
buff_V_lapack, &m,
buff_taup,
buff_W,
&lwork,
&info );
FLA_Obj_create( datatype, m, m, 0, 0, &B_lapack ); FLA_Set( FLA_ZERO, B_lapack );
{
FLA_Obj BTL, BTR, BBL, BBR;
FLA_Part_2x2( B_lapack, &BTL, &BTR, &BBL, &BBR, 1,1, FLA_BL );
FLA_Set_diagonal_matrix( d_lapack, B_lapack );
FLA_Set_diagonal_matrix( e_lapack, BTR );
}
FLA_Obj_free( &W );
if (1) {
fprintf( stdout, " - LAPACK ----------\n");
FLA_Obj_fshow( stdout, " - Given A - ", A, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - A - ", A_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - U - ", U_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - V - ", V_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - d - ", d_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - e - ", e_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - B - ", B_lapack, "% 6.4e", "------");
}
}
{
testtype dummy;
int zero = 0, one = 1;
FLA_Obj D_lapack;
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, A, &D_lapack ); FLA_Set( FLA_ZERO, D_lapack );
if ( is_flame ) {
buff_d_lapack = (testtype*)FLA_Obj_buffer_at_view( d_flame );
buff_e_lapack = (testtype*)FLA_Obj_buffer_at_view( e_flame );
buff_U_lapack = (testtype*)FLA_Obj_buffer_at_view( U_flame );
buff_V_lapack = (testtype*)FLA_Obj_buffer_at_view( V_flame );
}
FLA_Obj_create( datatype, 4*m, 1, 0, 0, &W );
buff_W = (testtype*)FLA_Obj_buffer_at_view( W );
sbdsqr_( "U", &m, &m, &m, &zero,
buff_d_lapack, buff_e_lapack,
buff_V_lapack, &m,
buff_U_lapack, &m,
&dummy, &one,
buff_W, &info );
FLA_Obj_free( &W );
if (info != 0)
printf( " Error info = %d\n", info );
if ( is_flame )
FLA_Set_diagonal_matrix( d_flame, D_lapack );
else
FLA_Set_diagonal_matrix( d_lapack, D_lapack );
if ( is_flame ) {
fprintf( stdout, " - FLAME ----------\n");
FLA_Obj_fshow( stdout, " - U - ", U_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - V - ", V_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - d - ", d_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - e - ", e_flame, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - D - ", D_lapack, "% 6.4e", "------");
} else {
fprintf( stdout, " - LAPACK ----------\n");
FLA_Obj_fshow( stdout, " - U - ", U_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - V - ", V_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - d - ", d_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - e - ", e_lapack, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - D - ", D_lapack, "% 6.4e", "------");
}
FLA_Obj_free( &D_lapack );
}
if ( is_flame ) {
FLA_Obj_free( &TU );
FLA_Obj_free( &TV );
FLA_Obj_free( &U_flame );
FLA_Obj_free( &V_flame );
FLA_Obj_free( &d_flame );
FLA_Obj_free( &e_flame );
FLA_Obj_free( &B_flame );
} else {
FLA_Obj_free( &tauq );
FLA_Obj_free( &taup );
FLA_Obj_free( &d_lapack );
FLA_Obj_free( &e_lapack );
FLA_Obj_free( &U_lapack );
FLA_Obj_free( &V_lapack );
FLA_Obj_free( &B_lapack );
}
FLA_Obj_free( &A );
FLA_Obj_free( &A_flame );
FLA_Obj_free( &A_lapack );
FLA_Obj_free( &C );
FLA_Finalize_safe( init_result );
}
void time_Bidiag_UT(
int param_combo, int type, int nrepeats, int m, int n,
FLA_Obj A, FLA_Obj tu, FLA_Obj tv, FLA_Obj TU, FLA_Obj TV,
double *dtime, double *diff, double *gflops )
{
int
irep;
double
dtime_old = 1.0e9;
FLA_Obj
A_save, norm;
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, A, &A_save );
FLA_Obj_create( FLA_Obj_datatype_proj_to_real( A ), 1, 1, 0, 0, &norm );
FLA_Copy_external( A, A_save );
for ( irep = 0 ; irep < nrepeats; irep++ ){
FLA_Copy_external( A_save, A );
*dtime = FLA_Clock();
switch( param_combo ){
case 0:
{
switch( type )
{
case FLA_ALG_REFERENCE:
REF_Bidiag_UT( A, tu, tv );
break;
case FLA_ALG_FRONT:
FLA_Bidiag_UT( A, TU, TV );
break;
}
break;
}
}
*dtime = FLA_Clock() - *dtime;
dtime_old = min( *dtime, dtime_old );
}
{
FLA_Obj AL, AR;
FLA_Obj ATL, ATR,
ABL, ABR;
FLA_Obj QU;
FLA_Obj QV, QVL, QVR;
FLA_Obj E, EL, ER;
FLA_Obj F;
FLA_Obj WU, WV, eye;
FLA_Obj tvT,
tvB;
dim_t m_A, n_A, m_TU;
//FLA_Obj_show( "A_save", A_save, "%10.3e", "" );
m_A = FLA_Obj_length( A );
n_A = FLA_Obj_width( A );
m_TU = FLA_Obj_length( TU );
FLA_Obj_create( FLA_Obj_datatype( A ), m_A, m_A, 0, 0, &QU );
FLA_Obj_create( FLA_Obj_datatype( A ), n_A, n_A, 0, 0, &QV );
FLA_Obj_create( FLA_Obj_datatype( A ), m_TU, m_A, 0, 0, &WU );
FLA_Obj_create( FLA_Obj_datatype( A ), m_TU, n_A, 0, 0, &WV );
FLA_Set_to_identity( QU );
FLA_Set_to_identity( QV );
FLA_Part_1x2( QV, &QVL, &QVR, 1, FLA_LEFT );
FLA_Part_1x2( A, &AL, &AR, 1, FLA_LEFT );
FLA_Part_2x2( A, &ATL, &ATR,
&ABL, &ABR, 1, 1, FLA_BL );
FLA_Part_2x1( tv, &tvT,
&tvB, 1, FLA_BOTTOM );
if ( type == FLA_ALG_REFERENCE )
{
if ( FLA_Obj_is_real( A ) )
FLA_Apply_Q_blk_external( FLA_LEFT, FLA_TRANSPOSE, FLA_COLUMNWISE, A, tu, QU );
else
FLA_Apply_Q_blk_external( FLA_LEFT, FLA_CONJ_TRANSPOSE, FLA_COLUMNWISE, A, tu, QU );
//FLA_Apply_Q_blk_external( FLA_RIGHT, FLA_NO_TRANSPOSE, FLA_ROWWISE, AR, tv, QVR );
//
// Need to apply backwards transformation, since vectors are stored columnwise.
// QL? RQ?
//
//FLA_Apply_Q_blk_external( FLA_RIGHT, FLA_NO_TRANSPOSE, FLA_ROWWISE, ATR, tvT, QVR );
//FLA_Apply_Q_blk_external( FLA_RIGHT, FLA_CONJ_TRANSPOSE, FLA_ROWWISE, ATR, tvT, QVR );
//FLA_Apply_Q_blk_external( FLA_RIGHT, FLA_CONJ_TRANSPOSE, FLA_ROWWISE, AR, tvT, QVR );
}
else
{
FLA_Apply_Q_UT( FLA_LEFT, FLA_CONJ_TRANSPOSE, FLA_FORWARD, FLA_COLUMNWISE, A, TU, WU, QU );
FLA_Apply_Q_UT( FLA_RIGHT, FLA_NO_TRANSPOSE, FLA_FORWARD, FLA_ROWWISE, AR, TV, WV, QVR );
}
/*
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, A, &eye );
FLA_Set_to_identity( eye );
//FLA_Gemm( FLA_NO_TRANSPOSE, FLA_CONJ_TRANSPOSE,
// FLA_ONE, QV, QV, FLA_MINUS_ONE, eye );
FLA_Gemm( FLA_NO_TRANSPOSE, FLA_CONJ_TRANSPOSE,
FLA_ONE, QU, QU, FLA_MINUS_ONE, eye );
FLA_Obj_show( "eye", eye, "%10.3e", "" );
FLA_Norm_frob( eye, norm );
FLA_Obj_extract_real_scalar( norm, diff );
FLA_Obj_free( &eye );
*/
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, A, &E );
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, A, &F );
FLA_Gemm( FLA_NO_TRANSPOSE, FLA_NO_TRANSPOSE,
FLA_ONE, A_save, QV, FLA_ZERO, E );
FLA_Gemm( FLA_NO_TRANSPOSE, FLA_NO_TRANSPOSE,
FLA_ONE, QU, E, FLA_ZERO, F );
//FLA_Obj_show( "A_save", A_save, "%10.3e", "" );
FLA_Copy( A, E );
FLA_Triangularize( FLA_UPPER_TRIANGULAR, FLA_NONUNIT_DIAG, E );
FLA_Part_1x2( E, &EL, &ER, 1, FLA_LEFT );
FLA_Triangularize( FLA_LOWER_TRIANGULAR, FLA_NONUNIT_DIAG, ER );
//FLA_Obj_show( "B", E, "%10.3e", "" );
//FLA_Obj_show( "Q'AV", F, "%10.3e", "" );
//FLA_Obj_show( "B", E, "%10.3e + %10.3e ", "" );
//FLA_Obj_show( "Q'AV", F, "%10.3e + %10.3e ", "" );
*diff = FLA_Max_elemwise_diff( E, F );
FLA_Obj_free( &E );
FLA_Obj_free( &F );
FLA_Obj_free( &QU );
FLA_Obj_free( &QV );
FLA_Obj_free( &WU );
FLA_Obj_free( &WV );
}
*gflops = 4.0 * n * n * ( m - n / 3.0 ) /
dtime_old / 1e9;
if ( FLA_Obj_is_complex( A ) )
*gflops *= 4.0;
*dtime = dtime_old;
FLA_Copy_external( A_save, A );
FLA_Obj_free( &A_save );
FLA_Obj_free( &norm );
}
FLA_Error FLA_Svd_ext_u_unb_var1( FLA_Svd_type jobu, FLA_Svd_type jobv,
dim_t n_iter_max,
FLA_Obj A, FLA_Obj s, FLA_Obj U, FLA_Obj V,
dim_t k_accum,
dim_t b_alg )
{
FLA_Error r_val = FLA_SUCCESS;
FLA_Datatype dt;
FLA_Datatype dt_real;
FLA_Datatype dt_comp;
FLA_Obj scale, T, S, rL, rR, d, e, G, H, C; // C is dummy.
dim_t m_A, n_A, min_m_n;
dim_t n_GH;
double crossover_ratio = 17.0 / 9.0;
FLA_Bool u_is_formed = FALSE,
v_is_formed = FALSE;
int apply_scale;
n_GH = k_accum;
m_A = FLA_Obj_length( A );
n_A = FLA_Obj_width( A );
min_m_n = min( m_A, n_A );
dt = FLA_Obj_datatype( A );
dt_real = FLA_Obj_datatype_proj_to_real( A );
dt_comp = FLA_Obj_datatype_proj_to_complex( A );
// Create matrices to hold block Householder transformations.
FLA_Bidiag_UT_create_T( A, &T, &S );
// Create vectors to hold the realifying scalars.
if ( FLA_Obj_is_complex( A ) )
{
FLA_Obj_create( dt, min_m_n, 1, 0, 0, &rL );
FLA_Obj_create( dt, min_m_n, 1, 0, 0, &rR );
}
// Create vectors to hold the diagonal and sub-diagonal.
FLA_Obj_create( dt_real, min_m_n, 1, 0, 0, &d );
FLA_Obj_create( dt_real, min_m_n-1, 1, 0, 0, &e );
// Create matrices to hold the left and right Givens scalars.
FLA_Obj_create( dt_comp, min_m_n-1, n_GH, 0, 0, &G );
FLA_Obj_create( dt_comp, min_m_n-1, n_GH, 0, 0, &H );
// Create a real scaling factor.
FLA_Obj_create( dt_real, 1, 1, 0, 0, &scale );
// Scale matrix A if necessary.
FLA_Max_abs_value( A, scale );
apply_scale =
( FLA_Obj_gt( scale, FLA_OVERFLOW_SQUARE_THRES ) == TRUE ) -
( FLA_Obj_lt( scale, FLA_UNDERFLOW_SQUARE_THRES ) == TRUE );
if ( apply_scale )
FLA_Scal( apply_scale > 0 ? FLA_SAFE_MIN : FLA_SAFE_INV_MIN, A );
if ( m_A < crossover_ratio * n_A )
{
// Reduce the matrix to bidiagonal form.
// Apply scalars to rotate elements on the superdiagonal to the real domain.
// Extract the diagonal and superdiagonal from A.
FLA_Bidiag_UT( A, T, S );
if ( FLA_Obj_is_complex( A ) )
FLA_Bidiag_UT_realify( A, rL, rR );
FLA_Bidiag_UT_extract_real_diagonals( A, d, e );
// Form U and V.
if ( u_is_formed == FALSE )
{
switch ( jobu )
{
case FLA_SVD_VECTORS_MIN_OVERWRITE:
if ( jobv != FLA_SVD_VECTORS_NONE )
FLA_Bidiag_UT_form_V_ext( FLA_UPPER_TRIANGULAR, A, S, FLA_NO_TRANSPOSE, V );
v_is_formed = TRUE; // For this case, V should be formed here.
U = A;
case FLA_SVD_VECTORS_ALL:
case FLA_SVD_VECTORS_MIN_COPY:
FLA_Bidiag_UT_form_U_ext( FLA_UPPER_TRIANGULAR, A, T, FLA_NO_TRANSPOSE, U );
u_is_formed = TRUE;
break;
case FLA_SVD_VECTORS_NONE:
// Do nothing
break;
}
}
if ( v_is_formed == FALSE )
{
if ( jobv == FLA_SVD_VECTORS_MIN_OVERWRITE )
{
FLA_Bidiag_UT_form_V_ext( FLA_UPPER_TRIANGULAR, A, S, FLA_CONJ_TRANSPOSE, A );
v_is_formed = TRUE; /* and */
V = A; // This V is actually V^H.
// V^H -> V
FLA_Obj_flip_base( &V );
FLA_Obj_flip_view( &V );
if ( FLA_Obj_is_complex( A ) )
FLA_Conjugate( V );
}
else if ( jobv != FLA_SVD_VECTORS_NONE )
{
FLA_Bidiag_UT_form_V_ext( FLA_UPPER_TRIANGULAR, A, S, FLA_NO_TRANSPOSE, V );
v_is_formed = TRUE;
}
}
// For complex matrices, apply realification transformation.
if ( FLA_Obj_is_complex( A ) && jobu != FLA_SVD_VECTORS_NONE )
{
FLA_Obj UL, UR;
FLA_Part_1x2( U, &UL, &UR, min_m_n, FLA_LEFT );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_CONJUGATE, rL, UL );
}
if ( FLA_Obj_is_complex( A ) && jobv != FLA_SVD_VECTORS_NONE )
{
FLA_Obj VL, VR;
FLA_Part_1x2( V, &VL, &VR, min_m_n, FLA_LEFT );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_NO_CONJUGATE, rR, VL );
}
// Perform a singular value decomposition on the upper bidiagonal matrix.
r_val = FLA_Bsvd_ext_opt_var1( n_iter_max,
d, e, G, H,
jobu, U, jobv, V,
FALSE, C, // C is not referenced
b_alg );
}
else // if ( crossover_ratio * n_A <= m_A )
{
FLA_Obj TQ, R;
FLA_Obj AT,
AB;
// Perform a QR factorization on A.
FLA_QR_UT_create_T( A, &TQ );
FLA_QR_UT( A, TQ );
// Set the lower triangle of R to zero and then copy the upper
// triangle of A to R.
FLA_Part_2x1( A, &AT,
&AB, n_A, FLA_TOP );
FLA_Obj_create( dt, n_A, n_A, 0, 0, &R );
FLA_Setr( FLA_LOWER_TRIANGULAR, FLA_ZERO, R );
FLA_Copyr( FLA_UPPER_TRIANGULAR, AT, R );
// Form U; if necessary overwrite on A.
if ( u_is_formed == FALSE )
{
switch ( jobu )
{
case FLA_SVD_VECTORS_MIN_OVERWRITE:
U = A;
case FLA_SVD_VECTORS_ALL:
case FLA_SVD_VECTORS_MIN_COPY:
FLA_QR_UT_form_Q( A, TQ, U );
u_is_formed = TRUE;
break;
case FLA_SVD_VECTORS_NONE:
// Do nothing
break;
}
}
FLA_Obj_free( &TQ );
// Reduce the matrix to bidiagonal form.
// Apply scalars to rotate elements on the superdiagonal to the real domain.
// Extract the diagonal and superdiagonal from A.
FLA_Bidiag_UT( R, T, S );
if ( FLA_Obj_is_complex( R ) )
FLA_Bidiag_UT_realify( R, rL, rR );
FLA_Bidiag_UT_extract_real_diagonals( R, d, e );
if ( v_is_formed == FALSE )
{
if ( jobv == FLA_SVD_VECTORS_MIN_OVERWRITE )
{
FLA_Bidiag_UT_form_V_ext( FLA_UPPER_TRIANGULAR, R, S, FLA_CONJ_TRANSPOSE, AT );
v_is_formed = TRUE; /* and */
V = AT; // This V is actually V^H.
// V^H -> V
FLA_Obj_flip_base( &V );
FLA_Obj_flip_view( &V );
if ( FLA_Obj_is_complex( A ) )
FLA_Conjugate( V );
}
else if ( jobv != FLA_SVD_VECTORS_NONE )
{
FLA_Bidiag_UT_form_V_ext( FLA_UPPER_TRIANGULAR, R, S, FLA_NO_TRANSPOSE, V );
v_is_formed = TRUE;
}
}
// Apply householder vectors U in R.
FLA_Bidiag_UT_form_U_ext( FLA_UPPER_TRIANGULAR, R, T, FLA_NO_TRANSPOSE, R );
// Apply the realifying scalars in rL and rR to U and V, respectively.
if ( FLA_Obj_is_complex( A ) && jobu != FLA_SVD_VECTORS_NONE )
{
FLA_Obj RL, RR;
FLA_Part_1x2( R, &RL, &RR, min_m_n, FLA_LEFT );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_CONJUGATE, rL, RL );
}
if ( FLA_Obj_is_complex( A ) && jobv != FLA_SVD_VECTORS_NONE )
{
FLA_Obj VL, VR;
FLA_Part_1x2( V, &VL, &VR, min_m_n, FLA_LEFT );
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_NO_CONJUGATE, rR, VL );
}
// Perform a singular value decomposition on the bidiagonal matrix.
r_val = FLA_Bsvd_ext_opt_var1( n_iter_max,
d, e, G, H,
jobu, R, jobv, V,
FALSE, C,
b_alg );
// Multiply R into U, storing the result in A and then copying back
// to U.
if ( jobu != FLA_SVD_VECTORS_NONE )
{
FLA_Obj UL, UR;
FLA_Part_1x2( U, &UL, &UR, min_m_n, FLA_LEFT );
if ( jobu == FLA_SVD_VECTORS_MIN_OVERWRITE ||
jobv == FLA_SVD_VECTORS_MIN_OVERWRITE )
{
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, UL, &C );
FLA_Gemm( FLA_NO_TRANSPOSE, FLA_NO_TRANSPOSE,
FLA_ONE, UL, R, FLA_ZERO, C );
FLA_Copy( C, UL );
FLA_Obj_free( &C );
}
else
{
FLA_Gemm( FLA_NO_TRANSPOSE, FLA_NO_TRANSPOSE,
FLA_ONE, UL, R, FLA_ZERO, A );
FLA_Copy( A, UL );
}
}
FLA_Obj_free( &R );
}
// Copy the converged eigenvalues to the output vector.
FLA_Copy( d, s );
// No sort is required as it is applied on FLA_Bsvd.
if ( apply_scale )
FLA_Scal( apply_scale < 0 ? FLA_SAFE_MIN : FLA_SAFE_INV_MIN, s );
// When V is overwritten, flip it again.
if ( jobv == FLA_SVD_VECTORS_MIN_OVERWRITE )
{
// Always apply conjugation first wrt dimensions used; then, flip base.
if ( FLA_Obj_is_complex( V ) )
FLA_Conjugate( V );
FLA_Obj_flip_base( &V );
}
FLA_Obj_free( &scale );
FLA_Obj_free( &T );
FLA_Obj_free( &S );
if ( FLA_Obj_is_complex( A ) )
{
FLA_Obj_free( &rL );
FLA_Obj_free( &rR );
}
FLA_Obj_free( &d );
FLA_Obj_free( &e );
FLA_Obj_free( &G );
FLA_Obj_free( &H );
return r_val;
}
int main( int argc, char** argv ) {
FLA_Datatype datatype = TESTTYPE;
FLA_Datatype realtype = REALTYPE;
FLA_Obj
A, TU, TV,
A_copy, A_recovered,
U, V, Vb, B, Be, d, e,
DU, DV;
FLA_Obj
ATL, ATR,
ABL, ABR, Ae;
FLA_Uplo uplo;
dim_t m, n, min_m_n;
FLA_Error init_result;
double residual_A = 0.0;
if ( argc == 3 ) {
m = atoi(argv[1]);
n = atoi(argv[2]);
min_m_n = min(m,n);
} else {
fprintf(stderr, " \n");
fprintf(stderr, "Usage: %s m n\n", argv[0]);
fprintf(stderr, " m : matrix length\n");
fprintf(stderr, " n : matrix width\n");
fprintf(stderr, " \n");
return -1;
}
if ( m == 0 || n == 0 )
return 0;
FLA_Init_safe( &init_result );
// FLAME Bidiag setup
FLA_Obj_create( datatype, m, n, 0, 0, &A );
FLA_Bidiag_UT_create_T( A, &TU, &TV );
// Rand A and create A_copy.
FLA_Random_matrix( A );
{
scomplex *buff_A = FLA_Obj_buffer_at_view( A );
buff_A[0].real = 4.4011e-01; buff_A[0].imag = -4.0150e-09; buff_A[2].real = -2.2385e-01; buff_A[2].imag = -1.5546e-01; buff_A[4].real = -6.3461e-02; buff_A[4].imag = 2.7892e-01; buff_A[6].real = -1.3197e-01; buff_A[6].imag = 5.0888e-01;
buff_A[1].real = 3.3352e-01; buff_A[1].imag = -6.6346e-02; buff_A[3].real = -1.9307e-01; buff_A[3].imag = -8.4066e-02; buff_A[5].real = -6.0446e-03; buff_A[5].imag = 2.2094e-01; buff_A[7].real = -2.3299e-02; buff_A[7].imag = 4.0553e-01;
}
//FLA_Set_to_identity( A );
//FLA_Scal( FLA_MINUS_ONE, A );
if ( m >= n ) {
uplo = FLA_UPPER_TRIANGULAR;
FLA_Part_2x2( A, &ATL, &ATR,
&ABL, &ABR, min_m_n - 1, 1, FLA_TL );
Ae = ATR;
} else {
uplo = FLA_LOWER_TRIANGULAR;
FLA_Part_2x2( A, &ATL, &ATR,
&ABL, &ABR, 1, min_m_n - 1, FLA_TL );
Ae = ABL;
}
FLA_Obj_create_copy_of( FLA_NO_TRANSPOSE, A, &A_copy );
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, A, &A_recovered );
// Bidiag test
{
FLA_Obj norm;
FLA_Bool apply_scale;
FLA_Obj_create( realtype, 1,1, 0,0, &norm );
FLA_Max_abs_value( A, norm );
apply_scale = FLA_Obj_gt( norm, FLA_OVERFLOW_SQUARE_THRES );
if ( apply_scale ) FLA_Scal( FLA_SAFE_MIN, A );
FLA_Bidiag_UT( A, TU, TV );
if ( apply_scale ) FLA_Bidiag_UT_scale_diagonals( FLA_SAFE_INV_MIN, A );
FLA_Obj_free( &norm );
}
// Orthonomal basis U, V.
FLA_Obj_create( datatype, m, min_m_n, 0, 0, &U ); FLA_Set( FLA_ZERO, U );
FLA_Obj_create( datatype, min_m_n, n, 0, 0, &V ); FLA_Set( FLA_ZERO, V );
FLA_Bidiag_UT_form_U_ext( uplo, A, TU, FLA_NO_TRANSPOSE, U );
FLA_Bidiag_UT_form_V_ext( uplo, A, TV, FLA_CONJ_TRANSPOSE, V );
if ( FLA_Obj_is_complex( A ) ){
FLA_Obj rL, rR;
FLA_Obj_create( datatype, min_m_n, 1, 0, 0, &rL );
FLA_Obj_create( datatype, min_m_n, 1, 0, 0, &rR );
FLA_Obj_fshow( stdout, " - Factor no realified - ", A, "% 6.4e", "------");
FLA_Bidiag_UT_realify( A, rL, rR );
FLA_Obj_fshow( stdout, " - Factor realified - ", A, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - rL - ", rL, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - rR - ", rR, "% 6.4e", "------");
FLA_Apply_diag_matrix( FLA_RIGHT, FLA_CONJUGATE, rL, U );
FLA_Apply_diag_matrix( FLA_LEFT, FLA_CONJUGATE, rR, V );
FLA_Obj_free( &rL );
FLA_Obj_free( &rR );
}
// U^H U
FLA_Obj_create( datatype, min_m_n, min_m_n, 0, 0, &DU );
FLA_Gemm_external( FLA_CONJ_TRANSPOSE, FLA_NO_TRANSPOSE,
FLA_ONE, U, U, FLA_ZERO, DU );
// V^H V
FLA_Obj_create( datatype, min_m_n, min_m_n, 0, 0, &DV );
FLA_Gemm_external( FLA_NO_TRANSPOSE, FLA_CONJ_TRANSPOSE,
FLA_ONE, V, V, FLA_ZERO, DV );
// Recover the matrix
FLA_Obj_create( datatype, min_m_n, min_m_n, 0, 0, &B );
FLA_Set( FLA_ZERO, B );
// Set B
FLA_Obj_create( datatype, min_m_n, 1, 0, 0, &d );
FLA_Set_diagonal_vector( A, d );
FLA_Set_diagonal_matrix( d, B );
FLA_Obj_free( &d );
if ( min_m_n > 1 ) {
FLA_Obj_create( datatype, min_m_n - 1 , 1, 0, 0, &e );
FLA_Set_diagonal_vector( Ae, e );
if ( uplo == FLA_UPPER_TRIANGULAR ) {
FLA_Part_2x2( B, &ATL, &ATR,
&ABL, &ABR, min_m_n - 1, 1, FLA_TL );
Be = ATR;
} else {
FLA_Part_2x2( B, &ATL, &ATR,
&ABL, &ABR, 1, min_m_n - 1, FLA_TL );
Be = ABL;
}
FLA_Set_diagonal_matrix( e, Be );
FLA_Obj_free( &e );
}
// Vb := B (V^H)
FLA_Obj_create_copy_of( FLA_NO_TRANSPOSE, V, &Vb );
FLA_Trmm_external( FLA_LEFT, uplo, FLA_NO_TRANSPOSE,
FLA_NONUNIT_DIAG, FLA_ONE, B, Vb );
// A := U Vb
FLA_Gemm_external( FLA_NO_TRANSPOSE, FLA_NO_TRANSPOSE,
FLA_ONE, U, Vb, FLA_ZERO, A_recovered );
residual_A = FLA_Max_elemwise_diff( A_copy, A_recovered );
if (1) {
FLA_Obj_fshow( stdout, " - Given - ", A_copy, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - Factor - ", A, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - TU - ", TU, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - TV - ", TV, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - B - ", B, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - U - ", U, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - V - ", V, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - Vb - ", Vb, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - U'U - ", DU, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - VV' - ", DV, "% 6.4e", "------");
FLA_Obj_fshow( stdout, " - Recovered A - ", A_recovered, "% 6.4e", "------");
fprintf( stdout, "lapack2flame: %lu x %lu: ", m, n);
fprintf( stdout, "recovery A = %12.10e\n\n", residual_A ) ;
}
FLA_Obj_free( &A );
FLA_Obj_free( &TU );
FLA_Obj_free( &TV );
FLA_Obj_free( &B );
FLA_Obj_free( &U );
FLA_Obj_free( &V );
FLA_Obj_free( &Vb );
FLA_Obj_free( &DU );
FLA_Obj_free( &DV );
FLA_Obj_free( &A_copy );
FLA_Obj_free( &A_recovered );
FLA_Finalize_safe( init_result );
}