FLA_Error FLA_LQ_UT_form_Q( FLA_Obj A, FLA_Obj T, FLA_Obj Q )
{
FLA_Error r_val = FLA_SUCCESS;
// Flip a base once.
FLA_Obj_flip_base( &A );
if ( FLA_Obj_is( A, Q ) == FALSE )
FLA_Obj_flip_base( &Q );
// Dimensions of the both matrices should be flipped.
FLA_Obj_flip_view( &A );
FLA_Obj_flip_view( &Q );
// Run the QR utility function.
r_val = FLA_QR_UT_form_Q( A, T, Q );
// Apply conjugation on Q as we use QR_UT_form_Q.
if ( FLA_Obj_is_complex( Q ) )
FLA_Conjugate( Q );
// Recover the base object.
if ( FLA_Obj_is( A, Q ) == FALSE )
FLA_Obj_flip_base( &Q );
FLA_Obj_flip_base( &A );
return r_val;
}
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;
}
void time_Apply_Q_UT_lnfc(
int variant, int type, int n_repeats, int m, int n, int nb_alg,
FLA_Obj A, FLA_Obj A_orig, FLA_Obj t, FLA_Obj T, FLA_Obj s, FLA_Obj S, FLA_Obj B,
double *dtime, double *diff, double *gflops )
{
int
irep;
double
dtime_old = 1.0e9;
FLA_Obj
A_save, A_orig_save, B_save, norm;
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, A, &A_save );
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, A, &A_orig_save );
FLA_Obj_create_conf_to( FLA_NO_TRANSPOSE, B, &B_save );
FLA_Obj_create( FLA_Obj_datatype_proj_to_real( A ), 1, 1, 0, 0, &norm );
FLA_Copy_external( A, A_save );
FLA_Copy_external( A, A_orig_save );
FLA_Copy_external( B, B_save );
for ( irep = 0 ; irep < n_repeats; irep++ ){
FLA_Copy_external( A_save, A );
FLA_Copy_external( A_orig_save, A_orig );
FLA_Copy_external( B_save, B );
*dtime = FLA_Clock();
switch( variant )
{
case 0:
REF_Apply_Q_UT_lnfc( A, t, B );
//REF_Bidiag_form_U_blk_external( FLA_LEFT, FLA_NO_TRANSPOSE, A, t, B );
//FLA_Bidiag_blk_external( A_orig, t, s );
//REF_Bidiag_form_U_blk_external( FLA_LEFT, FLA_NO_TRANSPOSE, A_orig, t, B );
break;
case 1:
{
// Time variant 1
switch( type ){
case FLA_ALG_BLOCKED:
//FLA_Apply_Q_UT( FLA_LEFT, FLA_NO_TRANSPOSE, FLA_FORWARD, FLA_COLUMNWISE, A, T, W, B );
FLA_QR_UT_form_Q( A, T, B );
//FLA_Bidiag_UT_form_U( A, T, B );
//FLA_Bidiag_UT( A_orig, T, S );
//FLA_Bidiag_UT_form_U( A_orig, T, B );
break;
}
break;
}
}
*dtime = FLA_Clock() - *dtime;
dtime_old = min( *dtime, dtime_old );
}
/*
if ( variant == 0 )
{
FLA_Copy_external( b, b_ref );
if ( FLA_Obj_is_real( A ) )
FLA_Apply_Q_blk_external( FLA_LEFT, FLA_TRANSPOSE, FLA_COLUMNWISE, A, t, b );
else
FLA_Apply_Q_blk_external( FLA_LEFT, FLA_CONJ_TRANSPOSE, FLA_COLUMNWISE, A, t, b );
FLA_Trsm_external( FLA_LEFT, FLA_UPPER_TRIANGULAR, FLA_NO_TRANSPOSE,
FLA_NONUNIT_DIAG, FLA_ONE, A, b );
FLA_Gemv_external( FLA_NO_TRANSPOSE, FLA_MINUS_ONE, A_save, b, FLA_ONE, b_ref );
FLA_Nrm2_external( b_ref, norm );
if ( FLA_Obj_is_single_precision( A ) )
*diff = *(FLA_FLOAT_PTR(norm));
else
*diff = *(FLA_DOUBLE_PTR(norm));
}
else
*/
{
FLA_Obj_set_to_identity( A );
//FLA_Obj_show( "B", B, "%8.1e %8.1e ", "" );
FLA_Gemm_external( FLA_NO_TRANSPOSE, FLA_CONJ_TRANSPOSE,
FLA_ONE, B, B, FLA_MINUS_ONE, A );
FLA_Norm_frob( A, norm );
FLA_Obj_extract_real_scalar( norm, diff );
}
/*
*gflops = 2.0 * n * n * ( m - n / 3.0 ) /
dtime_old / 1e9;
if ( FLA_Obj_is_complex( A ) )
*gflops *= 4.0;
*/
*gflops = ( 4.0 * ( 2.0 * m * n * n - 2.0 / 3.0 * n * n * n ) +
4.0 * ( 4.0 / 3.0 * m * m * m ) +
4.0 * ( 4.0 / 3.0 * n * n * n ) +
( 13.0 * 2 * m * m ) +
2.0 * ( 3.0 * 2 * m * m * m ) +
2.0 * ( 3.0 * 2 * n * n * n ) ) /
dtime_old / 1e9;
*dtime = dtime_old;
FLA_Copy_external( A_save, A );
FLA_Copy_external( A_orig_save, A_orig );
FLA_Copy_external( B_save, B );
FLA_Obj_free( &A_save );
FLA_Obj_free( &A_orig_save );
FLA_Obj_free( &B_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;
}
