FLA_Svd_ext.h File Reference

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Functions
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 V, FLA_Obj U, dim_t k_accum, dim_t b_alg)

Function Documentation

FLA_Svd_ext_u_unb_var1()

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	V,
		FLA_Obj	U,
		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;
}

References FLA_Apply_diag_matrix(), FLA_Bidiag_UT(), FLA_Bidiag_UT_create_T(), FLA_Bidiag_UT_extract_real_diagonals(), FLA_Bidiag_UT_form_U_ext(), FLA_Bidiag_UT_form_V_ext(), FLA_Bidiag_UT_realify(), FLA_Bsvd_ext_opt_var1(), FLA_Conjugate(), FLA_Copy(), FLA_Copyr(), FLA_Gemm(), FLA_Max_abs_value(), FLA_Obj_create(), FLA_Obj_create_conf_to(), FLA_Obj_datatype(), FLA_Obj_datatype_proj_to_complex(), FLA_Obj_datatype_proj_to_real(), FLA_Obj_flip_base(), FLA_Obj_flip_view(), FLA_Obj_free(), FLA_Obj_gt(), FLA_Obj_is_complex(), FLA_Obj_length(), FLA_Obj_lt(), FLA_Obj_width(), FLA_ONE, FLA_OVERFLOW_SQUARE_THRES, FLA_Part_1x2(), FLA_Part_2x1(), FLA_QR_UT(), FLA_QR_UT_create_T(), FLA_QR_UT_form_Q(), FLA_SAFE_INV_MIN, FLA_SAFE_MIN, FLA_Scal(), FLA_Setr(), FLA_UNDERFLOW_SQUARE_THRES, and FLA_ZERO.

Referenced by FLA_Svd(), and FLA_Svd_ext().