FLA_Svd_uv.h File Reference

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Functions
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	FLA_Svd_uv_unb_var2 (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)

Function Documentation

FLA_Svd_uv_unb_var1()

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;
}

References FLA_Apply_diag_matrix(), FLA_Bidiag_UT(), FLA_Bidiag_UT_create_T(), FLA_Bidiag_UT_extract_real_diagonals(), FLA_Bidiag_UT_form_U(), FLA_Bidiag_UT_form_V(), FLA_Bidiag_UT_realify(), FLA_Bsvd_v_opt_var1(), FLA_Copy(), FLA_Copyr(), FLA_Gemm(), FLA_Inv_scal(), FLA_Obj_create(), FLA_Obj_datatype(), FLA_Obj_datatype_proj_to_complex(), FLA_Obj_datatype_proj_to_real(), FLA_Obj_equals(), FLA_Obj_free(), FLA_Obj_length(), FLA_Obj_min_dim(), FLA_Obj_width(), FLA_ONE, FLA_Part_1x2(), FLA_Part_2x1(), FLA_QR_UT(), FLA_QR_UT_create_T(), FLA_QR_UT_form_Q(), FLA_Scal(), FLA_Setr(), FLA_Sort_svd(), FLA_Svd_compute_scaling(), FLA_ZERO, and i.

FLA_Svd_uv_unb_var2()

FLA_Error FLA_Svd_uv_unb_var2	(	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, 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;
 
    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 );
 
    // 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 >= n_A )
    {
        if ( m_A < crossover_ratio * n_A )
        {
            // 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_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_var2( n_iter_max, d, e, G, H, RG, RH, W, 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_var2( n_iter_max, d, e, G, H, RG, RH, W, 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 );
        }
    }
    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 );
 
    // 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 );
    FLA_Obj_free( &RG );
    FLA_Obj_free( &RH );
    FLA_Obj_free( &W );
 
    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(), FLA_Bidiag_UT_form_V(), FLA_Bidiag_UT_realify(), FLA_Bsvd_v_opt_var2(), FLA_Copy(), FLA_Copyr(), FLA_Gemm(), FLA_Inv_scal(), FLA_Obj_create(), FLA_Obj_datatype(), FLA_Obj_datatype_proj_to_complex(), FLA_Obj_datatype_proj_to_real(), FLA_Obj_equals(), FLA_Obj_free(), FLA_Obj_length(), FLA_Obj_min_dim(), FLA_Obj_width(), FLA_ONE, FLA_Part_1x2(), FLA_Part_2x1(), FLA_QR_UT(), FLA_QR_UT_create_T(), FLA_QR_UT_form_Q(), FLA_Scal(), FLA_Set_to_identity(), FLA_Setr(), FLA_Sort_svd(), FLA_Svd_compute_scaling(), FLA_ZERO, and i.