FLA_Bsvd_ext_opt_var1.c File Reference

(r)

Functions
FLA_Error	FLA_Bsvd_ext_opt_var1 (dim_t n_iter_max, FLA_Obj d, FLA_Obj e, FLA_Obj G, FLA_Obj H, FLA_Svd_type jobu, FLA_Obj U, FLA_Svd_type jobv, FLA_Obj V, FLA_Bool apply_Uh2C, FLA_Obj C, dim_t b_alg)
FLA_Error	FLA_Bsvd_ext_ops_var1 (int m_d, int m_U, int m_V, int m_C, int n_C, int n_GH, int n_iter_max, float *buff_d, int inc_d, float *buff_e, int inc_e, scomplex *buff_G, int rs_G, int cs_G, scomplex *buff_H, int rs_H, int cs_H, float *buff_U, int rs_U, int cs_U, float *buff_V, int rs_V, int cs_V, float *buff_C, int rs_C, int cs_C, int b_alg)
FLA_Error	FLA_Bsvd_ext_opd_var1 (int m_d, int m_U, int m_V, int m_C, int n_C, int n_GH, int n_iter_max, double *buff_d, int inc_d, double *buff_e, int inc_e, dcomplex *buff_G, int rs_G, int cs_G, dcomplex *buff_H, int rs_H, int cs_H, double *buff_U, int rs_U, int cs_U, double *buff_V, int rs_V, int cs_V, double *buff_C, int rs_C, int cs_C, int b_alg)
FLA_Error	FLA_Bsvd_ext_opc_var1 (int m_d, int m_U, int m_V, int m_C, int n_C, int n_GH, int n_iter_max, float *buff_d, int inc_d, float *buff_e, int inc_e, scomplex *buff_G, int rs_G, int cs_G, scomplex *buff_H, int rs_H, int cs_H, scomplex *buff_U, int rs_U, int cs_U, scomplex *buff_V, int rs_V, int cs_V, scomplex *buff_C, int rs_C, int cs_C, int b_alg)
FLA_Error	FLA_Bsvd_ext_opz_var1 (int m_d, int m_U, int m_V, int m_C, int n_C, int n_GH, int n_iter_max, double *buff_d, int inc_d, double *buff_e, int inc_e, dcomplex *buff_G, int rs_G, int cs_G, dcomplex *buff_H, int rs_H, int cs_H, dcomplex *buff_U, int rs_U, int cs_U, dcomplex *buff_V, int rs_V, int cs_V, dcomplex *buff_C, int rs_C, int cs_C, int b_alg)

Function Documentation

FLA_Bsvd_ext_opc_var1()

FLA_Error FLA_Bsvd_ext_opc_var1	(	int	m_d,
		int	m_U,
		int	m_V,
		int	m_C,
		int	n_C,
		int	n_GH,
		int	n_iter_max,
		float *	buff_d,
		int	inc_d,
		float *	buff_e,
		int	inc_e,
		scomplex *	buff_G,
		int	rs_G,
		int	cs_G,
		scomplex *	buff_H,
		int	rs_H,
		int	cs_H,
		scomplex *	buff_U,
		int	rs_U,
		int	cs_U,
		scomplex *	buff_V,
		int	rs_V,
		int	cs_V,
		scomplex *	buff_C,
		int	rs_C,
		int	cs_C,
		int	b_alg
	)

{
    scomplex  one        = bl1_c1();
    float     rzero      = bl1_s0();
 
    int       maxitr     = 6;
 
    float     eps;
    float     tolmul;
    float     tol;
    float     thresh;
 
    scomplex* G;
    scomplex* H;
    float*    d1;
    float*    e1;
    int       r_val;
    int       done;
    int       m_GH_sweep_max;
    int       ij_begin;
    int       ijTL, ijBR;
    int       m_A11;
    int       n_iter_perf;
    int       n_UV_apply;
    int       total_deflations;
    int       n_deflations;
    int       n_iter_prev;
    int       n_iter_perf_sweep_max;
 
    // Compute some convergence constants.
    eps    = FLA_Mach_params_ops( FLA_MACH_EPS );
    tolmul = max( 10.0F, min( 100.0F, powf( eps, -0.125F ) ) );
    FLA_Bsvd_compute_tol_thresh_ops( m_d,
                                     tolmul,
                                     maxitr,
                                     buff_d, inc_d,
                                     buff_e, inc_e,
                                     &tol,
                                     &thresh );
 
    // Initialize our completion flag.
    done = FALSE;
 
    // Initialize a counter that holds the maximum number of rows of G
    // and H that we would need to initialize for the next sweep.
    m_GH_sweep_max = m_d - 1;
 
    // Initialize a counter for the total number of iterations performed.
    n_iter_prev = 0;
 
    // Iterate until the matrix has completely deflated.
    for ( total_deflations = 0; done != TRUE; )
    {
 
        // Initialize G and H to contain only identity rotations.
        bl1_csetm( m_GH_sweep_max,
                   n_GH,
                   &one,
                   buff_G, rs_G, cs_G );
        bl1_csetm( m_GH_sweep_max,
                   n_GH,
                   &one,
                   buff_H, rs_H, cs_H );
 
        // Keep track of the maximum number of iterations performed in the
        // current sweep. This is used when applying the sweep's Givens
        // rotations.
        n_iter_perf_sweep_max = 0;
 
        // Perform a sweep: Move through the matrix and perform a bidiagonal
        // SVD on each non-zero submatrix that is encountered. During the
        // first time through, ijTL will be 0 and ijBR will be m_d - 1.
        for ( ij_begin = 0; ij_begin < m_d;  )
        {
            // Search for the first submatrix along the diagonal that is
            // bounded by zeroes (or endpoints of the matrix). If no
            // submatrix is found (ie: if the entire superdiagonal is zero
            // then FLA_FAILURE is returned. This function also inspects
            // superdiagonal elements for proximity to zero. If a given
            // element is close enough to zero, then it is deemed
            // converged and manually set to zero.
            r_val = FLA_Bsvd_find_submatrix_ops( m_d,
                                                 ij_begin,
                                                 buff_d, inc_d,
                                                 buff_e, inc_e,
                                                 &ijTL,
                                                 &ijBR );
 
            // Verify that a submatrix was found. If one was not found,
            // then we are done with the current sweep. Furthermore, if
            // a submatrix was not found AND we began our search at the
            // beginning of the matrix (ie: ij_begin == 0), then the
            // matrix has completely deflated and so we are done with
            // Francis step iteration.
            if ( r_val == FLA_FAILURE )
            {
                if ( ij_begin == 0 )
                {
                    done = TRUE;
                }
 
                // Break out of the current sweep so we can apply the last
                // remaining Givens rotations.
                break;
            }
 
            // If we got this far, then:
            //   (a) ijTL refers to the index of the first non-zero
            //       superdiagonal along the diagonal, and
            //   (b) ijBR refers to either:
            //       - the first zero element that occurs after ijTL, or
            //       - the the last diagonal element.
            // Note that ijTL and ijBR also correspond to the first and
            // last diagonal elements of the submatrix of interest. Thus,
            // we may compute the dimension of this submatrix as:
            m_A11 = ijBR - ijTL + 1;
 
            // Adjust ij_begin, which gets us ready for the next submatrix
            // search in the current sweep.
            ij_begin = ijBR + 1;
 
            // Index to the submatrices upon which we will operate.
            d1 = buff_d + ijTL * inc_d;
            e1 = buff_e + ijTL * inc_e;
            G  = buff_G + ijTL * rs_G;
            H  = buff_H + ijTL * rs_H;
 
            // Search for a batch of singular values, recursing on deflated
            // subproblems whenever a split occurs. Iteration continues as
            // long as
            //   (a) there is still matrix left to operate on, and
            //   (b) the number of iterations performed in this batch is
            //       less than n_G.
            // If/when either of the two above conditions fails to hold,
            // the function returns.
            n_deflations = FLA_Bsvd_iteracc_v_ops_var1( m_A11,
                                                        n_GH,
                                                        ijTL,
                                                        tol,
                                                        thresh,
                                                        d1, inc_d,
                                                        e1, inc_e,
                                                        G,  rs_G, cs_G,
                                                        H,  rs_H, cs_H,
                                                        &n_iter_perf );
            
            // Record the number of deflations that were observed.
            total_deflations += n_deflations;
 
            // Update the maximum number of iterations performed in the
            // current sweep.
            n_iter_perf_sweep_max = max( n_iter_perf_sweep_max, n_iter_perf );
 
            // Store the most recent value of ijBR in m_G_sweep_max.
            // When the sweep is done, this value will contain the minimum
            // number of rows of G and H we can apply and safely include all
            // non-identity rotations that were computed during the
            // singular value searches.
            m_GH_sweep_max = ijBR;
 
            // Make sure we haven't exceeded our maximum iteration count.
            if ( n_iter_prev >= n_iter_max * m_d )
            {
                FLA_Abort();
                //return FLA_FAILURE;
            }
        }
 
        // The sweep is complete. Now we must apply the Givens rotations
        // that were accumulated during the sweep.
 
        // Recall that the number of columns of U and V to which we apply
        // rotations is one more than the number of rotations.
        n_UV_apply = m_GH_sweep_max + 1;
 
        // Apply the Givens rotations. Note that we only apply k sets of
        // rotations, where k = n_iter_perf_sweep_max. Also note that we only
        // apply to n_UV_apply columns of U and V since this is the most we
        // need to touch given the most recent value stored to m_GH_sweep_max.
 
        if ( buff_U != NULL ) // ( rs_U == 0 && cs_U == 0 ) // if U is requested
          FLA_Apply_G_rf_blc_var3( n_iter_perf_sweep_max,
                                   m_U,
                                   n_UV_apply,
                                   buff_G, rs_G, cs_G,
                                   buff_U, rs_U, cs_U,
                                   b_alg );
        
        if ( buff_V != NULL ) // ( rs_V == 0 && cs_V == 0 ) // if V is requested
          FLA_Apply_G_rf_blc_var3( n_iter_perf_sweep_max,
                                   m_V,
                                   n_UV_apply,
                                   buff_H, rs_H, cs_H,
                                   buff_V, rs_V, cs_V,
                                   b_alg );
                
        // lf with non-flipped cs/rs m/n should be used; 
        // but the same operation can be achieved by using rf
        // ( U^H C )^H = C^H U where U is G_0 G_1 G_2 .... G_{k-1}
        if ( buff_C != NULL ) // ( rs_C == 0 && cs_C == 0 ) // if C is requested, flip
        {
          FLA_Apply_G_rf_blc_var3( n_iter_perf_sweep_max,
                                   n_C,
                                   m_C,
                                   buff_G, rs_G, cs_G,
                                   buff_C, cs_C, rs_C,
                                   b_alg );
        }
 
        // Increment the total number of iterations previously performed.
        n_iter_prev += n_iter_perf_sweep_max;
    }
 
    // Make all the singular values positive.
    {
        int    i;
        float  minus_one = bl1_sm1();
 
        for ( i = 0; i < m_d; ++i )
        {
            if ( buff_d[ (i  )*inc_d ] < rzero )
            {
                buff_d[ (i  )*inc_d ] = -buff_d[ (i  )*inc_d ];
 
                // Scale the right singular vectors.
                if ( buff_V != NULL )
                  bl1_csscalv( BLIS1_NO_CONJUGATE,
                               m_V,
                               &minus_one,
                               buff_V + (i  )*cs_V, rs_V );
            }
        }
    }
 
    return FLA_SUCCESS;
}

References bl1_c1(), bl1_csetm(), bl1_csscalv(), bl1_s0(), bl1_sm1(), BLIS1_NO_CONJUGATE, FLA_Abort(), FLA_Apply_G_rf_blc_var3(), FLA_Bsvd_compute_tol_thresh_ops(), FLA_Bsvd_find_submatrix_ops(), FLA_Bsvd_iteracc_v_ops_var1(), FLA_Mach_params_ops(), and i.

Referenced by FLA_Bsvd_ext_opt_var1().

FLA_Bsvd_ext_opd_var1()

FLA_Error FLA_Bsvd_ext_opd_var1	(	int	m_d,
		int	m_U,
		int	m_V,
		int	m_C,
		int	n_C,
		int	n_GH,
		int	n_iter_max,
		double *	buff_d,
		int	inc_d,
		double *	buff_e,
		int	inc_e,
		dcomplex *	buff_G,
		int	rs_G,
		int	cs_G,
		dcomplex *	buff_H,
		int	rs_H,
		int	cs_H,
		double *	buff_U,
		int	rs_U,
		int	cs_U,
		double *	buff_V,
		int	rs_V,
		int	cs_V,
		double *	buff_C,
		int	rs_C,
		int	cs_C,
		int	b_alg
	)

{
    dcomplex  one        = bl1_z1();
    double    rzero      = bl1_d0();
 
    int       maxitr     = 6;
 
    double    eps;
    double    tolmul;
    double    tol;
    double    thresh;
 
    dcomplex* G;
    dcomplex* H;
    double*   d1;
    double*   e1;
    int       r_val;
    int       done;
    int       m_GH_sweep_max;
    int       ij_begin;
    int       ijTL, ijBR;
    int       m_A11;
    int       n_iter_perf;
    int       n_UV_apply;
    int       total_deflations;
    int       n_deflations;
    int       n_iter_prev;
    int       n_iter_perf_sweep_max;
 
    // Compute some convergence constants.
    eps    = FLA_Mach_params_opd( FLA_MACH_EPS );
    tolmul = max( 10.0, min( 100.0, pow( eps, -0.125 ) ) );
    FLA_Bsvd_compute_tol_thresh_opd( m_d,
                                     tolmul,
                                     maxitr,
                                     buff_d, inc_d,
                                     buff_e, inc_e,
                                     &tol,
                                     &thresh );
 
    // Initialize our completion flag.
    done = FALSE;
 
    // Initialize a counter that holds the maximum number of rows of G
    // and H that we would need to initialize for the next sweep.
    m_GH_sweep_max = m_d - 1;
 
    // Initialize a counter for the total number of iterations performed.
    n_iter_prev = 0;
 
    // Iterate until the matrix has completely deflated.
    for ( total_deflations = 0; done != TRUE; )
    {
 
        // Initialize G and H to contain only identity rotations.
        bl1_zsetm( m_GH_sweep_max,
                   n_GH,
                   &one,
                   buff_G, rs_G, cs_G );
        bl1_zsetm( m_GH_sweep_max,
                   n_GH,
                   &one,
                   buff_H, rs_H, cs_H );
 
        // Keep track of the maximum number of iterations performed in the
        // current sweep. This is used when applying the sweep's Givens
        // rotations.
        n_iter_perf_sweep_max = 0;
 
        // Perform a sweep: Move through the matrix and perform a bidiagonal
        // SVD on each non-zero submatrix that is encountered. During the
        // first time through, ijTL will be 0 and ijBR will be m_d - 1.
        for ( ij_begin = 0; ij_begin < m_d;  )
        {
            // Search for the first submatrix along the diagonal that is
            // bounded by zeroes (or endpoints of the matrix). If no
            // submatrix is found (ie: if the entire superdiagonal is zero
            // then FLA_FAILURE is returned. This function also inspects
            // superdiagonal elements for proximity to zero. If a given
            // element is close enough to zero, then it is deemed
            // converged and manually set to zero.
            r_val = FLA_Bsvd_find_submatrix_opd( m_d,
                                                 ij_begin,
                                                 buff_d, inc_d,
                                                 buff_e, inc_e,
                                                 &ijTL,
                                                 &ijBR );
 
            // Verify that a submatrix was found. If one was not found,
            // then we are done with the current sweep. Furthermore, if
            // a submatrix was not found AND we began our search at the
            // beginning of the matrix (ie: ij_begin == 0), then the
            // matrix has completely deflated and so we are done with
            // Francis step iteration.
            if ( r_val == FLA_FAILURE )
            {
                if ( ij_begin == 0 )
                {
                    done = TRUE;
                }
 
                // Break out of the current sweep so we can apply the last
                // remaining Givens rotations.
                break;
            }
 
            // If we got this far, then:
            //   (a) ijTL refers to the index of the first non-zero
            //       superdiagonal along the diagonal, and
            //   (b) ijBR refers to either:
            //       - the first zero element that occurs after ijTL, or
            //       - the the last diagonal element.
            // Note that ijTL and ijBR also correspond to the first and
            // last diagonal elements of the submatrix of interest. Thus,
            // we may compute the dimension of this submatrix as:
            m_A11 = ijBR - ijTL + 1;
 
            // Adjust ij_begin, which gets us ready for the next submatrix
            // search in the current sweep.
            ij_begin = ijBR + 1;
 
            // Index to the submatrices upon which we will operate.
            d1 = buff_d + ijTL * inc_d;
            e1 = buff_e + ijTL * inc_e;
            G  = buff_G + ijTL * rs_G;
            H  = buff_H + ijTL * rs_H;
 
            // Search for a batch of singular values, recursing on deflated
            // subproblems whenever a split occurs. Iteration continues as
            // long as
            //   (a) there is still matrix left to operate on, and
            //   (b) the number of iterations performed in this batch is
            //       less than n_G.
            // If/when either of the two above conditions fails to hold,
            // the function returns.
            n_deflations = FLA_Bsvd_iteracc_v_opd_var1( m_A11,
                                                        n_GH,
                                                        ijTL,
                                                        tol,
                                                        thresh,
                                                        d1, inc_d,
                                                        e1, inc_e,
                                                        G,  rs_G, cs_G,
                                                        H,  rs_H, cs_H,
                                                        &n_iter_perf );
 
            // Record the number of deflations that were observed.
            total_deflations += n_deflations;
 
            // Update the maximum number of iterations performed in the
            // current sweep.
            n_iter_perf_sweep_max = max( n_iter_perf_sweep_max, n_iter_perf );
 
            // Store the most recent value of ijBR in m_G_sweep_max.
            // When the sweep is done, this value will contain the minimum
            // number of rows of G and H we can apply and safely include all
            // non-identity rotations that were computed during the
            // singular value searches.
            m_GH_sweep_max = ijBR;
 
            // Make sure we haven't exceeded our maximum iteration count.
            if ( n_iter_prev >= n_iter_max * m_d )
            {
                FLA_Abort();
                //return FLA_FAILURE;
            }
        }
 
        // The sweep is complete. Now we must apply the Givens rotations
        // that were accumulated during the sweep.
 
        // Recall that the number of columns of U and V to which we apply
        // rotations is one more than the number of rotations.
        n_UV_apply = m_GH_sweep_max + 1;
 
        // Apply the Givens rotations. Note that we only apply k sets of
        // rotations, where k = n_iter_perf_sweep_max. Also note that we only
        // apply to n_UV_apply columns of U and V since this is the most we
        // need to touch given the most recent value stored to m_GH_sweep_max.
        
        if ( buff_U != NULL ) // ( rs_U == 0 && cs_U == 0 ) // if U is requested
          FLA_Apply_G_rf_bld_var3( n_iter_perf_sweep_max,
                                   m_U,
                                   n_UV_apply,
                                   buff_G, rs_G, cs_G,
                                   buff_U, rs_U, cs_U,
                                   b_alg );
 
        if ( buff_V != NULL ) // ( rs_V == 0 && cs_V == 0 ) // if V is requested
          FLA_Apply_G_rf_bld_var3( n_iter_perf_sweep_max,
                                   m_V,
                                   n_UV_apply,
                                   buff_H, rs_H, cs_H,
                                   buff_V, rs_V, cs_V,
                                   b_alg );
        
        
        // lf with non-flipped cs/rs m/n should be used; 
        // but the same operation can be achieved by using rf
        // ( U^H C )^H = C^H U where U is G_0 G_1 G_2 .... G_{k-1}
        if ( buff_C != NULL ) // ( rs_C == 0 && cs_C == 0 ) // if C is requested, flip
        {
          FLA_Apply_G_rf_bld_var3( n_iter_perf_sweep_max,
                                   n_C,
                                   m_C,
                                   buff_G, rs_G, cs_G,
                                   buff_C, cs_C, rs_C,
                                   b_alg );
        }
 
        // Increment the total number of iterations previously performed.
        n_iter_prev += n_iter_perf_sweep_max;
    }
 
    // Make all the singular values positive.
    {
        int    i;
        double minus_one = bl1_dm1();
 
        for ( i = 0; i < m_d; ++i )
        {
            if ( buff_d[ (i  )*inc_d ] < rzero )
            {
                buff_d[ (i  )*inc_d ] = -buff_d[ (i  )*inc_d ];
 
                // Scale the right singular vectors.
                if ( buff_V != NULL )
                  bl1_dscalv( BLIS1_NO_CONJUGATE,
                              m_V,
                              &minus_one,
                              buff_V + (i  )*cs_V, rs_V );
            }
        }
    }
 
    return n_iter_prev;
}

References bl1_d0(), bl1_dm1(), bl1_dscalv(), bl1_z1(), bl1_zsetm(), BLIS1_NO_CONJUGATE, FLA_Abort(), FLA_Apply_G_rf_bld_var3(), FLA_Bsvd_compute_tol_thresh_opd(), FLA_Bsvd_find_submatrix_opd(), FLA_Bsvd_iteracc_v_opd_var1(), FLA_Mach_params_opd(), and i.

Referenced by FLA_Bsvd_ext_opt_var1().

FLA_Bsvd_ext_ops_var1()

FLA_Error FLA_Bsvd_ext_ops_var1	(	int	m_d,
		int	m_U,
		int	m_V,
		int	m_C,
		int	n_C,
		int	n_GH,
		int	n_iter_max,
		float *	buff_d,
		int	inc_d,
		float *	buff_e,
		int	inc_e,
		scomplex *	buff_G,
		int	rs_G,
		int	cs_G,
		scomplex *	buff_H,
		int	rs_H,
		int	cs_H,
		float *	buff_U,
		int	rs_U,
		int	cs_U,
		float *	buff_V,
		int	rs_V,
		int	cs_V,
		float *	buff_C,
		int	rs_C,
		int	cs_C,
		int	b_alg
	)

{
    scomplex  one        = bl1_c1();
    float     rzero      = bl1_s0();
 
    int       maxitr     = 6;
 
    float     eps;
    float     tolmul;
    float     tol;
    float     thresh;
 
    scomplex* G;
    scomplex* H;
    float*    d1;
    float*    e1;
    int       r_val;
    int       done;
    int       m_GH_sweep_max;
    int       ij_begin;
    int       ijTL, ijBR;
    int       m_A11;
    int       n_iter_perf;
    int       n_UV_apply;
    int       total_deflations;
    int       n_deflations;
    int       n_iter_prev;
    int       n_iter_perf_sweep_max;
 
    // Compute some convergence constants.
    eps    = FLA_Mach_params_ops( FLA_MACH_EPS );
    tolmul = max( 10.0F, min( 100.0F, powf( eps, -0.125F ) ) );
    FLA_Bsvd_compute_tol_thresh_ops( m_d,
                                     tolmul,
                                     maxitr,
                                     buff_d, inc_d,
                                     buff_e, inc_e,
                                     &tol,
                                     &thresh );
 
    // Initialize our completion flag.
    done = FALSE;
 
    // Initialize a counter that holds the maximum number of rows of G
    // and H that we would need to initialize for the next sweep.
    m_GH_sweep_max = m_d - 1;
 
    // Initialize a counter for the total number of iterations performed.
    n_iter_prev = 0;
 
    // Iterate until the matrix has completely deflated.
    for ( total_deflations = 0; done != TRUE; )
    {
 
        // Initialize G and H to contain only identity rotations.
        bl1_csetm( m_GH_sweep_max,
                   n_GH,
                   &one,
                   buff_G, rs_G, cs_G );
        bl1_csetm( m_GH_sweep_max,
                   n_GH,
                   &one,
                   buff_H, rs_H, cs_H );
 
        // Keep track of the maximum number of iterations performed in the
        // current sweep. This is used when applying the sweep's Givens
        // rotations.
        n_iter_perf_sweep_max = 0;
 
        // Perform a sweep: Move through the matrix and perform a bidiagonal
        // SVD on each non-zero submatrix that is encountered. During the
        // first time through, ijTL will be 0 and ijBR will be m_d - 1.
        for ( ij_begin = 0; ij_begin < m_d;  )
        {
 
            // Search for the first submatrix along the diagonal that is
            // bounded by zeroes (or endpoints of the matrix). If no
            // submatrix is found (ie: if the entire superdiagonal is zero
            // then FLA_FAILURE is returned. This function also inspects
            // superdiagonal elements for proximity to zero. If a given
            // element is close enough to zero, then it is deemed
            // converged and manually set to zero.
            r_val = FLA_Bsvd_find_submatrix_ops( m_d,
                                                 ij_begin,
                                                 buff_d, inc_d,
                                                 buff_e, inc_e,
                                                 &ijTL,
                                                 &ijBR );
 
            // Verify that a submatrix was found. If one was not found,
            // then we are done with the current sweep. Furthermore, if
            // a submatrix was not found AND we began our search at the
            // beginning of the matrix (ie: ij_begin == 0), then the
            // matrix has completely deflated and so we are done with
            // Francis step iteration.
            if ( r_val == FLA_FAILURE )
            {
                if ( ij_begin == 0 )
                {
                    done = TRUE;
                }
 
                // Break out of the current sweep so we can apply the last
                // remaining Givens rotations.
                break;
            }
 
            // If we got this far, then:
            //   (a) ijTL refers to the index of the first non-zero
            //       superdiagonal along the diagonal, and
            //   (b) ijBR refers to either:
            //       - the first zero element that occurs after ijTL, or
            //       - the the last diagonal element.
            // Note that ijTL and ijBR also correspond to the first and
            // last diagonal elements of the submatrix of interest. Thus,
            // we may compute the dimension of this submatrix as:
            m_A11 = ijBR - ijTL + 1;
 
            // Adjust ij_begin, which gets us ready for the next submatrix
            // search in the current sweep.
            ij_begin = ijBR + 1;
 
            // Index to the submatrices upon which we will operate.
            d1 = buff_d + ijTL * inc_d;
            e1 = buff_e + ijTL * inc_e;
            G  = buff_G + ijTL * rs_G;
            H  = buff_H + ijTL * rs_H;
 
            // Search for a batch of singular values, recursing on deflated
            // subproblems whenever a split occurs. Iteration continues as
            // long as
            //   (a) there is still matrix left to operate on, and
            //   (b) the number of iterations performed in this batch is
            //       less than n_G.
            // If/when either of the two above conditions fails to hold,
            // the function returns.
            n_deflations = FLA_Bsvd_iteracc_v_ops_var1( m_A11,
                                                        n_GH,
                                                        ijTL,
                                                        tol,
                                                        thresh,
                                                        d1, inc_d,
                                                        e1, inc_e,
                                                        G,  rs_G, cs_G,
                                                        H,  rs_H, cs_H,
                                                        &n_iter_perf );
            
            // Record the number of deflations that were observed.
            total_deflations += n_deflations;
 
            // Update the maximum number of iterations performed in the
            // current sweep.
            n_iter_perf_sweep_max = max( n_iter_perf_sweep_max, n_iter_perf );
 
            // Store the most recent value of ijBR in m_G_sweep_max.
            // When the sweep is done, this value will contain the minimum
            // number of rows of G and H we can apply and safely include all
            // non-identity rotations that were computed during the
            // singular value searches.
            m_GH_sweep_max = ijBR;
 
            // Make sure we haven't exceeded our maximum iteration count.
            if ( n_iter_prev >= n_iter_max * m_d )
            {
                //FLA_Abort();
                return n_iter_prev;
            }
        }
 
        // The sweep is complete. Now we must apply the Givens rotations
        // that were accumulated during the sweep.
 
        // Recall that the number of columns of U and V to which we apply
        // rotations is one more than the number of rotations.
        n_UV_apply = m_GH_sweep_max + 1;
 
        // Apply the Givens rotations. Note that we only apply k sets of
        // rotations, where k = n_iter_perf_sweep_max. Also note that we only
        // apply to n_UV_apply columns of U and V since this is the most we
        // need to touch given the most recent value stored to m_GH_sweep_max.
 
        if ( buff_U != NULL ) // ( rs_U == 0 && cs_U == 0 ) // if U is requested
          FLA_Apply_G_rf_bls_var3( n_iter_perf_sweep_max,
                                   m_U,
                                   n_UV_apply,
                                   buff_G, rs_G, cs_G,
                                   buff_U, rs_U, cs_U,
                                   b_alg );
 
        if ( buff_V != NULL ) // ( rs_V == 0 && cs_V == 0 ) // if V is requested
          FLA_Apply_G_rf_bls_var3( n_iter_perf_sweep_max,
                                   m_V,
                                   n_UV_apply,
                                   buff_H, rs_H, cs_H,
                                   buff_V, rs_V, cs_V,
                                   b_alg );
        
        // lf with non-flipped cs/rs m/n should be used; 
        // but the same operation can be achieved by using rf
        // ( U^H C )^H = C^H U where U is G_0 G_1 G_2 .... G_{k-1}
        if ( buff_C != NULL ) // ( rs_C == 0 && cs_C == 0 ) // if C is requested, flip
        {
          FLA_Apply_G_rf_bls_var3( n_iter_perf_sweep_max,
                                   n_C,
                                   m_C,
                                   buff_G, rs_G, cs_G,
                                   buff_C, cs_C, rs_C,
                                   b_alg );
        }
 
        // Increment the total number of iterations previously performed.
        n_iter_prev += n_iter_perf_sweep_max;
 
    }
 
    // Make all the singular values positive.
    {
        int    i;
        float  minus_one = bl1_sm1();
 
        for ( i = 0; i < m_d; ++i )
        {
            if ( buff_d[ (i  )*inc_d ] < rzero )
            {
                buff_d[ (i  )*inc_d ] = -buff_d[ (i  )*inc_d ];
 
                // Scale the right singular vectors.
                if ( buff_V != NULL )
                  bl1_sscalv( BLIS1_NO_CONJUGATE,
                              m_V,
                              &minus_one,
                              buff_V + (i  )*cs_V, rs_V );
            }
        }
    }
 
    return n_iter_prev;
}

References bl1_c1(), bl1_csetm(), bl1_s0(), bl1_sm1(), bl1_sscalv(), BLIS1_NO_CONJUGATE, FLA_Apply_G_rf_bls_var3(), FLA_Bsvd_compute_tol_thresh_ops(), FLA_Bsvd_find_submatrix_ops(), FLA_Bsvd_iteracc_v_ops_var1(), FLA_Mach_params_ops(), and i.

Referenced by FLA_Bsvd_ext_opt_var1().

FLA_Bsvd_ext_opt_var1()

FLA_Error FLA_Bsvd_ext_opt_var1	(	dim_t	n_iter_max,
		FLA_Obj	d,
		FLA_Obj	e,
		FLA_Obj	G,
		FLA_Obj	H,
		FLA_Svd_type	jobu,
		FLA_Obj	U,
		FLA_Svd_type	jobv,
		FLA_Obj	V,
		FLA_Bool	apply_Uh2C,
		FLA_Obj	C,
		dim_t	b_alg
	)

{
    FLA_Error    r_val = FLA_SUCCESS;
    FLA_Datatype datatype;
    int          m_U, m_V, m_C, n_C, n_GH;
    int          m_d, inc_d;
    int          inc_e;
    int          rs_G, cs_G;
    int          rs_H, cs_H;
    int          rs_U, cs_U;
    int          rs_V, cs_V;
    int          rs_C, cs_C;
 
    if      ( jobu != FLA_SVD_VECTORS_NONE )
      datatype = FLA_Obj_datatype( U );
    else if ( jobv != FLA_SVD_VECTORS_NONE )
      datatype = FLA_Obj_datatype( V );
    else if ( apply_Uh2C == TRUE )
      datatype = FLA_Obj_datatype( C );
    else 
      datatype = FLA_Obj_datatype( d );
 
    m_d        = FLA_Obj_vector_dim( d );
    inc_d      = FLA_Obj_vector_inc( d );
    inc_e      = FLA_Obj_vector_inc( e );
 
    n_GH       = FLA_Obj_width( G );
 
    rs_G       = FLA_Obj_row_stride( G );
    cs_G       = FLA_Obj_col_stride( G );
 
    rs_H       = FLA_Obj_row_stride( H );
    cs_H       = FLA_Obj_col_stride( H );
 
    if ( jobu == FLA_SVD_VECTORS_NONE )
    {
        m_U    = 0;
        rs_U   = 0;
        cs_U   = 0;
    }
    else
    {
        m_U    = FLA_Obj_length( U );
        rs_U   = FLA_Obj_row_stride( U );
        cs_U   = FLA_Obj_col_stride( U );
    }
    if ( jobv == FLA_SVD_VECTORS_NONE )
    {
        m_V    = 0;
        rs_V   = 0;
        cs_V   = 0;
    }
    else
    {
        m_V    = FLA_Obj_length( V );
        rs_V   = FLA_Obj_row_stride( V );
        cs_V   = FLA_Obj_col_stride( V );
    }
    if ( apply_Uh2C == FALSE )
    {
        m_C    = 0;
        n_C    = 0;
        rs_C   = 0;
        cs_C   = 0;
    }
    else
    {
        m_C    = FLA_Obj_length( C );
        n_C    = FLA_Obj_width( C );
        rs_C   = FLA_Obj_row_stride( C );
        cs_C   = FLA_Obj_col_stride( C );
    }
 
    switch ( datatype )
    {
    case FLA_FLOAT:
    {
        float*    buff_d = FLA_FLOAT_PTR( d );
        float*    buff_e = FLA_FLOAT_PTR( e );
        scomplex* buff_G = FLA_COMPLEX_PTR( G );
        scomplex* buff_H = FLA_COMPLEX_PTR( H );
        float*    buff_U = ( jobu == FLA_SVD_VECTORS_NONE ? NULL : FLA_FLOAT_PTR( U ) );
        float*    buff_V = ( jobv == FLA_SVD_VECTORS_NONE ? NULL : FLA_FLOAT_PTR( V ) );
        float*    buff_C = ( apply_Uh2C == FALSE          ? NULL : FLA_FLOAT_PTR( C ) );
 
        r_val = FLA_Bsvd_ext_ops_var1( m_d,
                                       m_U,
                                       m_V,
                                       m_C,
                                       n_C,
                                       n_GH,
                                       n_iter_max,
                                       buff_d, inc_d,
                                       buff_e, inc_e,
                                       buff_G, rs_G, cs_G,
                                       buff_H, rs_H, cs_H,
                                       buff_U, rs_U, cs_U,
                                       buff_V, rs_V, cs_V,
                                       buff_C, rs_C, cs_C,
                                       b_alg );
 
        FLA_Sort_bsvd_ext_b_ops( m_d, buff_d, inc_d,
                                 m_U, buff_U, rs_U, cs_U,
                                 m_V, buff_V, rs_V, cs_V,
                                 n_C, buff_C, rs_C, cs_C );
        break;
    }
 
    case FLA_DOUBLE:
    {
        double*   buff_d = FLA_DOUBLE_PTR( d );
        double*   buff_e = FLA_DOUBLE_PTR( e );
        dcomplex* buff_G = FLA_DOUBLE_COMPLEX_PTR( G );
        dcomplex* buff_H = FLA_DOUBLE_COMPLEX_PTR( H );
        double*   buff_U = ( jobu == FLA_SVD_VECTORS_NONE ? NULL : FLA_DOUBLE_PTR( U ) );
        double*   buff_V = ( jobv == FLA_SVD_VECTORS_NONE ? NULL : FLA_DOUBLE_PTR( V ) );
        double*   buff_C = ( apply_Uh2C == FALSE          ? NULL : FLA_DOUBLE_PTR( C ) );
 
        r_val = FLA_Bsvd_ext_opd_var1( m_d,
                                       m_U,
                                       m_V,
                                       m_C,
                                       n_C,
                                       n_GH,
                                       n_iter_max,
                                       buff_d, inc_d,
                                       buff_e, inc_e,
                                       buff_G, rs_G, cs_G,
                                       buff_H, rs_H, cs_H,
                                       buff_U, rs_U, cs_U,
                                       buff_V, rs_V, cs_V,
                                       buff_C, rs_C, cs_C,
                                       b_alg );
 
        FLA_Sort_bsvd_ext_b_opd( m_d, buff_d, inc_d,
                                 m_U, buff_U, rs_U, cs_U,
                                 m_V, buff_V, rs_V, cs_V,
                                 n_C, buff_C, rs_C, cs_C );
        break;
    }
 
    case FLA_COMPLEX:
    {
        float*    buff_d = FLA_FLOAT_PTR( d );
        float*    buff_e = FLA_FLOAT_PTR( e );
        scomplex* buff_G = FLA_COMPLEX_PTR( G );
        scomplex* buff_H = FLA_COMPLEX_PTR( H );
        scomplex* buff_U = ( jobu == FLA_SVD_VECTORS_NONE ? NULL : FLA_COMPLEX_PTR( U ) );
        scomplex* buff_V = ( jobv == FLA_SVD_VECTORS_NONE ? NULL : FLA_COMPLEX_PTR( V ) );
        scomplex* buff_C = ( apply_Uh2C == FALSE          ? NULL : FLA_COMPLEX_PTR( C ) );
 
        r_val = FLA_Bsvd_ext_opc_var1( m_d,
                                       m_U,
                                       m_V,
                                       m_C,
                                       n_C,
                                       n_GH,
                                       n_iter_max,
                                       buff_d, inc_d,
                                       buff_e, inc_e,
                                       buff_G, rs_G, cs_G,
                                       buff_H, rs_H, cs_H,
                                       buff_U, rs_U, cs_U,
                                       buff_V, rs_V, cs_V,
                                       buff_C, rs_C, cs_C,
                                       b_alg );
 
        FLA_Sort_bsvd_ext_b_opc( m_d, buff_d, inc_d,
                                 m_U, buff_U, rs_U, cs_U,
                                 m_V, buff_V, rs_V, cs_V,
                                 n_C, buff_C, rs_C, cs_C );        
        break;
    }
 
    case FLA_DOUBLE_COMPLEX:
    {
        double*   buff_d = FLA_DOUBLE_PTR( d );
        double*   buff_e = FLA_DOUBLE_PTR( e );
        dcomplex* buff_G = FLA_DOUBLE_COMPLEX_PTR( G );
        dcomplex* buff_H = FLA_DOUBLE_COMPLEX_PTR( H );
        dcomplex* buff_U = ( jobu == FLA_SVD_VECTORS_NONE ? NULL : FLA_DOUBLE_COMPLEX_PTR( U ) );
        dcomplex* buff_V = ( jobv == FLA_SVD_VECTORS_NONE ? NULL : FLA_DOUBLE_COMPLEX_PTR( V ) );
        dcomplex* buff_C = ( apply_Uh2C == FALSE          ? NULL : FLA_DOUBLE_COMPLEX_PTR( C ) );
 
        r_val = FLA_Bsvd_ext_opz_var1( m_d,
                                       m_U,
                                       m_V,
                                       m_C,
                                       n_C,
                                       n_GH,
                                       n_iter_max,
                                       buff_d, inc_d,
                                       buff_e, inc_e,
                                       buff_G, rs_G, cs_G,
                                       buff_H, rs_H, cs_H,
                                       buff_U, rs_U, cs_U,
                                       buff_V, rs_V, cs_V,
                                       buff_C, rs_C, cs_C,
                                       b_alg );
 
        FLA_Sort_bsvd_ext_b_opz( m_d, buff_d, inc_d,
                                 m_U, buff_U, rs_U, cs_U,
                                 m_V, buff_V, rs_V, cs_V,
                                 n_C, buff_C, rs_C, cs_C );
        break;
    }
    }
 
    return r_val;
}

References FLA_Bsvd_ext_opc_var1(), FLA_Bsvd_ext_opd_var1(), FLA_Bsvd_ext_ops_var1(), FLA_Bsvd_ext_opz_var1(), FLA_Obj_col_stride(), FLA_Obj_datatype(), FLA_Obj_length(), FLA_Obj_row_stride(), FLA_Obj_vector_dim(), FLA_Obj_vector_inc(), FLA_Obj_width(), FLA_Sort_bsvd_ext_b_opc(), FLA_Sort_bsvd_ext_b_opd(), FLA_Sort_bsvd_ext_b_ops(), FLA_Sort_bsvd_ext_b_opz(), and i.

Referenced by FLA_Bsvd(), FLA_Bsvd_ext(), and FLA_Svd_ext_u_unb_var1().

FLA_Bsvd_ext_opz_var1()

FLA_Error FLA_Bsvd_ext_opz_var1	(	int	m_d,
		int	m_U,
		int	m_V,
		int	m_C,
		int	n_C,
		int	n_GH,
		int	n_iter_max,
		double *	buff_d,
		int	inc_d,
		double *	buff_e,
		int	inc_e,
		dcomplex *	buff_G,
		int	rs_G,
		int	cs_G,
		dcomplex *	buff_H,
		int	rs_H,
		int	cs_H,
		dcomplex *	buff_U,
		int	rs_U,
		int	cs_U,
		dcomplex *	buff_V,
		int	rs_V,
		int	cs_V,
		dcomplex *	buff_C,
		int	rs_C,
		int	cs_C,
		int	b_alg
	)

{
    dcomplex  one        = bl1_z1();
    double    rzero      = bl1_d0();
 
    int       maxitr     = 6;
 
    double    eps;
    double    tolmul;
    double    tol;
    double    thresh;
 
    dcomplex* G;
    dcomplex* H;
    double*   d1;
    double*   e1;
    int       r_val;
    int       done;
    int       m_GH_sweep_max;
    int       ij_begin;
    int       ijTL, ijBR;
    int       m_A11;
    int       n_iter_perf;
    int       n_UV_apply;
    int       total_deflations;
    int       n_deflations;
    int       n_iter_prev;
    int       n_iter_perf_sweep_max;
 
    // Compute some convergence constants.
    eps    = FLA_Mach_params_opd( FLA_MACH_EPS );
    tolmul = max( 10.0, min( 100.0, pow( eps, -0.125 ) ) );
    FLA_Bsvd_compute_tol_thresh_opd( m_d,
                                     tolmul,
                                     maxitr,
                                     buff_d, inc_d,
                                     buff_e, inc_e,
                                     &tol,
                                     &thresh );
 
    // Initialize our completion flag.
    done = FALSE;
 
    // Initialize a counter that holds the maximum number of rows of G
    // and H that we would need to initialize for the next sweep.
    m_GH_sweep_max = m_d - 1;
 
    // Initialize a counter for the total number of iterations performed.
    n_iter_prev = 0;
 
    // Iterate until the matrix has completely deflated.
    for ( total_deflations = 0; done != TRUE; )
    {
 
        // Initialize G and H to contain only identity rotations.
        bl1_zsetm( m_GH_sweep_max,
                   n_GH,
                   &one,
                   buff_G, rs_G, cs_G );
        bl1_zsetm( m_GH_sweep_max,
                   n_GH,
                   &one,
                   buff_H, rs_H, cs_H );
 
        // Keep track of the maximum number of iterations performed in the
        // current sweep. This is used when applying the sweep's Givens
        // rotations.
        n_iter_perf_sweep_max = 0;
 
        // Perform a sweep: Move through the matrix and perform a bidiagonal
        // SVD on each non-zero submatrix that is encountered. During the
        // first time through, ijTL will be 0 and ijBR will be m_d - 1.
        for ( ij_begin = 0; ij_begin < m_d;  )
        {
            // Search for the first submatrix along the diagonal that is
            // bounded by zeroes (or endpoints of the matrix). If no
            // submatrix is found (ie: if the entire superdiagonal is zero
            // then FLA_FAILURE is returned. This function also inspects
            // superdiagonal elements for proximity to zero. If a given
            // element is close enough to zero, then it is deemed
            // converged and manually set to zero.
            r_val = FLA_Bsvd_find_submatrix_opd( m_d,
                                                 ij_begin,
                                                 buff_d, inc_d,
                                                 buff_e, inc_e,
                                                 &ijTL,
                                                 &ijBR );
 
            // Verify that a submatrix was found. If one was not found,
            // then we are done with the current sweep. Furthermore, if
            // a submatrix was not found AND we began our search at the
            // beginning of the matrix (ie: ij_begin == 0), then the
            // matrix has completely deflated and so we are done with
            // Francis step iteration.
            if ( r_val == FLA_FAILURE )
            {
                if ( ij_begin == 0 )
                {
                    done = TRUE;
                }
 
                // Break out of the current sweep so we can apply the last
                // remaining Givens rotations.
                break;
            }
 
            // If we got this far, then:
            //   (a) ijTL refers to the index of the first non-zero
            //       superdiagonal along the diagonal, and
            //   (b) ijBR refers to either:
            //       - the first zero element that occurs after ijTL, or
            //       - the the last diagonal element.
            // Note that ijTL and ijBR also correspond to the first and
            // last diagonal elements of the submatrix of interest. Thus,
            // we may compute the dimension of this submatrix as:
            m_A11 = ijBR - ijTL + 1;
 
            // Adjust ij_begin, which gets us ready for the next submatrix
            // search in the current sweep.
            ij_begin = ijBR + 1;
 
            // Index to the submatrices upon which we will operate.
            d1 = buff_d + ijTL * inc_d;
            e1 = buff_e + ijTL * inc_e;
            G  = buff_G + ijTL * rs_G;
            H  = buff_H + ijTL * rs_H;
 
            // Search for a batch of singular values, recursing on deflated
            // subproblems whenever a split occurs. Iteration continues as
            // long as
            //   (a) there is still matrix left to operate on, and
            //   (b) the number of iterations performed in this batch is
            //       less than n_G.
            // If/when either of the two above conditions fails to hold,
            // the function returns.
            n_deflations = FLA_Bsvd_iteracc_v_opd_var1( m_A11,
                                                        n_GH,
                                                        ijTL,
                                                        tol,
                                                        thresh,
                                                        d1, inc_d,
                                                        e1, inc_e,
                                                        G,  rs_G, cs_G,
                                                        H,  rs_H, cs_H,
                                                        &n_iter_perf );
 
            // Record the number of deflations that were observed.
            total_deflations += n_deflations;
 
            // Update the maximum number of iterations performed in the
            // current sweep.
            n_iter_perf_sweep_max = max( n_iter_perf_sweep_max, n_iter_perf );
 
            // Store the most recent value of ijBR in m_G_sweep_max.
            // When the sweep is done, this value will contain the minimum
            // number of rows of G and H we can apply and safely include all
            // non-identity rotations that were computed during the
            // singular value searches.
            m_GH_sweep_max = ijBR;
 
            // Make sure we haven't exceeded our maximum iteration count.
            if ( n_iter_prev >= n_iter_max * m_d )
            {
                FLA_Abort();
                //return FLA_FAILURE;
            }
        }
 
        // The sweep is complete. Now we must apply the Givens rotations
        // that were accumulated during the sweep.
 
        // Recall that the number of columns of U and V to which we apply
        // rotations is one more than the number of rotations.
        n_UV_apply = m_GH_sweep_max + 1;
 
        // Apply the Givens rotations. Note that we only apply k sets of
        // rotations, where k = n_iter_perf_sweep_max. Also note that we only
        // apply to n_UV_apply columns of U and V since this is the most we
        // need to touch given the most recent value stored to m_GH_sweep_max.
 
        if ( buff_U != NULL ) // ( rs_U == 0 && cs_U == 0 ) // if U is requested
          FLA_Apply_G_rf_blz_var3( n_iter_perf_sweep_max,
                                   m_U,
                                   n_UV_apply,
                                   buff_G, rs_G, cs_G,
                                   buff_U, rs_U, cs_U,
                                   b_alg );
        
        if ( buff_V != NULL ) // ( rs_V == 0 && cs_V == 0 ) // if V is requested
          FLA_Apply_G_rf_blz_var3( n_iter_perf_sweep_max,
                                   m_V,
                                   n_UV_apply,
                                   buff_H, rs_H, cs_H,
                                   buff_V, rs_V, cs_V,
                                   b_alg );
        
        // lf with non-flipped cs/rs m/n should be used; 
        // but the same operation can be achieved by using rf
        // ( U^H C )^H = C^H U where U is G_0 G_1 G_2 .... G_{k-1}
        if ( buff_C != NULL ) // ( rs_C == 0 && cs_C == 0 ) // if C is requested, flip
        {
          FLA_Apply_G_rf_blz_var3( n_iter_perf_sweep_max,
                                   n_C,
                                   m_C,
                                   buff_G, rs_G, cs_G,
                                   buff_C, cs_C, rs_C,
                                   b_alg );
        }
 
        // Increment the total number of iterations previously performed.
        n_iter_prev += n_iter_perf_sweep_max;
    }
 
    // Make all the singular values positive.
    {
        int    i;
        double minus_one = bl1_dm1();
 
        for ( i = 0; i < m_d; ++i )
        {
            if ( buff_d[ (i  )*inc_d ] < rzero )
            {
                buff_d[ (i  )*inc_d ] = -buff_d[ (i  )*inc_d ];
 
                // Scale the right singular vectors.
                if ( buff_V != NULL )
                  bl1_zdscalv( BLIS1_NO_CONJUGATE,
                               m_V,
                               &minus_one,
                               buff_V + (i  )*cs_V, rs_V );
            }
        }
    }
 
    return n_iter_prev;
}

References bl1_d0(), bl1_dm1(), bl1_z1(), bl1_zdscalv(), bl1_zsetm(), BLIS1_NO_CONJUGATE, FLA_Abort(), FLA_Apply_G_rf_blz_var3(), FLA_Bsvd_compute_tol_thresh_opd(), FLA_Bsvd_find_submatrix_opd(), FLA_Bsvd_iteracc_v_opd_var1(), FLA_Mach_params_opd(), and i.

Referenced by FLA_Bsvd_ext_opt_var1().