I need to implement these updates into the algorithm

So I write this code below:

format long
m = 5;
A = rand( m, m );
t = rand( m, 1 );
r = rand( m, 1 );
[ B, t, r ] = BiRed( A, t, r );
Bi = BiFromB( B );
disp('Singular values of B')
svd( Bi )
disp('Singular values of A')
svd( A )

function [ A_out, t_out, r_out ] = BiRed( A, t, r )
  [ ATL, ATR, ...
    ABL, ABR ] = FLA_Part_2x2( A, ...
                               0, 0, 'FLA_TL' );
  [ tT, ...
    tB ] = FLA_Part_2x1( t, ...
                         0, 'FLA_TOP' );
  [ rT, ...
    rB ] = FLA_Part_2x1( r, ...
                         0, 'FLA_TOP' );                    
  while ( size( ATL, 1 ) < size( A, 1 ) )
    [ A00,  a01,     A02,  ...
      a10t, alpha11, a12t, ...
      A20,  a21,     A22 ] = FLA_Repart_2x2_to_3x3( ATL, ATR, ...
                                                    ABL, ABR, ...
                                                    1, 1, 'FLA_BR' );
    [ t0, ...
      tau1, ...
      t2 ] = FLA_Repart_2x1_to_3x1( tT, ...
                                    tB, ...
                                    1, 'FLA_BOTTOM' );
    [ r0, ...
      rho1, ...
      r2 ] = FLA_Repart_2x1_to_3x1( rT, ...
                                    rB, ...
                                    1, 'FLA_BOTTOM' );                                           

    %-----implement the updates here-----% 
    [u21, tau] = Housev1(a21);      
    n = size(A22, 1); 
    u = [1; u21];
    H = eye(n+1) - tau * (u * u');
    updated_block = H * [a12t; A22];
    a12t = updated_block(1, :);
    A22 = updated_block(2:end, :);  

    [u12, rho] = Housev1(a12t');  
    H_right = eye(size(A22, 2)) - rho * (u12 * u12');
    A22 = A22 * H_right;
    %------------------------------------% 
          
    [ ATL, ATR, ...
      ABL, ABR ] = FLA_Cont_with_3x3_to_2x2( A00,  a01,     A02,  ...
                                             a10t, alpha11, a12t, ...
                                             A20,  a21,     A22, ...
                                             'FLA_TL' );
    [ tT, ...
      tB ] = FLA_Cont_with_3x1_to_2x1( t0, ...
                                       tau1, ...
                                       t2, ...
                                       'FLA_TOP' );
    [ rT, ...
      rB ] = FLA_Cont_with_3x1_to_2x1( r0, ...
                                       rho1, ...
                                       r2, ...
                                       'FLA_TOP' );                                   
  end
  A_out = [ ATL, ATR
            ABL, ABR ];
  t_out = [ tT
            tB ];       
  r_out = [ rT
            rB ];
end

function [ B ] = BiFromB( B )
    B = tril( triu( B ), 1 );
end

function [ rho, ...
           u2, tau ] = Housev( chi1, ...
                               x2 )
  normx = sqrt( chi1 * chi1 + x2' * x2 );
  rho = - sign( chi1 ) * normx;
  nu1 = chi1 - rho;
  u2 = x2 / nu1;
  tau = ( 1 + u2' * u2 ) / 2;
end

function [ u, tau ] = Housev1( x )
  n = size( x, 1 );
  if ( n == 0 )
      error('Housev1:InputError', 'Error calling Housev1: input vector x is empty.');
  end  
  [ rho, ...
     u2, tau ] = Housev( x(1,1), ...
                         x(2:n, 1) );                     
  u = [ rho
         u2 ];
end

When I run it, it shows error input vector x is empty.

I try to add some debug code:

while (size(ATL, 1) < size(A, 1))
    [A00, a01, A02, ...
      a10t, alpha11, a12t, ...
      A20, a21, A22] = FLA_Repart_2x2_to_3x3(ATL, ATR, ABL, ABR, ...
                                             1, 1, 'FLA_BR');

    disp(['Sizes - A00: ', mat2str(size(A00)), ', A22: ', mat2str(size(A22))]);

    if isempty(a21)
        disp('a21 is empty, ending loop.');
        break;  % Break the loop if no more updates can be done.
    end

    [u21, tau] = Housev1(a21);

    n = size([a12t; A22], 1);
    H = eye(n) - tau * [1; u21] * [1; u21]';  
    updated_block = H * [a12t; A22];  
    a12t = updated_block(1, :);
    A22 = updated_block(2:end, :);

    if isempty(a12t)
        disp('a12t is empty, ending loop.');
        break;  % Break the loop if no more updates can be done.
    end

    [u12, rho] = Housev1(a12t');  
    H_right = eye(size(A22, 2)) - rho * (u12 * u12');  
    A22 = A22 * H_right;

    [ATL, ATR, ...
      ABL, ABR] = FLA_Cont_with_3x3_to_2x2(A00, a01, A02, ...
                                            a10t, alpha11, a12t, ...
                                            A20, a21, A22, ...
                                            'FLA_TL');

    disp(['Size of ATL after update: ', mat2str(size(ATL))]);
    if size(ATL, 1) == size(A, 1)
        break;  % Explicitly break if ATL has grown to full size.
    end
end
,

this is the output:

Sizes - A00: [0 0], A22: [4 4]
Size of ATL after update: [1 1]
Sizes - A00: [1 1], A22: [3 3]
Size of ATL after update: [2 2]
Sizes - A00: [2 2], A22: [2 2]
Size of ATL after update: [3 3]
Sizes - A00: [3 3], A22: [1 1]
Size of ATL after update: [4 4]
Sizes - A00: [4 4], A22: [0 0]
a21 is empty, ending loop.
Singular values of B

ans =

   1.106627568873231
   0.857960978170602
   0.413228870344143
   0.301257867048619
   0.013521687400442

Singular values of A

ans =

   2.301262463484019
   0.820665505888531
   0.455426251555126
   0.287366178798951
   0.046120083989295

I don’t understand, why the input is empty? How to implement it correctly?

notes:

1. If you want to see the detail description, it is on https://www.cs.utexas.edu/users/flame/laff/alaff/chapter11-reduction-to-bidirectional-form.html or you can google “ALAFF Reduction to bidiagonal form” and pick the top result.
2. If you want to see the skeleton code, initial code, and flame library, it is on https://github.com/ULAFF/ALAFF/blob/master/Assignments/Week11/matlab or you can google “assignments/week11/matlab/ github” and pick the top result then choose the “week 11” folder.