seed = 1641329179
...
jk: 7 8 0
jk: 7 8 0
jk: 7 8 0
... infinite loop

>> V([8 9],:)*V([8 9],:)'
ans =

                     1  3.45776883297926e-16
  3.45776883297926e-16                     1

>> ans-eye(2)
ans =

  4.44089209850063e-16  3.45776883297926e-16
  3.45776883297926e-16  -4.44089209850063e-16

>> Q=nash( V(:,[8 9]))
p =   -7.5460e-17
q =  1
r =  1.00000
swap = 0
p =   -7.5460e-17
r =    1.1102e-16
v =    1.8736e-16
Q =

  0.89235  0.45135
  -0.45135  0.89235

>> e2 = 10*m*eps*eps, tol = 0.01*eps*eps
e2 =    4.9304e-30
tol =    4.9304e-34
>> S(1)
ans =  2.9467
>> e2*S(1)
ans =    1.4528e-29
>> p=V(:,8)'*V(:,9)
p =   -7.5460e-17
>> p*p
ans =    5.6943e-33
>> tol
tol =    4.9304e-34
>> 0.1*eps
ans =    2.2204e-17
>> abs(p)
ans =    7.5460e-17
>>

code with convergence problems, using Nash tolerances and convergence tests:

    Real e2 = 10*m*eps*eps, tol = 0.01*eps*eps, q, r, v, c; Complex p, s;
    int done; unsigned long count = 0;
    if( limit == 0) { limit = n/2; } if( limit < 12) limit = 12; // twice Nash values
    do { done = 1; ++count; // make rows j & k of V orthogonal:
      for( j = 0, Vj = V, Qj = Q; j < n-1; ++j, Vj += m, Qj += n)
        for( k = j+1, Vk = Vj+m, Qk = Qj+n; k < n; ++k, Vk += m, Qk += n) {
          q = S[j]; r = S[k]; p = prod(Vj,Vk,m); int swap = 0;
          if( q >= r) {
            // if both norms or inner product are too small then skip
            if( q <= e2*S[0] || abs2(p) <= tol*q*q) continue;
          } else { // rows out of order, incorporate swap with rotate
            swap = 1; Real t = q; q = r; r = t; p = Conj(p); }
          // std::cout << q << " " << r << " " << p << " "; // debug
          done = 0; p = p/q; r = 1-r/q; v = sqrt(4*abs2(p) + r*r);
          c = sqrt(0.5*(1+r/v)); s = Conj(p)/(v*c);
          // std::cout << c << " " << s << "\n"; // debug
          rotate( Vj, Vk, m, c, s, swap); rotate( Qj, Qk, n, c, s, swap);
          S[j] = abs2( Vj, m); S[k] = abs2( Vk, m); } // update norms
    } while( !done && count < limit);
    if( !done && count >= limit)
      std::cerr << "svd: reached sweep limit (" << limit << ")" << std::endl;