[petsc-users] KSPLSQR convergence criterion

[Chetan Jhurani]

Hello!

Could someone help me figure out the correct way to
to use KSPLSQR?  Maybe I'm missing something simple.

I'm trying to use it to solve an over-determined set
of equations.  It seems like the KSP iterations keep on
going because the residual is not small enough.  But
since the system is over-determined, in general, the
exact residual Ax - b will not be zero.

For example, to solve the system

min_x ||Ax - b||, with A = [1;2], b = [3;4],

x must be inv(A'A)A'b = 2.2.  Relative residual is 0.17888.
See the Matlab code below too, which gives the correct
value in 1 iteration.

KSPLSQR, however, stagnates after 1 iteration.  At that
point, the relative residual and true residual norm
match the correct values exactly.

Is there an assumption in KSPLSQR that the system is always
invertible or under-determined?  This is petsc-3.1p2 if
it matters.

Thanks,

Chetan

----------------------------------------------------
KSPLSQR:

  0 KSP preconditioned resid norm 5.000000000000e+000 true resid norm 5.000000000000e+000 ||Ae||/||Ax|| 1.000000000000e+000
  1 KSP preconditioned resid norm 8.944271909999e-001 true resid norm 8.944271909999e-001 ||Ae||/||Ax|| 1.788854382000e-001
  2 KSP preconditioned resid norm 8.944271909999e-001 true resid norm 8.944271909999e-001 ||Ae||/||Ax|| 1.788854382000e-001
  3 KSP preconditioned resid norm 8.944271909999e-001 true resid norm 8.944271909999e-001 ||Ae||/||Ax|| 1.788854382000e-001
  4 KSP preconditioned resid norm 8.944271909999e-001 true resid norm 8.944271909999e-001 ||Ae||/||Ax|| 1.788854382000e-001
  5 KSP preconditioned resid norm 8.944271909999e-001 true resid norm 8.944271909999e-001 ||Ae||/||Ax|| 1.788854382000e-001

Matlab:

>> a = [1;2];
>> b = [3;4];
>> [x,FLAG,RELRES,ITER] = lsqr(a,b, 1e-10, 10)
x =
   2.200000000000000
FLAG =
     1
RELRES =
   0.178885438199983
ITER =
     1
>> norm(a*x-b)
ans =
   0.894427190999916

----------------------------------------------------