A note on assignment 3, question 1(d):
When we say that you have software implementing stable linear solvers,
we refer to the numerical stability of the methods (algorithms)
for computing the solution of a linear system.
So, you can assume that a stable linear solver
gives rise to stable operations
(operations with errors at the level of machine epsilon),
so that the residual is at the level of machine epsilon.
A few more notes, if you want to read further:
Recall that numerical stability of an algorithm
is a different issue from the conditioning of the problem
(which is solved by the algorithm).
In the case of a linear system, the most stable solvers
are LU/GE with complete pivoting, or QR factorization
(though, in practice, due to efficiency considerations,
we often choose to use LU/GE with partial pivoting
which is more stable than GE/LU without pivoting,
but not as stable as LU/GE with complete pivoting, or QR factorization).
Again in the case of a linear system, the conditioning
of the problem of solving it, is governed by the condition number
of the matrix, and this is an inherent property of the particular matrix
You can have a stable solver (and small residual),
but a badly conditioned matrix, in which case
the solution error may be large.
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