CS205 Midterm 1
Multipleanswer multiplechoice questions [7pts]
1. Which of the following statements about the solution to the linear least
squares problem are true?
(a) It satisfies
Ax
=
b
.
r
=
0
, in general.
(b) It can be found by solving the linear system
A
T
Ax
=
A
T
x
.
It can be found by solving the linear system
A
T
Ax
=
A
T
b
.
(c) It cannot be computed by the normal equations.
The normal equations give the solution.
(d)
Its associated residual lies in the nullspace of
A
T
.
This is a consequence the leastsquares solution corresponding to an
orthogonal projection.
(e)
It can be computed by performing a
QR
decomposition via
Householder transformations.
This is, in fact, the preferred method.
2. For which of the following reasons do iterative methods typically use the
residual to measure convergence?
(a)
Absolute error is difficult to compute directly.
In fact, it is often impossible to do so.
(b) Relative error does not indicate how close a given iterate is to the
exact solution.
This is directly contradictory to the definition of relative error.
(c)
The residual will be zero if the current iterate is a solution.
This follows from the definition of residual.
(d) Using the residual is equivalent to using the absolute error.
The residual and the absolute error are completely different entities.
1
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 Fall '07
 Fedkiw
 Linear Algebra, following statements, Singular value decomposition, singular values, condition number

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