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\begin_layout Section*
18.335 Problem Set 2
\end_layout

\begin_layout Standard
Due Friday March 1, 2019, at 3pm via Stellar.
\end_layout

\begin_layout Subsection*
Problem 1: Stability
\end_layout

\begin_layout Enumerate
Trefethen, exercise 15.1.
 [In parts (e) and (f), assume that 
\begin_inset Formula $\frac{1}{k!}$
\end_inset

 can be computed to 
\begin_inset Formula $O(\epsilon_{\mbox{machine}})$
\end_inset

 and concentrate on the accumulation of errors in the summations.]
\end_layout

\begin_layout Enumerate
Trefethen, exercise 16.1.
\end_layout

\begin_layout Subsection*
Problem 2: Norms
\end_layout

\begin_layout Enumerate
Derive Trefethen eq.
 (3.10) (for which Trefethen only writes 
\begin_inset Quotes eld
\end_inset

by much the same argument
\begin_inset Quotes erd
\end_inset

).
 Find the code that computes the induced 
\begin_inset Formula $\Vert A\Vert_{\infty}$
\end_inset

 norm in Julia, the 
\family typewriter
opnorm(A, Inf)
\family default
 function, on 
\begin_inset Flex URL
status open

\begin_layout Plain Layout

github.com/JuliaLang/julia
\end_layout

\end_inset

 in stdlib/LinearAlgebra/src/generic.jl and satisfy yourself that it is equivalen
t to (3.10).
 
\end_layout

\begin_layout Enumerate
Trefethen, problem 3.4.
 Check your result for a random 
\begin_inset Formula $10\times7$
\end_inset

 matrix 
\begin_inset Formula $A$
\end_inset

 in Julia, constructed by 
\family typewriter
A=randn(10,7)
\family default
 with the induced 
\begin_inset Formula $p=2$
\end_inset

 norm as computed by 
\family typewriter
opnorm(A)
\family default
 in Julia.
\end_layout

\begin_layout Subsection*
Problem 3: Conditioning
\end_layout

\begin_layout Standard
In class, we found the condition number of 
\begin_inset Formula $f(x)=Ax$
\end_inset

, in the 
\begin_inset Formula $L_{2}$
\end_inset

 norm, to be 
\begin_inset Formula $\Vert A\Vert_{2}\Vert x\Vert_{2}/\Vert Ax\Vert_{2}$
\end_inset

 for some 
\begin_inset Formula $A\in\mathbb{C}^{m\times n}$
\end_inset

 and 
\begin_inset Formula $x\in\mathbb{C}^{n}$
\end_inset

 .
 Instead, compute the condition number of 
\begin_inset Formula $g(A)=Ax$
\end_inset

, using any convenient choice of norm (e.g.
 the Frobenius norm for the input matrices 
\begin_inset Formula $A$
\end_inset

).
 
\end_layout

\begin_layout Subsection*
Problem 4: SVD and low-rank approximations
\end_layout

\begin_layout Enumerate
Trefethen, probem 4.5.
\end_layout

\begin_layout Enumerate
Trefethen, problem 5.2.
\end_layout

\begin_layout Enumerate
Trefethen, problem 5.4.
\end_layout

\begin_layout Standard

\end_layout

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