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\centerline{Mathematics 372 -- Numerical Linear Algebra}
\centerline{Problem Set 2 -- {\it due:} February 2, 2007}
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\noindent
I.  For this problem, you will work in a {\it decimal} 
floating point number system, where exactly 5 decimal digits
are stored, and exponents lie in the range $-12 \le e \le 12$.
All nonzero numbers will be represented in normalized form as
$$ \pm .d_1 d_2 d_3 d_4 d_5 \times 10^e,$$
where $d_i \in \{0, 1, 2, \ldots, 9\}$ and $d_1 \ne 0$.
All computations are done with {\it rounding} to 5 digits.
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\item{A)}  Compute $(.32113 \times 10^3) - (.97532 \times 10^{-1})$
\item{B)}  Compute $(.63784 \times 10^4)/(.56734 \times 10^8)$
\item{C)}  What are the largest and smallest numbers that can 
be represented in normalized form?
\item{D)}  For the purposes of this part, let's say the system also
contains the one non-normalized number $0$ as well as all the
numbers in normalized form.  How many {\it different} numbers in all 
are there in this system?  
\item{E)}  What can you say about the {\it spacing} between ``consecutive''
numbers in the system?  Is it always the same?  
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\noindent
II.  
\item{A)}
Show that if $A = (a_{ij})$ is a lower-triangular
$n\times n$ matrix, then $\det(A) = a_{11} a_{22} \cdots a_{nn}$.  (Hint: Use
induction on $n$.)
\item{B)}  Show the same formula for $\det(A)$ if $A$ is upper-triangular.
(Hint: recall that a determinant can be expanded along any row or column
of the matrix, if you have the correct signs.)
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\noindent
III.  Do problem 1.3.4 from the text, but changing the right hand side
to 
$$\pmatrix{3 & 4 & 6 & 5}^t.$$
Do the arithmetic in two different ways:
\item{A)}  ``By hand'' using exact rational number (``fraction'') arithmetic.
\item{B)}  ``By hand (with calculator)'', but using  
the floating point number system from problem I above.  
To make effects of this type of arithmetic more apparent, when you 
do a computation using several additions, multiplications, etc., group
the terms and do the rounding to 5 digits and normalization 
after {\it each} addition, multiplication,
etc.  For instance if you are computing $((a*b) + c)/d$, you would first
multiply $a$ and $b$, round that result and normalize, then add $c$ and round 
and normalize that result, 
then divide by $d$ and round and normalize again.  
\item{C)} Compare your results in parts A and B.
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\noindent
IV.  Do problem 1.3.17a from the text, but change
the right hand side to 
$$\pmatrix{1 & -3 & 5 & 5}^t.$$
Do the calculations in the same two ways as in 
the last problem, and compare your results again.
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\noindent
V. Let $A$ be a real symmetric $n\times n$ matrix.  In this 
problem you will show that $A$ is positive definite if and only
if all the eigenvalues of $A$ are strictly positive.  You 
will need to use the following version of the Spectral Theorem:
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\noindent
{\it There is an invertible matrix $Q$ with $Q^t = Q^{-1}$ and
$$A = Q^t{\rm diag}(\lambda_1, \ldots, \lambda_n)Q$$
where ``diag'' means the diagonal matrix with those entries on the 
main diagonal.  (Indeed, $Q^t$ here is the matrix
whose columns are the orthonormal basis of ${\bf R}^n$ consisting
of eigenvectors of $A$, but you should not need to use
that fact.)}
\bigskip
\item{A)}  Show that if $\lambda_i > 0$ for all $i$, then 
$A$ is positive definite.  
\item{B)}  Show conversely that if $A$ is positive definite, 
then $\lambda_i > 0$ for all $i$.    
\bye
