% Solve for the unknowns in the following problems.
%
% - $$\begin{eqnarray}
% x - y & = & 5 \\
% 3x + 5y & = & 1
% \end{eqnarray}
% $$
% - $$\begin{eqnarray}
% x - y + 3z & = & 5 \\
% 3x + 5y - z & = & 1 \\
% 5y - 3z & = & -22 \\
% \end{eqnarray}
% $$
% -
% $$\begin{eqnarray}
% a & = & 4(b - c) \\
% \frac{b-a}{2c} & = & 1 \\
% a + b & = & c-2
% \end{eqnarray}
% $$
% - Solve the linear system
% $$\begin{eqnarray}
% ax_1^2 + b x_1 + c & = & y_1 \\
% ax_2^2 + b x_2 + c & = & y_2 \\
% ax_3^2 + b x_3 + c & = & y_3
% \end{eqnarray}
% $$
% for unknown parameters
% $a$,
% $b$ and
% $c$,
% where points
% $(x_1,y_1)$,
% $(x_2,y_2)$,
% $(x_3,y_3)$ are known and are given by
% $(1,3)$,
% $(2,5)$,
% $(3,-1)$, respectively.
%
% - Solve the linear system $A{\bf x}
% = {\bf b}$, where $A$ is a $5 \times 5$ matrix with ones on the sub- and
% super-diagonal, and -2 on the diagonal, i.e.
% $$
% A = \left[\begin{array}{rrrrr}
% -2 & 1 & 0 & 0 & 0 \\
% 1 & -2 & 1 & 0 & 0 \\
% 0 & 1 & -2 & 1 & 0 \\
% 0 & 0 & 1 & -2 & 1 \\
% 0 & 0 & 0 & 1 & -2 \\
% \end{array}\right].
% $$
% The right hand side vector ${\bf b}$ is a
% $5 \times 1$ column vector of ones.
% Create the system using Matlab functions (1) diag and (2) spdiag.
% -
% Often, linear systems arise in the context of solving
% recurrence relations. For example, the famous
% Fibonacci sequence, named after the 13th century Italian mathematician
% Fibonacci, is generated from the recurrence relation
% $$
% x_{j+1} = x_j + x_{j-1}, \qquad j = 1,2,3,4,...
% $$
% where $x_0 = 0$ and $x_1 = 1$. The first few numbers
% in the sequence are $[0,1,1,2,3,5,8,13,21,34,...]$. Construct a linear
% system that relates values $x_j$,
% $j = 1,2,3,4,5...$ to the known
% starting values $x_0$ and
% $x_1$. Solve
% the linear system to generate the first 10 Fibonacci numbers.
% Try this using both diag and
% spdiag.
%
% Hint: Write out the equations
% for enough values of $j$ so that you
% can see a pattern in the linear system. Then construct the system.
% - Suppose you only knew $x_0$
% and $x_{11}$. Could you still
% generate the Fibonacci sequence? Modify the system you found above to
% solve for the first 10 values of the sequence, using only the known
% values $x_0 = 0$ and
% $x_{11} = 89$.
%
%
%
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%
%