HW #5, due Thursday April 23 (I may add more problems ...)
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Problem #1
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We want to solve numerically the following boundary value
problem:

y"(x) - y(x) = 0,  -1 < x <1
y(1)  = 1
y(-1) = 1

Introduce n equispaced nodes such that
x(0) = -1, x(n) = 1

At each node use approximation

y"(x(k)) = ( y(x(k-1)) + y(x(k+1)) - 2*y(x(k)) ) /(h*h)

where h is the discretisation step, h = (1 - (-1) )/(n-1).

This gives n linear equations, including y(x(0)) = 1
and  y(x(n)) = 1 .

Re-write this in the form A*y = b.
We will use the Jacobi iterations to solve this
system. Here are the steps that need to be made:


1) Analyze the matrix that will be iterated
during Jacobi iterations, and prove theoretically
that the iterations will converge.

2) Solve the problem analytically (Hint: exp(x)
and exp(-x) both solve the equation; find
such a combination of these two functions that
satisfies the boundary conditions).

3) For n = 11, 21, 41, 81, 161 solve the
problem numerically, using Jacobi iterations.
In each case stop the iterations when the next
approximation differs from the previous by less
than 1.e-8 in infinity norm. Compare your answer
with the exact solution and fill the following
three-column table:


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n         Number of iterations        Max difference with the exact solution
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11

21

41

81

161