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There was an exam in the class. The course is "High Performance Scientific Computing". One of the question in the exam is as follows:

Consider the linear system

$$ \begin{bmatrix} a & b \\ b & a \end{bmatrix} \times \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$$ with $a,b>0$.

a) If $a$ is very similar to $b$, what is the numerical difficulty in solving this linear system?

b) Suggest a numerically stable formula for computing $z = x + y$ given $a$ and $b$.

This is a Computer Engineering course, however I am not able to answer these questions. What is the keyword to find a solution on the issue?

Thanks in advance.

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The $2\times 2$ matrix has determinant close to zero, and so its condition number is very large, causing numerical instability.

The explicit solution of your system is $$ x=\frac{a}{a^2-b^2}, y=-\frac{b}{a^2-b^2}. $$ Therefore $$ x+y = \frac{a-b}{a^2-b^2} = \frac{1}{a+b}, $$ thus avoiding the numerical issues.

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