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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.

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.

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.

added 87 characters in body; edited title
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Yuval Filmus
Yuval Filmus
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Implementing a Matrix Multiplication that matrix elements are Very Similar to Each Other Numerical issues in solving linear systems

There was an exam in the class. The course is "High Performance Scientific Computing". One of the question in the exam is as follows: There is a multiplication of 2x2 matrix ab (M1) and 1x1 martix xy (M2) and it results to 1x1 10 matrix (M3).

Consider the linear system

(M1) * (M2) = (M3)

-a b - x - 1

-b a - y - 0

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

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

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

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.

Implementing a Matrix Multiplication that matrix elements are Very Similar to Each Other

There was an exam in the class. The course is "High Performance Scientific Computing". One of the question in the exam is as follows: There is a multiplication of 2x2 matrix ab (M1) and 1x1 martix xy (M2) and it results to 1x1 10 matrix (M3).

Consider the linear system

(M1) * (M2) = (M3)

-a b - x - 1

-b a - y - 0

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.

Numerical issues in solving linear systems

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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tahasozgen
tahasozgen
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Implementing a Matrix Multiplication that matrix elements are Very Similar to Each Other

There was an exam in the class. The course is "High Performance Scientific Computing". One of the question in the exam is as follows: There is a multiplication of 2x2 matrix ab (M1) and 1x1 martix xy (M2) and it results to 1x1 10 matrix (M3).

Consider the linear system

(M1) * (M2) = (M3)

-a b - x - 1

-b a - y - 0

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.