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## Teacher Elias Jarlebring edited 11 December 2015

Hej,

Homework 3 is now corrected and all master students who handed in homework 3 passed. I am still expecting a revision of homework 2 from 6 students.

The ~~course is ~~lectures of this course are almost over. I have enjoyed teaching this course. If you enjoyed learning this material, you are very welcome to write so in the course evaluation. More importantly, if you did not enjoy it and you think there are things to improve in this course, I very much appreciate such comments, such that the course improves next year:

We already have two improvements that we will implement next year:

* One more lecture on preconditioning

* Further written material on convergence of the QR-method

Elias

## Teacher Elias Jarlebring edited 8 November 2015

Hej!

I hope the first homework is going well.

Here is some more hint for HW1 problem 4c. The relationship between shift-and-invert Arnoldi and the standard Arnoldi method is completely analogous to the relationship between inverse iteration and the power method. So, instead of Arnoldi's method for tex:\displaystyle B you need to do Arnoldi's method for the matrix tex:\displaystyle (~~A~~B-\sigma I)^{-1} without explicitly computing the inverse, but only solve linear systems of equations. You will need to modify arnoldi.m. Don't forget to reverse the eigenvalue transformation. We will show an example of shift-and-inverse on the next lecture (tuesday).

Elias

## Teacher Elias Jarlebring edited 8 November 2015

Hej!

I hope the first homework is going well.

~~Here is some more hint for HW1~~Some hints / clarifications for HW1:¶

problem 2: The commands "rand('seed',0); A=gallery(’wathen’,nn,nn);" generate a sparse matrix of size nn^2 x nn^2. For the purpose of learning what you are expected to understand in the exercise, it is not so important to understand how the matrix is generated. (It is matrix with sparsity structure like a particular finite-element discretizations. The non-zero elements are generated randomly, so we need to do reset the random seed in order get the same results every time we run the command.")¶

problem 4c. The relationship between shift-and-invert Arnoldi and the standard Arnoldi method is completely analogous to the relationship between inverse iteration and the power method. So, instead of Arnoldi's method for tex:\displaystyle B you need to do Arnoldi's method for the matrix tex:\displaystyle (B-\sigma I)^{-1} without explicitly computing the inverse, but only solve linear systems of equations. You will need to modify arnoldi.m. Don't forget to reverse the eigenvalue transformation. We will show an example of shift-and-inverse on the next lecture (tuesday).

Elias

## Teacher Elias Jarlebring edited 8 November 2015

Hej!

I hope the first homework is going well.

Some hints / clarifications for HW1:

problem 2: On some computers 10 minutes of execution time is difficult to test due to limitations in RAM. It is not so important that you achieve 10 minutes in those situations, if you can show that you interpret the result correctly. The commands "rand('seed',0); A=gallery(’wathen’,nn,nn);" generate a sparse matrix of size nn^2 x nn^2. For the purpose of learning what you are expected to understand in the exercise, it is not so important to understand exactly how the matrix is generated. (It is matrix with sparsity structure like a particular finite-element discretizations. The non-zero elements are generated randomly, so we need to do reset the random seed in order get the same results every time we run the command.")

problem 4c. The relationship between shift-and-invert Arnoldi and the standard Arnoldi method is completely analogous to the relationship between inverse iteration and the power method. So, instead of Arnoldi's method for tex:\displaystyle B you need to do Arnoldi's method for the matrix tex:\displaystyle (B-\sigma I)^{-1} without explicitly computing the inverse, but only solve linear systems of equations. You will need to modify arnoldi.m. Don't forget to reverse the eigenvalue transformation. We will show an example of shift-and-inverse on the next lecture (tuesday).

Elias

## Teacher Elias Jarlebring edited 8 November 2015

Hej!

I hope the first homework is going well.

Some hints / clarifications for HW1 based on email questions:

problem 2: On some computers 10 minutes of execution time is difficult to test due to limitations in RAM. It is not so important that you achieve 10 minutes in those situations, if you can show that you interpret the result correctly. The commands "rand('seed',0); A=gallery(’wathen’,nn,nn);" generate a sparse matrix of size nn^2 x nn^2. For the purpose of learning what you are expected to understand in the exercise, it is not so important to understand exactly how the matrix is generated. (It is matrix with sparsity structure like a particular finite-element discretizations. The non-zero elements are generated randomly, so we need to do reset the random seed in order get the same results every time we run the command.")~~ ~~

problem 4c. The relationship between shift-and-invert Arnoldi and the standard Arnoldi method is completely analogous to the relationship between inverse iteration and the power method. So, instead of Arnoldi's method for tex:\displaystyle B you need to do Arnoldi's method for the matrix tex:\displaystyle (B-\sigma I)^{-1} without explicitly computing the inverse, but only solve linear systems of equations. You will need to modify arnoldi.m. Don't forget to reverse the eigenvalue transformation. We will show an example of shift-and-inverse on the next lecture (tuesday).

Elias

## Teacher Elias Jarlebring edited 17 December 2014

There will be an extra lecture on Friday Dec. 19 at 13:15-15:00 in the seminar room 3733 of the KTH Mathematics building. The seminar room is located on the top floor of Lindstedtsvägen 25. This lecture is not visible in the official "schedule".

## Teacher Elias Jarlebring edited 25 November 2014

There is a now a PDF-file with the pseudo-code for GMRES on the course web page. The pseudocode gives some more detail than TB.