[Math] Typing a large size block matrix on matlab

MATLAB

I am reading this paper The iterative methods for computing the generalized inverse of the bounded linear operator between Banach spaces

Where I came across a matrix $A \in \mathbb{C}^{58\times 57}$ (Example 4.2) given as follows:
$A =\left(
\begin{array}{ccc}
A_{11} & A_{12} & A_{13} \\
A_{21} & A_{22} & A_{23} \\
\end{array}
\right) $ where $A_{ij}\in \mathbb{C}^{29\times 19} $. Author has given each $A_{ij}$ in this paper. I want to ask how to write this matrix on matlab? I am finding it difficult to type this matrix because of its large size and also I don't know how to type block matrices on matlab? Is there any alternate way to write this matrix on matlab? Could anybody help me with this? I would be very much thankful.

Best Answer

Not really sure whether this is what you are asking, but inputting a matrix in Matlab is quite straightforward:

A = [1+3i 2+4i 12-1i;
2 2i 2+2i]

Combining matrices is also quite simple:

B = [A A+1 2*A;
A-1 A-2 A+1i]

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[Math] How to determine stopping criteria in iterative methods

For iterative methods the difference from one iteration to the other gets lower and lower as it is supposed to converge. Therefore absolute error or absolute squared error from one iteration to the other is usually defined as the stopping criterion. Of course there is one other way, that is to fix the number of iterations which is trivial. Important of choosing a threshold for the absolute error from one iteration to the next provides you a certain level of quality. If you set it to $10^{-8}$ it will take more time for the convergence but the quality of the approximation will be of course better than the case where you set it to $10^{-4}$. It is all related to the trade-off between speed of convergence and the quality of convergence which is determined by the user.

In your example, the thresholds are determined in a similar way. It is a subjective process but the quality can be objectively evaluated.

[Math] Generating Different types of Matrices in Matlab

I think that you're looking for the gallery function. It has over 60 types of test matrices. For the particular case of diagonally dominant matrices, you might look at the 'dorr' option:

A = full(gallery('dorr',5,0.01))

You can find the code for most of these matrix generation functions in the matlabroot/toolbox/matlab/elmat/private/ directory, where matlabroot is the root directory of your Matlab installation.

If you need random diagonally dominant matrices, then you might look at the answers to this StackOverflow question. I believe that this is equivalent Matlab code to the accepted answer (you'll have to check if the resultant matrices are indeed diagonally dominant):

rng(1)
n = 5;
mmax = 100;
mat1 = 2*mmax*rand(n)-mmax;
mat = mat1+diag(sum(abs(mat1),2).*sign(diag(mat1)))

Best Answer

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[Math] How to determine stopping criteria in iterative methods

[Math] Generating Different types of Matrices in Matlab

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