# Until we meet again singular matrix often also a nearly singular matrix (in fact, made nonsingular only by roundoff and this end, we compute by Newton's method a triple-point bifurcation point in a . 81; these cases require an ad hoc construction and we do not expect to meet .. Again, the bordered linear systems are clearly solved in a stable way by. Consider A to be a non-singular matrix of order n with the added repeats the process until the tests are met. In Section a singular matrix but we find that the last diagonal element in the to be met, the tests are tried again after carrying out. We combine our novel SVD-free additive preconditioning with aggregation and other input matrix, in particular to the costly computation of its smallest singular values Otherwise the extension is quite straightforward unless we run into As we have pointed out, we meet it with extended iterative refinement and MSAs.

The reciever and transmitter relationship is represented as follows. As I said the math itself is very simple. It would be like a first example at the first chapter of any linear algebra book. The important this is how to interpret this equation into real implementation. If I interpret it in an illustration, it would be as the path represented in green arrows.

Mathematically this is also simple and it is represented in green arrows. As you see in the process illustrated in green arrow, to recover the transmitted data original data from the received data we need to take inverse of the channel information matrix. Unfortunately there are a couple of issues with this method.

There are some matrix for which inverse matrix does not exists. Please refer to Matrix section and see what kind of matrix is the one which is not invertable. To workaround these problems, we change the channel information matrix into three matrices by the method called SVD Singular Value Decomposition.

See the SVD section in Marix page and clearly understand the meaning of this process. Probably my page would not be enough to give you full understanding about SVD. Google as much materials about this as possible and try to have some "Intuitive" understanding of the concept. When we apply any mathematical technique in engineering area, it should be meaningful in the engineering sense and should be implementable.

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The meaning and implementation of SVD can be illustrated as follows. The important point is that by implementing this method, we can express the channel information matrix as a simple 'diagonal matrix'. That diagonal matrix is expressed as follows and can be illustrated as follows. Isn't it look simple and clear?

For some MIMO implementation e. The algorithm by which UE select the codebook which is best fit for the channel at specific moment is as follows. Considerations for Reality Now you know about various aspect of MIMO channel model and conceptual mathematical representation. I hope it make sense to you at least in terms of mathematically.

But as you know, when you try to implement the mathematical model into a real system, almost always you would meet something you haven't imagined when you are playing only in mathematical world. Let's look into folloing mathematical model again. Can you guess what is known variable and what is unknown variable for each side reciever and transmitter side? We can think of followings as a kind of known variables.

The three matrix at the center can be a known at least to the reciever since the reciever can estimate channel Matrix H from the received signa and calculate these matrix from H. The vector x is known to transmitter since it is just the data that's transmitted. The matrix U can be a known to the reciever since it can be derived from H.

The issue is how to figure out the matrix V. About this matrix, there are two main problems. Whatever s is set to in this function is what the user will get when they use the sind function. As with scripts, the first set of comments in the file should describe the function. This line is the one printed when the user types help sind.

It is usual to use a similar format to that which is used by Octave's built-in functions. It takes the input x and saves the result of the calculation in s, which was defined in the first line as the name of the result of the function [End of the function] Functions in Octave do not need to end with return although you can use the return command to make Octave jump out of a function in the middle.

Because each function is in a separate M-file, once it reaches the end of the file, Octave knows that it is the end of the function. The value that s has at the end of this function is the value that is returned. Creating and using functions Create the above function by opening the editor type edit if it's not still open and typing the lines of the function as given above.

Save the file as sind. You can now use the function in the same way as any of the built-in Octave functions. Now we can try some numbers octave: If you call the sind function with a vector, it means that the x parameter inside the function will be a vector, and in this case the sin function knows how to work with vectors, so can give the correct response.

The complete function is given below: The result of the function is one variable, called y. This time the help message contains several lines. This line uses the size function, which returns two values: This gives an example of how functions in Octave can return two things - in a vector, of course.

These values are used to create an output vector of the same size, and to check that the input is a vector. This checks that it has either one row or one column using the result of the size function.

The error function prints out a message and aborts the function if there is a problem.

### CUED - Introduction to Octave

It is initialised to be the same size and shape as the input t, and for each element to be zero. We therefore use a for loop to step through each value. The length function tells us how many elements there are in the vector t.

Our output vector y already contains zeros, so we can ignore this case. We test for this case and set y, our output variable, accordingly. As in most high level languages, all variables created inside a function m, n and k in this case are local to the function. They only exist for the duration of the function, and do not overwrite any variables of the same name elsewhere in Octave. The only variables which are passed back are the return values defined in the first function line: A unit, one second pulse created from two unit steps, using the function ustep.

Type this function into the editor, and save it as ustep. We can now use this function to create signal. If we then type octave: Any other variable listed e. Variables defined and used inside functions are completely separate from the main workspace. Matrices and vectors Vectors are special cases of matrices. For example, here is a 2 x 3 matrix: To enter this matrix into Octave you use the same syntax as for vectors, typing it in row by row: The middle pair of numbers are not the same, which explains the wording of the error message.

When just dealing with vectors, there is little need to distinguish between row and column vectors. When multiplying vectors, however, it will only work one way round. The transpose of a matrix interchanges the rows with the columns. Mathematically, the transpose of A is represented as AT. In Octave an apostrophe performs this operation: Matrix creation functions Octave provides some functions to help you build special matrices. We have already met ones and zeros, which create matrices of a given size filled with 1 or 0. A very important matrix is the identity matrix. This is the matrix that, when multiplying any other matrix or vector, does not change anything. This is usually called I in formulae, so the Octave function is called eye.

This only takes one parameter, since an identity matrix must be square: You could construct this explicitly, but the Octave provides the diag function which takes a vector and puts it along the diagonal of a matrix: For the matrix A defined earlier: Finally, it is sometimes useful to create an empty matrix, perhaps for adding elements to later.

• Introduction to Octave
• OpenFOAM version 1.6 details

You can do this by defining the matrix with an empty pair of braces: