1. Kaczmarz\'s algorithm Consider a square system of linear equations y Hx with

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1. Kaczmarz's algorithm Consider a square system of linear equations y Hx with a full-rank matrix H E RMN and y = (yn) 0 E RNx1 we can look for the solution x = (xn-in two ways. concentrating on either the columns or the rows of H. When concentrating on the rows N-1 0 we see the solution x as the vector that has all the correct inner products: "N-1 (la) Kaczmarz's algorithm uses the row-based view. Normalize the rows h to be of unit norm, " hn/llhnll, where H stands for the Euclidean (12) norm. Then, (la) becomes (1b) All x that satisfy (la) or (1b) make a hyperplane S, also known as an affine subspace. (Note that S is not necessarily a subspace because it may not include 0, hence the name affine subspace.) The idea of Kaczmarz's algorithm Kaczmarz 1937. Vetterli et al. 2014, Ex. 2.66l is to iteratively satisfy the constraints (1b). Starting with an initial guesshe first update step is N computations (n-1) T-(n-1) called a sweep. With the update (2), x(n) satisfies (3a) or, equivalently 3b Kn i.e., r e S At the end of this sweep, i is likely that l no satisfy all N constraints in (la); for example, it is likely that x(N-1) Sol and thus, further sweeps are required. (a) Show that the intersection of the hyperplanes Sa is unique and specifies the solution (2) of the linear system y- Hx. Hint: Use the fact that H is square and has full rank Hence,x-1) X(N-1) will not satisfy 0x(N-11-yo.

Explanation / Answer

Xarn=yn

Xn and xn-1

And

So it variables depends on the

Xn-1/xn

Xa xb variables

so

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