How many arithmetic operations does matlab require to determine the Schur decomposition?

Consider a matrix A of size n.

I would like to determine the square root of this matrix which can be done like this:

n = 10; % variable size 
A = rand(n); % random square matrix A of size n by n
A_sqrt = sqrtm(A); % square root of matrix A

Inside the sqrtm command, matlab requires the Schur decomposition for determining the square root of a matrix.

[S, T] = schur(A); % with A = S*T*S and S*S' = I and T = (quassi) upper triangular matrix

To determine the speed of sqrtm, I would like an expression of the amount of required distinguisable operations (summation, subtraction, multiplication, division and square root) expressed in the matrix size n. To get this expression, I would like to know how Matlab determines the Schur decomposition of a matrix.

I read that the first step is to determine the upper Hessenberg form H by means of

[G,H] = hess(A); % with A = G*H*G' and G*G' = I

After this, a QR decomposition of H is executed

[Q,R] = qr(H); % with H = Q*R and Q*Q' = I and R = upper triangular matrix

How does one continue from here to a Schur decomposition?

If this is not the way how Matlab determines a Schur decomposition, what is it?

NOTE:-

Matlabsolutions.com provide latest MatLab Homework Help,MatLab Assignment Help for students, engineers and researchers in Multiple Branches like ECE, EEE, CSE, Mechanical, Civil with 100% output.Matlab Code for B.E, B.Tech,M.E,M.Tech, Ph.D. Scholars with 100% privacy guaranteed. Get MATLAB projects with source code for your learning and research.

the names are very similar). The Hessenberg form is computed in advance to allow the QR algorithm to be applied implicitly. There are other ways of computing the Schur decomposition, but the QR algorithm is the most standard and simplest.

It’s not very practical to count the number of operations in this algorithm, particularly because the QR algorithm is iterative, so the number of passes through the data depends on the convergence speed, which again depends on the eigenvalues that are used. In practice, the locality of the code (how many times a block of memory is accessed) is also more relevant to performance than simply counting how many times each operation is done.

SEE COMPLETE ANSWER CLICK THE LINK

--

--

Get the Medium app

A button that says 'Download on the App Store', and if clicked it will lead you to the iOS App store
A button that says 'Get it on, Google Play', and if clicked it will lead you to the Google Play store
Technical Source

Technical Source

Simple! That is me, a simple person. I am passionate about knowledge and reading. That’s why I have decided to write and share a bit of my life and thoughts to.