hgd7833

The question is number (38.2) in page 302 in Numerical Linear Algebra (Trefethen & Bau) We have a real symmetric matrix with eigenvalues: 1, 1.01, 1.02,..., 8.98, 8.99, 9.00 and 10, 12, 16, 24.

How many steps of the conjugate gradient iteration must you take to reduce the initial error by a factor of 10^6 ??

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The answer is easy: The condition number of the matrix is

k = lamda max / lamda min = 24/1 = 24.

So, 1/10^6 = ||e_n|| / || e_0|| < = 2(sqrt(k)-1/sqrt(k)+1)^n .

Solving the equation: We get n>= 35.03 . So, we need at least 36 steps of the CG iteration.

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How does the distribution of the eigenvalues affect the rate

of convergence ??

In other words, what if the eigenvalues were: 1,2,3,...,24 for example. what does that afect the convergence ?

The whole calculations are based on the fact that the condition number k = lamda max / lamda min.

It seems to me that the only thing that would affect that are only the maximum and the minimum eigenvalues, NOT all eigenvalues ?

which implies that the rate of convergence will not change if we have a different kind of distribution of the eigenvalues.

Is this right ? Any ideas ??

Thank you

Hi Cap'n Refsmmat

Actually the problem is:

Find the mean value of the function ln(r ) on the circle (x-a)^2 + (y-b)^2 = R^2

I am sorry, the function is a function of r, so f® = lnr , and i was trying to change in polar coordinates, r = (x^2 + y^2)^1/2 , and double integral

but i couldn't do it

How to integrate f(x) = lnx on the circle (x-a)^2 + (y-b)^2 = R^2 ???

I will have ti use complex analysis

Thank you

This the out put after one step of L'Hopital's rule: cos(tanz).sec^2(z) - sec^2(sinz).cosz / 1/sqrt(1+z^2). sqrt( 1- (arctanz)^2)) - 1/ sqrt(1-z^2) . 1+(arcsinz)^2

So, we don't have any cancellation, and when plug with 0 it'll give you 1-1/1-1 = 0/0 !!

The second step is even much more tedious.

I suggested that in the class, but the professor said: never attempt to do. He said you need an elegant way !

Actually, i tried but i got a pretty nasty and unbelievable calculations.

Isn't correct that i-e^x / 1+e^x = - tanh(x/2) ???

If yes, then how can we proceed ??

Have you tired L'Hopital's rule?

Yes i did. You will see that it gives 0/0 infinitely many times. So it doesn't work.

Yes , that's what the professor said.

How to find the imporper integral from - 00 to 00 of f(x) = (e^(ikx)).(1-e^x / 1+e^x) where k is a real fixed number ??

I tried to write e^ikx = coskx + i sinkx but i don't know what to do with the term multiplied by e^ikx which is (1-e^x / 1+e^x) ??

Thank you

How to find the limit of the following complex function as z --> 0

f(z)= [ sin(tanz)- tan(sinz)] / [ arcsin(arctanz)- arctan(arcsinz)]