- #1
pinodk
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I have two excercises which have been causing me to tear my hair off for some time now.
(a) the power method to find largest eigenvalue of A is defined as x(k+1) = Ax(k)
(b) the inverse power method is to solve Ax(k+1) = x(k) to find smallest eigenvalue of A
(c) the smallest/largest eigenvalue is then found by the Rayleigh quotient of A and x : R_A(x) = (Ax,x)/(x,x)
The eigenvalues of A are arranged as lambda_n >= lambda_n-1 >=... lambda_2 >= lambda_1
Henceforth lambda_i will be written as Li, so
Ln >= Ln-1 ... >= L2 >= L1
exercise 1.)
The convergence of x(k) in (a) and (b) is of order 1 with rate of convergence |Ln-1/Ln| and |L1/L2| respectively.
State conditions for this to be true for each coordinate.
that was the first, my question is then, what does that question mean (its translated from danish, hope i did it right :-S)? I read it as:
give the conditions for the eigenvalues that must be fulfilled in order for the above to be true, and show that it applies to each coordinate in x(k)...
If that makes sense to you, please help me on how to get on with this exercise, because I am still stucked...
exercise 2.)
Assume that A is symmetrical, and that x(k) converges towards (a multiple of) the eigenvector, so that each coordinate of x(k) converges with rate of convergence c.
Show that L(k) converges by order 1 with rate of convergence c^2.
Again I am baffled... please get me started :-)
(a) the power method to find largest eigenvalue of A is defined as x(k+1) = Ax(k)
(b) the inverse power method is to solve Ax(k+1) = x(k) to find smallest eigenvalue of A
(c) the smallest/largest eigenvalue is then found by the Rayleigh quotient of A and x : R_A(x) = (Ax,x)/(x,x)
The eigenvalues of A are arranged as lambda_n >= lambda_n-1 >=... lambda_2 >= lambda_1
Henceforth lambda_i will be written as Li, so
Ln >= Ln-1 ... >= L2 >= L1
exercise 1.)
The convergence of x(k) in (a) and (b) is of order 1 with rate of convergence |Ln-1/Ln| and |L1/L2| respectively.
State conditions for this to be true for each coordinate.
that was the first, my question is then, what does that question mean (its translated from danish, hope i did it right :-S)? I read it as:
give the conditions for the eigenvalues that must be fulfilled in order for the above to be true, and show that it applies to each coordinate in x(k)...
If that makes sense to you, please help me on how to get on with this exercise, because I am still stucked...
exercise 2.)
Assume that A is symmetrical, and that x(k) converges towards (a multiple of) the eigenvector, so that each coordinate of x(k) converges with rate of convergence c.
Show that L(k) converges by order 1 with rate of convergence c^2.
Again I am baffled... please get me started :-)