Solving Matrix/Error Proof? Get a Hint Here!

[Gameowner]

Homework Statement

Not too sure what this proof would be under, but am pretty sure it has to do with approximation errors, can anyone give me a hint as to how to even start this?

Homework Equations

The Attempt at a Solution

For the first one, my guess at the solution would be...

1)take the norm of both sides of x = Tx + b
2) then substitute in x^k = Tx^(k-1) + b into the x's
3)since x^k = Tx^(k-1) + b, we get norm( x^k = T(Tx^(k-1) + b) + b), which then we get infinite multiples of T. Hence T^k?

That's really wild guess at the solution, any help would be much appreciated

 

[qbert]

for the first part you might look at something like
|x^k+1 - x| = |(Tx^k + b) - (Tx + b)| = |T(x^k -x)|
and continue till k->0.


for the second, maybe try to establish something like
|x^k - x| <= ||T|| |x^k-1 - x| <= ||T|| ( |x^k-1 - x^k| + |x^k - x|).

Then solve for |x^k - x| in terms of |x^k-1 - x^k|.

And then try to figure out how to go from
|x^k-1 - x^k| to |x⁰ - x¹|.

 

[kudaparadza]

for the first part you might look at something like
|x^k+1 - x| = |(Tx^k + b) - (Tx + b)| = |T(x^k -x)|
and continue till k->0.for the second, maybe try to establish something like
|x^k - x| <= ||T|| |x^k-1 - x| <= ||T|| ( |x^k-1 - x^k| + |x^k - x|).

Then solve for |x^k - x| in terms of |x^k-1 - x^k|.

And then try to figure out how to go from
|x^k-1 - x^k| to |x⁰ - x¹|.

Dear qbert

For the 2nd part of the question. Are we meant to figure out how to go from |x^k - x^k-1| to |x¹ - x⁰| rather than the given above |x^k-1 - x^k| to |x⁰ - x¹|

my attempt at the question is as follows

||x(k) - x|| <= ||T|| ||x(k-1) - x(k)|| + ||T|| ||x(k) - x||

||x(k) - x|| - ||T|| ||x(k) - x|| <= ||T|| ||x(k-1) - x(k)||

||x(k) - x|| (1 - ||T||) <= ||T|| ||x(k-1) - x(k)||

solving for ||x(k) - x|| <= (||T||/(1 - ||T||)) ||x(k-1) - x(k)||

hence my query above.

Kind regards,
kcp

 

[kudaparadza]

for the first part you might look at something like
|x^k+1 - x| = |(Tx^k + b) - (Tx + b)| = |T(x^k -x)|
and continue till k->0.


for the second, maybe try to establish something like
|x^k - x| <= ||T|| |x^k-1 - x| <= ||T|| ( |x^k-1 - x^k| + |x^k - x|).

Then solve for |x^k - x| in terms of |x^k-1 - x^k|.

And then try to figure out how to go from
|x^k-1 - x^k| to |x⁰ - x¹|.

Dear qbert

For the 2nd part of the question. Are we meant to figure out how to go from |x^k - x^k-1| to |x¹ - x⁰| rather than the given above |x^k-1 - x^k| to |x⁰ - x¹|

my attempt at the question is as follows

||x(k) - x|| <= ||T|| ||x(k-1) - x(k)|| + ||T|| ||x(k) - x||

||x(k) - x|| - ||T|| ||x(k) - x|| <= ||T|| ||x(k-1) - x(k)||

||x(k) - x|| (1 - ||T||) <= ||T|| ||x(k-1) - x(k)||

solving for ||x(k) - x|| <= (||T||/(1 - ||T||)) ||x(k-1) - x(k)||

hence my query above.

Kind regards,
kcp