Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 41C / c

Table of contents
- Solution c

Solution

The gradient descent iteration is

x^{(k + 1)} = (I - α A) x^{(k)} + α b .

For the error

e^{(k)} = x^{(k)} - A^{- 1} b

,

e^{(k + 1)} = (I - α A) e^{(k)} .

Part (a) shows that convergence for every initial vector is equivalent to

∣1 - α λ_{i} ∣ < 1

for every eigenvalue of

A

. Since all eigenvalues are positive,

0 < α < \frac{2}{λ _{m a x} ( A )} .

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version