Introduction to Scientific Computing - Tutorial 13: Recapitulation
Introduction to Scientific Computing - Tutorial 13: Recapitulation
Introduction to Scientific Computing - Tutorial 13: Recapitulation
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Fixed-point iterations<br />
Banach fixed point<br />
Lipshitz continuity (for q 0):<br />
If this holds when 0 q < 1 then<br />
‖Φ(x) − Φ(y)‖ q‖x − y‖<br />
Φ(x ∗ ) = x ∗ has unique solution<br />
for any initial value sequence x n+1 = Φ(x n ) converges <strong>to</strong><br />
solution x ∗ (this means that iterative method is convergent and<br />
gives the solution)<br />
speed of convergence are given by apriori and aposteriori<br />
estimates<br />
1. Februar 20<strong>13</strong> Bojana Rosic <strong>Introduction</strong> <strong>to</strong> <strong>Scientific</strong> <strong>Computing</strong> Seite 20
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