In this course, we try to create sequences
that converge to some desired point (most often a local minimizer). The fundamental question is how fast the convergence is.

Assume . Convergence is -linear if there exists a constant such that

**Example: **
. Note
.

Then,

We say that the convergence of a sequence is -superlinear if

About -linear convergence: It could be so bad that numerically we can barely see it; e.g.,

We say that convergence is -quadratic if

**Note: ** -quadratic convergence implies -superlinear convergence (to see this, multiply both sides of the equation by
.

We need -linear convergences for examples like the following:

Usually, we will just drop the in -linear and say linear.

*Friday, 1-28-2005*