Definition:Direct Iteration

Definition

Direct iteration is iteration of the form:

$x_{n + 1} = \map \phi {x_n}$

for $n = 0, 1, 2, \ldots$, where $x_0$ is chosen as a first approximation to the desired root.

Also known as

Direct iteration is also known as fixed-point iteration.

Examples

Example: $x = \cos x$

The following is a graph of the direct iteration $x_{n + 1} = \map \cos {x_n}$ for $n = 0, 1, 2, 3, 4$ where $x_0$ is an arbitrary point such that $1 < x_0 < \dfrac \pi 2$:

Also see

• Results about direct iteration can be found here.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): direct iteration
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): iteration
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): direct iteration
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): iteration
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): fixed-point iteration