1.2: What is iteration?
- Page ID
- 143380
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Have you ever idly repeatedly pressed one of the buttons on your calculator? Consider for example the \(\sqrt{ }\) button. You could compute
\[
\sqrt{2}, \sqrt{ } \sqrt{2}, \sqrt{ } \sqrt{ } \sqrt{2}, \sqrt{ } \sqrt{ } \sqrt{2}, \ldots
\nonumber \]
successively by entering the number 2 and pressing the \(\sqrt{ }\) button repeatedly. This is an example of iteration, that is, repeated application of the same function given some initial value. Numbers obtained in this way are called iterates of the starting value.
If you carry out the process above you will be rewarded with a sequence of numbers
\[
1.414 \ldots, 1.189 \ldots, 1.090 \ldots, 1.044 \ldots, 1.021 \ldots, 1.010 \ldots, \ldots
\nonumber \]
which are getting nearer and nearer to 1 . We say that the sequence is converging to 1 . Indeed, you will find the same behavior if you start with any other positive number. For example, beginning instead with 5, you obtain the sequence
\[
2.236 \ldots, 1.495 \ldots, 1.222 \ldots, 1.105 \ldots, 1.051 \ldots, 1.025 \ldots, \ldots
\nonumber \]
approaching 1 from above, and beginning with 0.2 , you will get
\[
0.447 \ldots, 0.669 \ldots, 0.818 \ldots, 0.904 \ldots, 0.951 \ldots, 0.975 \ldots, \ldots
\nonumber \]
which approaches 1 from below.
Even a moderately complicated function \(f(x)\) can give rise to a highly complicated sequence of iterates
\[
x_0, x_1=f\left(x_0\right), x_2=f\left(x_1\right), x_3=f\left(x_2\right), \ldots
\nonumber \]
Indeed iteration is very much a field of active current research.
Iteration is also of some practical importance. Your education to date has been rather misleading. You may think that most equations can be solved exactly. Nothing could be further from the truth!
Most equations can only be solved approximately, and, as we may show in a later investigation, iteration is a useful method for obtaining ever more accurate approximate solutions.