## Quick Reference

*K* such that |*f*(*x*)| ≤ *K*ϕ(*x*) for all *x* ≥ *x*_{1}, then we say that *f*(*x*) is order ϕ(*x*) as *x* tends to infinity, and we write *f*(*x*) = O(ϕ(*x*) For example, 100*x*^{2} + 100*x* + 2 = O(*x*^{2}) as *x* → ∞ If then we write *f*(*x*) = o(*g*(*x*) For example, *x* = o(*x*^{2}) as *x* → ∞ Both these notations are statements about maximum magnitude and do not exclude *f* from being of smaller magnitude. For example, *x* = O(*x*^{2}) is perfectly valid, but equally *x* = O(*x*) If then we write *f*(*x*) ≅ *k**g*(*x*) as *x* → *a* For example, 10*x*^{2} + *x* + 1 ≅ 10*x*^{2} as *x* → ∞ The term order and the O notation is used in numerical analysis, particularly in discretization methods. In ordinary differential equations, if *h* denotes the stepsize, then a method (or formula) has order *p* (a positive integer) if the global discretization error is O(*h** ^{p}*). This means that as the step size

*h*is decreased, the error goes to zero at least as rapidly as

*h*

*. Similar considerations apply to partial differential equations. High-accuracy formulas (order up to 12 or 13) are sometimes used in methods for ordinary differential equations. For reasons of computational cost and stability, low-order formulas tend to be used in methods for partial differential equations.*

^{p}|*f*(*x*)| ≤ *K*ϕ(*x*)

*f*(*x*) = O(ϕ(*x*)

100*x*^{2} + 100*x* + 2 = O(*x*^{2})

as *x* → ∞

*f*(*x*) = o(*g*(*x*)

*x* = o(*x*^{2}) as *x* → ∞

*x* = O(*x*^{2})

*x* = O(*x*)

*f*(*x*) ≅ *k**g*(*x*) as *x* → *a*

10*x*^{2} + *x* + 1 ≅ 10*x*^{2}

as *x* → ∞

The term is also used to refer to the speed of convergence of iteration schemes, for example Newton's method for computing the zero of a function *f*(*x*). Subject to appropriate conditions, Newton's method converges quadratically (or has order of convergence 2), i.e. an approximate squaring of the error is obtained in each iteration.

*Another name for* operation code.

**From:**
order
in
A Dictionary of Computing »

*Subjects:*
Computing.

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