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NormFunction

is an option for functions such as FindFit and NDSolve which gives a function to be minimized in generating results.

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NormFunction

NormFunction

is an option for functions such as FindFit and NDSolve which gives a function to be minimized in generating results.

Details

  • NormFunction->f specifies that f[data] should be minimized in generating results.

Examples

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Basic Examples  (2)

Find a "best" fit for data using different norms:

Wolfram Language code: data = Table[Prime[x], {x, 20}]

The default is to find the best least-squares fit:

Wolfram Language code: FindFit[data, a x Log[b + cx], {a, b, c}, x]

Use the -norm instead:

Wolfram Language code: FindFit[data, a x Log[b + cx], {a, b, c}, x, NormFunction -> (Norm[#, Infinity]&)]

Use the 1-norm:

Wolfram Language code: FindFit[data, a x Log[b + cx], {a, b, c}, x, NormFunction -> (Norm[#, 1]&)]

Use a 2-norm for estimating local error in solving an ODE:

Wolfram Language code: sol2 = First[x /. NDSolve[{x''[t] + x[t] == 0, x[0] == 1, x'[0] == 0}, x, {t, 0, 100π}, NormFunction -> (Norm[#, 2]&)]]

Use an -norm:

Wolfram Language code: soli = First[x /. NDSolve[{x''[t] + x[t] == 0, x[0] == 1, x'[0] == 0}, x, {t, 0, 100π}, NormFunction -> (Norm[#, ∞]&)]]

The error measure makes a difference in the solution quality:

Wolfram Language code: {sol2[100π], soli[100π]} - 1

Scope  (1)

The norm is used for space and time in PDE solutions:

Wolfram Language code: sol[p_] := NDSolve[{D[u[t, x], t, t] == D[u[t, x], x, x], u[0, x] == Exp[-x ^ 2], Derivative[1, 0][u][0, x] == -Exp[-x ^ 2], u[t, -L] == u[t, L]}, u, {t, 0, 4L}, {x, -L, L}, NormFunction -> (Norm[#, p]&)]

Plot the actual solution error, when using different error estimation norms:

Wolfram Language code: L = 10;Table[ Plot[Evaluate[(u[40, x] /. sol[p]) - (Exp[-x ^ 2] - 2Sqrt[π])], {x, -L, L}, PlotRange -> All], {p, {1, 2, 3, ∞}}]//Quiet

A plot of the best solution:

Wolfram Language code: Plot3D[Evaluate[u[t, x] /. sol[1]], {t, 0, 40}, {x, -10, 10}, Mesh -> False, PlotPoints -> 20]
Wolfram Research (2003), NormFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/NormFunction.html.

Text

Wolfram Research (2003), NormFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/NormFunction.html.

CMS

Wolfram Language. 2003. "NormFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NormFunction.html.

APA

Wolfram Language. (2003). NormFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NormFunction.html

BibTeX

@misc{reference.wolfram_2026_normfunction, author="Wolfram Research", title="{NormFunction}", year="2003", howpublished="\url{https://reference.wolfram.com/language/ref/NormFunction.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_normfunction, organization={Wolfram Research}, title={NormFunction}, year={2003}, url={https://reference.wolfram.com/language/ref/NormFunction.html}, note=[Accessed: 13-June-2026]}