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RootMeanSquare

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

RootMeanSquare[list] gives the root mean square of values in list.

Examples

All examples below are verified against the current Mathilda build.

In[1]:= RootMeanSquare[{a, b, c, d}]
Out[1]= 1/2 Sqrt[a^2 + b^2 + c^2 + d^2]

In[2]:= RootMeanSquare[{{1, 2}, {5, 10}, {5, 2}, {4, 8}}]
Out[2]= {1/2 Sqrt[67], Sqrt[43]}

In[3]:= RootMeanSquare[{1, 2, 3, 4}]
Out[3]= Sqrt[15/2]

In[4]:= RootMeanSquare[{Pi, E, 2}]
Out[4]= Sqrt[1/3 (4 + E^2 + Pi^2)]

In[5]:= RootMeanSquare[{1., 2., 3., 4.}]
Out[5]= 2.73861

Implementation notes

Algorithm. builtin_rootmeansquare computes Sqrt[Mean[x^2]]. It reduces matrices column-wise via apply_columnwise and requires a List. For data containing a real, it sums squares in double and returns expr_new_real(sqrt(sum_sq/n)). For exact/symbolic data it builds Plus[x_i^2 ...], then carefully distributes the square root to keep results exact: if the summed result is non-numeric and n is a perfect square it factors out 1/Sqrt[n]; if the mean-square is a rational with a perfect-square denominator it pulls that root out before applying Power[..., 1/2]. Otherwise it returns Power[meanSq, 1/2] for the evaluator to simplify. ATTR_PROTECTED.

  • Protected.
  • Gives the square root of the second sample moment.
  • For a list {x1, x2, ...}, it computes Sqrt[1/n Total[{x1^2, x2^2, ...}]].
  • Handles both numerical and symbolic data.
  • Works column-wise on matrices.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= RootMeanSquare[{3, 4}]
Out[1]= 5/Sqrt[2]

In[1]:= RootMeanSquare[{a, b}]
Out[1]= Sqrt[1/2 (a^2 + b^2)]

In[1]:= RootMeanSquare[Range[10]]
Out[1]= Sqrt[77/2]

In[2]:= N[RootMeanSquare[{1, 2, 3, 4, 5}], 30]
Out[2]= 3.316624790355399849114932736672

In[1]:= N[RootMeanSquare[Table[Sin[n], {n, 1, 1000}]], 20]
Out[1]= 0.707242937053949660224

Notes

RootMeanSquare[list] returns Sqrt[Mean[list^2]] and stays exact on exact input — a list of two symbols yields the closed form Sqrt[(a^2 + b^2)/2]. The final example is a numerical demonstration of an analytic fact: the RMS of Sin sampled over many integer arguments approaches 1/Sqrt[2] ≈ 0.70711, the continuous RMS of a sinusoid.