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Re: [Phys-l] Significant figures -- again



On 03/30/2012 11:21 AM, David Bowman wrote:
Of course sometimes it *is* necessary to expand a function to
preserve significance in the difference involving a subtraction of
nearly equal quantities.

Agreed.

And sometimes it is convenient, even if it is not strictly
necessary.

.... I would recommend the
series expansion method only *after* one has decided that there is no
discernible way to rewrite things to avoid the subtraction.

IMHO that's going a bit too far.

Sometimes champagne is nice, but sometimes I'd really rather have
a glass of water.

1) For example, in many cases it is preferable to write the kinetic
energy as
KE = p·v/2 [1]
even though special relativity tells us eq. [1] is only the lowest
term in an expansion in powers of v/c. Equation [1] is plenty
accurate in a wide range of situations, and is easy to evaluate,
easier than any of the fully-relativistic formulas.

2) As another example: Sometimes you want a full description
of a pendulum in terms of elliptic integrals. On the other hand,
sometimes it is preferable to approximate it as a harmonic oscillator.
This is tantamount to making a first-order approximation to the sine
function in the equation of motion. On the third hand, sometimes
it makes sense to include the lowest-order anharmonic term without
bothering with the elliptic integrals.

*) et cetera.

====================

As for the "big root, small root" trick: That is easy to remember,
and easy to generalize to polynomials of all orders, as follows:
Write the polynomial in standard form:

Σ k_i x^i = 0 [2]

then k_0/k_N is the product of the roots ... as you can easily
verify by induction (or otherwise). If you have found N-1 of
the roots, you can always find the last one by dividing k_0/k_N
by all of the others. No subtraction required.

==

That will not, however, save you from all problems with the
quadratic formula.

-- In the case of the characteristic polynomial for the
eigenvalues of a 2x2 matrix, as we discussed yesterday, you
can incur catastrophic roundoff errors *before* you even get
the polynomial into standard form.

-- In addition to the "big root, small root" problem, you can
also get into trouble if the discriminant is almost zero,
i.e. if there are two roots close together. If there is any
appreciable uncertainty in the coefficients of the polynomial,
this can lead to a bizarrely non-normal distribution of roots,
including complex roots ... which may /or may not/ be unphysical.

The widely-taught pseudo-sophisticated "propagation of uncertainty"
techniques fail miserably in this case. They fail without giving
any indication that they have failed, which all-too-often causes
people to have full confidence in the wrong answer.
http://www.av8n.com/physics/uncertainty.htm#sec-quad-roots