CS 205 – class 1
Types of Errors
Covered in class: 4, 6, 7
1.
When doing integer calculations one can many times proceed exactly, except of course in
certain situations, e.g. division 5/2=2.5. However, when doing floating point calculations
rounding errors are the norm, e.g. 1./3.=.3333333… cannot be expressed on the computer.
Thus the computer commits rounding errors to express numbers with machine precision, e.g.
1./3.=.3333333. Machine precision is
7
10
−
for single precision and
16
10
−
for double precision.
Rounding errors are only one source of approximation error when considering floating point
calculations. Some others are listed below.
2.
Approximation errors come in many forms:
a.
empirical constants
– Some numbers are unknown and measured in a laboratory only
to limited precision. Others may be known more accurately but limited precision
hinders the ability to express these numbers on a finite precision computer. Examples
include Avogadro’s number, the speed of light in a vacuum, the charge on an electron,
Planck’s constant, Boltzmann’s constant, pi, etc. Note that the speed of light is
299792458 m/s exactly, so we are ok for double precision but not single precision.
b.
modeling errors
– Parts of the problem under consideration may simply be ignored.
For example, when simulating solids or fluids, sometimes frictional or viscous effects
respectively are not included.
c.
truncation errors
– These are also sometimes called discretization errors and occur in
the mathematical approximation of an equation as opposed to the mathematical
approximation of the physics (i.e. as in modeling errors). We will see later that one
cannot take a derivative or integral exactly on the computer so we approximate these
with some formula (recall Simpson’s rule from your Calculus class).
d.
inaccurate inputs
– Many times we are only concerned with part of a calculation and
we receive a set of input numbers and produce a set of output numbers. It is important
to realize that the inputs may have been previously subjected to any of the errors listed
above and thus may already have limited accuracy. This can have implications for
algorithms as well, e.g. if the inputs are only accurate to 4 decimal places, it makes little
sense to carry out the algorithm to an accuracy of 8 decimal places. This issue
commonly resurfaces in scientific visualization or physical simulation where
experimental engineers can be unhappy with visualization algorithms that are “lossy”,
meanwhile forgetting that the part that is lost may contain no useful, or accurate
information whatsoever.

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