MATH 521 Additional Online Reading List
Prof. Zachariah B. Etienne
Note 1: All links to Wikipedia reference versions of the article
that have been vetted by the instructor, and may be considered trustworthy.
Note 2: This reading list should be considered as a
complement to the lecture notes as well as the suggested text for
this course: (Numerical Recipes, second edition or
higher)
Note 3: For many terms in Numerical Analysis, multiple definitions
may be found. In such a case, only the lecture notes' definition
will be accepted.
Preliminaries:
All material in MATH 521 builds on basic undergraduate
mathematics, and when you start this course, I assume you are
already well-versed in undergraduate mathematics.
If you need to brush up, here are some resources that may
help:
Tips for determining how many digits of
significance you can expect from a double-precision calculation (note
that this
neglects guard
digits):
Evaluating the expression according to proper order-of-operation,
check for arithmetic steps that go out of bounds for double precision
arithmetic. (Note that the smallest nonzero number is roughly plus or minus
1e-308; largest non-infinite number is roughly plus or minus
1e+308). If this happens, evaluate to zero or infinity as appropriate. You might still
retain some digits of significance.
Check for catastrophic cancellation.
Check for numbers that are exactly representable by double
precision. If these exist, they are
known to an infinite number of significant digits. If not, they are
generally known to only 15--16 significant digits.
Dividing or subtracting two numbers that are identical to all
significant digits will yield exactly one or zero, respectively.
Please contact the instructor if you would like to contribute further tips.
Example problems:
When the following expressions are evaluated by the computer,
to how many significant decimal digits will the numerical result
agree with the exact result? Your answer to this part will consist of
a single integer (infinity is an acceptable answer), and the numerical
answer will be accepted so long as it is correct to within 1 decimal
digit.
For the purpose of this problem, apply IEEE 754 standard-compliant
double precision arithmetic, assuming all given
integers represented in integer or floating point format between
-253 and 253 (i.e., integers between ≅
-9×1015
and ≅ 9×1015
inclusive) are exact (for
example, 2.01e2 is exact), as well as powers of
1/2. Otherwise, assume that in double precision the number
is only known to 16 significant digits, and that machine epsilon is
4e-16. We use computer scientific notation, such that, e.g.,
5.63e22 = 5.63\times 10^{22}$.