Quiz and Formula Sheet Philosophy:

Numerical computation is a motley collection of tools, many of which
require you to use a lot of equations.  Memorizing complex equations
is not an effective use of your time, hence the formula sheet.  This
sheet is not meant to be an exhaustive recipe for each method; rather,
it contains the truly dense, hard-to-remember formulas that are used
in SOME of the methods.  Here's the way to use it: if you needed to
flip back in the book to check something (method algorithm, equation,
minus sign) while you're doing a homework problem, and what you looked
up is not either in the list below or easily derivable by ten seconds
of thought, you should spend some time understanding where it comes
from and/or committing it to memory.  If you feel that some equation
that is missing from the sheet is not worth this time, send me mail.

Most quiz problems are derived directly from the homework problems,
with an occasional "concept" problem thrown in, which you'll be able
to do if you EITHER went to class OR did the reading.  An example of
the latter is "what's the difference between the secant method and the
false position method?"

There will be a concept-level exam during the final exam period.  It
will consist of conceptual questions like

- here's a problem.  what methods solve it?
- here's a method.  what kinds of problems does it solve?
- what's the idea behind method x - how does it work?
- what's the conceptual difference between methods x and y?
- what breaks method x?
- how would you parallelize method x?
- what's the effect of changing parameter x in method y?

This will not require any algebra, arithmetic, or manipulation/use of
formulae.  It may require some manipulation of symbols and is likely
to require several drawings.

I would suggest that you study for the test as follows.  Review all of
the assigned problems, quizzes, and programming assignments.  Think
about how you would know when to use each method and how the solutions
would change with different numbers, more or fewer points, points in
different order, etc.  Make yourself a study sheet: a list of problem
types and methods, and make sure you know the mapping between the two,
as well as the differences between different methods, and what kinds
of problems break each method.  Finally, go over the reading and class
notes and highlight important concepts.

============================================================
	Formula Sheet for CSCI3656 
============================================================

TAYLOR SERIES:

The second equation from the bottom of page A-3 in appendix A, plus
the remainder equation at the top of page A-4.

These also appear in my Taylor Series notes: the first equations on
pages 2 and 4, respectively.

SECANT METHOD:

The lower equation on p44

FIXED-POINT METHOD:

The boxed error convergence condition in the middle of p58

The Aitken acceleration formula at the bottom of p58, including the
definitions of D1 and D2.  (This combines the two equations I put up
in class.  I'll also give you those two equations, in case you prefer
working the algorithm that way.)

MULTIPLE ROOTS:

The boxed equation on p79

CROUT REDUCTION:

eqns 2.20 and 2.21 on p138

SOLVING NONLINEAR EQUATIONS:

eqn 2.40 on p175
eqn 2.42 on p176 

LAGRANGIAN POLYNOMIALS:

- eqn 3.1 on p225

CUBIC SPLINES:

- the matrix equation at the bottom of p243
- the equations for the a_i, b_i, c_i, and d_i at the top of p245
- the definition of divided difference at the bottom of p229
  (note: this is a section that we skipped, but we need this one
  piece of notation for the formulae in the cubic spline section)

BEZIER CURVES AND B-SPLINES:

- the boxed eqns on p250
- the top boxed eqn on p251
- eqn 3.17 and the eqns for x_i(u) and y_i(u) on p254

LEAST SQUARES:

- the "normal eqns" for lines (eqn 3.23) 
- the "normal eqns" for degree-n polynomials (eqns 3.25 and 3.26) 

DERIVATIVES FROM DATA:

- the "formulas for computing derivatives" on pp373-374, !EXCEPT FOR!
  the first two (that is, you should memorize the forward, backward,
  and center difference ones.  note that the backward difference
  formula isn't in this textbook.  use your lecture notes.)

- eqn 5.21, for use in extrapolation

INTEGRALS FROM DATA:

- the boxed "newton-cotes" equations on p376.  note that these three
  equations have other names: trapezoidal, simpson's 1/3, and simpson's
  3/8, respectively.

- the composite trapezoidal (eqn 5.30), simpson's 1/3 (eqn 5.35)
  and simpson's 3/8 (eqn 5.36) rules

- the versions of those same three rules, complete with error terms,
  that appear in the shaded box on p387

- eqn 5.21, for use in extrapolation

- table 5.14, for use in computing integrals by Gaussian
  quadrature, along with the change-of-variable formula at the 
  bottom of p390

ODE SOLVERS:

- the RK4 equations

- the Adams(-Bashforth) formula on p466, and its 2nd-order cousin,
  which I gave in class: 
      X(t_{n+1}) = X(t_n) + h/2 [3 F(X_n,t_n) - F(X_{n-1},t_{n-1})]
  here, I'm using capitals (X,F) to show quantities that may be vectors.
  the former has O(h^4) error, while the latter has O(h^2) error.