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.