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gibson:teaching:fall-2016:math753:finalexam

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 gibson:teaching:fall-2016:math753:finalexam [2016/12/12 18:33]gibson gibson:teaching:fall-2016:math753:finalexam [2016/12/12 19:00] (current)gibson Both sides previous revision Previous revision 2016/12/12 19:00 gibson 2016/12/12 18:33 gibson 2016/12/12 18:21 gibson created 2016/12/12 19:00 gibson 2016/12/12 18:33 gibson 2016/12/12 18:21 gibson created Line 33: Line 33: * Horner'​s method: be able to rearrange a polynomial into Horner'​s form, and understand why you'd do that * Horner'​s method: be able to rearrange a polynomial into Horner'​s form, and understand why you'd do that * Lagrange interpolating polynomial: be able to write down the Lagrange interpolating polynomial passing through a set of data points $x_i, y_i$, and understand why the formula works * Lagrange interpolating polynomial: be able to write down the Lagrange interpolating polynomial passing through a set of data points $x_i, y_i$, and understand why the formula works - ​* ​ + ​* ​Newton divided differences:​ know how to use this technique to find the interpolating polynomial through a set of data points $x_i, y_i$ + * Chebyshev points: what are they, what are they good for, why do we need them? + + Least-squares models + * Understand how to set up least-squares $Ax=b$ problems to find the best fit for functions of the following forms to $m$ pairs of datapoints $t_i, y_i$ + * an $n$th order polynomial + * an exponential $y=c e^{at}$ + * a power law $y=c t^a$ + * a curve of the form $y = c t e^{at}$ + + Finite differencing and quadrature + * be able to approximate the first & second derivatives of a function $y(x)$ from the values $y_i = y(x_i)$ where the $x_i$ are evenly spaced gridpoints $x_i = x_0 + i h$ + * provide error estimates of those approximate derivatives + * be able to approximate the integral $\int_a^b y(x) dx$ of the function $y(x)$ from evenly space gridpoint values $y_i = y(x_i)$, using the Trapeziod Rule and Simpson'​s rule + * provide error estimates for those approximate integrals + + Ordinary differential equations + * what is an initial value problem? + * why do we need to solve initial value problems numerically?​ + * what are the timestepping formulae for + * Forward Euler + * Midpoint Method (a.k.a. 2nd order Runge-Kutta) + * 4th-order Runge-Kutta + * Backwards Euler + * Adams-Moulton + * what are the global error estimates of the above timestepping formulae? + * what is a global error estimate versus a local error estimate, and how are the two related? + * what's the difference between an explicit method and an implicit method? + * what's a stiff differential equation? what kind of method do you use for a stiff  equation? + * how do you convert an $n$th order differential equation in one variable to a system of first order differential equations in $n$ variables?​ +