gibson:teaching:fall-2016:math753:finalexam

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* 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? | ||

+ | | ||

gibson/teaching/fall-2016/math753/finalexam.txt · Last modified: 2016/12/12 19:00 by gibson