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gibson:teaching:fall-2016:math753:finalexam

Math 753/853 final exam topics

Wed, Dec 14, 2016 10:30am-12:30pm Kingsbury N343

Floating point numbers

• binary representation
• how #s of bits in mantissa and exponent lead to # digits in same
• floating point arithmetic: expected accuracy of arithmetic operations
• what is machine epsilon?

Solving 1d nonlinear equations

• bisection: the algorithm, the required conditions, the convergence rate
• Newton: the algorithm, the required conditions, the convergence rate
• when to use bisection, when to use Newton

Gaussian elimination / LU decomposition

• the LU algorithm: what are the formulae for computing the multipliers of ?
• be able to compute the LU decomp of a small matrix by hand
• backsubstitution, forward substitution
• using LU to solve
• pivoting –what is it, why is it a practical necessity?
• what form does the LU decompostion take with pivoting? How do you use this form to solve ?

QR decomposition

• what is a QR decomposition?
• what algorithm do you know for computing the QR decomposition?
• what are the formulae for the elements of and the column vectors of ?
• what is an orthogonal matrix?
• how to use QR decomp to solve a square problem
• how to use QR decomp to find a least-squares solution to an oblong problem ( matrix , with )

Polynomials

• 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 , 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
• 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 problems to find the best fit for functions of the following forms to pairs of datapoints
• an th order polynomial
• an exponential
• a power law
• a curve of the form

• be able to approximate the first & second derivatives of a function from the values where the are evenly spaced gridpoints
• provide error estimates of those approximate derivatives
• be able to approximate the integral of the function from evenly space gridpoint values , 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