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====== 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 $\ell_ij$ of $L$? * be able to compute the LU decomp of a small matrix by hand * backsubstitution, forward substitution * using LU to solve $Ax=b$ * 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 $Ax=b$? QR decomposition * what is a QR decomposition? * what algorithm do you know for computing the QR decomposition? * what are the formulae for the elements $r_ij$ of $R$ and the column vectors $q_j$ of $Q$? * what is an orthogonal matrix? * how to use QR decomp to solve a square $Ax=b$ problem * how to use QR decomp to find a least-squares solution to an oblong $Ax=b$ problem ($m \time n$ matrix $A$, with $M>n$) 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 $x_i, y_i$, and understand why the formula works *
