Study Guide for the final.
Math. 417
The final is cumulative, with rough weighting, 40% for the first half and
60% for the second. From the first half, you will be responsible for basic
polynomial interpolation, finite differences, and basic numerical integration
(excludes Gaussian quadrature).
LU-factorization and Direct Methods. Given a factorization of a matrix A into LU form, know how to reduce the solution to Ax = b into back
and forward solves. Know how to implement the back solve (or forward
solve) algorithms. Know the difference between an LU factorization and a
Choleski factorization. I will not ask you to develop algorithms for either
LU or Choleski decomposition.
Linear iterative methods for linear systems. Given a linear iterative
method for a matrix problem Ax = b, with A an n × n real valued matrix,
know how to find the reduction matrix, i.e., the matrix G satisfying
ei+1 = Gei .
Rn
Know how to use norms on
and the definition of the matrix norm induced by a vector norm. Know how a bound for an induced matrix norm of
the reducer G implies rates of convergence of the iterative method. Know
the definitions of the Jacobi, Gauss-Seidel and SOR methods and how to
compute reduction matrices for Richardson’s method, Jacobi’s method, and
the Gauss-Seidel method.
Nonlinear iterative methods for linear systems. Understand the
derivation of the steepest descent method. It is based on changing xj by
a multiple of residual so that the new error is orthogonal (in the A-inner
product) to the residual. Know how to use the properties of the inner product to develop the coefficient appearing in the iteration. Know how to show
that the resulting error is the minimum error over all such multiples.
Numerical ODE’s. We considered several techniques for developing numerical approximation to ODE’s in this class. In all cases, the schemes are
based frist on an equation involving the scheme applied to the solution of
the ODE plus an error term, e.g., forward Euler comes from
(0.1)
y(tj+1 ) = y(tj ) + ky ′ (tj ) + O(k 2 )
1
2
where k = tj+1 − tj . You should be able to derive such an equation (with
an explicit error term) either from Taylor series or using a result for the
forward difference approximation to the derivative. Given an equation such
as (0.1), you should be able to write down the corresponding numerical ODE
approximation and state its approximation order. You should also be able to
derive Adams-Bashford and Adams-Moulton schemes, based on quadrature
approximation to the integral:
Z tj+1
µ+m
X
f (t, y(t)) dt ≈
wl f (tj+1−ℓ , y(tj+1−ℓ )) + O(k m+2 )
tj
l=µ
where the weights are chosen so that the scheme is exact on Pm .