Lecture Topics in 421, Spring 01
- Tuesday January 16: introduction, approximations in scientific computing, data error,
model errors, computational errors, sensitivity of functions and conditioning
- Thursday January 18: floating point systems, scientific notation, single and double precision IEEE numbers
- Tuesday January 23: IEEE floating point systems, rounding of numbers, binary operations, effects of cancellation on accuracy, solving quadratic equations
- Thursday January 25: cancellation and truncation error when replacing derivatives with difference quotients, review of Taylor's formula, computing exponential function values using a series.
- Tuesday January 30: Recurrence relations, stable and unstable. Computing Fibonacci
numbers forwards and backwards.
- Thursday February 1: Review of Gaussian elimination. Triangular matrices, LU
factorization of matrices. Partial pivoting.
- Tuesday February 13: Vector and matrix norms. Conditioning of linear
systems of algebraic equations.
- Thursday February 15: Creating matrices in MATLAB. Vectorizing forward
elimination and backward substitution. Cholesky's algorithm. Sparsity. Solving
tridiagonal linear systems.
- Tuesday February 20: Introduction to least squares problems. The normal
equations. Augmented systems. Using the QR factorization to solve
least squares problems.
- Thursday February 22: Solving underdetermined systems. Householder
transforms and their use in computing QR factorizations of matrices.
- Tuesday February 27: Givens transforms, Gram-Schmidt orthogonalization.
- Thursday March 1: Midterm exam.
- Tuesday March 6: Discussion of problems 2 and 3 of the midterm.
Introduction to polynomial interpolation: Horner's rule, interpolation
by using Vandermonde matrices, Lagrange's method, Newton's formula.
- Thursday March 8: Newton's interpolation formula; an example and a
proof of correctness. The symmetry of the divided difference
formulas. Hermite cubic interpolation: solvability and uniqueness.
- Tuesday March 20: Hermite cubic interpolation computed by Newton's
interpolation formula. Cubic spline interpolation; derivation of
a tridiagonal linear system of equations to determine the values
of the first derivative of the spline function at the interior nodes
from the requirement that the spline has two continuous derivatives.
- Thursday March 22: Demonstration of Runge's example for equidistant
interpolation points and the Chebyshev set of points. Proof that the
tridiagonal linear system for the spline interpolant can be solved
without using pivoting.
- Tuesday March 27: Numerical integration. Rectangular, midpoint, trapezoidal
and Simpson's rule. Error estimates of the different quadrature rules by
using the error estimate of polynomial interpolation and the second mean
value theorem from calculus. Adaptive quadrature, in particular adaptive
- Thursday March 29: How Archimedes and many others computed approximations
of pi. Huygens's idea. Richardson extrapolation to determine pi and to
derive Simpson's rule from the trapezoidal rule.
- Tuesday April 3: Finding roots of nonlinear equations. The bisection
method, Newton's method, the secant method, and regula falsi. Inverse
- Thursday April 5: Convergence rates of Newton's and the secant methods.
The Illinois method. Solving systems of nonlinear equations with Newton's
- Tuesday April 10: Computing the square root using Newton's method and a
table of initial values. Finding the minimum of a function by Newton's
method, succesive parabolic interpolation, and Golden Section search.
- Thursday April 12: Ordinary differential equations. Conditions for
uniqueness and solvability; examples of equations with several solutions
and blow up of the solution after a finite time. Euler's method and its
convergence. Higher order methods based on Taylor series and multiple
evaluations of the function in each time step. Runge-Kutta schemes with
two and four function evaluations.
- Tuesday April 17: Numerical solution of initial value problems for ODE.
Turning higher order equations into first order. A study of different
methods when applied to a simple linear equation with constant
coefficients. Stability diagrams. Implicit methods including the
trapezoidal scheme and backward Euler.
- Thursday April 19: A detailed analysis of the midpoint rule. Using
polynomial interpolation formulas to derive multistep methods. Adapative
- Tuesday April 24: Review of floating point, solving linear systems, and
least squares problems.
Attendance at all lectures is REQUIRED. If you are unable to attend,
you must send me an email with an explanation.