Classes for David Stewart
(Schedule subject to change without notice.)
Because of FERPA (i.e., privacy) requirements, all courses are ICON courses.
Spring 2008
This is a 1 s.h. seminar type course.
You will hear about many different areas of mathematics and the
research work done by faculty members. So be prepared to find out
about algebra, number theory, topology, differential equations,
geometry of curves and surfaces, knots, chaos, fractals, wavelets,
quantum mechanics, and logic. Sophomore level background assumed.
This course covers numerical methods for ordinary differential
equations dx/dt = f(t,x), how to solve linear systems A x
= b, solve least-squares
problems, find eigenvalues and eigenvectors, and possibly other matrix
computations. It is a companion course to 22M:170/22C:170. They can be taken in either order.
Fall 2007
- 22M:170/22C:170 Numerical Analysis I (Nonlinear Equations &
Approximation Theory) ICON Syllabus
This course is one of two foundational
courses in Numerical Analysis. This one covers similar material to
22M:072/22C:072 but in much greater depth. Topics covered
include: Floating point
arithmetic: round-off error, error analysis, catastrophic
cancelation; Solution of nonlinear
equations: bisection, secant, Newton's methods, multivariate
versions; Interpolation:
polynomial interpolation, divided differences, error estimates,
trigonometric interpolation, spline interpolation; Approximation theory: minimax and
least-squares approximation, equioscillation theorem, orthogonal
polynomials.
Spring 2007
- 22M:072/22C:072 Elementary Numerical
Analysis (Section 231) ICON
This is the starting point for
numerical analysis, which is about how
we use computers to solve mathematical problems (which might come from
engineering, physics, biology, economics etc., etc., etc.). You
will
learn about roundoff error (because we can't represent the
infinitely
many digits of ``pi'', for example), and how to estimate the growth
or decay of errors. We will see how to solve equations, how
to approximate functions, compute integrals, and solve differential
equations. This course is an ICON course.
- 22M:171/22C:171 Numerical Analysis II
(Differential Equations, Solving Linear Systems and related problems)
ICON
This course covers numerical methods for ordinary differential
equations dx/dt = f(t,x), how to solve linear systems A x
= b, solve least-squares
problems, find eigenvalues and eigenvectors, and possibly other matrix
computations. It is a companion course to 22M:170/22C:170.
This course is an ICON course.
Fall 2006
This course is about mathematical
models of things that have continuous variation in time and/or space,
like fluid flow (air and water: rivers, weather, airplane flight),
solid mechanics (stress and strain, why buildings stay up (or not)),
electromagnetism (radio waves, static electricity), and biology (animal
migrations and population dynamics). We will talk about processes
and concepts like diffusion, momentum, convection, and energy.
There will even be a field trip (or two)!
- 22M:170/22C:170 Numerical Analysis I (Nonlinear Equations &
Approximation Theory)
This course is one of two foundational
courses in Numerical Analysis. This one covers similar material to
22M:072/22C:072 but in much greater depth. Topics covered
include: Floating point
arithmetic: round-off error, error analysis, catastrophic
cancelation; Solution of nonlinear
equations: bisection, secant, Newton's methods, multivariate
versions; Interpolation:
polynomial interpolation, divided differences, error estimates,
trigonometric interpolation, spline interpolation; Approximation theory: minimax and
least-squares approximation, equioscillation theorem, orthogonal
polynomials.
Spring 2006
- 22M:174 Optimization Techniques ICON
Optimization:
Unconstrained and constrained optimization; necessary and sufficient
conditions; when optima exist; convex functions and sets; local vs.
global minima; development of reliable, efficient algorithms.
- 22M:072/22C: Elementary Numerical
Analysis ICON
This is the starting point for
numerical analysis, which is about how
we use computers to solve mathematical problems (which might come from
engineering, physics, biology, economics etc., etc., etc.). You
will
learn about roundoff error (because we can't represent the
infinitely
many digits of ``pi'', for example), and how to estimate the growth
or decay of errors. We will see how to solve equations, how
to approximate functions, compute integrals, and solve differential
equations.
Fall 2005
Fundamental methods and concepts of
the differential and integral
calculus. Limits, tangents and chords, derivatives. Differentiation.
How to differentiate common functions, sums, products, ratios, etc. Techniques
of integration. Integration as ``anti-differentiation'',
integration
as ``the area under the curve'', how to compute integrals, the
fundamental
theorem of calculus, tips & tricks. Applications of the calculus.
How to compute velocities and accelerations, areas, volumes, averages.
- 22M:072/22C: Elementary Numerical
Analysis ICON
This is the starting point for
numerical analysis, which is about how
we use computers to solve mathematical problems (which might come from
engineering, physics, biology, economics etc., etc., etc.). You
will
learn about roundoff error (because we can't represent the
infinitely
many digits of ``pi'', for example), and how to estimate the growth
or decay of errors. We will see how to solve equations, how
to approximate functions, compute integrals, and solve differential
equations.
Spring 2005
- 22M:174 Optimization Techniques
(Syllabus) Blackboard
Optimization:
Unconstrained and constrained optimization; necessary and sufficient
conditions; when optima exist; convex functions and sets; local vs.
global minima; development of reliable, efficient algorithms.
- 22M:321 Topics in Applied
Mathematics: Mathematics and
mechanics of contact and impact (Syllabus) Blackboard
In this course we will develop mathematical models to describe and
simulate contact and impact situations where solid bodies touch and
impact each other. This has applications to robotics,
biomechanics (e.g., How do we walk, run, jump?), computer graphics
(e.g., games), and manufacturing (e.g., getting parts in the right
position). These models are based on differential equations, but
we have to incorporate contact conditions and impulsive forces into
these equations.
Fall 2004
Fundamental methods and concepts of
the differential and integral
calculus. Limits, tangents and chords, derivatives. Differentiation.
How to differentiate common functions, sums, products, ratios, etc. Techniques
of integration. Integration as ``anti-differentiation'',
integration
as ``the area under the curve'', how to compute integrals, the
fundamental
theorem of calculus, tips & tricks. Applications of the calculus.
How to compute velocities and accelerations, areas, volumes, averages.
Spring 2003
- 22M:025 Calculus I (Liberal Arts) sections
BBB
and CCC. Syllabus and Resources.
Fundamental methods and concepts of the differential and integral
calculus. Limits, tangents and chords, derivatives. Differentiation.
How to differentiate common functions, sums, products, ratios, etc. Techniques
of integration. Integration as ``anti-differentiation'',
integration
as ``the area under the curve'', how to compute integrals, the
fundamental
theorem of calculus, tips & tricks. Applications of the calculus.
How to compute velocities and accelerations, areas, volumes, averages.
- 22M:171/22C:171 Numerical Analysis II:
Differential equations
and linear algebra.
Syllabus and Resources.
Numerical solution of differential equations: Euler method,
implicit Euler method, midpoint rule, Runge-Kutta methods, multistep
methods; solution
of linear systems: Gaussian eleimination and LU factorization,
error
estimates, condition numbers, tridiagonal and other sparse systems of
linear
equations, Cholesky factorization; eigenvalues and eigenvectors:
power method, inverse power method, other methods for eigenvalues and
eigenvectors.
Fall 2002
- 22M:170/22C:170
Numerical Analysis
I: Solving equations and approximation theory
Floating point arithmetic; nonlinear equations:
bisection,
Newton, secant, multivariate Newton; polynomial
interpolation: error estimates, choice of interpolation nodes; approximation
of functions; numerical
integration.
Spring 2002
No teaching. I was away on sabbatical (Faculty Scholar award).
Fall 2001
- 22M:026 Calculus II (Liberal Arts)
This is the second of a two-semester sequence in the Calculus.
We will talk about techniques of integration, improper
integrals, applications
of integration (arclength, area, volume, hydrostatic pressure,
centers
of mass), differential equations (which are vital for modeling
most
things physical, biological, chemcial etc.), parametric equations
and
polar coordinates (for finding your way around curves and curved
surfaces), sequences
and series and their convergence, and how to approximate functions
with power series.
- 22M:042 Multivariate
Calculus for Engineering
Most physical quantities (e.g., temperature, stress, density) depend
on several coordinates and many of these are vector functions
(e.g.,
velocity, electric and magnetic fields), so to really model these
things
we need functions of several variables, and how to do calculus with
them.
So, after reviewing lines, planes and vectors, we introduce curves
in space before going on to partial derivatives of
functions
of several variables and their applications (like finding minima and
maxima).
Then we look at multiple integrals for finding areas, volumes,
masses
etc. Along the way we use polar coordinates which help when we
have
nice shapes like circles and spheres to deal with. Finally we come to vector
calculus and we discover the very important Green's, Stoke's and
divergence
theorems. If you go on to do advanced modeling and simulation
(using
partial differential equations) you will be using this stuff just about
all the time.
- 22M:072/22C:036 Elementary
Numerical Analysis
This is the starting point for numerical analysis, which is about how
we use computers to solve mathematical problems (which might come from
engineering, physics, biology, economics etc., etc., etc.). You
will
learn about roundoff error (because we can't represent the
infinitely
many digits of ``pi'', for example), and how to estimate the growth
or decay of errors. We will see how to solve equations, how
to approximate functions, compute integrals, and solve differential
equations.
Summer 2001
- 22C:34 Discrete Structures
This course is about the structures needed to design and understand
algorithms and data structures. We start with logic,
quantifiers
(``for all ...'' and ``there is a ...'') and mathematical induction,
and then talk about sets, sequences, relations, functions and
algorithms.
After that we will investigate counting methods and some
simple recurrence
relations. The last part of the course will look at graph
(or network) theory which is used to describe many problems and
algorithms
in Computer Science.
Spring 2001
- 22M:174/22C:174 Optimization
Techniques
This course is about theoretical and computational techniques of
optimization,
covering both unconstrained and constrained optimization: First
and second order necessary and sufficient conditions for a (local)
minimum/maximum; techniques for unconstrained optimization:
steepest
descent, Newton's method, quasi-Newton methods, line-searching and
trust-region
methods for globalization; Kuhn-Tucker conditions and techniques
for constrained optimization: quadratic programming, SQP methods; convex
functions and convex programs. If time permits: dynamic
oprimization
and optimal control.
Fall 2000
- 22M:72/22C:36 Elementary Numerical
Analysis
(Section 002)
This course is about basic topics in numerical analysis and scientific
computing. In it you will learn about roundoff error and
how
to minimize its effects, how to solve a nonlinear equation in
one
variable, how to approximate functions (especially by using interpolation),
how to numerically approximate integrals and solve differential
equations.
- 22M:270 Theoretical Numerical Analysis
This course is about using the tools of mathematical analysis to
understand
numerical methods; in particular to understand issues like rates of
convergence
and the error analysis of methods. This is particularly useful
for
solving problems such as partial differential equations (PDE's),
integral
equations (IE's) and variational inequalities (VI's) that commonly
arise
in many applications.
- 22M:36 Engineering Calculus II
(Section
131)
This course continues 22M:35 (Eng. Calculus I). Topics: Inverse
functions, exponential and logarithmic functions, hyperbolic
functions
(sinh, cosh, tanh), l'Hôpital's rule, differential equations; Integration
techniques: integration by parts, trig integrals and substitutions,
partial fractions, numerical approximations, improper integrals; Applications
of integrals to finding areas, volumes, arc lengths, moments and
centers
of gravity; Infinite sequences and series (sums): convergence
tests,
power series and Taylor series; Curves in the plane in
Cartesian
and polar coordinates; Vectors, lines and planes in space.
Spring 2000
22C:174/22M:174
Techniques of
Optimization
This course will cover a number of aspects of unconstrained and
constrained
optimization and the numerical methods needed to compute minima and
maxima.
First and second order necessary and sufficient conditions for a
(local)
minimum/maximum; techniques for unconstrained optimization:
steepest
descent, Newton's method, quasi-Newton methods, line-searching and
trust-region
methods for globalization; Kuhn-Tucker conditions and techniques
for constrained optimization: quadratic programming, SQP methods; convex
functions and convex programs.
Fall 1999
22M:036 Engineering calculus II (Sections 101 and 121)
Inverse functions, exponential and logarithmic functions,
hyperbolic
functions (sinh, cosh, tanh), l'Hôpital's rule, differential
equations; Integration
techniques: integration by parts, trig integrals and substitutions,
partial fractions, numerical approximations, improper integrals; Applications
of integrals to finding areas, volumes, arc lengths, moments and
centers
of gravity; Infinite sequences and series (sums): convergence
tests,
power series and Taylor series; Curves in the plane in
Cartesian
and polar coordinates; Vectors, lines and planes in space.
Spring 1999
22C:171/22M:171
Numerical Analysis
II: Differential
equations and matrix computations
Solution of ordinary differential equations by Euler's method,
implicit Euler's method, mid-point rule, Runge-Kutta methods, multistep
methods, error analysis; solution of linear equations: Gaussian
Elimination (LU factorization), condition numbers, effect of
roundoff
errors; least squares problems: normal equations, Cholesky
factorization, QR
factorization; eigenvalue and eigenvector problems: power
method,
inverse power method, and introduction to the QR algorithm.
Fall 1998
- 22C:170/22M:170 Numerical Analysis I: Solving equations and
approximation
theory
Floating point arithmetic; nonlinear equations:
bisection,
Newton, secant, multivariate Newton; polynomial interpolation:
error
estimates, choice of interpolation nodes; approximation of functions; numerical
integration.
- 22M:176 Finite
Element
Methods
Introduction to finite element methods; two-point boundary value
problems
in one dimension; weak form and variational forms of PDE's; Galerkin
method;
natural vs. essential boundary conditions; elliptic PDE's in two and
more
dimensions; parabolic and hyperbolic equations (if time permits).
Spring 1998
- 22C:174/22M:174 Optimization techniques.
Introduction to optimization; 1st and 2nd order necessary and
sufficient
conditions for optimality; techniques for unconstrained
optimization:
steepest descent, Newton's method, conjugate gradient methods,
quasi-Newton
methods; globablization issues: line-searching, trust region methods;
convergence:
rates and robustness; constrained optimization; Kuhn-Tucker
conditions
and constraint qualification; convexity and convex programming.
Spring 1997
MATH2224 Multivariate Calculus (at
Virginia
Tech)
Introduction to multivariate calculus; partial derivatives;
differentiability; Taylor's theorem to 2nd order; multiple
integrals:
areas, volumes, centers of mass, etc.; power series and radii
of
convergence.
MATH4446 Numerical Methods and
Analysis
(at
Virginia Tech)
Polynomial interpolation; Chebyshev interpolation vs.
uniformly
spaced interpolation; Runge's phenomenon; approximation techniques; numerical
integration: mid-point, trapezoidal, Simpson's rules and Gaussian
quadrature; ordinary
differential equations.
Fall 1996
MATH2214 Ordinary Differential
Equations
(at
Virginia Tech)
Introduction to ODE's; 1st order ODE's; separable equations;
linear equations: integrating factors, particular integrals and
variation
of constants formula; 2nd order linear equations; pendulum
equation; nth
order linear ODE's; systems of linear ODE's.
MATH4445 Numerical Methods and
Analysis:
Matrix Computations
(at Virginia Tech)
Floating point arithmetic; linear equations; Gaussian
elimination; conditioning and error bounds; backward error bounds; least
squares; Cholesky factorization; QR factorization; eigenvalues
- theory (incl. perturbation theory); power method; Jacobi method;
intro
to QR algorithm.
Spring 1996
- MATH417 Numerical Analysis (at
Texas
A&M)
This was a general undergraduate introduction to numerical
analysis.
It essentially covered the topics of MATH4445 and MATH4446 above, but
at
a lower level.
Fall 1995
- MATH141 Business Mathematics I (at
Texas A&M)
Linear programming: simplex tableau method; combinatorics;
probability.
Other courses
Courses taught by David Stewart include ``Unix
tools''
for graduate students at Virginia Tech with Prof. C. Beattie
(Mathematics)
in 1997; ``Dynamical Systems: Theory and
Computation'',
at the Australian National University with Prof. R.L. Dewar (Plasma
Research
Lab, ANU) in 1993; ``Introduction to Scientific
Computing''
(3 times), ``Linear Programming'', ``Mathematics
on Microcomputers'' (twice), ``Numerical
Linear
Algebra'' (undergraduate), ``Numerical
Linear
Algebra'' (graduate), ``Numerical
Optimization''
at the University of Queensland, Australia, during the period
1986-1991.
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