Classes for David Stewart

Schedule (Fall 2016)

Important information for students

Fall 2016

I am teaching:

• MATH:3700/CS:3700 Elementary Numerical Analysis (sections 0002 and 0004)
  
  This course will cover basic numerical analysis and particularly issues such as roundoff error (i.e., you can't store infinitely many digits in a computer), approximating functions (you can't store $y=f\left(x\right)$ for all values of $x$), how to solve equations, integrate functions (numerically), and even solve some differential equations.
  

Spring 2016

• MATH:4820 Optimization Techniques
  
  This course covers the theory and practice of numerical optimization: numerically estimating the maximum or minimum of a function, possibly subject to a number of either equality or inequality constraints. On the theoretical side we will talk about necessary and sufficient condition for local minima and maxima with or without constraints, convex sets and functions, and the analysis of algorithms. On the practical side we will consider a wide range of algorithms and their implementation, including line search and trust region techniques.
  
  
• MATH:6860 Theoretical Numerical Analysis II
  
  This is the second half of the course on the theoretical side of numerical analysis. In this half we will concentrate on Sobolev spaces, which are used for understanding partial differential equations. These will be used in the weak and variational formulation of PDE's. Based on these formulations, we will set up and analyze finite element methods for elliptic, parabolic and hyperbolic PDE's. Also considered: variational inequalities, mixed methods, and "variational crimes".
  

Fall 2015

• MATH:4610 Continuous Mathematical Models
  
  This course will be on mathematical models for a wide range of phenomena. The models will mainly be based on differential equations (ordinary and partial), but will also include random processes. Phenomena that will be modeled will include basic mechanics, traffic flow, population dynamics, solid and fluid mechanics, epidemics, and may include other topics such as optimization, statistical mechanics, and quantum mechanics.
  
  
• MATH:6850 Theoretical Numerical Analysis I
  
  
  This is the first half on the theoretical side of numerical analysis. As such we will be dealing with Banach and Hilbert spaces as a way of formulating questions and methods in numerical analysis. As always in numerical analysis, we can only store a finite amount of data, and so we are dealing with finite-dimensional approximations to the objects we wish to compute. The main questions are: Will our approximations converge to the desired solution? How quickly do the computed objects converge to the desired solution(s)?
  
  
  A central issue is the problem of solving partial differential equations, where the solution necessarily belongs to an infinite dimensional space. Related questions such as multidimensional approximation theory (such as approximating functions on triangles, spheres, cubes, etc.) become very important.
  
  
  

Summer 2015

• MATH:3800 Elementary Numerical Analysis
  
  This course will cover basic numerical analysis and particularly issues such as roundoff error (i.e., you can't store infinitely many digits in a computer), approximating functions (you can't store $y=f\left(x\right)$ for all values of $x$), how to solve equations, integrate functions (numerically), and even solve some differential equations.

Spring 2015

• MATH:4820 Optimization Techniques
  
  This course covers the theory and practice of numerical optimization: numerically estimating the maximum or minimum of a function, possibly subject to a number of either equality or inequality constraints. On the theoretical side we will talk about necessary and sufficient condition for local minima and maxima with or without constraints, convex sets and functions, and the analysis of algorithms. On the practical side we will consider a wide range of algorithms and their implementation, including line search and trust region techniques.
  
  
• MATH:5810 Numerical Analysis (Differential Equations and Linear Algebra)
  
  This is the second half of a comprehensive course in numerical analysis. Numerical analysis is what connects continuous mathematics with computers and computing. Numerical analysis is about how to solve equations, compute derivatives and integrals, and solve differential equations. The difference is that computers can only store real numbers to a finite (in fact, very limited) number of decimal (or binary) places. Whenever a computation is done, there is almost always some error in the result.
  
  In this course we will discuss methods for numerically (thus approximately) differential equations, systems of linear equations, (linear) least squares problems, and eigenvalue/eigenvector problems.

Fall 2014

• 22M:170/CS:170 (MATH:5800/CS:5710) Numerical Analysis (Nonlinear Equations & Approximation Theory)
  
  This course is part of a two semester sequence with 22M:171/22C:171 (MATH:5810/CS:5720) giving a thorough grounding in Numerical Analysis. This course will cover roundoff error and numerical methods for solving nonlinear equations, interpolation and approximation, and numerical integration and differentiation.
  

• 22M:072/22C:072 (MATH:3800/CS:3700) Elementary Numerical Analysis
  
  This course gives a one-semester introduction to numerical analysis, covering roundoff error and floating point arithmetic, numerical solution of linear and nonlinear equations, interpolation and approximation, and numerical solution of differential equations.
  
  This will be a computer-lab based version of Elementary Numerical Analysis involving hands-on in-class group work.
  

Spring 2014

• 22M:171/22C:171 (MATH:5810/CS:5720) Numerical Analysis (Differential Equations & Linear Algebra)
  
  This course is part of a two semester sequence with 22M:170/22C:170 (MATH:5800/CS:5710) giving a thorough grounding in Numerical Analysis. This course will cover numerical methods for solving ordinary differential equations and methods for solving linear systems, linear least squares problems, and computing eigenvalues and eigenvectors.
  
  
  
• 22M:174/22C:174 (MATH:4820/CS:4720) Optimization Techniques
  
  This course is about continuous optimization with and without constraints. That is, it is about finding minima and maxima of smooth functions of real variables. The emphasis will be on practical numerical methods, but it is vitally important that the underlying theory is also understood.
  

Fall 2013

• 22M:140 (MATH:4610) Continuous Mathematical Models
  
  This course shows you how to develop continuous mathematical models in a wide variety of subject areas including mechanics, biology, fluid dynamics (water and air flow), electromagnetism, traffic behavior, and possibly some behavioral models as well.

• 22M:142 (MATH:5600) Nonlinear Dynamics with Numerical Methods
  
  This course will be about nonlinear differential equations and their long-time behavior.  Topics include one- and two-dimensional flows, equilibria, stability, limit cycles, bifurcations, chaos, fractals and numerical methods including Euler's method.
  

Spring 2013

• 22M:26 (MATH:1860) Calculus II
  
  This course further develops the differential and integral calculus following 22M:25 (MATH: 1850) in the following ways: further techniques for computing integrals are given including integration by parts, trig integrals, trig substitutions, partial fractions, improper integals, and approximate (numerical techniques).  These are applied to the computation of various areas, volumes, surface areas, forces due to hydrostatic pressure, and probabilities.  They are also applied to the solution of differential equations (which are used to model a huge number of dynamic phenomena in physics, chemistry, biology, economics, etc.).  Parametric representations of curves are developed, so that instead of writing x² + y² = 1 for a unit circle, we can write x and y in terms of another parameter: x = cos(t), y = sin(t).  Then we see how to compute arclengths, enclosed areas, and other quantities in terms of parameterized curves.  Finally, we deal with infinite sequences and series (i.e., infinite sums), up to and including Taylor series for representing functions in terms of power series.
  
  
• 22M:72/22C:72 (MATH:3800/CS:3700) Elementary Numerical Analysis
  
  This course gives a basic foundation in numerical analysis, looking at algorithms to solve continuous mathematical problems and their performance in practice.  In short, we develop algorithms, implement them, test them, develop theory to explain their performance, and compare theory with practice.  This course is a "TILE" course, which involves hands-on interaction which shows just what the theory means in practice.

Fall 2012

• 22M:72/22C:72 (MATH:3800/CS:3700) Elementary Numerical Analysis
  
  This course gives a basic foundation in numerical analysis, looking at algorithms to solve continuous mathematical problems and their performance in practice.  In short, we develop algorithms, implement them, test them, develop theory to explain their performance, and compare theory with practice.  This course is a "TILE" course, which involves hands-on interaction which shows just what the theory means in practice.
  

• 22M:140 (MATH:4610)  Continuous Mathematical Models
  
  This course shows you how to develop continuous mathematical models in a wide variety of subject areas including mechanics, biology, fluid dynamics (water and air flow), electromagnetism, traffic behavior, and possibly some behavioral models as well.
  

Spring 2012

• 22M:174 (MATH:4820) Optimization Techniques
  
  This course is about the theory and practice of (computational) optimization.  We review not only the basic mathematical theory of both unconstrained and constrained optimization, and go on to study algorithms for solving optimization problems.
  
  
• 22M:271 (MATH:6860) Theoretical Numerical Analysis II
  
  This continues 22M:270.  In particular, we study Sobolev spaces, the finite element method and its convergence, and deal with some useful topics such as interpolation spaces.
  

Fall 2011

• 22M:142 Nonlinear Dynamics with Numerical Methods  ICON  ISIS course link
  
  This course will be about nonlinear differential equations and their long-time behavior.  Topics include one- and two-dimensional flows, equilibria, stability, limit cycles, bifurcations, chaos, fractals and numerical methods including Euler's method.
  
  
  
• 22M:270 Theoretical Numerical Analysis I  ICON  ISIS course link
  
  This course provides theoretical foundations for numerical analysis based on functional analysis.  Applications include approximation theory, partial differential equations (finite difference and finite element methods), integral equations.
  
  This is the first part of a two-semester sequence; the second part is 22M:271.
  

Spring 2011

• 22M:027 Introduction to Linear Algebra  ICON 
  
  Linear algebra starts with the study of linear systems of equations.  These are ubiquitous in the uses of mathematics, in computing, and in developing mathematics itself.  From this we develop the algebra and properties of matrices and vectors including topics such as: (reduced) row echelon form, linear dependence and independence of vectors, matrix inverses, singular vs. non-singular matrices, determinants, dimension, rank, nullity, eigenvalues and eigenvectors, characteristic equation.  Special classes of matrices, such as symmetric matrices are important, and their properties are described.  This is an ICON course.
  
  
• 22M:171/22C:171 Numerical Analysis II: Differential Equations and Matrix Computations  ICON
  
  This is the second of the two foundational courses in Numerical Analysis.  This covers similar material to 22M:072/22C:072 but in much greater depth. Topics covered include: numerical solution of differential equations (Euler method, multistep methods, Runge--Kutta methods, stability, stiff equations), matrix computations (solving linear systems, LU factorization, condition number, linear least squares, Cholesky factorization, QR factorization, eigenvalues and eigenvectors, power method, inverse iteration, Schur decomposition, QR algorithm for eigenvalues/eigenvectors).  This is an ICON course.

Fall 2010

• 22M:072/22C:072 Elementary Numerical Analysis  ICON  Syllabus
  
  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:170/22C:170 Numerical Analysis I: Nonlinear Equtions and 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 2009--Spring 2010

• On leave.
  

Fall 2008

• 22M:072 Elementary Numerical Analysis  ICON  Syllabus  Schedule

Spring 2008

• 22M:095 Introduction to Mathematical Research Opportunities  ICON  Syllabus  Schedule  

• 22M:171/22C:171 Numerical Analysis II (Differential Equations & Linear Algebra)  ICON  Syllabus

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.

• 22M:096 Introduction to Mathematics Research (Mathematical modeling)  ICON  Syllabus
  
  This course is an introduction to mathematical modeling with examples from mechanics, geology, biology & economics. There will be lots of projects to work on as well as regular homework.  The modeling will involve differential equations (ordinary and partial) and ideas from optimization.

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

• 22M:140 Continuous Mathematical Models
  

• 22M:170/22C:170 Numerical Analysis I (Nonlinear Equations & Approximation Theory)
  

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

Fall 2005

• 22M:025 Calculus I (Liberal Arts & Sciences) section JJJ ICON

• 22M:072/22C: Elementary Numerical Analysis ICON

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

• 22M:025 Calculus I (Liberal Arts & Sciences) section DDD (Syllabus and Resources).
• 22M:025 Calculus I (Liberal Arts & Sciences) section EEE (Syllabus and Resources).
  

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

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

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