Instructor: Michael Weingart, weingart@math.rutgers.edu
Class meeting times: MTTh 6:15-8:45pm in Hill 425.
Office hours: MTTH 5-6pm in Hill 523 unless otherwise stated, and by appointment.
Office phone: 732-445-8204
You will be given a formula sheet along with your exam; you may not bring any notes with you.
There will be extra office hours on Sunday August 10 from 5:00pm to 7:00pm
Anything that's been on a homework (including non-homework 9) or quiz/in-class-exercise, plus a few extras from lecture:
-The geometric derivation of Newton's method and the secant method.
-The argument that numerical differentiation is vulnerable to roundoff error as h is decreased, but numerical integration is not.
-The arguments from the Gaussian quadrature lecture, which just require you to understand what it means to have a certain degree of precision, and what it means to be a quadrature rule. For example: the argument that if a quadrature rule on n points {x1, x2, ..., xn} has degree of precision 2n-1, then the polynomial P(x)=(x-x1)(x-x2)...(x-xn) is orthogonal to all polynomials of degree strictly less than n.
-The argument that in the composite integration rules, the n different "mystery points" can be replaced by a single one.
-The geometric meaning of a solution of an initial value problem , i.e. as a certain curve in a direction field.
Same comment as for the midterm: You should understand what the advantages and disadvantages are of the various methods we've discussed; there are certain ones which have really been emphasized in lecture. The same holds for certain key pictures (e.g. for the midpoint and trapezoid methods).
The final will be closed book; no calculators will be allowed. The problems won't require you to do intensive computation. But you really must understand _how_ to use the methods and those error formulas which we have discussed (Taylor, bisection, fixed point iteration, Lagrange interpolation, differentiation rules, integration rules).
homework 1 due Thursday July 10 solution 1
homework 2 due Monday July 14 solution 2
homework 3 due Tuesday July 15 solution 3
homework 4 due Thursday July 17 solution 4
homework 5 due Tuesday July 22 solution 5
homework 6 due Thursday July 31 solution 6
homework 7 due Monday August 4 solution 7
homework
8 due Thursday August 7 solution
quiz 1
solution
The problems on homework 5 are exactly the kind from chapter 3 (interpolation) which you should be able to do on the midterm. Similarly, the homework and quizzes are the main guideline for the material from chapters 1 and 2, although there are a few ideas from lecture which did not make their way into the homework and which you are expected to understand. In particular, you should be able to derive Newton's method, both by the Taylor series method of quiz 2, and by the geometric argument presented in lecture (drawing a line tangent to the graph of f(x) and seeing where it crosses the x-axis,etc.). You should understand what the secant method is, and how to write down the function g(x) for that method (slightly different than for Newton). For each rootfinding method you have seen the "key picture" explaining what's going on drawn on the blackboard; you should not be surprised to see one on the test, or to be asked straightforward questions about it. You should know the Mean Value Theorem and Generalized Rolle's Theorem.
You should understand what the advantages and disadvantages are of the various methods we've discussed; there are certain ones which have really been emphasized in lecture.
The midterm will be closed book; no calculators will be allowed.
As explained in lecture, the problems won't require you to do intensive
computation. But you really must understand _how_ to use the methods and
those error formulas which we have discussed (Taylor, bisection, fixed
point iteration, Lagrange interpolation).
Numerical analysis is the art and science of approximate computation. How do you find the precise point at which a graph crosses the x-axis, when it is impossible to solve algebraically? Or integrate a function whose antiderivative is impossible to express in terms of the standard functions which you know? Or solve a differential equation for which you don't know an explicit technique? Solving a problem "numerically" means carrying out a finite procedure in finite time on a finite machine which produces an approximate solution with tolerably small error. Numerical analysis studies such methods, their efficiency and reliability, and the ways in which their performance can be improved. The overwhelming number of real-life mathematical problems require numerical solutions, because they are too difficult or even impossible to solve precisely.
You have had at least a glimpse of this subject already; Newton's method, Taylor's remainder theorem, the trapezoid rule and Simpson's method, and Euler's method for differential equations are all mentioned in the calculus sequence at Rutgers, but not discussed in detail. We will examine methods for finding solutions of equations in one variable, interpolating and approximating functions, differentiating and integrating numerically, and approximating solutions of differential equations. Each topic has geometric and analytic aspects, and you will become familiar with both.
This is a course in understanding how and why certain ideas of computational mathematics work, and how they can be used in practice. You will do some computation in order to understand the methods, but this is not a course in implementation or programming. That's what engineering and computer science courses are for.
The prerequisites are calculus through differential equations, i.e. 244, 252, or 292. If through some administrative error you have succeeded in registering without having taken a class in differential equations, you will experience great pain during a major component of this course.
There will be frequent homework assignments, which are an important
part of the course. There will also be short quizzes, a midterm, and a
final; the final will be on August 12 from 6-9pm. The official text is
the seventh edition of Numerical Analysis by Richard L. Burden and
J. Douglas Faires.