1.1 Review of Calculus
Theorems: Mean Value Theorem and its variations (Rolle's Thm., Generalized Rolle's Thm., MVT for Integrals), Intermediate Value Theorem, Extreme Value Theorem, Taylor's Theorem
Exercises: 4, 7, 9, 11
1. Show that if a < xk < b for k = 1,...,N and f is continous on (a,b), then
f(x1) + f(x2) + .. + f(xN)= N f(c) for some c in the interval (a,b).
2. If f is in C1[0,1] show that
I01 f(x)(x-1/2)dx = (1/12)f'(c) for some c in [0,1].
Hint: write f(x) = f(1/2) + (f(x) - f(1/2))/(x-1/2)*(x-1/2)

1.2 Roundoff Error & Computer Arithmetic
Key Ideas: Floating point number, Chopping, Rounding, Round-off error, Absolute error, Relative error, Significant figures, Computer arithmetic
Exercises: 3, 4, 5, 13, 17, 22, 26
1. Let x* be an approximaton to x, then f(x*) is an approximation to f(x). If f is in C1, estimate the absolute error in f(x*) in terms of the absolute error in x* and f'(x).
2. Find the smallest number e on a k-digit-base-10-with-rounding computer such that 1 + e > 1. This number e is called the machine epsilon or unit roundoff.
1.3 Algorithms & Convergence
Key Ideas: Stable, Unstable, Rate of Convergence
Exercises: 6, 7, 8, 15
1. Let S = 1 - 1/2 + 1/3 - 1/4 + ... - 1/1000.
Rank the following four methods for computing S from the most accurate to the least accurate. Justify your answer.
1. From largest to smallest (left to right)
2. From smallest to largest (right to left)
3. From largest to smallest, summing the positive and negative terms separately
4. From smallest to largest, summing the positive and negative terms separately
2. Show that if xn = x + O(an), with x not 0, and yn = y + O(an), with an converging to 0, then
xn+ yn = x + y + O(an), and
xn yn = xy + O(an).

3.1 Interpolation/ Lagrange Polynomial
Key Ideas: Interpolation, Lagrange interpolating polynomial
Algorithms: Horner's Algorithmi, also known as Nested Multiplication and Synthetic Division (Thm 2.18/Algo 2.7), Lagrange interpolating polynomial (Thm 3.2)
Theorems: Error Estimate (Thm 3.3)
Exercises: 1, 2, 3, 9, 10, 15, 17, 24
3.2 Divided Differences/ Newton's Polynomial
Key Ideas: Divided difference table (Table 3.8)
Theorems: Theorem 3.6
Algorithms: Newton's Interpolatory Divided-Difference (Algo 3.2)
Exercises: 1, 4, 6, 7, 10
1. Show that if f is a polynomial of degree m then if n > m, then
f[x0,x1,...,xn] = 0
for any choice of x0, x1,..., xn.

3.3 Hermite Interpolation
Key Ideas: Osculating polynomials, Hermite polynomials (Lagrange-Type and Divided Difference-Type)
Algorithms: Hermite Interpolation (Algo 3.3)
Theorems: Error Estimate (Thm 3.9)
Exercises: 1, 3, 4, 7
3.4 Cubic Splines
Key Ideas: Piecewise polynomial, Interpolating cubic spline, Natural spline, Clamped spline, Knot
Algorithms: Natural Cubic Spline (Algo 3.4)
Theorems: Error Estimate (Thm 3.13)
Exercises: 1, 2, 3cd, 7, 9, 24
8.1 Discrete Least-Squares Approximation
Key Ideas: Least squares, Normal equations
Algorithms: Normal equations (8.3)
Exercises: 4, 7, 8
8.2 Orthogonal Polynomials
Key Ideas: Linearly independent, Orthogonal, Orthonormal
Algorithms: Least-Squares for Orthogonal polynomials (Thm 8.5)
8.4 Rational Function Approximation (SKIP)

8.5 Trigonometric Polynomial Approximation
Key Ideas: Trigonometric Polynomial, Fourier Series (Discrete and Continuous)
Algorithms: Formulas for the Fourier Coefficients (Discrete and Continuous)
Exercises: 1, 2, 5, 7ab, 13, 14 (thus odd functions have only sine terms, even functions only cosine terms)
4.1 Numerical Differentiation
Key Ideas: Truncation error
Algorithms: n-point formulas (Eqn 4.2, 4.12-15, 4.20), General formula (Eqn 4.4)
Theorems: Truncation errors for $n$-point formulas
Exercises: 1, 2, 3, 10, 16
1. Estimate f'(0) = 1 for f(x) = ex by computing
(eh - e-h)/(2h)
for h = 10-N, N = 1, ..., 20. Comment on the results.
2. Estimate f'(0.3) as best as possible, using the data f(0.0) = 1, f(0.2) = 1.2 and f(0.3) = 1.2.

4.2 Richardson's Extrapolation
Algorithms: Richardson extrapolation (in Class), On-line error estimates
Exercises: 6, 8
1. Assume we have an algorithm T(h) which approximates T with the error estimate
T = T(h) + Kh2 + O(h3).
If T(0.1) = 1.326 and T(0.01) = 1.417 (1) use Richardson extrapolation to produce an improved approximation of T and (2) estimate the value of h needed to have an error of less than 10-6.

4.3 Elements of Numerical Integration
Algorithms: Trapezoid rule (Eqn 4.40), Simpson's rule (Eqn 4.41), Midpoint rule (Eqn 4.46)
Theorems: Truncation errors
Exercises: 1abe, 2, 9
4.4 Composite Numerical Integration
Algorithms: Composite Simpson's (Algo 4.1), Composite Trapezoid (Thm 4.5), Composite Midpoint (Thm 4.6)
Theorems: Truncation errors (Thm 4.4-4.6)
Exercises: 123abcf, 6, 13, 22, 26ab
4.5 Romberg Integration
Algorithms: Romberg integration (Algo 4.3)
Exercises: 1abe, 12
Exercises:
1. Use adaptive quadrature with the Trapezoid Rule and the exact error to estimate \int_0^1 x3 dx to an accuracy of 10-2.

Key Ideas: Orthogonal functions, Weight function
Algorithms: Gaussian quadrature (Eqn 4.67), Change of interval (Eqn 4.72)
Theorems: Degree of accuracy (Thm 4.8)
Exercises: 1abc, 2abc
1. Show that the coefficients given in Table 4.11 for n=3 are the weights obtained from integrating over the interval [-1,1] the functions Lk(x) (Lagrange interpolating polynomials) for the roots given.

5.1 Theory of Initial-Value Problems
Key Ideas: Initial-Value Problem (IVP), Well-posed IVP
5.2 Euler's Method
Key Ideas: Mesh points, Step size
Algorithms: Euler's (Algo 5.1)
Theorems: Error estimate (Eqn 5.20)
Exercises: 1abc, 4
5.3 Higher-Order Taylor's Methods
Key Ideas: Local truncation error
Algorithms: Taylor method of order $n$ (Eqn 5.27)
Exercises: 1abc, 6abc
5.4 Runge-Kutta Methods
Algorithms: Midpoint (5.35), Modified Euler (5.37), Order-Four (Classic R-K) (Algo 5.2)
Theorems: Truncation errors
Exercises: 1ab, 2ab, 3ab, 6abc, 11
5.5 Error Control and Runge-Kutta-Fehlberg Method
Key Ideas: Error Control using two methods
Algorithms: Runge-Kutta-Fehlberg (5.3)
Exercises: 2a (program, not by hand)

5.6 Multistep Methods
Key Ideas: Multistep, Explicit, Implicit, Predictor-Corrector method
Algorithms: Adams-Bashforth Four-Step Method (Eqn 5.34), Adams-Moulton Three-Step Method (Eqn 5.37), Predictor-Corrector (Algo 5.4)
Theorems: Truncation errors
Exercises: 1b, 8a, 9a
5.9 Higher-Order Equations and Systems of Differential Equations
Key Ideas: Order of an equation, Systems, Conversion between the two
Exercises: 1ac, Convert #2abc to Systems
5.10 Stability
Key Ideas: Consistent, Stable, Convergent, Root Condition, Absolute Stability (5.11, Def 5.24)
Exercises: 4, 7
11.1 The Linear Shooting Method
Key Ideas: Boundary Value Problem, Linear DE, Shooting Method
Algorithms: Linear Shooting (11.1)
Exercises: 1, 2, 4, 5
11.2 The Shooting Method for Nonlinear Problems
Key Ideas: Nonlinear DE, Shooting Method
Algorithms: Non-Linear Shooting (11.2)
Exercises: 1, 2
11.3 Finite-Difference Methods for Linear Problems
Key Ideas: Finite Differences, Centered Difference
Algorithms: Linear Finite-Difference (11.3)
Exercises: 1, 2, 3, 5

ccollins@math.utk.edu