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).

    6.1 Linear Systems of Equations
    Key Ideas: Row operations, Triangular matrix, Augmented matrix, Flop count, Pivot element
    Algorithms: Backward-Substitution, Gaussian Elimination (6.1)
    Exercises: 2, 3, 5, 7
    6.2 Pivoting Strategies
    Key Ideas: Pivoting
    Algorithms: Pivoting Strategies: GE w/Parial Pivoting (6.2), Scaled PP (6.3)
    Exercises: 1, 2, 3, 6, 8, 16
    6.5 Matrix Factorization
    Key Ideas: Gaussian Transformation Matrix
    Algorithms: LU (6.4) w/permutation
    Exercises: 1, 2, 4, 5
    6.6 Special Types of Matrices
    Key Ideas: Diagonal Dominant, Positive Definite, Bandwidth, Band Matrix, Tridiagonal Matrix
    Algorithms: Choleski (6.6), Crout (6.7)
    Theorems: Cor. 6.27
    Exercises: 1, 2, 3, 6, 10, 15
    7.1 Norms of Vectors and Matrices
    Key Ideas: Vector norm, Matrix norm, Natural (Induced) matrix norm, l2 and linfty norms, Convergence
    Algorithms: Infinity and 2
    Theorems: Equivalence of Norms (7.7 and paragraph after), Cor. 7.10, Thm 7.11
    Exercises: 1, 2, 3, 4, 5
    7.2 Eigenvalues and Eigenvectors
    Key Ideas: Eigenvalue, Eigenvector, Spectral radius, Convergent matrix
    Algorithms: Characteristic Polynomial
    Theorems: Spectral Radius and Norms (Thm. 7.15), Thm. 7.17
    Exercises: 1, 2, 3, 4, 5, 10
    7.3 Iterative Techniques for Solving Linear Systems
    Key Ideas: Iteration Matrix, Residual vector
    Algorithms: Jacobi (7.1), Gauss-Seidel (7.2), SOR (7.3)
    Theorems: Iteration Convergence (Thm 7.19), Convergence bound (7.20), Stein-Rosenberg (7.22), Ostrowski-Reich (7.25)
    Exercises: 1, 2, 5, 8, 13
    7.4 Error Estimates and Iterative Refinement
    Key Ideas: Condition Number, Ill and Well conditioned
    Algorithms: Iterative Refinement (7.4)
    Theorems: Error bounds (7.27 and 7.28)
    Exercises: 1, 2, 3
    9.1 Linear Algebra and Eigenvalues
    Key Ideas: Eigenvalues and Eigenvectors; Similar matrices; Linearly independent; Orthogonal
    Theorems: Theorem 9.9 (Schur Decompositon), Theorem 9.10 (Symmetric Decomposition)
    Exercises: 1, 6
    9.2 The Power Method
    Key Ideas: Power method
    Algorithms: Power Method (9.1), Inverse Power Method (9.3)
    Exercises: 1, 2
    9.3 Householder's Method
    Key Ideas: Householder Transformation, Similarity transformation
    Algorithms: Householder's (9.5)
    Exercises: 1
    2.1 The Bisection Method
    Key Ideas: Root, Zero, Stopping procedure
    Algorithms: Bisection (2.1)
    Theorems: Convergence Bound (Thm. 2.1)
    Exercises: 4, 8, 10, 13, 14, 15, 16
    2.2 Fixed-Point Iteration
    Key Ideas: Fixed-Point, Fixed-Point iteration
    Theorems: Fixed-Point Theorem (Thm 2.3), Convergence Bound (Cor 2.4,2.5)
    Exercises: 1, 4, 9, 11, 20
    2.3 The Newton-Raphson Method
    Algorithms: Newton's (Algo 2.3), Secant (Algo 2.4), Method of False-Position (Algo 2.5)
    Exercises: 6, 7, 8, 11, 12, 15, 24, 29
    2.4 Error Analysis for Iterative Methods
    Key Ideas: Convergence of order alpha, Linear convergence, Quadratic convergence, Root of multiplicity m, Simple zero
    Theorems: Thm. 2.7, Thm. 2.8
    Exercises: 1, 2, 5, 8, 11
    2.5 Accelerating Convergence
    Key Ideas: Aitken's Delta-Squared Method
    Algorithms: Steffensen's Method (2.6)
    Exercises: 1, 4, 9, 12, 13
    2.6 Zeros of Polynomials and Muller's Method
    Key Ideas: Deflation
    Algorithms: Horner's (Algo 2.7), Muller's (Algo 2.8)
    Exercises: 2adg, 4adg, 9, 10
    10.1 Fixed Points for Functions of Several Variables
    Key Ideas:
    Algorithms:
    Theorems:
    Exercises:
    10.2 Newton's Method
    Key Ideas:
    Algorithms:
    Theorems:
    Exercises:
    10.3 Quasi-Newton Methods
    Key Ideas:
    Algorithms:
    Theorems:
    Exercises:
    10.4 Steepest Descent Techniques
    Key Ideas:
    Algorithms:
    Theorems:
    Exercises:

    ccollins@math.utk.edu
    Last Modified: January 3, 2001