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)
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:

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