The catalog description for Numerical Analysis (MATH 4257/5257) is: "Presents floating point arithmetic and error propagation, numerical solution to functions of a single variable and functional approximation, numerical differentiation and integration, program design, coding, debugging, and execution of numerical procedures." The formal prerequisites are Calculus 2 (MATH 1920) and Linear Algebra (MATH 2010).

Copies of the classnotes are on the internet in PDF format as given below. The notes and supplements may contain hyperlinks to posted webpages; the links appear in red fonts. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes have not been classroom tested and may have typographical errors.

1. Chapter 1. Mathematical Preliminaries and Error Analysis.
2. Chapter 2. Solutions of Equations in One Variable.
3. Chapter 3. Interpolation and Polynomial Approximation.
4. Chapter 4. Numerical Differentiation and Integration.
5. Chapter 5. Initial-Value Problems for Ordinary Differenital Equations.
6. Additional Chapters.

1. Mathematical Preliminaries and Error Analysis.

• 1.1. Review of Calculus.
• 1.2. Round-off Errors and Computer Arithmetic.
• 1.3. Algorithm and Convergence.
• 1.4. Numerical Software.
• Study Guide 1.

2. Solutions of Equations in One Variable.

• 2.1. The Bisection Method. Section 2.1 notes
• Supplement. Examples, Exercises, and Proofs from Section 2.1. Beamer file of Section 2.1 examples and proofs
• Supplement. Printouts of Examples, Exercises, and Proofs from Section 2.1. Print file of Section 2.1 examples and proofs
• 2.2. Fixed-Point Iteration. Section 2.2 notes
• Supplement. Examples, Exercises, and Proofs from Section 2.2. Beamer file of Section 2.2 examples and proofs
• Supplement. Printouts of Examples, Exercises, and Proofs from Section 2.2. Print file of Section 2.2 examples and proofs
• 2.3. Newton's Method and Its Extensions.
• 2.4. Error Analysis for Iterative Methods.
• 2.5. Accelerating Convergence.
• 2.6. Zeros of Polynomials and Müller's Method.
• 2.7. Numerical Software and Chapter Review.
• Study Guide 2.

3. Interpolation and Polynomial Approximation.

• 3.1. Interpolation and the Lagrange Polynomial. Section 3.1 notes
• Supplement. Examples, Exercises, and Proofs from Section 3.1. Beamer file of Section 3.1 examples and proofs
• Supplement. Printouts of Examples, Exercises, and Proofs from Section 3.1. Print file of Section 3.1 examples and proofs
• 3.2. Data Approximation and Neville's Method.
• 3.3. Divided Differences.
• 3.4. Hermite Interpolation.
• 3.5. Cubic Spline Interpolation.
• 3.6. Parametric Curves.
• 3.7. Numerical Software and Chapter Review.
• Study Guide 3.

4. Numerical Differentiation and Integration.

• 4.1. Numerical Differentiation.
• 4.2. Richardson's Extrapolation.
• 4.3. Elements of Numerical Integration.
• 4.4. Composite Numerical Integration.
• 4.5. Romberg Integration.
• 4.6. Adaptive Quadrature Methods.
• 4.7. Gaussian Quadrature.
• 4.8. Multiple Integrals.
• 4.9. Improper Integrals.
• 4.10. Numerical Software and Chapter Review.
• Study Guide 4.

5. Initial-Value Problems for Ordinary Differenital Equations.

• 5.1. The Elementary Theory of Initial-Value Problems.
• 5.2. Euler's Method.
• 5.3. Higher-Order Taylor Methods.
• 5.4. Runge-Kutta Methods.
• 5.5. Error Control and the Runge-Kutta-Fehlberg Method.
• 5.6. Multistep Methods.
• 5.7. Variable Step-Size Multistep Methods.
• 5.8. Extrapolation Methods.
• 5.9. Higher-Order Equations and systems of Differenital Equations.
• 5.10. Stability.
• 5.11. Stiff Differential Equations.
• 5.12. Numerical Software.
• Study Guide 5.

Additional Chapters.

• Chapter 6. Direct Methods for Solving Linear Systems.
• Chapter 7. Iterative Techniques in Matrix Algebra.
• Chapter 8. Approximation Theory.
• Chapter 9. Approximating Eigenvalues.
• Chapter 10. Numerical Solutions of Nonlinear Systems of Equations.
• Chapter 11. Boundary-Value Problems for Ordinary Differential Equations.
• Chapter 12. Numerical Solutions to Partial Differential Equations.

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