Math 128A Lecture Videos

Lecture Videos - UC Berkeley Math 128A, Numerical Analysis

Per-Olof Persson (persson@berkeley.edu)

Google Drive directory with all videos and notes

Errata: Errors in the videos and notes

Chapter 1: Mathematical Preliminaries and Error Analysis

Math 128A Lecture Slides Chapter 1 (PDF)
Math 128A Lecture Notes Chapter 1 (PDF)

1.1: Review of Calculus
- Part 1: Limits (11:32)
- Part 2: Derivatives (10:33)
- Part 3: Integrals (13:27)
- Part 4: Taylor Polynomials (14:23)
1.2: Round-off Errors and Computer Arithmetic
- Part 1: Floating Point Numbers (16:30)
- Part 2: Binary Numbers (7:16)
- Part 3: Double Precision 1 (15:29)
- Part 4: Double Precision 2 (11:44)
- Part 5: Round-off Errors (17:04)
1.3: Algorithms and Convergence
- Part 1: Errors Growth (8:58)
- Part 2: Rate of Convergence (10:28)

Chapter 2: Solutions of Equations in One Variable

Math 128A Lecture Slides Chapter 2 (PDF)
Math 128A Lecture Notes Chapter 2 (PDF)

2.1: The Bisection Method
- Part 1: The Bisection Method (9:00)
- Part 2: MATLAB Demo (9:27)
- Part 3: Convergence (8:54)
2.2: Fixed-Point Iteration
- Part 1: Fixed Points (10:42)
- Part 2: Existence and Uniqueness Proof (11:06)
- Part 3: Fixed-Point Iteration (10:16)
- Part 4: Fixed-Point Theorem (8:49)
- Part 5: Examples (8:23)
2.3: Newton’s Method and Its Extensions
- Part 1: Newton Graphically (10:01)
- Part 2: Newton from Taylor and MATLAB Demo (12:17)
- Part 3: Newton Convergence Proofs (15:40)
- Part 4: The Secant Method (5:31)
2.4: Error Analysis for Iterative Methods
- Part 1: Order of Convergence (9:10)
- Part 2: Order of Convergence Proofs (10:14)
- Part 3: Newton as Fixed Point Method (4:29)
- Part 4: Modified Newton’s Method (10:59)
2.5: Accelerating Convergence
- Part 1: Aitken and Steffensen’s Methods (11:21)
2.6: Zeros of Polynomials and Müller’s Method
- Part 1: Polynomials (7:52)
- Part 2: Horner’s Method (11:50)
- Part 3: Deflation (8:08)
- Part 4: Müller’s Method (9:33)

Chapter 3: Interpolation and Polynomial Approximation

Math 128A Lecture Slides Chapter 3 (PDF)
Math 128A Lecture Notes Chapter 3 (PDF)

3.1: Interpolations and the Lagrange Polynomial
- Part 1: Lagrange Polynomials 1 (13:51)
- Part 2: Lagrange Polynomials 2 (12:12)
- Part 3: Error Bounds (13:13)
3.3: Divided Differences
- Part 1: Divided Differences (11:43)
- Part 2: Example (9:11)
- Part 3: Derivatives (9:59)
- Part 4: Equal Spacing (9:59)
- Part 5: Forward Differences Example (7:11)
3.4: Hermite Interpolation
- Part 1: Osculating Polynomials (10:26)
- Part 2: Hermite Polynomials (7:27)
- Part 3: Hermite Theorem Proof (6:44)
- Part 4: Divided Differences Example (12:43)
3.5: Cubic Spline Interpolation
- Part 1: Spline Interpolation Motivation (14:13)
- Part 2: Cubic Spline Definition (14:33)
- Part 3: Spline Example (14:54)
- Part 4: MATLAB Demo (13:06)

Chapter 4: Numerical Differentiation and Integration

Math 128A Lecture Slides Chapter 4 (PDF)
Math 128A Lecture Notes Chapter 4 (PDF)

4.1: Numerical Differentiation
- Part 1: Forward and Backward Differences (13:10)
- Part 2: Three-Point Formulas (18:57)
- Part 3: Second Derivatives (8:26)
- Part 4: Optimal h (6:13)
4.2: Richardson’s Extrapolation
- Part 1: Richardson’s Extrapolation (8:03)
- Part 2: MATLAB Demo (6:14)
4.3: Elements of Numerical Integration
- Part 1: Numerical Integration (7:20)
- Part 2: The Trapezoidal Rule (9:30)
- Part 3: Simpson’s Rule (9:44)
4.4: Composite Numerical Integration
- Part 1: Composite Trapezoidal Rule (12:09)
- Part 2: Composite Simpson’s Rule (6:09)
- Part 3: Error Term (9:41)
- Part 4: Numerical Example (9:49)
4.5: Romberg Integration
- Part 1: Romberg Integration (8:53)
- Part 2: MATLAB Demo (9:41)
4.6: Adaptive Quadrature Methods
- Part 1: Adaptive Quadrature (7:01)
- Part 2: Error Estimation (13:11)
4.7: Gaussian Quadrature
- Part 1: Gaussian Quadrature (8:23)
- Part 2: Two-point Rule (6:01)
- Part 3: Legendre Polynomials (6:30)
- Part 4: Proof of Gaussian Quadrature (12:28)
- Part 5: Arbitrary Intervals (13:17)
4.8: Multiple Integrals
- Part 1: Double Integrals (13:57)
- Part 2: Example Simpson’s and Gaussian (13:26)
- Part 3: Non-Rectangular Regions in MATLAB (10:05)
4.9: Improper Integrals
- Part 1: Improper Integrals (17:40)
- Part 2: Infinite Limits Example (4:39)

Chapter 5: Initial-Value Problems for Ordinary Differential Equations

Math 128A Lecture Slides Chapter 5 (PDF)
Math 128A Lecture Notes Chapter 5 (PDF)

5.1: The Elementary Theory of Initial-Value Problems
- Part 1: Initial Value Problems (14:33)
- Part 2: Lipschitz Continuity (12:23)
- Part 3: Examples (14:19)
- Part 4: Well-Posedness (6:30)
5.2: Euler’s Method
- Part 1: Euler’s Method (9:26)
- Part 2: Error Bound Proof, Part 1 (8:23)
- Part 3: Error Bound Proof, Part 2 (9:14)
- Part 4: MATLAB Demo (7:53)
5.3: Higher-Order Taylor Methods
- Part 1: Taylor Methods (14:51)
- Part 2: MATLAB Demo (9:23)
5.4: Runge-Kutta Methods
- Part 1: Midpoint Method (12:43)
- Part 2: MATLAB Demo (10:07)
- Part 3: Graphical Interpretation (10:21)
- Part 4: Higher-Order Runge-Kutta Methods (9:46)
5.6: Multistep Methods
- Part 1: Multistep Methods (12:02)
- Part 2: Adams-Bashforth (11:20)
- Part 3: Adams-Moulton (6:47)
5.9: Higher-Order Equations and Systems of Differential Equations
- Part 1: Systems of Equations (12:48)
- Part 2: Higher-Order Equations (11:59)
5.10: Stability
- Part 1: One-Step Method Convergence (14:29)
- Part 2: An Unstable Multistep Method (11:58)
- Part 3: The Root Condition (7:36)
5.11: Stiff Differential Equations
- Part 1: Stiff Differential Equations (10:45)
- Part 2: Stability Regions (10:45)

Chapter 6: Direct Methods for Solving Linear Systems

Math 128A Lecture Slides Chapter 6 (PDF)
Math 128A Lecture Notes Chapter 6 (PDF)

6.1: Linear Systems of Equations
- Part 1: Gaussian Elimination (10:53)
- Part 2: Implementation (11:28)
- Part 3: Backward Substitution (7:21)
- Part 4: Operation Count (11:50)
6.2: Pivoting Strategies
- Part 1: Partial Pivoting (11:40)
6.3: Linear Algebra and Matrix Inversion
- Part 1: Matrix Multiplication (14:19)
- Part 2: Matrix Inverse (6:33)
- Part 3: Matrix Transpose (5:39)
- Part 4: Finding Inverses (6:37)
6.4: The Determinant of a Matrix
- Part 1: Determinants (11:06)
6.5: Matrix Factorization
- Part 1: Matrix Factorization (10:34)
- Part 2: LU Factorization (12:03)
- Part 3: Permutation Matrices (6:16)
- Part 4: MATLAB Demo (6:22)
6.6: Special Types of Matrices
- Part 1: Positive Definite Matrices (9:30)
- Part 2: LDL Factorization (12:20)

Exam Reviews

Math 128A Midterm Review (PDF)

Midterm exam review
- Part 1: Problem 1.2.6(c) (6:28)
- Part 2: Problem 1.3.9(a) (4:35)
- Part 3: Problem 2.2.9 (6:24)
- Part 4: Problem 2.5.17(a) (6:18)
- Part 5: Problem 3.5.11 (7:25)
- Part 6: Problem 4.1.28 (10:08)

Math 128A Final Exam Review (PDF)

Final exam review
- Part 1: Taylor Method (7:25)
- Part 2: Problem 5.4.27 (7:27)
- Part 3: Multistep Convergence (10:20)
- Part 4: Operation Counts (11:55)
- Part 5: Problem 4.2.10 (6:00)