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Lecture 1: Course Overview, Newton's Method for Root-Finding
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Square Roots via Newton's Method (PDF)
Square Roots
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Problem Set 1 (PDF)
Problem set 1 notebook
Solutions to Problem Set 1 (PDF)
Solutions to Problem Set 1 notebook
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| Lecture 2: Floating-Point Arithmetic |
Floating-Point Arithmetic, the IEEE Standard (PDF) (Courtesy of Per-Olof Persson. Used with permission.)
Floating-Point Arithmetic
Some Myths about Floating-Point Arithmetic (PDF)
Julia & IJulia Cheat-Sheet (PDF)
Introduction to Julia (PDF)
Julia for Numerical Computation in MIT Courses
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[No problem set] |
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Lecture 3: Floating-Point Summation and Backwards Stability
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 Notes on the Accuracy of Naive Summation (PDF)
Backwards Stability of Recursive Summation (PDF)
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[No problem set]
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| Lecture 4: Norms on Vector Spaces |
Notes on the Equivalence of Norms (PDF) |
[No problem set] |
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Lecture 5: Condition Numbers
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[No handout/notebook]
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Problem set 2 (PDF)
Solutions to Problem Set 2 (PDF)
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| Lecture 6: Numerical Methods for Ordinary Differential Equations |
Modern Differential Equations Solver Software: Where We Are and Where We're Headed (PDF - 2.4MB) (Courtesy of Christopher Rackauckas. Used with permission.) |
[No problem set] |
| Lecture 7: The SVD, its Applications, and Condition Numbers |
[No handout/notebook] |
[No problem set]
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| Lecture 8: Linear Regression and the Generalized SVD |
Many Viewpoints on Linear Regression |
[No problem set] |
| Lecture 9: Solving the Normal Equations by QR and Gram-Schmidt |
[No handout/notebook] |
[No problem set] |
| Lecture 10: Modified Gram-Schmidt and Householder QR |
Householder Reflectors and Givens Rotations (PDF) (Courtesy of Per-Olof Persson. Used with permission.)
Classical vs. Modified Gram-Schmidt
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[No problem set] |
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Lecture 11: Matrix Operations, Caches, and Blocking
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Performance Experiments with Matrix Multiplication (PDF)
Ideal-Cache Terminology (PDF)
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Problem set 3 (PDF)
Solutions to Problem Set 3 (PDF)
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| Lecture 12: Cache-Oblivious Algorithms and Spatial Locality |
Experiments with Cache-Oblivious Matrix Multiplication (PDF)
Experiments with Memory Access and Matrices
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[No problem set] |
| Lecture 13: LU Factorization and Partial Pivoting |
[No handout/notebook] |
[No problem set] |
| Lecture 14: Cholesky Factorization and other Specialized Solvers. Eigenproblems and Schur Factorizations |
[No handout/notebook] |
[No problem set] |
| Lecture 15: Eigensolver Algorithms: Companion Matrices, Ill-Conditioning, and Hessenberg Factorization |
Hessenberg Factorization (PDF) |
[No problem set] |
| Lecture 16: The Power Method and the QR Algorithm |
[No handout/notebook] |
[No problem set] |
| Lecture 17: Shifted QR and Rayleigh Quotients |
[No handout/notebook] |
[No problem set] |
| Lecture 18: Krylov Methods and the Arnoldi Algorithm |
[No handout/notebook] |
[No problem set] |
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Lecture 19: Arnoldi and Lanczos with Restarting
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Why Restarting Arnoldi/Lanczos is not Trivial (PDF)
Experiments with Arnoldi Iterations
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[No problem set] |
| Lecture 20: The GMRES Algorithm and Convergence of GMRES and Arnoldi |
[No handout/notebook] |
Problem set 4 (PDF)
Solutions to Problem Set 4 (PDF)
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| Lecture 21: Preconditioning Techniques. The Conjugate-Gradient Method |
Large-Scale Linear Algebra: Dense Matrix Methods |
[No problem set] |
| Lecture 22: Convergence of Conjugate Gradient |
[No handout/notebook]
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[No problem set] |
| Lecture 23: Biconjugate Gradient Algorithms |
[No handout/notebook] |
[No problem set] |
| Lecture 24: Sparse-Direct Solvers |
Sparse Matrix Algorithms (PDF) (Courtesy of Per-Olof Persson. Used with permission.)
Sparse-Direct Solvers in Julia
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[No problem set] |
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Lecture 25: Overview of Optimization Algorithms
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A Brief Overview of Optimization Problems (PDF)
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Midterm exam (PDF)
Solutions to midterm exam (PDF)
Midterm exams and solutions from previous years
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| Lecture 26: Adjoint Methods |
Adjoint Methods (PDF)
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[No problem set] |
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Lecture 27: Adjoint Methods for Eigenproblems and Recurrence Relations
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Adjoint Methods and Sensitivity Analysis for Recurrence Relations (PDF)
Recurrence Relation
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[No problem set] |
| Lecture 28: Trust-Regions Methods and the CCSA Algorithm |
[No handout/notebook]
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[No problem set] |
| Lecture 29: Lagrange Dual Problems |
Lagrangian, Lagrange Dual Function and Dual Problem |
[No problem set] |
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Lecture 30: Quasi-Newton Methods and the BFGS Algorithm
Lecture 31: Derivation of the BFGS Update
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Quasi-Newton Optimization: Origin of the BFGS Update (PDF)
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[No problem set] |
| Lecture 32: Derivative-Free Optimization by Linear and Quadratic Approximations |
[No handout/notebook] |
[No problem set] |
| Lecture 33: Numerical Integration and the Convergence of the Trapezoidal Rule |
Numerical Integration and the Redemption of the Trapezoidal Rule (PDF)
Fourier Cosine Series Examples (PDF)
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[No problem set] |
| Lecture 34: Clenshaw-Curtis Quadrature |
[No handout/notebook] |
[No problem set] |
| Lecture 35: Chebyshev Approximation |
[No handout/notebook] |
[No problem set] |
| Lecture 36: Integration with Weight Functions, and Gaussian Quadrature |
[No handout/notebook] |
[No problem set] |
| Lecture 37: Adaptive and Multidimensional Quadrature |
[No handout/notebook] |
[No problem set] |
| Lecture 38: The Discrete Fourier Transform (DFT) and FFT Algorithms |
Fast Fourier Transform Algorithms (PDF) |
[No problem set] |
| Lecture 39: FFT Algorithms and FFTW |
Fast Fourier Transform and Fast Fourier Transform in the West (PDF - 2.6MB) |
[No problem set] |
