Introduction to Numerical Methods

Resource Index

This resource index gives users access to most of the course resources in a single location.

LecTURES	LECTURE HANDOUTS & ACCOMPANYING NOTEBOOKS	PROBLEM SETS/EXAM AND SOLUTIONS
Lecture 1: Course Overview, Newton’s Method for Root-Finding	Square Roots via Newton’s Method (PDF) Square Roots	Problem Set 1 (PDF) Problem set 1 notebook Solutions to Problem Set 1 (PDF) Solutions to Problem Set 1 notebook
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	[No problem set]
Lecture 3: Floating-Point Summation and Backwards Stability	Notes on the Accuracy of Naive Summation (PDF) Backwards Stability of Recursive Summation (PDF)	[No problem set]
Lecture 4: Norms on Vector Spaces	Notes on the Equivalence of Norms (PDF)	[No problem set]
Lecture 5: Condition Numbers	[No handout/notebook]	Problem set 2 (PDF) Solutions to Problem Set 2 (PDF)
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]
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	[No problem set]
Lecture 11: Matrix Operations, Caches, and Blocking	Performance Experiments with Matrix Multiplication (PDF) Ideal-Cache Terminology (PDF)	Problem set 3 (PDF) Solutions to Problem Set 3 (PDF)
Lecture 12: Cache-Oblivious Algorithms and Spatial Locality	Experiments with Cache-Oblivious Matrix Multiplication (PDF) Experiments with Memory Access and Matrices	[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]
Lecture 19: Arnoldi and Lanczos with Restarting	Why Restarting Arnoldi/Lanczos is not Trivial (PDF) Experiments with Arnoldi Iterations	[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)
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]	[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	[No problem set]
Lecture 25: Overview of Optimization Algorithms	A Brief Overview of Optimization Problems (PDF)	Midterm exam (PDF) Solutions to midterm exam (PDF) Midterm exams and solutions from previous years
Lecture 26: Adjoint Methods	Adjoint Methods (PDF)	[No problem set]
Lecture 27: Adjoint Methods for Eigenproblems and Recurrence Relations	Adjoint Methods and Sensitivity Analysis for Recurrence Relations (PDF) Recurrence Relation	[No problem set]
Lecture 28: Trust-Regions Methods and the CCSA Algorithm	[No handout/notebook]	[No problem set]
Lecture 29: Lagrange Dual Problems	Lagrangian, Lagrange Dual Function and Dual Problem	[No problem set]
Lecture 30: Quasi-Newton Methods and the BFGS Algorithm Lecture 31: Derivation of the BFGS Update	Quasi-Newton Optimization: Origin of the BFGS Update (PDF)	[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)	[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]