CS450 Calendar

Note: Some calendar entries are clickable and link to entries below.

Lecture 29 (Dec. 7, 2021, 11 a.m. - Dec. 7, 2021, 12:15 p.m.)

Quiz for Lecture 29

BVP conditioning, solution uniqueness
Shooting method, finite difference method
Collocation method
Weak solutions, weighted residual method
Galerkin/finite element method

Scribbles

Lecture 28 (Dec. 2, 2021, 11 a.m. - Dec. 2, 2021, 12:15 p.m.)

Quiz for Lecture 28

Runge-Kutta methods: Heun's method, RK4
Multi-step methods (BDFs)
Stability regions
More advanced methods for IVPs
BVP problem setup, solution existence

Scribbles

Lecture 27 (Nov. 30, 2021, 11 a.m. - Nov. 30, 2021, 12:15 p.m.)

Quiz for Lecture 27

Stability of a method
- Forward Euler
- Systems
- Nonlinear ODEs
- Backward Euler
Stiff ODEs
Predictor-corrector methods
Runge-Kutta methods, Butcher tableau, properties

Scribbles

Lecture 26 (Nov. 18, 2021, 11 a.m. - Nov. 18, 2021, 12:15 p.m.)

Quiz for Lecture 26

Properties of ODEs: linear, homogeneous, constant-coefficient, autonomous
Existence and uniqueness, conditioning
Euler's method (forward and backward)
Global and local error, order of accuracy
Stability of a method

Scribbles

Lecture 25 (Nov. 16, 2021, 11 a.m. - Nov. 16, 2021, 12:15 p.m.)

Quiz for Lecture 25

Differentiation matrices
Finite differences and Taylor expansion
Richardson extrapolation and Romberg integration
Examples of IVPs and BVPs
IVP for ODEs, reducing ODEs to first-order form

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Lecture 24 (Nov. 11, 2021, 11 a.m. - Nov. 11, 2021, 12:15 p.m.)

Quiz for Lecture 24

Gaussian quadrature continued
Composite quadrature
Numerical differentiation
Finite differences (forward difference)

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Lecture 23 (Nov. 9, 2021, 11 a.m. - Nov. 9, 2021, 12:15 p.m.)

Quiz for Lecture 23

Quadrature, conditioning
Interpolatory quadrature
- Newton-Cotes
  - Midpoint rule
  - Trapezoidal rule
  - Simpson's rule
  - Simpson's second (3/8) rule
- Clenshaw-Curtis
Accuracy and stability of interpolatory quadrature
Gaussian quadrature

Scribbles

Lecture 22 (Nov. 4, 2021, 11 a.m. - Nov. 4, 2021, 12:15 p.m.)

Quiz for Lecture 22

Truncation error in interpolation
Connection to Chebyshev nodes, convergence of order $n$
Piecewise linear interpolation
Piecewise cubic interpolation (splines)

Scribbles

Lecture 21 (Nov. 2, 2021, 11 a.m. - Nov. 2, 2021, 12:15 p.m.)

Quiz for Lecture 21

Modes and nodes
Lagrange interpolation
Newton interpolation
Orthogonal polynomials
- Legendre polynomials
- Chebyshev polynomials
Chebyshev interpolation

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Lecture 20 (Oct. 28, 2021, 11 a.m. - Oct. 28, 2021, 12:15 p.m.)

Quiz for Lecture 20

Gauss-Newton and Levenberg-Marquardt
Constrained optimization
- Lagrange multipliers
- KKT conditions
Intro to Interpolation

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Lecture 19 (Oct. 26, 2021, 11 a.m. - Oct. 26, 2021, 12:15 p.m.)

Quiz for Lecture 19

Improving steepest descent: momentum, heavy ball method
Brief introduction to: SGD, ADAM, conjugate gradient method, Nelder-Mead method
Newton's method and quasi-Newton methods ($n$ dimension)
Nonlinear least squares: setup and Gauss-Newton method

Scribbles

Lecture 18 (Oct. 21, 2021, 11 a.m. - Oct. 21, 2021, 12:15 p.m.)

Quiz for Lecture 18

Convexity
Optimality conditions
Sensitivity and conditioning
Unimodality, golden section search
Newton's method
Steepest descent, convergence

Scribbles

Lecture 17 (Oct. 19, 2021, 11 a.m. - Oct. 19, 2021, 12:15 p.m.)

Quiz for Lecture 17

Topics:

Discussion on Newton's method
Muller's method, inverse quadratic interpolation
Strategies to achieve global convergence
Systems in $n$ dimension (fixed point iteration, Newton's method, Broyden's method)
Optimization: problem statement
Optimization: existence and uniqueness, coercivity

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Lecture 16 (Oct. 14, 2021, 11 a.m. - Oct. 14, 2021, 12:15 p.m.)

Quiz for Lecture 16

Topics:

Stopping criteria
Bisection method
Fixed point iteration
Newton's method
Secant method

Scribbles

Lecture 15 (Oct. 12, 2021, 11 a.m. - Oct. 12, 2021, 12:15 p.m.)

Quiz for Lecture 15

Topics:

SVD (kiddy version)
Nonlinear equations, existence of a root
Sensitivity and multiplicity
Rates of convergence

Scribbles

Lecture 14 (Oct. 7, 2021, 11 a.m. - Oct. 7, 2021, 12:15 p.m.)

Quiz for Lecture 14

Topics:

Computational cost of QR iteration
QR iteration with Hessenberg matrices
Krylov space methods, Arnoldi iteration and Lanczos algorithms
Ritz values

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Lecture 13 (Oct. 5, 2021, 11 a.m. - Oct. 5, 2021, 12:15 p.m.)

Quiz for Lecture 13

Topics:

Schur form in details
Eigenvalues
Eigenvectors
Computing multiple eigenvalues
Simultaneous iteration
Orthogonal iteration
QR iteration (with shift)

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Lecture 12 (Sept. 30, 2021, 11 a.m. - Sept. 30, 2021, 12:15 p.m.)

Quiz for Lecture 12

Topics:

In practice
- Power iteration
- Inverse iteration
- Rayleigh quotient iteration
- Convergence of power iteration
- Failure modes of power iteration
- Cost of power iteration
Schur form

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Lecture 11 (Sept. 28, 2021, 11 a.m. - Sept. 28, 2021, 12:15 p.m.)

Quiz for Lecture 11

Topics:

Similar matrices, diagonalizability
Sensitivity of eigenvalues
Power iteration, inverse iteration, Rayleigh quotient iteration

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Lecture 10 (Sept. 23, 2021, 11 a.m. - Sept. 23, 2021, 12:15 p.m.)

Quiz for Lecture 10

Topics:

Rank-deficient matrix and least-squares
SVD, Eckart-Young-Mirsky theorem
Total least-squares, pseudoinverse (rank-deficient generalization)
Eigenvalues, algebraic and geometric multiplicity, defective matrix

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Lecture 9 (Sept. 21, 2021, 11 a.m. - Sept. 21, 2021, 12:15 p.m.)

Quiz for Lecture 9

Topics:

Solve linear least squares using QR
(Modified) Gram-Schmidt
Householder reflectors
Givens rotations

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Lecture 8 (Sept. 16, 2021, 11 a.m. - Sept. 16, 2021, 12:15 p.m.)

Quiz for Lecture 8

Topics:

Data fitting with polynomials
Least squares
Normal equation, orthogonal projection, pseudoinverse
Sensitivity and conditioning of least squares
QR factorization (the very beginning)

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Lecture 7 (Sept. 14, 2021, 11 a.m. - Sept. 14, 2021, 12:15 p.m.)

Quiz for Lecture 7

Topics:

Partial and complete pivoting
BLAS and LAPACK, LU complexity
LU for non-invertible or rectangular matrices
Round-off error in LU and pivoting
Schur complement, Sherman-Morrison formula

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Lecture 6 (Sept. 9, 2021, 11 a.m. - Sept. 9, 2021, 12:15 p.m.)

Quiz for Lecture 6

Topics:

Error when changing the matrix, Left/right preconditioning
SVD, relation to 2-norm and Frobenius norm
Back-substitution, Gaussian elimination, LU factorization

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Lecture 5 (Sept. 7, 2021, 11 a.m. - Sept. 7, 2021, 12:15 p.m.)

Quiz for Lecture 5

Topics:

Linear system, matrix norms
Condition number of a matrix
Residual and error

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Examlet 0 (tentative) (Sept. 2, 2021 - Sept. 5, 2021)

As you may have seen in our class policies, our "examlets" will take place in a computer-based testing facility ("CBTF") in Grainger Library.

You must schedule your test appointment with the Computer-Based Testing Facility at this link.

Find out more about the testing facility, such as:

where it is
when to show up
what to bring (and not to bring)

The exam will be 110 minutes in length, and it will cover the material of the first two lectures, plus linear algebra basics and Python coding basics.

Lecture 4 (Sept. 2, 2021, 11 a.m. - Sept. 2, 2021, 12:15 p.m.)

Quiz for Lecture 4

Topics: