# 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.)

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

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

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

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

- Stability of a method
- Forward Euler
- Systems
- Nonlinear ODEs
- Backward Euler

- Stiff ODEs
- Predictor-corrector methods
- Runge-Kutta methods, Butcher tableau, properties

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

- 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

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

- 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

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

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

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

- Quadrature, conditioning
- Interpolatory quadrature
- Newton-Cotes
- Midpoint rule
- Trapezoidal rule
- Simpson's rule
- Simpson's second (3/8) rule

- Clenshaw-Curtis

- Newton-Cotes
- Accuracy and stability of interpolatory quadrature
- Gaussian quadrature

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

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

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

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

- Chebyshev interpolation

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

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

- Intro to Interpolation

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

- 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

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

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

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

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

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

Topics:

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

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

Topics:

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

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

Topics:

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

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

Topics:

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

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

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

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

Topics:

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

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

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

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

Topics:

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

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

Topics:

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

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

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

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

Topics:

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

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

Topics:

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

**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.)

Topics:

- Rounding modes, density of floating point numbers
- Unit roundoff (machine epsilon)
- Floating point arithmetics

**Lecture 3**(Aug. 31, 2021, 11 a.m. - Aug. 31, 2021, 12:15 p.m.)

Topics:

- Floating point
- Fw/bw error in-class activity
- Underflow, overflow, rounding, subnormals

**Lecture 2**(Aug. 26, 2021, 11 a.m. - Aug. 26, 2021, 12:15 p.m.)

Topics:

- Forward/backward error
- Conditioning
- Example: Conditioning of function evaluation

- Accuracy and stability

**Lecture 1**(Aug. 24, 2021, 11 a.m. - Aug. 24, 2021, 12:15 p.m.)

Topics:

- Introduction
- Class Policies
- Truncation and Rounding
- Well-posedness
- Norms (the very beginning)