As you may have seen in our class policies, our "midterms" and "finals" 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.
This examlet is now available for scheduling.

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 170 minutes in length, and it will cover the material of from chapter 1 through 12.

Lecture Notes Lecture Live Scribbles

- The Poisson model problem
- Jacobi vs Conjugate Gradient
- Multigrid
- Discrete Fourier Transform
- FFT algorithm

Lecture Notes Lecture Live Scribbles

- Kronecker Products and Sparse Matrices
- Direct Methods for Sparse Linear Systems
- Fill and Elimination in Graphs
- Iterative Methods by Splitting
- Jacobi, Gauss-Seidel, and Successive Overrelaxation

Lecture Notes Lecture Live Scribbles

- Introduction to PDEs
- Classification of PDEs
- Method of lines
- Stability of PDEs
- Finite difference methods for PDEs
- Collocation and finite element methods for PDEs

Lecture Notes Lecture Live Scribbles

- Finite differences methods for ODE BVPs
- Collocation method
- Weighted residual and Galerkin method
- Weak form and differentiability of bases
- Finite element methods for ODE BVPs
- Eigenvalue ODE BVP problems

Lecture Notes Lecture Live Scribbles

- Boundary conditions and boundary value problems
- Solutions to linear nonhomogeneous ODEs
- Green's functions and conditioning of BVPs
- Shooting method
- Finite difference methods for ODE BVPs

Lecture Notes Lecture Live Scribbles

- Review ODEs and IVPs
- Multistage and multistep methods
- Runge Kutta methods

Lecture Notes Lecture Live Scribbles

- Ordinary differential equations basic definitions
- Error and stability in ODEs
- Forward and backward Euler method
- Order of accuracy
- Stiffness

As you may have seen in our class policies, our "midterms" and "finals" 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.
This examlet is now available for scheduling.

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 from chapter 5 through 7. There will be a few questions from the first 4 chapters too.

Lecture Notes Lecture Live Scribbles

- Review of quadrature rules
- Gaussian quadrature
- Integral equations
- Numerical differentiation
- Extrapolation

Lecture Notes Lecture Live Scribbles

- Review of piecewise interpolation and B-splines (lecture 20 notes/scribbles updated)
- Integration and quadrature
- Conditioning and stability
- Newton-Cotes quadrature rules
- Error in quadrature methods
- Quadrature with Chebyshev nodes

Lecture Notes Lecture Live Scribbles

- Orthogonal polynomials, normalization
- Lagrange interpolation
- Chebyshev nodes
- Chebyshev interpolation
- Piecewise interpolation
- Splines

Lecture Notes Lecture Live Scribbles

- Introduction to interpolation
- Existence and uniqueness of interpolants
- Monomial, Newton, and Lagrange bases

Lecture Notes Lecture Live Scribbles

- Krylov subspace methods for least squares
- Conjugate gradient method
- Penalty and barrier functions

Quiz 18: Conjugate Gradient and Constrained Numerical Optimization

Lecture Notes Lecture Live Scribbles

- Constrained optimization problems
- Lagrange function and the dual problem
- Sequential quadratic programming
- Convergence of first order methods for quadratic programming
- Conjugate gradient method

Lecture Notes Lecture Live Scribbles

- Successive parabolic interpolation
- Convergence of multidimensional Newton's method
- Basics of Quasi-Newton methods (BFGS)
- Nonlinear least squares problems
- Gauss-Newton method
- Levenberg-Marquardt (Tykhonov regularization)

Quiz 16: Algorithms for Unconstrained Numerical Optimization

Lecture Notes Lecture Live Scribbles

- Broyden's method for nonlinear equations
- Optimization problems
- Characterization of solutions and conditioning in optimization
- Golden section search
- Newton's method for optimization

Lecture Notes Lecture Live Scribbles

- Review 1D nonlinear equations
- Inverse interpolation
- Newton's method in higher dimension
- Convergence of Newton's iteration
- Convergence of fixed-point iterations

Lecture Notes Lecture Live Scribbles

- Nonlinear equations
- Conditioning of solving nonlinear equations
- Rates of convergence
- Fixed point iteration
- Bisection
- Newton's method

As you may have seen in our class policies, our "midterms" and "finals" 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.
This examlet is now available for scheduling.

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 four chapters.

Lecture Notes Lecture Live Scribbles

- Krylov subspaces
- Ritz vectors and values
- Arnoldi iteration
- Lanczos iteration
- Convergence of iterative methods
- Matrix functions
- Solving ordinary differential equations by matrix diagonalization

Lecture Notes Lecture Live Scribbles

- Introduction to field of values
- Review of Jordan/Schur forms
- Review of overall structure of eigenvalue algorithms
- Methods for diagoanlizing tridiagonal symmetric matrices
- Krylov subspace methods, Ritz values and Ritz vectors

Lecture Notes Lecture Live Scribbles

- Perturbation analysis of eigenvalue problems, Gerhsgorin circles
- Relationship between orthogonal iteration and QR iteration
- Introduction to Krylov subspace methods

Quiz 10: Sensitivity of Eigenvalue Problems and Krylov Subspace Methods

- Computing many eigenvalues at once, QR iteration
- Reduction to Hessenberg and tridiagonal form
- Iterative methods for computing extremal eigenvalues

Quiz 9: Solving Eigenvalue Problems by Similarity Transformations

- Eigenvalue problems
- Matrix properties in the context of eigenvalue problems
- Conditioning of eigenvalue problems
- Iterative methods for computing extremal eigenvalues

- Conditioning of linear least squares problems
- Stability in Normal equations and Gram Schmidt
- Householder and Givens QR and their stability
- QR with column pivoting

- Solving banded linear systems
- Linear least squares problem
- Solving least squares problems via SVD
- Gram-Schmidt orthogonalization
- Basics of QR factorization

- Solving triangular systems of equations
- LU factorization existence
- LU with partial pivoting
- Error in LU factorization

- Orthogonal matrices
- Singular values and conditioning
- Peturbation analysis of linear systems
- Solving simple linear systems

- Error in floating point arithmetic
- Vector and matrix norms
- Matrix condition number

- Floating point representation
- Floating point arithmetic
- Roundoff error analysis

- Course administration
- Motivation
- Applications
- Error
- Posedness
- Conditioning