Office: Siebel Center 0207

Office: Siebel Center 0207

Office: Siebel Center 0207

Office: Siebel Center 0207

Office: Siebel Center 0207

Office: Siebel Center 0207

Office: Siebel Center 0207

- Introduction to interpolation
- Vandermonde systems and conditioning
- Polynomial bases
- Chebyshev nodes

- Nonlinear least squares and Gauss-Newton
- Constrained optimization optimality
- Sequential quadratic programming
- Active set methods
- Penalty and barrier methods

Quiz 17: Multidimensional Unconstrained Optimization Algorithms

- Golden section search
- Steepest descent
- Convergence of steepest descent and extrapolation methods
- Conjugate gradient
- Newton's method for multidimensional unconstrained optimization

Quiz 16: Multidimensional Unconstrained Optimization Algorithms

- Numerical optimization introduction
- Conditions of optimal solutions
- Golden section search
- Newton's method and quasi-Newton methods for 1D problems
- Safeguarding techniques
- Steepest descent

Quiz 15: Optimization Problems and Algorithms for 1D Optimization

- Secant method
- Convergence of fixed point iteration and Newton's method for multidimensional nonlinear solve
- Broyden's method
- Safeguarding techniques

Quiz 14: Cost and Robustness of Methods for Solving Nonlinear Equations

As you may have seen in our class policies, our "examlets" 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 50 minutes in length, and it will cover the material of the first four chapters.

- Introduction to nonlinear function solve problems
- Existence, uniqueness, and conditioning of roots
- Multiplicity of roots and conditioning
- Convergence rates
- Bisection algorithms
- Fixed-point functions and convergence
- Newton's method

- Review orthogonal iteration and QR iteration
- Divide and conquer for tridiagonal eigenproblem
- Krylov subspace methods
- Ritz values and vectors

Activity: Orthogonal Iteration

Quiz 12: Tridiagonal Eigenproblems and Krylov Subspace Methods

- Deflation
- Simultaneous and Orthogonal Iteration
- QR Iteration

Quiz 11: Schur Decomposition and QR Iteration

- Similarity and matrix types
- Canonical forms: Schur and Jordan forms
- Obtaining eigenvalues from triangular matrices
- Conditioning and sensitivity of eigenvalue decomposition and eigenpairs
- Convergence of power iteration
- Deflation

Quiz 10: Canonical Forms for Eigenvalue Problems and Deflation

- Eigenvalues and similarity
- Eigenvalue decomposition
- Power iteration, inverse iteration, Rayleigh-Quotient iteration
- Conditioning, Gershgorin theorem

Quiz 9: Conditioning and Basic Algorithms for Eigenvalue Problems

- Review lecture for eigenvalue problems

- Householder QR
- Givens QR
- Rank-deficient least squares
- Truncated SVD and Eckart-Young-Mirsky theorem
- Tykhonov regularization
- QR with column pivoting

- Review lecture for least squares

As you may have seen in our class policies, our "examlets" 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 50 minutes in length, and it will cover the material of the first two chapters.

- Linear least squares motivation and introduction
- Conditioning of linear least squares
- QR factorization
- Normal equations and Cholesky QR
- Gram-Schmidt methods
- Householder QR

- Pivoting in LU
- Multiplier growth and stability
- Cholesky and other specialized factorizations
- Sherman-Morrison-Woodbury formula

Quiz 6: Stability and Matrix Structure in Solving Linear Systems

- 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

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

Quiz 3: Floating Point, Matrix Norms, and Matrix Condition Number

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

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