Study guide for Midterm 2

Here is a non-exhaustive list of questions you should be able to answer as you prepare for the first midterm. The midterm will cover chapters 5-7 with some minor review questions from chapters 1-4.

Nonlinear Equations

What is a nonlinear equation and how is it different from a linear equation? Give examples
In general, what form do we want for our linear equation and what exactly are we looking for in terms of solutions?
Given a nonlinear system of equations, how many solutions do you have? Can this be estimated a priori?
What is a fixed point of a function?
Define multiplicity and state it mathematically?
What is the condition number of root finding? Can you use any form of conditioning (i.e. relative or absolute)? Explain
What is the Jacobian? Evaluated at the root what can we expect the Jacobian to be?
Define convergence rate and state the formula. Clasify the different kinds of convergence rates and give examples
What is interval bisection? What is its convergence rate? Given a tolerance can you tell in advance how many iterations it needs?
For a given function, how many fixed point problems can you use to find a root?
What condition do you need for your fixed point function so that it is locally convergent?
What happenes to a fixed point scheme if the previous condition is zero at the root?
What is Newton's method? State it mathematically. Define its convergence rate. Is it always this rate?
What is the secant method? State it mathematically. Define its convergence rate. Is it always this rate?
When would you prefer to use the secant method over Newton's method and viceversa?
What is inverse interpolation? Is it always a good idea to fit a higher order polynomial in this fashion?
Explain what a safeguarded method is
Describe some methods to find the roots of a polynomial
State Newton's method in multiple dimensions and define conditions for convergence
State Broyden's method in multiple dimensions and define conditions for convergence

Optimization

What is the definition difference between linear/nonlinear programming?
What is a convex function? What is a local/global minmum? What is a coercive function?
What is a critical point? What is the relation between critical point and local/global minmum?
Civen a unconstrained Optimization problem, what is requirement for critical to be a minmum?
Civen a constrained Optimization problem, what is the necessary condition for minmum?
What is the Lagrange function (form) of a constrained problem? What is the necessary condition of critical point for Lagrange function? For inequality constrains, what is the n KKT optimality conditions?
What is the general sensitivity and conditioning of optimization problem?
What is the golden section search method? In which condition it can be used? What is its convergence rate?
What is the Successive Parabolic Interpolation? What is its convergence rate?
What is the Steepest Descent Method? How to compute alpha? What is its convergence rate? What is its advantages and disadvantages?
What is the Safeguarded Methods?
What is the Newton’s Method (1D and $n$D)? What is its convergence rate? What is its restriction?
What is the Quasi-Newton Methods? What is its advantages over Newton method in general? What is its convergence rate in general?
What is the BFGS Method in high level? What is its advantages? What is its convergence rate in general?
What is the Conjugate Gradient Method in high level? What is Levenberg-Marquardt Method? What advantages do they have?
What is the Gauss-Newton Method? What is the Levenberg-Marquardt Method? What are their advantages? What properties does they have?
What is a Sequential Quadratic Programming (SQP method)?
What is Quadratic Programming?
What is the Active set Strategy?
What is the Penalty Methods? What is the Barrier Methods?

Interpolation

State the interpolation problem and some of its applications
What do basis functions mean in a general sense? Think of basis vectors or axes
How many different smooth interpolants can you have for a given set of data? What about piecewise interpolants?
What defines the conditioning of the interpolation problem as well as its existence? Is this true for any basis?
Define the monomial basis and use it to interpolate a set of points
Define the Lagrange basis and use it to interpolate a set of points
Define the Newton basis and use it to interpolate a set of points
What are divided differences? Use them to interpolate a set of points
What are orthogonal polynomials? Can you use any method learned in Chapter 3 to orthogonalize a set of polynomials? Pick a set and prove it
State the advantages and disadvantages of each basis (i.e. monomial, Lagrange, Newton, orthogonal) for interpolation
What are Chebyshev points? Are they preferable to equispaced points? Explain
In interpolating a continous function what is the maximum error you can attain? Can you provide a bound?
What is the difference between Hermite interpolation and cubic spline interpolation?
Using a quadratic spline define the equations and continuity conditions? How many conditions are left free?