1. Some project suggestionsMATH 5610/6860PROJECTSYou may work in pairs, or suggest your own project. Please choosea project by Nov 2 and notify me by email or in person that youhave done so. The following is expected from you, and both will beequally weighted to get the project grade.• A report due Fri 12/11. This doesn’t need to be long (2 pagesis OK) and can be handwritten. You should explain with yourown words the problem and the method. If your project hasa computational part, you should also include some numericalresults and the code you used to generate them. You do notneed to do proofs for the suggested problems.• A short (say 10min) presentation on either 12/3, 12/7 or12/8. If you plan to use the projector you can either bringyour laptop, or (preferably) bring your presentation on a flashdrive. You can also give a presentation on the chalkboard.1. Some project suggestionsI expect the amount of work for each of these projects to be betweenone and two homeworks. If you feel the project is taking longer, pleasecome see me as it probably means you are not approaching the problemcorrectly. If you have doubts about something or would like to knowwhat I am expecting from each project, please ask! I have personalcopies of the books that are cited, and you can borrow them.1. Newton’s method in n dimensions to solve the system of nonlinearequations F(x) = 0, whereF(x) =f1(x)f2(x)...fn(x)and x = (x1, x2, . . . , xn)T. I would expect you to solve a system forn = 2 or n = 3. Ref: B&F §10.2.2. Bairstow’s method to find the roots of a polynomial. Ref K&C 3.53. Laguerre’s iteration to find the roots of a polynomial. Ref K&C 3.54. Hermite interpolation when f(xi) and f0(xi) are specified at all n+1distinct nodes x0, x1, . . . , xn. Ref. B&F 3.312 MATH 5610/6860 PROJECTS5. Clamped cubic splines (and comparison to natural cubic splines wesaw in class). Ref B&F 3.46. B´ezier curves. Ref. B&F 3.57. Multiple integrals of functions of two variables on regions not neces-sarily rectangular. Implement Simpson double integral. Ref. B&F4.88. Multiple integrals of functions of two variables on regions not neces-sarily rectangular. Implement Gaussian double integral. Ref. B&F4.89. Use the Hermite family of orthogonal polynomials to generate Gaussian-Hermite integration formulas to approximateZ∞−∞f(x)e−x2dx.Ref. Abramowitz and Stegun, “Handbook of Mathematical Func-tions” §25.4.46 p890. This book is public domain and is availableonline to download.10. Generate a table with the first 20 Gaussian quadrature nodes andweights on the interval [-1,1], preferably using Jacobi matrices (see§37 in Trefethen and Bau, “Numerical Linear