short projects using numerical analysis methods(like, linear system of equations, differential equations, numerical integration,numerical root finding methods and so on)
Which will be suitable for small project based assignments for bachelor students?
For Numerical Analysis projects at undergraduate level students can be given various animations projects for Root Finding, Approximation Functions, Numerical Differentiation, Numerical Integration, etc as the value is found the point is plotted and so on. Use different PL..
For Bachelor's level, I can suggest two:
1) The algorithms to determine complex roots of a equation are very few. You can make the equation more complex to solve by having a non-integer order, and also by having complex coefficients. So why not work in that problem?
2) Code higher order (> 4 point) Gaussian-quadrature integration formula with the derivation of the respective weights.
Your own ideas are already very good (numerical root finding, systems of linear equations, numerical integration, etc.) I might also add numerical optimization in one dimension.
When I was going through my bachelor's degree (engineering), in the first year we faced many problems that involved solving N Equations in N Unknowns, root-finding, or finding the maximum/minimum of transcendental functions. But we only took a course on numerical methods in the second year. Personally, I think that course should have been given to students in the first year; it would have given them the tools to easier solve problems in first year--and all the following years.
My suggestion:
Start with a numerical routine for solving the Quadratic Equation for both real and complex solutions. Teach them how to create a robust solver that deals with possible underflow and overflow. Explain how rearranging the equation in certain ways can minimize computer round-off errors. Introduce students to computer quantities like Machine Epsilon and show them how these computer quantities have to be dealt with to quantify errors in the output.
Next, examine routines for solving N Equations in N Unknowns. Show them how pivoting a matrix appropriately can dramatically improve the accuracy of results. Introduce the concept of poorly-conditioned matrices, etc. Again, show students how errors in a solution can be quantified; otherwise, results may be meaningless.
Then extend these concepts to other root-finding problems, optimization routines, numerical integration, etc.
These routines are probably the ones used most often, so having them introduced in first year would be the most useful for students.
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