Here we list questions related to this course that a student should be able to answer at the final state examinations.
Topic: Fundamentals of numerical mathematics.
• What types of errors we recognize (in general, for a precise
number x and its approximation x̂) and what information they tell
us? How do those errors appear in calculations? What impact does it have on
numerical computing?
• Describe some method (good candidates are bisection or Newton
method) for finding roots (that is, solving equations) numerically. Discuss
how we recognize when to stop the algorithm.
• Describe some method (most likely the Euler method) for
solving initial value problems (an ODE with initial condition) numerically.
Discuss the notion of the order of method.
[error of method depends on some quality indicator, typically step of the
method h, as power h p, larger p
better - why? Good examples: solving ODEs numerically, evaluating definite
integrals numerically, approximating derivative of a function numerically].
• Discuss the notion of computational complexity and show how it
applies to solving large systems of linear equations using elimination.
[Runtime of algorithm proportional to number of elementary calculations that
need to be performed, this depends on some parameter describing size of the
problem (dimension of matrix for elimination). We express it as formula,
interested in asymptotic growth (ignore less important terms). For
elimination n3, too much for large matrices, people prefer alternative
methods like iteration.]
Topic: Linear ordinary differential equations and systems of linear
ordinary differential equations.
• What is a general solution of an ODE? What is a particular
solution of an ODE? How can we determine (choose) a certain particular
solution?
• What can we say about the set of solutions of a homogeneous
linear ODE? How does it help us find a general solution? Where do we get the
fundamental system? (It is enough to know the basic answer with
characteristic numbers, no need to go into details about higher
multiplicity.)
• What can we say about solutions of a general (non-homogeneous)
linear ODE? How does it help us in solving such an equation?
• What can we say about the set solutions of a system of linear
ODEs? How can we find a general solution? (Again, the simplest case of
simple real eigenvalues is enough).