Università Roma Tre

1. Fundamentals of matrices and vector spaces
Operations between matrices and between matrices and vectors, properties of matrices. Determinant and its properties (Binet theorem), inverse matrix, rank of a matrix. Vector spaces: bases, base change, similar matrces, orthogonal matrices, Gram-Schmidt orthogonalization.

2. Eigenvalues and eigenvectors of matrices
Definitions; algebraic and geometric molteplicity. Properties of eigenvalues. Diagonalization of a matrix. Properties of symmetric and non-symmmetric matrices. Left eigenvectors, dual basis. Jordan normal form. Generalized eigenvalues and eigenvectors.
3. Solution of linear equation systems
Generalities. Iterative methods. Convergence algorithms. Direct methods: Gauss elimination. Least square solution for oversized systems.
4. Solution of nonlinear equation systems
Root separation. Iterative methods. Method of Newton-Raphson. Convergence criteria.
5. Method of solution for ordinary differential equation systems
Canonical form of ordinary differential equation systems. Methods of solution: (i) spectral representation, (ii) diagonalization, (iii) direct integration through convolution integral.

6. Frequency domain solution of ordinary differential equation systems
Laplace transform. Frequency response function. Impulse, indicial, harmonic responses.
7. Finite-difference methods for solution of ordinary differential equations
Fundamental concepts, first and second order finite differnce formulas. Backwards, forwards, central finite difference formulas. Accuracy order. Arbitrary point finite-difference formulas. Global and truncation errors Methods of Euler (implicit and explicit), and Crank-Nicholson.
8. Numerical integration
Generality; formulas from interpolating polynomials: Newton-Cotes, trapeziodal and Cavalieri-Simpson rules. Degree of precision of integration from interpolating polynomials. Gauss-Legendre quadrature rules and their degree of precision.

9. Calculus of variation
Definition of a functional. Eulero-Lagrange equations. The problem of the brachistocrone. The problem of a beam on elastic soil. Riccati equation for optimal control.

10. Methods for the solution of partial differential equation systems
Eigenfunctions and eigenvalues of a linear operator. Adjoint and self-adjoint operators. Eigenfunction method. Free vibration modes and frequencies. Rayleigh-Ritz and Galërkin methods.

11. Generalized curvilinear coordinates
Covariant and contravariant base vectors. Relations between dual bases. Differential operators on vectors expressed in covarian and contravariant components.

F.B. Hildebrand, “Methods of Applied Mathematics”, Dover Publications, NY, 1992.
A.D. Aleksandrov, A.N. Kolmogorov, M.A. Lavrent'ev, “Mathematics: Its Content, Methods and Meaning”, The MIT Press, Cambridge, Massachusetts, Sixth Edition, 1989.
J.D. Hoffman, ``Numerical Methods for Engineers and Scientists'', McGraw-Hill, Seconf Edition, 1992.