**Requirements/prerequisites:**

**Duration:** 24 weeks

**Number of lectures per week:** 3

**Assessment:** Examination and Projects (some years)

**End-of-year Examination:** One 3-hour examination

**Description: **

**Interpolation**

Lagrange interpolation

First order Hermite interpolation

Functionals

Interpolation in finite dimensional spaces

General Hermite interpolation

Divided differences

Newton's representation of the interpolant

Neville and Aitken algorithms

Pointwise error in Lagrange interpolation - Cauchy and Peano error
estimates

**Numerical Integration**

Errors in numerical intergration - Peano error estimate

Piecewise polynomial spaces

Gerschgorin theorem

Diagonal dominance

Newton-Cotes integration rules

Gaussian integration rules

Quadrature rules involving values of derivatives

Gaussian rules with some preassigned nodes

Chebyshev rules

Quadrature rules for periodic functions

Repeated quadrature rules

Richardson extrapolation

Romberg interpolation

**Numerical Linear Algebra**

Triangular systems

Gaussian elimination

LU decomposition

Gauss-Jordan algorithm

Conditioning and stability

Pivoting

Condition number of a matrix

LDU decomposition

Jacobi method

Gauss-Siedel method

Convergence of iterative methods

Direct method for tridiagonal matrices

Block tridiagonal matrices

**Nonlinear Systems**

Contraction mapping

Method of successive substitution

Jun 10, 1998