## Numerical Analysis I and Laboratory

Code no:

**(1001)**Semester:**3th**, Teaching hours: 4## Core syllabus

Computer arithmetic, approximation and errors, floating point arithmetic, order of approximation and convergence, propagation of the errors difference operators.

Workshop exercises and programming in the PC-Lab using Matlab and Mathematica.

*Linear systems:*Gauss elimination method and factorization type methods. Computation of the inverse and determinant of a matrix. Iterative methods, fixed point method, Jacobi, Gauss-Seidel and Relaxation methods. Computation of eigenvalues and eigenvectors, power method.*Non linear equations and systems:*Bisection method, General iterative method, Newton-Raphson and Quasi-Newton methods, Secant and Regula-Falsi methods, Newton-Raphson and Quasi-Newton methods for non linear systems.*Interpolation and Approximation:*The problem of interpolation, Lagrange interpolation, Newton interpolation, Hermite interpolation, Piecewise interpolation and Spline interpolation. Least Squares approximation, discrete Least Squares approximation, continuous Least Squares approximation.*Numerical Differentiation and Integration:*Approximation of derivatives, Approximation of finite integral, basic rules for numerical integration, Composite Trapezoidal and Simpson integration formulas, Hermite and Gauss integration.Workshop exercises and programming in the PC-Lab using Matlab and Mathematica.

## SUBJECT CONTENT IN OUTLINE

The course aims at acquiring the knowledge for:

-The solution of systems of linear equations, non-linear algebraic equations, ordinary Differential equations, interpolation and approximation of data and numerical approximation of integrals.

-The aim of the course is to understand the importance of numerical methods for solving scientific and technological problems for which either there is no analytical solution, or it is very difficult to calculate it.

-A secondary objective is to familiarize students with: (a) constructing iterative methods to approximate numerical solutions to problems; and (b) convergence of iterative methods.

On completion of this course, students should be armed with numerical and computational techniques for solving a wide variety of fundamental mathematical problems that arise in diverse scientific areas.

In particular, upon successful completion of the course, the student will

-The solution of systems of linear equations, non-linear algebraic equations, ordinary Differential equations, interpolation and approximation of data and numerical approximation of integrals.

-The aim of the course is to understand the importance of numerical methods for solving scientific and technological problems for which either there is no analytical solution, or it is very difficult to calculate it.

-A secondary objective is to familiarize students with: (a) constructing iterative methods to approximate numerical solutions to problems; and (b) convergence of iterative methods.

On completion of this course, students should be armed with numerical and computational techniques for solving a wide variety of fundamental mathematical problems that arise in diverse scientific areas.

In particular, upon successful completion of the course, the student will

- understand the basic methods of Numerical Analysis a) to solve linear systems, non-linear equations and differential equations; b) to interpolate and approximate data and c) to approximate integrals.
- be familiar with the tools and techniques of iterative methods and can effectively use the appropriate stopping criteria.
- be aware of the importance of using stable algorithms to ensure the reliability of the results derived by the numerical methods.
- be able to distinguish the differences between the methods and choose the most appropriate to solve each problem.
- be able to analyse a) the asymptotic properties and the behaviour of the approximate models b) the numerical stability of the numerical solutions and c) the algorithmic and computational properties corresponding to the numerical methods.
- understands the effect of round-off errors in the computations and truncation errors of the methods and be able to calculate error bounds of approximate solutions.