Topic

No. of hours

Lecture

Tutorial/Practical

Introduction :-

The MATLAB and Octave environments.

Real numbers.

How we represent them.

How we operate with floating-point numbers.

Complex numbers.

Matrices.

Vectors.

Real functions.

The zeros.

Polynomials.

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Integration and differentiation.

To err is not only human.

Talking about costs.

The MATLAB language.

MATLAB statements.

Programming in MATLAB.

Examples of differences between MATLAB.

and Octave languages.

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Nonlinear equations : - 

Some representative problems.

The bisection method.  

The Newton method.

How to terminate Newton's iterations .  

The Newton method for systems of nonlinear.

Equations.

Fixed point iterations.

How to terminate fixed point iterations.

Acceleration using Aitken's method.

Algebraic polynomials.

H¨orner's algorithm.

The Newton-H¨orner method.

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Approximation of functions and data :-

Some representative problems.

Approximation by Taylor's polynomials.

Interpolation.

Lagrangian polynomial interpolation.

Stability of polynomial interpolation.

Interpolation at Chebyshev nodes.

Trigonometric interpolation and FFT.

Piecewise linear interpolation.

Approximation by spline functions.

The least-squares method.

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Numerical differentiation and integration :-

Some representative problems.

Approximation of function derivatives.

Numerical integration.

Midpoint formula.

Trapezoidal formula.

Simpson formula..

Interpolatory quadratures.

Simpson adaptive formula.

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Linear systems :-

Some representative problems.

Linear system and complexity.

The LU factorization method . 

The pivoting technique.

How accurate is the solution of a linear system?

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How to solve a tridiagonal system.

Over determined systems.

What is hidden behind the MATLAB command \  

Iterative methods.

How to construct an iterative method.

Richardson and gradient methods.

The conjugate gradient method.

When should an iterative method be stopped?

To wrap-up: direct or iterative?

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Eigenvalues and eigenvectors :-

Some representative problems.

The power method.

Convergence analysis.

Generalization of the power method..

How to compute the shift.

Computation of all the eigenvalues.

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Ordinary differential equations :-

Some representative problems.

The Cauchy problem.

Euler methods.

Convergence analysis.

The Crank-Nicolson method.

Zero-stability.

Stability on unbounded intervals.

The region of absolute stability.

Absolute stability controls perturbations.

High order methods.

The predictor-corrector methods.

Systems of differential equations.

Some examples..

The spherical pendulum.

The three-body problem..

Some stiff problems.

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Numerical approximation of boundary-value problems :-

Some representative problems.

Approximation of boundary-value problems.

Finite difference approximation of the one-dimensional Poisson problem.

Finite difference approximation of aconvection-dominated problem.

Finite element approximation of the one-dimensional Poisson problem.

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Finite difference approximation of the two-dimensional Poisson problem.

Consistency and convergence of finite difference.

discretization of the Poisson problem.

Finite difference approximation of the one-dimensional heat equation.

Finite element approximation of the one-dimensional heat equation.

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Hyperbolic equations: a scalar pure advection problem.

Finite difference discretization of the scalar.

transport equation.

Finite difference analysis for the scalar transport equation.

Finite element space discretization of the scalar advection equation.

The wave equation.

Finite difference approximation of the wave equation.

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