Roots of equations

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Roots of Polynomials
Exercises Roots of Equations

DIGITAL METHODS


1. MODELING

• Components of a mathematical model.
• The differential equations as mathematical models
• Implications of the analytical solution.
• Engineering models.
• Theme applied: The Law of Darcy

2. NUMERICAL APPROACH

• Numerical Solution.
• Significant figures.
• Accuracy and precision.
• Contribution of the series to the numerical approximations.
• Definitions of error.
• Rounding errors.
• Truncation errors.
• Taylor series.Propagation of error.
• Total numerical error.
• Numerical simulation: concept, applications, tasks.
• Theme applied: Numerical Reservoir Simulation.

3. ROOTS OF EQUATIONS.

• Graphic methods.
• Closed methods: bisection, false position.
• Searches increase.
• Determination of initial values.
• Open methods: fixed point, Newton-Raphson, secant.
• Calculation of multiple roots.
• Basic operation of polynomials: conventional methods of Muller method, Bairstow method.
• Calculation of complex roots.
• Theme applied: Equations of State.

4. DIRECT METHODS FOR SOLUTION OF SYSTEMS OF LINEAR EQUATIONS

• Mathematical background.
• Conventional solution methods.
• Simple Gaussian elimination.Difficulties in the methods of disposal.
• Techniques for improving the solutions.
• Gauss-Jordan elimination.
• Complementary techniques: LU decomposition, matrix inverse, error analysis and condition of a system.
• Theme applied: Balance of Matter (Torres Separation).

5. INTERACTIVE METHODS FOR SOLUTION OF LINEAR EQUATIONS

• Special matrices.
• Jacobi Method.
• Gauss-Seidel.
• Gauss-Seidel relaxation.
• Special section: numerical solution of nonlinear systems of equations.

6. TREATMENT OF INFORMATION

• Basic statistical measures.
• Regression: linear, nonlinear, multivariable linear.
• Interpolation: linear, polynomial (Newton, Lagrange), Splines, TFI.
• Theme applied: Some Phenomena of Heat Transfer.

7. NUMERICAL DIFFERENTIATION AND INTEGRATION

• Trapezoidal Rule.
• Simpson's rules.
• Integration with unequal segments.
• Multiple Integrals.
• Finite difference approximation.
• Theme applied: Quantifying the Thermal Energy.

8. NUMERICAL TREATMENT OF ORDINARY DIFFERENTIAL EQUATIONS

• Euler method.
• Improved Euler method.
• Runge Kutta method.
• Systems of ordinary differential equations.
• Initial value problems.Boundary value problems.
• Eigenvalue problems.
• Theme applied: Balance of Matter (reactors).

9. NUMERICAL TREATMENT OF PARTIAL DIFFERENTIAL EQUATIONS

• Elliptic equations: Laplace equation, solving techniques.
• Hyperbolic equations: wave equation.
• Parabolic equations: the heat conduction equation, explicit methods, implicit methods, Crank-Nicholson method, use of two spatial dimensions.
• Theme applied: Simulation of Fluid Flow in Porous Media.