
At the end of this lecture learners will get interest in mathematics and find applications of numerical techniques to solve equations.
Explore how algebraic equations define relationships between variables, from linear to nonlinear forms, including quadratic and polynomial equations, multivariable equations, and systems of equations.
Explore how linear and nonlinear equations yield roots where the function crosses the x-axis, including real and complex roots, and use the minus B plus/minus formula to find x.
Descartes' rule of signs links sign changes in polynomial coefficients to the number of positive real roots and bounds negative real roots.
The intermediate value theorem shows how a continuous function on [A,B] with opposite signs at the endpoints indicates a root and guides root localization.
use the bisection method to locate roots by bracketing x1 and x2 with opposite signs. iterate with midpoint x0=(x1+x2)/2, using the intermediate value theorem and the error bound.
Apply the regula falsi method to solve f(x)=0 for a continuous function. Use two bracketing points and the line through them to obtain a root approximation and iterate until accuracy.
Explore the secant method for finding roots of transcendental equations using two initial guesses; approximate roots by intersecting the secant line with the x-axis and iterating until the desired accuracy.
This video explains the rate of convergence and comparison of bracketing and non bracketing methods.
With availability of digital computers it is simple to solve completed problems based on mathematics. Numerical methods are extremely powerful problem solving tools. They are capable of handling large systems of equations, nonlinearities and complicated geometries that are uncommon in engineering practice and often impossible to solve by traditional ways. Numerical methods enhance your problem solving skills. Transcendental equations are those which consist of terms like trigonometric, logarithmic and exponential terms. Solving such equation by traditional ways is time consuming and these are not solved with the help of digital computers.
A numerical method gives very simple ways to enhance your understanding of mathematics. Because one function of numerical methods is to reduce higher mathematics to basic arithmetic operations. Numerical methods can be implemented by writing algorithm and implanting these steps by using any language in the digital computers.
This course is a very basic for studying numerical methods for transcendental equations.
In this course we will study concept of root, Intermediate value theorem and bracketing and open ended methods to solve transcendental equations with graphical explanation. Also which methods converges fast is explained by solving one numerical by all methods. At the end comparison of all methods is done with various parameters.