Q) What is Numerical Analysis?A) Numerical Analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis.Applications/Uses:*It can be essential in making numerical weather prediction feasible*It can compute the trajectory of a spacecraft and missiles.*It can improve the crash saftey of cars by using computer simulations of car crashes.*Airlines use sophisticated optimization algorithms to decide ticket prices, airplane and crew assignments and fuel needs.*Insurance companies use numerical programs for actuarial analysis.

Q) What is Root-finding Method?A) A root-finding method is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. Such an x is called a root of the function f. Numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards a limit, which is a root.

TYPES OF ROOT-FINDING METHODS:1) Bracketing Methods: As the name implies, these are based on two initial guesses that bracket the rootthat is, are on either side of the rooti) Bisection Method: ii) False-Position Method

2) Open Methods: These methods can involve one or more initial guesses, but there is no need for them to bracket the root.i) Secant Methodii) Newton-Raphson Methodiii) Interpolation Methods

Q) What are Iterative Methods?A) The term "iterative method" refers to a wide range of techniques that use successive approximations to obtain more accurate solutions to a linear system at each step.JACOBI's METHOD and GAUSS-SEIDEL METHOD are examples of Iterative Methods.Q) What do you mean by convergence and divergence?A) For any iterative numerical technique, each successive iteration results in a solution that moves progressively closer to the true solution. This is known as convergence. A numerical method is not always guaranteed to produce converging results. Convergence is subject to satisfying certain conditions. If these conditions are not met, each successive iteration produces a result that progressively moves away from the true solution. This is known as divergence. The Gauss Seidel approach is a numerical technique used to solve a system of linear equations. It is an improvement of another similar numerical technique known as the Jacobi Method, which is also used to solve a system of linear equations. The Gauss Seidel approach is generally preferred over the Jacobi approach as it provides faster convergence i.e. moves towards the true solution in fewer iterations.

Q) Explain Bisection Method.A) The bisection method in mathematics is a root-finding method that repeatedly bisects an interval and then selects a subinterval in which a root must lie for further processing. It is a very simple and robust method, but it is also relatively slow. Because of this, it is often used to obtain a rough approximation to a solution which is then used as a starting point for more rapidly converging methods.THEORY: The method is applicable for numerically solving the equation f(x) = 0 for the real variable x, where f is a continuous function defined on an interval [a, b] and where f(a) and f(b) have opposite signs. In this case a and b are said to bracket a root since, by the intermediate value theorem, the continuous function f must have at least one root in the interval (a, b).

ADVANTAGES:1. Bisection can be shown to be an "optimal" algorithm for functions that change sigh in [a,b] in that it produces the smallest interval of uncertainty in a given # of iterations2. f(x) need not be continuous on [a,b]3. convergence is guarenteed (linearly)

DISADVANTAGES/DRAWBACKS:4. Two initial guesses are required,with f(a) f(b)