Numerical Methods Solved Problems

Numerical Methods Solved Problems-60
We know from the definition of the derivative at a given point that it is the slope of a tangent at that point.

This is where numerical analysis comes into the picture. Note: the error analysis only gives a bound approximation to the error; the actual error may be much smaller.

The false position method (sometimes called the regula falsi method) is essentially same as the bisection method -- except that instead of bisecting the interval, we find where the chord joining the two points meets the X axis.

Such problems originate generally from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously.

These problems occur throughout the natural sciences, social sciences, medicine, engineering, and business.

While roots can be found directly for algebraic equations of fourth order or lower, and for a few special transcendental equations, in practice we need to solve equations of higher order and also arbitrary transcendental equations.

As analytic solutions are often either too cumbersome or simply do not exist, we need to find an approximate method of solution.The most popular types of computable functions \(p(x)\) are polynomials, rational functions, and piecewise versions of them, for example spline functions.Trigonometric polynomials are also a very useful choice.Suppose f : [a, b] → R is a differentiable function defined on the interval [a, b] with values in the real numbers R.The formula for converging on the root can be easily derived. Then we can derive the formula for a better approximation, xn 1 by referring to the diagram on the right.However, if iterating each step takes 50% longer, due to the more complex formula, there is no net gain in speed.For this reason, methods such as this are seldom used.There are other perspectives which vary with the type of mathematical problem being solved.Linear systems arise in many of the problems of numerical analysis, a reflection of the approximation of mathematical problems using linearization.Beginning in the 1940's, the growth in power and availability of digital computers has led to an increasing use of realistic mathematical models in science, medicine, engineering, and business; and numerical analysis of increasing sophistication has been needed to solve these more accurate and complex mathematical models of the world.The formal academic area of numerical analysis varies from highly theoretical mathematical studies to computer science issues involving the effects of computer hardware and software on the implementation of specific algorithms.


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