THE BISECTION METHOD

the bisection method. Given a nonlinear function f(x), we seek a value of x for which

f(x) = 0

Such a solution value for x is called a root of the equation, and a zero of the

function f(x).

The essence of the bisection method lies in the fact that the sign of a function f(x) changes on opposite sides of a root. Suppose the function f(x) has one root in the interval between x = a and x = c, or [a,c], as shown in the Figure below.

 The bisection method is based on the fact that when an interval [a,c] contains a

root, the sign of the function at the two ends (f(a) and f(c)) are opposite each

other, namely

f(a)*f(c) < 0

The first step in the bisection method is to bisect the interval [a,c] into two

halves, namely [a,b] and [b,c], where

b=(a + b)/2

By checking the sign of

f(a)*f(b)

the half-interval containing the root can be identified. If

f(a)*f(b) < 0

then, the interval [a,b] has the root, otherwise the other interval [b,c] has the

root. Then the new interval containing the root is bisected again. As the procedure is repeated the interval becomes smaller and smaller. At each step, the midpoint of the interval containing the root is taken as the best approximation of the root. The iterative procedure is stopped when the halfinterval size is less than a prespecified size.

This method is illustrated in the following flowchart.

Roots of Equations

Roots of Equations

Darcy Law

Henry Darcy (1803 to 1858) invented the modern style Pitot tube, was the first researcher to suspect the existence of the boundary layer in fluid flow, contributed in the development of the Darcy-Weisbach equation for pipe flow resistance, made major contributions to open channel flow research and of course developed Darcy's Law for flow in porous media. His Law is a foundation stone for several fields of study including ground-water hydrology, soil physics, and petroleum engineering.

CIPHER SIGNIFICANT

We can define the concept of significant cipher as that which provides unambiguous information about a specific experimental measurement.

• All nonzero numbers expressing quantities equal to or greater than the experimental uncertainty are significant.
• All zeros between significant digits are significant.
• in a number with digits to the right of the decimal zeros to the right of last nonzero numbers are significant.

Mathematic Model

Mathematical models of reality are the vastly more important type of representation. Essentially, anything in the physical or biological world, whether natural or involving technology and human intervention, is subject to analysis by mathematical models if it can be described in terms of mathematical expressions. Thus, optimization and control theory may be used to model industrial processes, traffic patterns, sediment transport in streams, and other situations; information theory may be used to model message transmission, linguistic characteristics, and the like; and dimensional analysis and computer simulation may be used to model atmospheric circulation patterns, stress distribution in engineering structures, the growth and development of landforms, and a host of other processes in science and engineering.

Numerical Methods in Engineering

Numerical Analysis

Numerical Analysis is the study of algorithms that use numerical approximation for the problems of continuous mathematics .

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences,

but in the 21^st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in the movement of heavenly bodies; optimization occurs in portfolio management; numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Before the advent of modern computers numerical methods often depended on hand interpolation in large printed tables. Since the mid 20th century, computers calculate the required functions instead.

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