Introduction

Various numerical methods

Examples

Real life use

references

Index

Equations that can be cast in the form of a

polynomial are referred to as algebraic equations. Equations involving more complicated terms, such as trigonometric, hyperbolic, exponential, or logarithmic functions are referred to as transcendental equations. The methods presented in this section are numerical methods that can be applied to the solution of such equations, to which we will refer, in general, as non-linear equations. In general, we will we searching for one, or more, solutions to the equation, f(x) = 0.

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Introduction

Bisection method

Newton – Raph son method

Secant method

False position method, etc

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Various numerical methods find the roots

The History of the Bisection Method

Although there is little concrete knowledge of the development the bisection method, we can infer that it was developed a short while after the Intermediate Value Theorem was first proven by Bernard Bolzano in 1817 (Edwards 1979). It appears that it was used as a proof of an intermediate theorem to the general proof Bolzano was developing for the Intermediate Value Theorem

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Bisection method

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If a function f(x) is continuous and there is a point a

that is negative and a point b that is positive then there is a point c between (a,b) that equal zero. An interval is always chosen in [a,b] which includes the root somewhere within. That interval [a,b] is then cut in half, and the half that contains the root is then chosen. That new interval is then cut in half once again, and the half that contains the root is chosen once again. The bisection method repeats these steps numerous times until the approximation is within a certain degree

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procedure

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Example 1

Consider finding the root of f(x) = x2 - 3. Let εstep = 0.01, εabs = 0.01 and start with the interval [1, 2].Table 1. Bisection method applied to f(x) = x2 - 3.

a b f(a) f(b)c = (a +b)/2

f(c) Updatenew b − a

1.0 2.0 -2.0 1.0 1.5 -0.75 a = c 0.5

1.5 2.0 -0.75 1.0 1.75 0.062 b = c 0.25

1.5 1.75 -0.75 0.0625 1.625 -0.359 a = c 0.125

1.625 1.75 -0.3594 0.0625 1.6875 -0.1523 a = c 0.0625

1.6875 1.75 -0.1523 0.0625 1.7188 -0.0457 a = c 0.0313

1.7188 1.75 -0.0457 0.0625 1.7344 0.0081 b = c 0.0156

1.71988/td>

1.7344 -0.0457 0.0081 1.7266 -0.0189 a = c 0.0078

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2.

2. Find the root of x4-x-10 = 0

The graph of this equation is given in the figure.

Let a = 1.5 and b = 2

IterationNo.

a b c f(a) * f(c)

1 1.5 2 1.75 15.264 (+ve)

2 1.75 2 1.875 -1.149 (-ve)

3 1.75 1.875 1.812 2.419 (+ve)

4 1.812 1.875 1.844 0.303 (-ve)

5 1.844 1.875 1.86 -0.027 (-ve)

So one of the roots of x4-x-10 = 0 is approximately 1.86

One of the biggest advantages to the bisection

method is that it never diverges. Error also

decreases with each iteration. Therefore, as the

interval keeps splitting, the approximation gets closer and closer to the desired root

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advantages

The biggest disadvantage of the bisection method is that

it converges slower than other methods and it cannot

depict multiple roots. Furthermore, if two roots lie close to

each other then the bisection method makes it difficult to

find both roots simultaneously. In the specific case of

f(x)=x2, the bisection method fails to converge on the root

(0,0). If a point a is chosen to the left of the zero and the

same point is taken to the right of the zero then the root will not be found.

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Disadvantages

Shot Detection in Video Content for Digital Video Library -

The study presented the usage of bisection method for shot detection in video content for the Digital Video Library (DVL). DVL is a networked Internet application allowing for storage, searching, cataloguing, browsing, retrieval, searching and uni-casting video sequences. The browsing functionality can be significantly facilitated by a fast shot detection process. Experiments show that usage of the bisection method, allows for accelerating shot detection about 3÷150 times (related to the shot density). At the end of the paper two possible networked applications are presented: a medical DVL developed for elearning purposes and a hypothetical networked news application

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Real-Life Applications

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Locating and computing periodic orbits in molecular systems - The Characteristic Bisection Method for finding the roots of non-linear algebraic and/or transcendental equations is applied to Li NC/Li CN molecular system to locate periodic orbits and to construct the continuation/bifurcation diagram of the bend mode family. The algorithm is based on the Characteristic Poly hidra which define a domain in phase space where the topological degree is not zero. The results are compared with previous calculations obtained by the Newton Multiple Shooting algorithm. The Characteristic Bisection Method not only reproduces the old results, but also, locates new symmetric and asymmetric families of periodic orbits of high multiplicity.

Bisection method for determining an adequate population size

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The name "Newton's method" is derived from Isaac Newton's

description of a special case of the method in De analysi per aequationes numero terminorum infinitas (written in 1669, published in 1711 by William Jones) and in De metodis fluxionum et serierum infinitarum (written in 1671, translated and published as Method of Fluxions in 1736 by John Colson). However, his method differs substantially from the modern method given above: Newton applies the method only to polynomials.

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History

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• He does not compute the successive approximations x_n, but computes a sequence of polynomials, and only at the end arrives at an approximation for the root x. Finally, Newton views the method as purely algebraic and makes no mention of the connection with calculus. Newton may have derived his method from a similar but less precise method by Vieta. The essence of Vieta's method can be found in the work of the Persian mathematician Sharaf al-Din al-Tusi, while his successor Jamshīd al-Kāshī used a form of Newton's method to solve x^P - N = 0 to find roots of N (Ypma 1995).

• Newton's method was first published in 1685 in A Treatise of Algebra both Historical and Practical by John Wallis. In 1690, Joseph Raphson published a simplified description in Analysis aequationum universalis. Raphson again viewed Newton's method purely as an algebraic method and restricted its use to polynomials, but he describes the method in terms of the successive approximations xn instead of the more complicated sequence of polynomials used by Newton. Finally, in 1740, Thomas Simpson described Newton's method as an iterative method for solving general nonlinear equations using calculus, essentially giving the description above. In the same publication, Simpson also gives the generalization to systems of two equations and notes that Newton's method can be used for solving optimization problems by setting the gradient to zero.

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Unlike the earlier methods, this method requires only one appropriate starting point as an initial assumption of the root of

the function At a tangent to is drawn. Equation of this tangent is given by

• The point of intersection, say , of this tangent with x-axis (y = 0) is taken to be the next approximation to the root of f(x) = 0. So on

substituting y = 0 in the tangent equation we get

)( 00 xfy and

10

00

xx

yx

dx

dy

atWe have and we need to find .

1x

Then,

10

00

/ )()(

xx

xfxf

Rearranging:)(

)(

0/

010

xf

xfxx

)(

)(

0/

001

xf

xfxx

Using and in the formula isn’t very

convenient, so, since we have)(xfy

0 at xdx

dy0y

)( 0/

10

00 xf

xx

yx

dx

dy

at

)(

)(

0/

001

xf

xfxx So,

We just need to alter the subscripts to find : 2x

)(

)(

1/

112

xf

xfxx

Generalising gives

)(

)(/1

n

nnn

xf

xfxx

We don’t need a diagram to use this formula but we must know how to differentiate . )(xf

Convergence is often very fast.

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We will use the Newton-Raphson method to find the positive root of the equation sinx = x2,

correct to 3D.

It will be convenient to use the method of false position to obtain an initial approximation.

Tabulating, one finds

With numbers displayed to 4D, we see that there is a root in the interval 0.75 < x < 1 at approximately

Example: 1

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Next, we will use the Newton-Raphson method; we have

and

yielding

Consequently, a better approximation is

Repeating this step, we obtain

so that

Since f(x2) = 0.0000, we conclude that the root is 0.877 to 3D.

The method is very expensive - It needs the function evaluation and then the derivative evaluation.

If the tangent is parallel or nearly parallel to the x-axis, then the method does not converge.

Usually Newton method is expected to converge only near the solution.

The advantage of the method is its order of convergence is quadratic.

Convergence rate is one of the fastest when it does converge.

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Advantages and Disadvantages

Applying NR to the system of equations we find that at

iteration k+1: all the coefficients of KCL, KVL and of BCE of the linear

elements remain unchanged with respect to iteration k Nonlinear elements are represented by a linearization of

BCE around iteration k This system of equations can be interpreted as the STA of

a linear circuit (companion network) whose elements are specified by the linearized BCE.

APPLICATION OF NEWTON RAPHSON METHOD TO A FINITE BARRIER QUANTUM WELL (FBQW) SYSTEM

Real life uses

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References

• http://www2.lv.psu.edu/ojj/courses/cmpsc-

201/numerical/bisection.html

• http://en.wikipedia.org/wiki/Bisection_method#Pseudo-

code

• Bisection Method, Autar Kaw and Jai

Paul, http://numericalmethods.eng.usf.edu

• http://newtons.wikia.com/wiki/NewtonRaphson_Wiki

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