• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Newton-Raphson Method: This is a fixed-point estimation method.

Extracts from this document...

Introduction

Method 2: Newton Raphson

Newton Raphson

Newton-Raphson Method:

This is a fixed-point estimation method. The estimate starts at x 1,for a root of f(x) = 0. A tangent is then draw to the curve y = f(x) at the point (x 1, f(x 1)). The point at which the tangent cuts the x-axis then gives the next approximation for the root, and the process is repeated.

I am going to use the equation y = x³ - 3x + 1.

***

As you can see there are three roots in this graph, they are in the interval [-2, -1]

[0, 1]

[1, 2]

The gradient for the tangent to the curve at (x 1, f(x 1)) is f’(x 1) (meaning dy/ dx for x). The equation of the tangent is: y-y1 = m(x-x1). Therefore y-f(x1) = f’(x1) [x-x1]. This tangent passes through the point (x2, 0). Carrying on with this process, this will get closer and closer to the tangent. But there is a general formula for this process:

x n+1 = x n – f(Xn)/ f’(Xn)

Returning to my function: f(x)

Middle

0.28

0.007317

-4.9216

0.28148678

0.281487

6.2E-07

-4.92077

0.28148691

0.281487

4.22E-15

-4.92077

0.28148691

Therefore,

x b = 0.28148691 (8 d.p.)

1. Root c:
 x n f(x)` x n+1 -4.5 15.25 -4.07540984 -4.07541 11.60897 -4.00772671 -4.00773 11.06187 -4.00604832 -4.00605 11.04842 -4.00604730 -4.00605 11.04841 -4.00604730

Therefore,

x c = -4.00604730 (8 d.p.)

NEWTON-RAPHSON METHOD DOESN’T ALWAYS WORK!

This method will not work if:

1. If the initial value is not close to the root, or is near a turning point, the iteration may diverge or converge to another root!
1. This method can break down when the equation is discontinuous.

I will demonstrate this using the equation g(x) = x 3 – 5x 2 + 8x -4.1

In this case, if we use the value x=2 as the starting point we get a “DIVERGENT” pattern. So the iteration cannot go to the upper or lower root.

At x=2 the trial will fail because the derivative there is 0; and in the equation x n+1 = x n – g(x n)/g`(x n), g`(x n) can’t equal 0 as it will lead to an undefined quantity for x n+1.

Conclusion

 x y 1.1 -0.0034 1.111 0.016937 1.112 0.037361 1.113 0.05782

This table now shows that the root is in the interval [1.11,1.111].

## Error Bounds

X=

Failure of the Change of Sign Method

Although the Change of Sign method has been proved to work above, there are many examples of cases or equations which would be wrongly represented by this technique. Examples of these are shown below:

1. Repeated Roots - in an initial search using the Change of Sign method, the table of values for an equation such as y = (x-1.26)2 (x+1.4) would overlook the second root. As you can see from the graph below, there are two roots; one in the interval [-2,-1] and one in the interval [-1,0]. However, the table only shows one change of sign in the interval [-2,-1].

2.   The Change of Sign Method would also fail with an equation where all of the roots fall within the same interval, such as in the equation y = x3-1.7x2+0.84x-0.108. The table below shows only one change of sign in the interval [0,1]. This would indicate that there is only one root, rather than three, in this interval.

This student written piece of work is one of many that can be found in our AS and A Level Core & Pure Mathematics section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related AS and A Level Core & Pure Mathematics essays

1.  ## The Gradient Function

5 star(s)

9 729 2916 General proof - (x+h)^4 - x^4 = x^4 + h^4 +4hx� + 6x�h� + 4xh� - x^4 = x + h - x h h (h� + 4x� +6x�h +4xh�) = h� + 4x� +6x�h +4xh� h H tends to 0 again here, and every term contains an h except for (4x�).

2. ## Numerical solution of equations, Interval bisection---change of sign methods, Fixed point iteration ---the Newton-Raphson ...

-0.5 -0.0008 0 0.0142 -0.25 -0.002675 0.25 3 -0.5 -0.0008 -0.25 -0.002675 -0.38 0.00177813 0.125 4 -0.375 0.001778125 -0.25 -0.002675 -0.31 -0.000302 0.0625 5 -0.375 0.001778125 -0.3125 -0.000301953 -0.34 0.00086626 0.03125 6 -0.34375 0.00086626 -0.3125 -0.000301953 -0.33 0.00030275 0.015625 7 -0.328125 0.000302753 -0.3125 -0.000301953 -0.32 4.1191E-06 0.0078125 8 -0.320313 4.11911E-06

1. ## Change of Sign Method.

= 0.000013160499 and x= 0.867495 into f(x) = -0.000002560434 The above calculations illustrate that there is a change of sign. Therefore the root is 0.86749�0.000005. I will now find the other root of the equation that lies in the interval [1,2].

2. ## Numerical Method (Maths Investigation)

1 -0.0841 1.1 -0.0361 1.2 -0.0081 1.3 -1E-04 1.4 -0.0121 1.5 -0.0441 1.6 -0.0961 1.7 -0.1681 1.8 -0.2601 1.9 -0.3721 X f(x) 2 -0.5041 2.1 -0.6561 2.2 -0.8281 2.3 -1.0201 2.4 -1.2321 2.5 -1.4641 2.6 -1.7161 2.7 -1.9881 2.8 -2.2801 2.9 -2.5921 3 -2.9241 NEWTON-RAPHSON METHOD One of the Fixed

1. ## Different methods of solving equations compared. From the Excel tables of each method, we ...

0.00977173 0.68750000 -0.00091797 0.70312500 0.00791229 0.69531250 0.00327038 0.00441513 0.68750000 -0.00091797 0.69531250 0.00327038 0.69140625 0.00112058 0.00209417 0.68750000 -0.00091797 0.69140625 0.00112058 0.68945313 0.00008753 0.00101928 0.68750000 -0.00091797 0.68945313 0.00008753 0.68847656 -0.00041864 0.00050275 0.68847656 -0.00041864 0.68945313 0.00008753 0.68896484 -0.00016641 0.00025309 0.68896484 -0.00016641 0.68945313 0.00008753 0.68920898 -0.00003965 0.00012697 0.68920898 -0.00003965 0.68945313 0.00008753 0.68933105 0.00002389 0.00006359

2. ## The method I am going to use to solve x&amp;amp;#8722;3x-1=0 is the Change ...

Here is two of them (A&B). Rearrangement A: 3x^5+5x�-1=0 x^5= (1- 5x�)/3 x= [(1- 5x�)/3]^(1/5) I will draw the graph of the equation y=g(x)=[(1- 5x�)/3]^(1/5) on the Autograph software and the line y=x. This is shown below: And I am going to show how this method work and it is

1. ## newton raphson

Change of sign method has big advantage because they provide bounds (the two ends of the interval) within root lies. Knowing that the root lies in the interval [0.680760, 0.680770] means that you can take the root as 0.680765 with a maximum error of +/- 0.000005.

2. ## Change of Sign Method

However the integer search does not show a change of sign between x=2 and x=3 and therefore misses the 2 roots shown in the graph: x -6 -5 -4 -3 -2 -1 0 1 2 3 4 y -71 1 43 61 61 49 31 13 1 1 19 This is therefore a failure of the change of sign method. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to
improve your own work 