Difference between Newton Raphson Method and Regular Falsi Method
Last Updated :
07 Aug, 2024
Numerical computations play an important role in solving real-life mathematical problems. Numerical methods are the procedures by applying arithmetic operations one can formulate mathematical problems to find the approximate result. In this, there is no need for algorithms, because numerical methods require the programming logic concept for the implementation. Following is the process of the numerical method,
- Formulated mathematical problems in simple arithmetic operations. This formulation is called the numerical implementation of the problem.
- Programming logic is then developed for numerical implementation. The programming is usually done with some high-level languages like Fortran, Basic, etc.
- The programs are then executed on the computing tools like computers.
- The results are displayed on the screen.
There are different numerical methods for the solutions of the problem but the particular method depends on the situation from which the problem is taken. The following methods come under the concept of obtaining the roots of the equation.
Newton Raphson Method
The Newton Raphson Method is one of the fastest methods among the bisection and false position methods. In this method, take one initial approximation instead of two. It is the process for the determination of a real root of an equation f(x) = 0 given just one point close to the desired root.
Formula for Newton raphson method: x1 = x0 - f(x0)/f'(x0)
Example: Find a root of an equation f(x) = x3 - x - 1
Solution:
Given equation x3 - x - 1 = 0
Using differentiate method the equation is,
∴ f′(x) = 3x2 - 1
Here, f(1) = -1 < 0 and f(2) = 5 > 0
∴ Root lies between 1 and 2
x0 = (1 + 2)/2 = 1.5
f(x0) = f(1.5) = 0.875
f′(x0) = f′(1.5) = 5.75
x1 = x0 - f(x0) / f′(x0)
=1.5 - 0.875/ 5.75
x1 = 1.34783
f(x1) = f(1.34783) = 0.10068
f′(x1) = f′(1.34783) = 4.44991
x2 = x1 - f(x1)/f′(x1)
= 1.34783 - 0.10068/4.44991
x2 = 1.3252
f(x2) = f(1.3252) = 0.00206
f′(x2) = f′(1.3252) = 4.26847
x3 = x2 - f(x2)/f′(x2)
=1.3252 - 0.00206/4.26847
x3 = 1.32472
f(x3) = f(1.32472) = 0
f′(x3) = f′(1.32472) = 4.26463
x4 = x3 - f(x3)/f′(x3)
=1.32472 - 0/ 4.26463
x4 = 1.32472
The Approximate root of the equation x3 - x - 1 = 0 using Newton Raphson method is 1.32472.
Advantages of Newton Raphson method
- It is the best method to solve non-linear equations.
- It is also used to solve non-linear equations, non-linear differentials, and non-linear integral equations.
- The order of convergence is quadratic.
- Easy to implement on a computer.
Disadvantages of Newton Raphson method
- This method becomes complicated if the derivative of the function f(x) is not simple.
- This method also fails if f(x) = 0, for some value of x.
- In each iteration, we have to evaluate two quantities f(x) and f'(x) for some x.
Regular Falsi Method
This method is the same as bisection but it is must faster than the bisection method. This is one of the oldest methods to find the real root of an equation f(x) = 0 and closely resembles with Bisection Method.
Graphical Representation of Regula Falsi MethodFormula for regular falsi method: x_{n+1} = (x_{n-1}f(x_n) - \frac{x_nf(x_{n-1}))}{(f(x_n)-f(x_{n-1}))}
Example: Find a root of an equation f(x) = x3 - x - 1
Solution:
Given equation, x3 - x - 1 = 0
let x = 0, 1, 2
f(1) = -1 < 0 and f(2) = 5 > 0
Root lies between these two points x0 = 1 and x1 = 2
x2 = x0 - f(x0)
= x1 - x0
f(x1) - f(x0)
x2 = 1 - (-1)⋅
= 2 - 1
= 5 - (-1)
x2 = 1.16667
f(x2) = f(1.16667) = -0.5787 < 0
f(1.16667) = -0.5787 < 0 and f(2) = 5 > 0
Root lies between these two points x0 = 1.16667 and x1 = 2
x3 = x0 - f(x0)
= x1 - x0
f(x1) - f(x0)
x3 = 1.16667 - (-0.5787)
= 2 - 1.16667
= 5 - (-0.5787)
x3 = 1.25311
f(x3) = f(1.25311) = -0.28536 < 0
f(1.25311) = -0.28536 < 0 and f(2) = 5 > 0
Root lies between these two points x0 = 1.25311 and x1 = 2
x4 = x0 - f(x0)⋅
= x1 - x0
f(x1) - f(x0)
x4 = 1.25311 - (-0.28536)⋅
= 2 - 1.25311
= 5 - (-0.28536)
x4 = 1.29344
f(x4) = f(1.29344) = -0.12954 < 0
f(1.29344) = -0.12954 < 0 and f(2) = 5 > 0
Root lies between these two points x0 = 1.29344 and x1 = 2
x5 = x0 - f(x0)⋅
= x1 - x0
= f(x1) - f(x0)
x5 = 1.29344 - (-0.12954)⋅
= 2 - 1.29344
= 5 - (-0.12954)
x5 = 1.31128
f(x5) = f(1.31128) = -0.05659 < 0
f(1.31128) = -0.05659 < 0 and f(2) = 5 > 0
Root lies between these two points x0 = 1.31128 and x1 = 2
x6 = x0 - f(x0)⋅
= x1 - x0
= f(x1) - f(x0)
x6 = 1.31128 - (-0.05659)⋅
= 2 - 1.31128
= 5 - (-0.05659)
x6 = 1.31899
f(x6) = f(1.31899) = -0.0243 < 0
f(1.31899) = -0.0243 < 0 and f(2) = 5 > 0
Root lies between these two points x0 = 1.31899 and x1 = 2
x7 = x0 - f(x0)⋅
= x1 - x0
f(x1) - f(x0)
x7 = 1.31899 - (-0.0243)⋅
= 2 - 1.31899
= 5 - (-0.0243)
x7 = 1.32228
f(x7) = f(1.32228) = -0.01036 < 0
f(1.32228) = -0.01036 < 0 and f(2) = 5 > 0
Root lies between these two points x0 = 1.32228 and x1 = 2
x8 = x0 - f(x0)⋅
x1 - x0
f(x1) - f(x0)
x8 = 1.32228 - (-0.01036)⋅
= 2 - 1.32228
= 5 - (-0.01036)
x8 = 1.32368
The approximate root of the equation x3 - x - 1 = 0 using the Regula Falsi method is 1.32368.
Advantages of Regular falsi method
- Its convergence is faster than the Bisection Method.
- It is easy to implement on a computer.
- Before starting the next iteration; only need to find id f(xn + 1)
Disadvantages of Regular falsi method
- The formula is very complicated.
- Time-consuming as compared to other methods.
Comparison between Regular Falsi Method and Newton Raphson Method
| Sr. No. | Regular Falsi Method | Newton Raphson Method |
|---|
| 1. | The rate of convergence is superlinear. | Here, the rate of convergence is second-order or quadratic. |
|---|
| 2. | Formula is: x_{n+1} = (x_{n-1}f(x_n) - \frac{x_nf(x_{n-1}))}{(f(x_n)-f(x_{n-1}))} | Formula is: x1 = x0 - f(x0)/f'(x0) |
|---|
| 3. | In this method, two initial approximations of the root are taken in which the root is expected to lie. | In this method, one initial approximation of the root is taken. |
|---|
| 4. | The computation of function per iteration is 1. | The computation of function per iteration is 2. |
|---|
| 5. | The initial approximation is less sensitive. | The initial approximation is very sensitive. |
|---|
| 6. | There is no need to find derivatives. | There is a need to find derivatives. |
|---|
| 7. | It is not applicable for finding complex, multiple, and nearly equal two roots. | It is applicable for finding complex, multiple, and nearly equal two roots. |
|---|
Similar Reads
Discrete Mathematics Tutorial Discrete Mathematics is a branch of mathematics that is concerned with "discrete" mathematical structures instead of "continuous". Discrete mathematical structures include objects with distinct values like graphs, integers, logic-based statements, etc. In this tutorial, we have covered all the topic
3 min read
Inequalities in LaTeX LaTeX is a powerful typesetting system widely used for creating high-quality scientific and technical documents. LaTeX offers precise and flexible tools for displaying a variety of inequalities, from simple linear ones to more intricate systems. In mathematics, an inequality represents a comparison
2 min read
Engineering Mathematics Tutorials Engineering mathematics is a vital component of the engineering discipline, offering the analytical tools and techniques necessary for solving complex problems across various fields. Whether you're designing a bridge, optimizing a manufacturing process, or developing algorithms for computer systems,
3 min read
Propositional Logic Logic is the basis of all mathematical reasoning and all automated reasoning. The rules of logic specify the meaning of mathematical statements. These rules help us understand and reason with statements such as -\exists~x~such~that~x~\neq~a^2~+~b^2,~where~\:x,~a,~b\in~ZWhich in Simple English means
10 min read
Eigenvalues and Eigenvectors Eigenvalues and eigenvectors are fundamental concepts in linear algebra, used in various applications such as matrix diagonalization, stability analysis and data analysis (e.g., PCA). They are associated with a square matrix and provide insights into its properties.Eigen value and Eigen vectorTable
10 min read
Number Theory in Mathematics Number theory is a branch of mathematics that studies numbers, particularly whole numbers, and their properties and relationships. It explores patterns, structures, and the behaviors of numbers in different situations. Number theory deals with the following key concepts:Prime Numbers: Properties, di
4 min read
Matrices Matrices are key concepts in mathematics, widely used in solving equations and problems in fields like physics and computer science. A matrix is simply a grid of numbers, and a determinant is a value calculated from a square matrix.Example: \begin{bmatrix} 6 & 9 \\ 5 & -4 \\ \end{bmatrix}_{2
3 min read
Arrow Symbols in LaTeX Arrow symbols are a key feature of LaTeX, commonly used in academic papers, mathematical documents, and technical writing. LaTeX uses arrows to represent relationships, directions, and logical flows. It has dozens of special mathematical symbols. A few of them, such as +, -, <, and >, are prod
3 min read
Newton Raphson Method Newton Raphson Method or Newton Method is a powerful technique for solving equations numerically. It is most commonly used for approximation of the roots of the real-valued functions.Newton-Raphson Method is a numerical technique for approximating the roots of real-valued functions.It starts with in
8 min read
Rules of Inference Rules of Inference: Rules of inference are logical tools used to derive conclusions from premises. They form the foundation of logical reasoning, allowing us to build arguments, prove theorems, and solve problems in mathematics, computer science, and philosophy. Understanding these rules is crucial
9 min read