You must study Newton forward interpolation method using this Web application

Newton Forward Interpolation

Newton Forward Interpolation is a numerical method used to estimate the values of a function based on a set of known values. This technique is particularly useful when the function is tabulated, and we want to predict values for points that fall within the range of the known data. The method is based on the concept of finite differences and utilizes polynomial interpolation to achieve accurate estimates.

Key Concepts:

Finite Differences: The differences between successive values in a table of data are called finite differences. These differences are used to construct the polynomial interpolation.

Newton's Forward Difference Formula: This method uses forward differences, which involve the data points arranged in increasing order, to compute the interpolated value.

Interpolation Polynomial: The goal is to find a polynomial P(x) that passes through the given data points. Newton’s Forward Interpolation constructs this polynomial based on the first n points.

Finite Difference Table

x	y	Δ	Δ²	Δ³	Δ⁴
1	10
2	26
3	58
4	112
5	194

X: r: NFIF:

\({y_{NFIF}} = {y_0} + r \times \Delta {y_0} + \frac{{r \times \left( {r - 1} \right)}}{{2!}}{\Delta ^2}{y_0} + \frac{{r \times \left( {r - 1} \right) \times \left( {r - 2} \right)}}{{3!}}{\Delta ^3}{y_0} + \frac{{r \times \left( {r - 1} \right) \times \left( {r - 2} \right) \times \left( {r - 3} \right)}}{{4!}}{\Delta ^4}{y_0}\)

Quiz on introduction to Numerical Methods for Engineers

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Text and reference books.

1. JAIN M K_NUMERICAL METHODS FOR 8ed

2. Numerical Methods for Engineers

3.Numerical Methods

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Interpolation is the part numerical method of Engineering

Unequal intervals using Lagrange's Interpolation

Interpolation is a mathematical technique used to estimate values between known data points. It plays a crucial role in various fields such as numerical analysis, computer graphics, and engineering. One popular method for interpolation is the Lagrange interpolation method, named after the French mathematician Joseph-Louis Lagrange. The Lagrange interpolation method constructs a polynomial that passes through a given set of data points. Suppose you have a set of n+1 data points (x0, y0), (x1, y1), ..., (xn, yn). The Lagrange polynomial, denoted as P(x), is expressed as a linear combination of Lagrange basis polynomials, where each basis polynomial is associated with a specific data point. The general form of the Lagrange polynomial is: \[P(x) = L_0(x)y_0 + L_1(x)y_1 + \ldots + L_n(x)y_n\] Here, \(L_i(x)\) is the ith Lagrange basis polynomial, defined as: \[L_i(x) = \frac{(x-x_0)(x-x_1)\ldots(x-x_{i-1})(x-x_{i+1})\ldots(x-x_n)}{(x_i-x_0)(x_i-x_1)\ldots(x_i-x_{i-1})(x_i-x_{i+1})\ldots(x_i-x_n)}\] The Lagrange interpolation method provides a flexible and efficient way to approximate values between data points, making it a valuable tool in numerical analysis and scientific computing.

X	Y
0	-4
2	2
3	14
6	158

Lagrange Interpolation formula :

\[\begin{array}{l} y = f(x) = \frac{{\left( {x - {x_1}} \right)\left( {x - {x_2}} \right)\left( {x - {x_3}} \right){y_0}}}{{\left( {{x_0} - {x_1}} \right)\left( {{x_0} - {x_2}} \right)\left( {{x_0} - {x_3}} \right)}} + \\ \frac{{\left( {x - {x_0}} \right)\left( {x - {x_2}} \right)\left( {x - {x_3}} \right){y_1}}}{{\left( {{x_1} - {x_0}} \right)\left( {{x_1} - {x_2}} \right)\left( {{x_1} - {x_3}} \right)}} + \\ \frac{{\left( {x - {x_0}} \right)\left( {x - {x_1}} \right)\left( {x - {x_3}} \right){y_2}}}{{\left( {{x_2} - {x_0}} \right)\left( {{x_2} - {x_1}} \right)\left( {{x_2} - {x_3}} \right)}} + \\ \frac{{\left( {x - {x_0}} \right)\left( {x - {x_1}} \right)\left( {x - {x_2}} \right){y_3}}}{{\left( {{x_3} - {x_0}} \right)\left( {{x_3} - {x_1}} \right)\left( {{x_3} - {x_2}} \right)}} \end{array}\]

Input value to be interpolated in box below

Find/ Interpolate :

Click calculate button

The interpolated value is :

Interpolation formula| JNTUA| Numerical methods for engineers

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