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 :

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

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The interpolated value is :