Lagrange Interpolation Formula

Lagrange Interpolation Formula

Using the Lagrange Interpolation Formula, one may obtain the Lagrange polynomial. a polynomial that assumes certain values at each given position. Lagrange Interpolation Formula is a polynomial expression of the function f at the nth degree (x). The new data points are located using the Lagrange Interpolation Formula approach inside the bounds of a discrete set of existing data points.

There will be a polynomial P with real coefficients meeting the requirements P(xi) = yi, I = 1, 2, 3,…, n, and the degree of the polynomial P must be smaller than the count of the real values, i.e., degree(P) n, given a small number of real values, x1, x2, x3,…, xn and y1, y2, y3,…,

The Lagrange Interpolation Formula may be used to locate a polynomial known as a Lagrange polynomial that assumes certain values at every location. Lagrange’s interpolation is an approximation to f using an Nth-degree polynomial (x). In the following sections, solved examples will help students comprehend the Lagrange Interpolation Formula.

What is Lagrange Interpolation Formula?

Lagrange Interpolation Formula is the process of locating additional data points within a range of a discrete set of data points. It is a method for estimating mathematical expressions that uses any possible middle value for the independent variable. To determine what more data could exist besides the data they have already acquired, Lagrange Interpolation Formula is mostly used. Lagrange Interpolation Formula is frequently used by experts such as photographers, scientists, mathematicians, and engineers. Interpolating the next position of a pixel based on the known positions of pixels in an image is frequently used when scaling images.

At Extramarks, students may find a description of the Lagrange Interpolation Theorem that has been written by subject specialists of Mathematics with an in-depth understanding of the subject and familiarity with the format of board exam question papers. The Lagrange Interpolation Theorem has been explained by Extramarks’ experts using the definitions of polynomials, interpolation, examples of polynomials, proof of the theorem, applications of the theorem, how to find interpolation, advantages, and disadvantages of interpolation, and frequently asked questions.

A polynomial is an expression with one or more indeterminates or variables, constants, and non-zero integer exponents. Mathematical operations including addition, subtraction, multiplication, and division are coupled with the expressions. Exponents cannot be negative or fractional, and there cannot be any division by a variable. A polynomial is something like x2 + 6x – 8. In reality, polynomials are a class of expressions. Polynomials may be thought of as a type of mathematics. They are used to express numbers in practically every area of mathematics. They are valued highly in several academic fields related to Mathematics, such as Calculus.

Extramarks offers a wide variety of study tools, including formulae, significant problems, sample papers, questions from prior years, revision notes, and much more, to assist students to improve their level of exam preparation and perform better on school and Mathematics board examinations. These study resources from Extramarks were compiled by a team of highly qualified Mathematics specialists at Extramarks while taking into account the format and scoring system of the question papers. Students will ultimately receive excellent marks on the Mathematics exam owing to the greatest accuracy in the preparation of these study materials.

Solved Examples Using Lagrange Interpolation Formula

This Lagrange Interpolation Formula may be used to create a polynomial that traverses a specified set of points and takes certain values at every given point. The Lagrange Interpolation Formula provides the approximation formula for nth-degree polynomials to the function f if a function f(x) is known at discrete places xi, I = 0, 1, 2,… (x). Additionally, it provides useful proof for the following Lagrange Interpolation Formula:

How can one describe point p(2,4) as a polynomial?

P(x) = 3

P(1) = 3

Similarly, how can one find a polynomial to represent a series of points like (2,3,4,5)?

P(x) = (x-4)/(2-4) * 3 + (x-2)/(4-2) * 5

P(2) = 3 and P(4) = 5

Using the aforementioned situations as examples, the general version of the Lagrange Interpolation theorem is as follows:

P(x) is equal to (x – x2)/(x1 – x2) (x-x3) (x1 – x3) * y1+(x-x1)(x-x3)/(x2-x1)(x2-x3) * y2+(x-x1)(x-x2)/(x3-x1)(x3-x2) * y3

Theorem: For n distinct real values, such as x1, x2, x3, x4, and xn, and n real values that might not be separate, such as y1, y2, y3, and y4, There is just one polynomial with real coefficients that satisfies the following formula:

P(xi) = yi, where I = 1, 2, 3,…, n, and deg(P) = n.

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FAQs (Frequently Asked Questions)

1. What are the applications of the Lagrange Interpolation Formula?

  • Lagrange Interpolation Formula Applications – In Science, it takes a lot of time and effort to solve a complex function. This makes conducting experiments challenging. The Lagrange Interpolation Formula approach is used to produce a little less intricate version of the original function.
  • The Lagrange Interpolation Formula generalises well-known mathematical principles like the statement that a line is uniquely specified by two points, the knowledge that the graph of a quadratic polynomial is uniquely determined by three points, and so on. There is a requirement that the points have distinct x coordinates.
  • The Lagrange Interpolation Formula is used in the picture enlargement approach to approximate the unknown data using interpolation polynomials in an effort to represent the tendency of image data. This aids in the enlarging of images.

2. What is the history of the Lagrange Interpolation Formula?

Waring created and initially published the Lagrange Interpolation Formula in 1779. Lagrange published the Lagrange Interpolation Formula in 1795 after another discovery by Euler in 1783. Lagrange interpolating polynomials have been implemented in the Wolfram Language as Interpolating Polynomial data, var. Newton-Cotes formulae are often built using Lagrange interpolating polynomials. When building interpolating polynomials, there is a trade-off between those that have a better fit and a smooth, well-behaved fitting function. The degree of the generated polynomial increases with the number of data points utilised in the interpolation, increasing the oscillation between the data points.

If a function, f(x), has some known values, one can calculate or guess what the function will be at additional data points. If the condition x0 x xn is valid and we also know that y0 = f(x0), y1 = f(x1), yn = f(xn), then interpolation is the estimated value of f. (x). Extrapolation refers to the estimated value of f(x) when x x0 or x > xn. The process of analysing a value between two points on a line or curve is referred to as Lagrange Interpolation Formula.