Polynomial Approximation and Interpolation

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Questions and Answers

What is one of the primary reasons for using polynomial approximation?

To reconstruct the function when it is not given explicitly. (correct)

To make functions easier to understand.

To derive analytical solutions for every polynomial.

To generate polynomial equations for all types of functions.

What phenomenon is described as the difference between the actual function and the approximating polynomial?

Root approximation

Error of approximation (correct)

Interpolation error

Polynomial deviation

Which interpolation method uses nodes that are not equally spaced?

Lagrange interpolation

Polynomial extrapolation

Newton's divided difference interpolation (correct)

Linear interpolation

What type of operation can be performed using the approximating polynomial P(x)?

Determining roots, differentiation, and integration Signup and view all the answers

Which option best describes 'nodes' in the context of polynomial approximation?

Distinct points where function values or derivatives are known. Signup and view all the answers

What action can be predicted at a non-tabular point using the polynomial approximation?

Predicted values of f(x) using P(x) Signup and view all the answers

What is the degree of the polynomial that can be constructed through n + 1 distinct points?

n Signup and view all the answers

Which of the following statements is true regarding Lagrange fundamental polynomials?

They evaluate to zero at all points except for one specific point. Signup and view all the answers

In Lagrange interpolation, how is the interpolation polynomial Pn(x) expressed?

As a sum of the function values multiplied by the fundamental polynomials. Signup and view all the answers

What is the relationship between the number of distinct points and the uniqueness of the polynomial in interpolation?

At least two points are necessary for uniqueness. Signup and view all the answers

Which equation represents the linear interpolation polynomial P1(x) for n = 1?

P1(x) = l0(x)f0 + l1(x)f1 Signup and view all the answers

For the Lagrange interpolating polynomial, what is necessary to ensure the proper functioning of the denominator of the fundamental polynomials?

It should not equal zero at any of the interpolation points. Signup and view all the answers

Which of the following is a characteristic of the interpolating polynomial in polynomial approximation?

It fits the given data exactly at the tabular points. Signup and view all the answers

What does the term 'non-equispaced' refer to concerning tabular points in polynomial approximation?

The distance between consecutive points varies. Signup and view all the answers

Study Notes

Introduction to Polynomial Approximation

Polynomial approximation is used to reconstruct a function when only values and derivatives at specific points (nodes) are known.
Approximation allows for operations like determining roots, differentiation, and integration of the function using the polynomial P(x).
The error of approximation is defined as the deviation of P(x) from f(x), expressed as f(x) - P(x).

Problem Definition

A continuous function f(x) is defined on an interval [a, b] with n + 1 distinct tabular points (x_0, x_1, \ldots, x_n) such that (a = x_0 < x_1 < \ldots < x_n = b).
These tabular points can be either equispaced or non-equispaced.
The goal is to find a polynomial (P_n(x)) of degree ≤ n that fits the data exactly, with conditions (P_n(x_i) = f(x_i)) for all i from 0 to n.
The polynomial (P_n(x)) is referred to as the interpolating polynomial, and the fitting conditions are known as interpolating conditions.

Constructing Polynomials

A unique polynomial of degree 1 (a straight line) can be constructed through two distinct points.
A unique polynomial of degree 2 (a parabola) or degree 1 can be derived from three distinct points.
In general, through n + 1 distinct points, a unique polynomial of degree ≤ n can be constructed.

Lagrange Interpolation

Lagrange Interpolation deals with data given at distinct unevenly spaced points (x_0, x_1, \ldots, x_n).
A unique polynomial (P_n(x)) of degree ≤ n is given by the formula: (P_n(x) = l_0(x)f(x_0) + l_1(x)f(x_1) + \ldots + l_n(x)f(x_n)).
The polynomials (l_i(x)) are known as Lagrange fundamental polynomials defined as: (l_i(x) = \frac{(x - x_0)(x - x_1)\cdots(x - x_{i-1})(x - x_{i+1})\cdots(x - x_n)}{(x_i - x_0)(x_i - x_1)\cdots(x_i - x_{i-1})(x_i - x_{i+1})\cdots(x_i - x_n)}).
The denominator is derived by substituting (x = x_i) in the numerator.

Lagrange Linear Interpolation

For n = 1, with data points (x_0) and (x_1) and their corresponding function values (f_0) and (f_1):
- The Lagrange fundamental polynomials are:
  - (l_0(x) = \frac{x - x_1}{x_0 - x_1})
  - (l_1(x) = \frac{x - x_0}{x_1 - x_0}).
The Lagrange linear interpolation polynomial is then expressed as: (P_1(x) = l_0(x)f_0 + l_1(x)f_1).

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Description

Explore the concepts of polynomial approximation and interpolation, focusing on techniques involving unevenly spaced points. This quiz covers essential topics like Lagrange Interpolation and divided differences, aimed at enhancing your understanding of these mathematical methods.