Lagrange's interpolation formula is a method for finding a polynomial that passes through a given set of data points. The formula does not require the data points to be equally spaced. The formula for Lagrange interpolation polynomial $P(x)$ for $n+1$ data points $(x_0, y_0), (x_1, y_1), \dots, (x_n, y_n)$ is: $$ P(x) = \sum_{j=0}^{n} y_j L_j(x) $$ where $L_j(x)$ are the Lagrange basis polynomials, defined as: $$ L_j(x) = \prod_{i=0, i\neq j}^{n} \frac{x-x_i}{x_j-x_i} $$ The derivation of these basis polynomials $L_j(x)$ involves constructing terms that are zero at all $x_i$ except for $x_j$, where $L_j(x_j)=1$. The construction of these terms, $\frac{x-x_i}{x_j-x_i}$, does not depend on the spacing between the $x_i$ values. The only requirement is that the $x_i$ values must be distinct.Therefore, Lagrange's interpolation formula can be used for both equal and unequal intervals between data points. Other interpolation methods, like Newton's forward or backward difference formulas, are specifically designed for equally spaced intervals.
Lagrange's interpolation formula can be used - - (5) Question not attempted
Asked in Asst. Statistical Officer (Agri. Deptt.) 2024 · Topic: Computer and Information Technology (Computational Physics: Numerical methods and algorithms for physical problems)
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Answer: C. for both equal and unequal intervals
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