For the following data values, construct the interpolating...

For the following data values, construct the interpolating polynomial in Lagrange form: (a) \begin{tabular}{c|cc} $t_i$ & -3 & 2 \\ \hline$y_i$ & 1 & 5 \end{tabular} (d) (b) \begin{tabular}{c|ccc} $t_i$ & 0 & 1 & 3 \\ \hline$y_i$ & 1 & .5 & .25 \end{tabular} (c) \begin{tabular}{c|ccc} $t_i$ & -1 & 0 & 1 \\ \hline$y_i$ & 1 & 2 & -1 \end{tabular} (e) \begin{tabular}{r|rrrrr} $t_i$ & -2 & -1 & 0 & 1 & 2 \\ \hline$y_i$ & -1 & -2 & 2 & 1 & 3 \end{tabular} (a) find the straight line $y=\alpha+\beta t$ that best fits the

For the following data values, construct the interpolating polynomial in Lagrange form:
(a)
\begin{tabular}{c|cc}
$t_i$ & -3 & 2 \\
\hline$y_i$ & 1 & 5
\end{tabular}
(d)
(b)
\begin{tabular}{c|ccc}
$t_i$ & 0 & 1 & 3 \\
\hline$y_i$ & 1 & .5 & .25
\end{tabular}
(c)
\begin{tabular}{c|ccc}
$t_i$ & -1 & 0 & 1 \\
\hline$y_i$ & 1 & 2 & -1
\end{tabular}
(e)
\begin{tabular}{r|rrrrr}
$t_i$ & -2 & -1 & 0 & 1 & 2 \\
\hline$y_i$ & -1 & -2 & 2 & 1 & 3
\end{tabular}
(a) find the straight line $y=\alpha+\beta t$ that best fits the

Key Concepts

Polynomial Interpolation

This is the process of determining a polynomial that passes exactly through a given set of data points. The degree of the polynomial is typically one less than the number of data points, ensuring that the polynomial fits the data exactly when the system has a unique solution.

Lagrange Interpolation

Lagrange interpolation is a specific method for polynomial interpolation that uses a weighted sum of basis polynomials. Each basis polynomial is constructed to be 1 at its corresponding data point and 0 at all others, which guarantees that the overall polynomial exactly reproduces the given data values.

Lagrange Basis Polynomials

These are the individual terms used in the Lagrange interpolation formula. Each basis polynomial is defined in such a way that it is 1 at one data point and 0 at all the others. They collectively form a basis for the space of polynomials of the required degree, ensuring that any set of data points can be interpolated uniquely.

Least Squares Linear Regression

This is a statistical method used to find the best-fitting straight line through a set of data by minimizing the sum of the squares of the deviations (errors) from the data points to the line. The resulting line is described by the equation y = ? + ?t, where ? and ? are determined to minimize these errors, providing the optimal linear approximation for the data set.

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