In Lagrange Interpolation,the interpolating polynomial is expressed as a linear, yields a single polynomial that passes through all the data points. It is a direct method for constructing the interpolating polynomial, does not involve divided differences explicitly.

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• Lagrange's form is more efficient when we have to interpolate several data sets on the same data points.

• Newton's form is more efficient when we have to interpolate data incrementally.

• The advantage of Newton interpolation is the use of nested multiplication and the relative easiness to add more data points for higher-order interpolating polynomials.

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2. In the case of evenly spaced points, Newton-Gregory forward and backward interpolation is the quickest application of the Divided difference formula.

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In floating point arithmetic this is very bad. It does have some small advantages: for instance, the Lagrange approach amounts to diagonalizing the problem of finding the coefficients, so it takes only linear time to find the coefficients. This is good if you need to use the same set of points repeatedly.

For Newton interpolation, you get the coefficients reasonably fast (quadratic time), the evaluation is much more stable (roughly because there is usually a single dominant term for a given x).

It is also fairly easy to see how to interpolate derivatives using the Newton framework.

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Newton Interpolation uses the divided difference formula to construct the interpolation polynomial, while Lagrange Interpolation uses Lagrange polynomials. In Newton Interpolation, the polynomial is represented as a sum of terms with divided differences and basis polynomials. It's efficient for adding new data points.

Lagrange Interpolation, on the other hand, uses a set of basis polynomials associated with each data point. The polynomial is the sum of these basis polynomials, multiplied by the data point values. It's straightforward but computationally expensive for adding new data points. The choice depends on factors like dataset size and specific requirements.

Lagrange interpolation employs Lagrange polynomials, whereas Newton interpolation uses the divided difference formula to get the interpolation polynomial. The polynomial is expressed in Newton Interpolation as the sum of terms with split differences and base polynomials. It works well for adding new information.

Conversely, Lagrange Interpolation makes use of a collection of basis polynomials that are connected to every data point. The polynomial is obtained by multiplying the values of the data points by the sum of these basic polynomials. Simple in theory, but computationally costly when adding additional data points. The decision is based on variables such as the size of the dataset and particular needs.

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From my understanding, Lagrange Interpolation

Use the sum of basis polynomial which related with the set of data, and this sum is related to the number of set data, which could require quite a lot of computational cost if its dealing with a large number of set data.

As for Newtons Interpolation,

Is use to find missing data in a set of data, there are three ways to use Newtons Interpolation depending which set of data are we want.

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Individual quiz

Newton's Interpolation Method:

Uses divided difference coefficients to construct the interpolating polynomial.

Requires the computation of finite differences between data points.

Suitable for incremental addition of data points without recalculating the entire polynomial.

Lagrange Interpolation:

Constructs the interpolating polynomial directly using Lagrange basis polynomials.

Doesn't require calculating differences between data points.

Simpler in terms of computation but may become less efficient for large datasets.

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Newton's Interpolation Method: Builds the interpolating polynomial using divided difference coefficients. demands that the finite differences between data points be calculated. appropriate for adding data points one at a time without having to recalculate the polynomial in its entirety. 

Lagrange Interpolation: This method uses Lagrange basis polynomials to directly construct the interpolating polynomial. does not necessitate computing the disparities between data points. simpler to calculate, however it might become less effective with bigger datasets.

Newton's form is better suited for incremental data interpolation.

Newton's interpolation involves nested multiplication, making it convenient to incorporate additional data points for constructing higher-order interpolating polynomials.