For the given data points matrix become singular, and it makes impossible to construct polynomial with order, where - number of data points, so we will use polynomial

Fig 3. Plot of the data with polynomial interpolation superimposed

Because the polynomial is forced to intercept every point, it weaves up and down.

1.2.3 Lagrange interpolating polynomial

The Lagrange interpolating polynomial is the polynomial of degree that passes through the points , , …, and is given by:

,

Where

Written explicitly

When constructing interpolating polynomials, there is a tradeoff between having a better fit and having a smooth well-behaved fitting function. The more data points that are used in the interpolation, the higher the degree of the resulting polynomial, and therefore the greater oscillation it will exhibit between the data points. Therefore, a high-degree interpolation may be a poor predictor of the function between points, although the accuracy at the data points will be "perfect."

Fig 4. Plot of the data with Lagrange interpolating polynomial interpolation superimposed

One can see, that Lagrange polynomial has a lot of oscillations due to the high order if polynomial.

1.2.4 Cubic spline interpolation

Remember that linear interpolation uses a linear function for each of intervals . Spline interpolation uses low-degree polynomials in each of the intervals, and chooses the polynomial pieces such that they fit smoothly together. The resulting function is called a spline. For instance, the natural cubic spline is piecewise cubic and twice continuously differentiable. Furthermore, its second derivative is zero at the end points.

Like polynomial interpolation, spline interpolation incurs a smaller error than linear interpolation and the interpolant is smoother. However, the interpolant is easier to evaluate than the high-degree polynomials used in polynomial interpolation. It also does not suffer from Runge's phenomenon.

Fig 5. Plot of the data with Lagrange interpolating polynomial interpolation superimposed

It should be noted that cubic spline curve looks like metal ruler fixed in the nodal points, one can see that such interpolation method could not be used for modeling sudden data points jumps.

1.3 Results and discussion

The following results were obtained by employing described interpolation methods for the points ; ; :

Linear interpolation	Least squares interpolation	Lagrange polynomial	Cubic spline	Root mean square
	0.148	0.209	0.015	0.14	0.146
	0.678	0.664	0.612	0.641	0.649
	1.569	1.649	1.479	1.562	1.566

Table 1. Results of interpolation by different methods in the given points.

In order to determine the best interpolation method for the current case should be constructed the table of deviation between interpolation results and root mean square, if number of interpolations methods increases then value of RMS become closer to the true value.

Linear interpolation	Least squares interpolation	Lagrange polynomial	Cubic spline
Average deviation from the RMS

Table 2. Table of Average deviation between average deviation and interpolation results.

One can see that cubic spline interpolation gives the best results among discussed methods, but it should be noted that sometimes cubic spline gives wrong interpolation, especially near the sudden function change. Also good interpolation results are provided by Linear interpolation method, but actually this method gives average values on each segment between values on it boundaries.

Problem 2

2.1 Problem definition

For the above mentioned data set, if you are asked to give the integration of between two ends and ? Please discuss the possibility accuracies of all the numerical integration formulas you have learned in this semester.

2.2 Problem solution

In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral.

There are several reasons for carrying out numerical integration. The integrand may be known only at certain points, such as obtained by sampling. Some embedded systems and other computer applications may need numerical integration for this reason.

A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function. An example of such an integrand is , the antiderivative of which cannot be written in elementary form.