سوال ۲۵ کنکور ارشد علوم کامپیوتر ۱۳۹۹

Read the following passage and answer the question by choosing the best choice.

PASSAGE 2

To put the idea of polynomial interpolation into a familiar framework, it is well-known, and perhaps obvious, that there is a unique straight line passing through any two points in the plane. This straight line is (the graph of) the linear, or first-degree, interpolating polynomial for this data. In precisely the same way it is easy to show that there is a unique parabola which passes through any three points: this is the graph of the quadratic, or second-degree, interpolating polynomial which agrees with the given data. In general, we will find that there is a unique polynomial of the minimum possible degree $n$ which agrees with a given function at $n+1$ data points, or nodes.

The drawbacks of this polynomial interpolation approach include the following. If the number of data points is large, then the degree of the interpolating polynomial will be high and therefore its evaluation will itself be computationally expensive. Second, the physical laws are of high-degree polynomial nature, and so the approximating function is likely to exhibit less stable behavior than the function or phenomenon being approximated. The use of a single interpolation polynomial over a wide range gives equal weight to function values distant from the point of interest as it does to those at nearby points.

It is likely to be more fruitful to use several lower-degree polynomials which utilize data at points close to the required one. This is achieved by the various difference schemes which we will discuss shortly or by the use of spline interpolation. The latter is based on the idea of a piecewise polynomial which interpolates the complete set of data but which is of low degree and therefore simple to evaluate in any subrange.

All these approaches and their associated errors are to be discussed in the next two chapters. We will see that the problem is often reduced to the necessity of solving a (sometimes large) system of linear equations. Before proceeding to the details of any single interpolation scheme, it is necessary to point out one further difficulty which may arise in practice. If the data is itself the result of practical experiment and therefore subject to its own errors, then it is probably unwise to insist that any approximation scheme agrees exactly with this erroneous data.

Question: Systems of linear equations ------------.