سوال ۱۹ کنکور ارشد ریاضی ۱۴۰۱

PART C: Reading Comprehension — PASSAGE 1 (excerpt)

"The interpolation problem for a function of a continuous argument, given discrete data, is closely related to the problem of constructing variational-difference schemes and continuous representations of solutions of difference problems. ... At the same time the solution of the differential problem usually represents an approximate solution of the original problem on a discrete set of points. ... An interpretation of this kind requires that certain conditions are taken into account: namely, if the solution of the difference equations has been obtained with a certain accuracy, then the accuracy of the interpolation must be at least as high. If we possess additional information regarding the error of the approximate solution, then the interpolation may be implemented, not from the exact data, but from those involving possible error at the net points. ...

As a rule, the interpolation algorithms for the exact data, specified on a discrete set of points, are based on Lagrange interpolation polynomials. ...

Another problem closely related to interpolation arises when the function values of ϕ at the net points x_k are known only up to a certain error, the maximum magnitude of which is a priori known at each of the points. In this case, we want to construct a curve which would be in some sense the best approximation of the function, the latter being specified at the net points up to random errors. Problems of this kind are usually solved by the least-squares method."

Q19. Least squares models ---------- of functions with known approximate values.