Topic 10. Interpolation. Parabolic interpolation. Spline interpolation

Interpolation by algebraic polynomials is called parabolic interpolation. The construction of a parabolic interpolation has the following form:

• for the given values ​​f (x _i ) = y _i , i = 0..n to construct the polynomial y = P (x) = a _o x ⁿ + a ₁ x ^n-1 + .. + a _n , where n is the power and which meets the requirements:
• at the points x _i , i = 0..n, the value of the polynomial P _n (x _i ) coincides with the value of the given function f (x _i ), that is P _n (x _i ) = f (x _i ) = y _i , i = 0 ..n;
• At any other point xÎ] x _o ; x _n [the equality P _n (x) »f (x) is approximated .

The geometric nature of parabolic interpolation is that the graph of one function f (x) is replaced by the graph of the polynomial P of degree n, and these two graphs have (n + 1) a common point.

For parabolic interpolation, the number of interpolation nodes generates a polynomial order per unit smaller.

Linear interpolation

Output data:

• given the function f (x) at two points: f (x _o ) = y _o and f (x ₁ ) = y ₁ ;
• the order of the sought interpolation polynomial n = 1.

The essence of the method. The graph of the linear function P ₁ (x) = a _o x + a ₁ must pass through two points (x _o ; y _o ) and (x ₁ ; y ₁ ). Therefore, the desired coefficients a _o and a ₁ can be determined from the system of linear equations:
 

The solution of this system relative to a _o and a _{1 is} obtained:

Then the interpolation polynomial can be written as:

Thus, the function f (x) can be rearranged by an approximate polynomial of the form P ₁ (x), which is called the linear interpolation formula. From the geometric point of view, the linear interpolation formulas replace the arc of the curve y = f (x) on the interval [x ₀ ; x ₁ ] segment of a straight line y = P ₁ (x), which passes through the point (x _o ; y _o ) and (x ₁ ; y ₁ ) type:

where:

Linear interpolation is often used when working with prepared tables when finding the value of a function for intermediate values ​​of arguments.

Quadratic interpolation

Output data:

• given the function f (x) at three points: f (x _o ) = y _o ; f (x ₁ ) = y ₁ ; and f (x ₂ ) = y ₂ ;
• the order of the sought interpolation polynomial n = 2.

The essence of the method. The graph of the quadratic function P ₂ (x) = a _o x ² + a ₁ x + a ₂ must pass through the maple three (x _o ; y _o ), (x ₁ ; y ₁ ) and (x ₂ ; y ₂ ). Therefore, the desired coefficients a _o , a ₁ and a ₂ can be determined from the system of linear equations:

After solving this system, we obtain an interpolation polynomial of the form

Thus, the function f (x) can be rearranged by an approximate polynomial of the form P ₂ (x), which is called the quadratic interpolation formula. From a geometrical standpoint, formula quadratic interpolation replace the arc of the curve y = f (x) on the interval [x ₀ ; x ₁ ; x ₂ ] parabolic curve y = P ₂ (x), which passes through the point (x _o ; y _o ) ; (x ₁ ; y ₁ ) and (x ₂ ; y ₂ ).