Theme 11. Approximation. Linear and quadratic approximations. Spline approximation
1. Approximation
The disadvantage of interpolation is that for many nodes a polynomial of high degree is formed. If the original data were not obtained accurately, then the interpolating function though accurately passes through nodal points, nevertheless is inaccurate. In such cases, an approximation should be used - the definition of the function that closest (best) approaches the nodal points.
Approximation - finding a function of a given type, which at points,,, .. took values as close as possible to the values,,, ..,.
Types of approximating functions:
Types of approximating functions:
- y = ax + b;
- y = ax ^ 2 + bx + c;
- y = ax ^ m;
- y = a ln (x) + b;
- y = x / (ax + b) and others.
Smallest squares method
For a function given by a table of values (approximation nodes), find a function of a predetermined form, for example, the polynomial is degree so that the sum of the squares of deviations from the nodal points was the smallest. In other words, find the coefficients of this polynomial from the condition that the line should as close as possible pass near the nodes of approximation.
Linear approximation
The essence of the method. Straight line P 2 (x) = a 0 + a 1 x be best to approach all these points (x o ; y o ), (x 1 ; y 1 ), (x 2 ; y 2 ) and (x 3 ; y 3 ).
Quadratic approximation
Output data:
- given the function f (x) at points and more points: f (x o ) = y o ; f (x 1 ) = y 1 ; f (x 2 ) = y 2 ; and f (x 3 ) = y 3 ;
- the order of the sought approximating polynomial n = 2.
The essence of the method. Figure quadratic function P 2 (x) = a 0 + a 1 x + a 2 x 2 should be the best to approach all these points (x o ; y o ), (x 1 ; y 1 ), (x 2 ; y 2 ) and (x 3 ; y 3 ).
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