In this work in a limited range located from points harvest was without a break 34 f ( x ) N (x) function for 𝐿 (𝑥) Lagrange and n In Newton Interpolation Methods 𝑛 approximation has been made, and their interpolation The errors were estimated

Аннотация

In this work in a limited range located from points harvest was without a break 34 f ( x ) N (x) function for 𝐿 (𝑥) Lagrange and n In Newton Interpolation Methods 𝑛 approximation has been made, and their interpolation The errors were estimated. At the same time, an improved algorithm for constructing Lagrange and Newton interpolation methods was developed. As a result, it was shown which method is more effective in approximating functions. Key words: Lagrange interpolation, Newton interpolation error, approximation. Introduction. The urgency of the problem is that in planning, weather forecasting, and determining land resources, it is necessary to use the values of the function at several points. Based on these aspects, it is urgent to derive the mathematical regularity of the problem on a computer using mathematical modeling and computer simulation. Most computational methods are based on replacing the functions involved in the formulation of the problem with functions that are close to it in some sense and simpler in structure. This article considers the simplest and most widely used part of the problem of approximation of functions, the problem of Lagrange and Newton interpolation of functions. Main part. In the Lagrange and Newton interpolation definition formula, the value of the coefficient is expressed by the distance between the nodes of the function and the nodes. y  f x   x y x Given a function, the value of is assigned to any possible value of. y It is not always easy to determine. For example, if, x is a parameter, y x it may be considered a solution to a complex problem, or y x if the values of have been determined as a result of expensive research. In this case, we can construct a table of values of the function, but this is not possible for very large values of the argument. Such in cases usually interpolation formulas is applied. a , b x ,x ,....,x in the cut 0 1 n of the argument n  1 to different values suitable visitor y  f  x  fx  y function values 0 0, f  x 1   y 1   f x  y ,…, n n given Let it be. x 0 , x 1 , . . . . , x n y  f x n given in knots function with same price reception to do and level not exceeding  х n plural construction demand be done, that is ILM-FAN VA INNOVATSIYA ILMIY-AMALIY KONFERENSIYASI in-academy.uz/index.php/si 35 p n х i  y i , i  1 , 2 , . . . , n . n  х  - interpolation often, x 0 , x 1 , . . . . , x n interpolation knots, breaking issue and interpolation issue because is conducted. Most of the time interpolation formulas of the argument interval values for y  f x function value in finding is used. In this х point х 0 , х n in the meantime while lying down the question of interpolation because is being conducted. Interpol or Zion of many general in appearance various expressions available : Newton, Lagrange, Gauss, Sterling, Bessel and others N Yuton and Lagrange formulas in calculations convenient computer also and in hand in calculations accuracy control to do provides the rest other forms interpolation knots location private in case appropriate. Optional located in knots in interpolation Lagrange interpolation from the formula is used. N Yuton interpolation formulas equal in the distance located knots for under consideration of the interval head and at the end at points function value calculation for convenient. The matter to be placed Faraz Let's do [a, b ] in the intersection. n at the point given Let it be. 𝑥 ,𝑥 ,𝑥 ,…,𝑥 0 1 2 𝑛 This points interpolation knots because is called. Someone f ( x ) function this in points value below equal let it be f ( x 0 )  y 0 , f ( x 1 )  y 1 , . . . . . . . . . . . . f ( x i )  y i , . . . . f ( x n )  y n Known to class relevant was and interpolation at the nodes f ( x ) function reception did values that is : F ( x 0 )  y 0 , F ( x 1 )  y 1 , . . . . . . . . . . . . F ( x i )  y i , . . . . F ( x n )  y n values reception doer F ( x ) function construction and of error determination demand Let it be done. The matter solution method F(x) Lagrange interpolation by the method using above cited issue for of the multiverse harvest we will do Every one interpolation knot (𝑥 ,𝑦 ) for separately multi-level we will fix it. 𝑖 𝑖 𝐹 (𝑥) = 𝐿 (𝑥) = 𝑦 ∙𝐹 (𝑥)+𝑦 ∙𝐹 (𝑥)+⋯+𝑦 ∙𝐹 (𝑥) (1) 𝑛 𝑛 0 0𝑛 1 1𝑛 𝑛 𝑛𝑛 𝐹 (𝑥)– of every one is of degree n polynomial, then ( 1) is also n- degree multi-level will 𝑖𝑛 be. 𝐹 (𝑥)– every one of them 𝑖𝑛 0, 𝑖 ≠ 𝑗 𝐹 (𝑥) = { 𝑖𝑛 1, 𝑖 = 𝑗 the conditions performant as we choose. 𝐹 (𝑥)- roots 𝑥 ,𝑥 ,𝑥 ,…,𝑥 ,𝑥 ,…,𝑥 – n -th degree It will be a polynomial. 𝑖𝑛 0 1 2 𝑖−1 𝑖+1 𝑛 Understandable multi-level 𝐹 (𝑥) = 𝐴∙(𝑥 −𝑥 )(𝑥−𝑥 )∙…∙(𝑥−𝑥 )(𝑥−𝑥 )∙…∙(𝑥−𝑥 ) 𝑖𝑛 0 1 𝑖−1 𝑖+1 𝑛 In appearance this is it where A is some constant. Condition according to 𝑃 (𝑥) = 1 𝑖𝑛 will be if i=j, then the constant A is we determine : 𝐴∙(𝑥 −𝑥 )(𝑥 −𝑥 )∙…∙(𝑥 −𝑥 )(𝑥 −𝑥 )∙…∙(𝑥 −𝑥 ) = 1 (2) 𝑗 0 𝑗 1 𝑗 𝑖−1 𝑗 𝑖+1 𝑗 𝑛 ILM-FAN VA INNOVATSIYA ILMIY-AMALIY KONFERENSIYASI in-academy.uz/index.php/si (2) A 's value harvest we will do 1