Lagrange interpolation polynomial pdf

Continue Lagrange Interpolation Polynomials If we want to describe all the ups and downs in the data set, and hit every point, we use what is called polynomial interpolation. This method is associated with Lagrange. Suppose the dataset consists of N data points: (x1, y1), (x2, y2), (x3, y3), ..., (xN, yN) Polynomial Interpolation will have a degree N - 1 . It is given: P (x) j1 (x) y1 y2 (x) y2 y3 (x) y3 ... It's easier than it looks. The main thing to note is that the ji numerator (x) contains the entire sequence of factors (x - x1), (x - x2), (x - x3), ... (x - xN), except for one factor (x - xi). Similarly, the denominator contains the entire sequence of factors (xi - x1), (xi - x2), (xi - x3), ... (xi - xN), except for one factor (xi - xi). Now note: ji (xi) No 1 (number and denominator), but ji (xj) 0 (number 0, as it contains the factor (xj - xj) ) for any j is not i. This means P (xi) and yi, which is exactly what we want! If this seems like smoke and mirrors, consider a simple example. Here's the data for g again: x g(x) 0 -250 10 0 20 50 30 -100 In this case there is a N 4 data point, so we will create a polynomial degree N - 1 and 3 . We have x1 y 0, x2 x 10, x3 x 20, x4 y 30, and y1 - 250, y2 y 0, y3 y 50, y4 and 100. So: Multiplying each of them appropriate yi (i q 1, 2, 3, 4) and adding together conditions of similar power, then gives: cubic conditions are canceled, and we come to a simple square description of the data. A quick data story along with a polynomic shows that it actually passes through each of the data points: For an interactive demonstration of Lagrange's polynomialization, showing how changes in data points affect the resulting curve, go here. This image shows four dots ((No9, 5), (No4, 2), (No1, No2), (7, 9)), (cubic) interpolation of polynomial L (x) (dashed, black), which is the sum of the scaled polynomial bases y0l0 (x), y1l1 (x), y2l2 (x) and y3l3 (x). Polynomial interpolation passes through all four checkpoints, and each scaled polynomial base passes through the appropriate checkpoint and is 0, where x corresponds to the other three checkpoints. In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For this set of dots (x j, y j) (display (x_ j',y_'j)) without two x j (display x_ j'j) values are equal, Lagrange-polynomial is the least polynomial, which assumes at each value x j 'displaystyle x_ j' appropriate value y j 'displaystyle y_'j' Although named after Joseph-Louis Lagrange, who in 1795, the method was first discovered in 1779 by Edward Waring, which is also a simple consequence of a formula published in 1783 by Leonhard Euler. The use of Lagrange polynomials includes the Newton-Cote numerical integration method and Shamir's secret cryptography exchange scheme. Lagrange interpolation is subject to the phenomenon of large swings of Runge. Because changing the x j'displaystyle points x_ requires a recalculation of the entire interpolant, it is often easier to use Newton's polynomials instead. Definition Here we are building Lagrange basic functions of the 1st, 2nd and 3rd order on a two-unit domain. The linear combinations of Lagrange's base functions are used to construct the polynomial interpolation lagrange. Lagrange's basic functions are commonly used in the analysis of the end elements as the basis for the function of the form of the element. In addition, two domain units are typically used as a natural space to determine the final item. Given the set of k 1 data points (x 0, y 0), ... , (x j, y j), ... ( x k, y k y_{0} x_{0}) (x_, y_ jj), ldots, (x_'y_'k) where there are no two x 'displaystyle x_ j'j), Interpolation polynomial in the form of Lagrange is a linear combination of L (x) : ∑ j y 0 k y j j ( x) : 'sum zj'y_ 'j' (x) of Lagrange basis polynomials l j (x) := ∏ 0 ≤ m ≤ k m ≠ j x − x m x j − x m = ( x − x 0 ) ( x j − x 0 ) ⋯ ( x − x j − 1 ) ( x j − x j − 1 ) ( x − x j + 1 ) ( x j − x j + 1 ) ⋯ ( x − x k ) ( x j − x k ) , displayell j'j (x): Prod start smallmatrix0'leq m'leq k'meq j'end'smallmatrix frac (x-x_'m'm'm'x_ yoya-x_mhamemerrak (x-x_{0}) (x_ jj x_{0} x_) (x_-y-x_ j x_-x_ x_1) frak (x-x_'k) (x_ j'-x_'k), where 0 ≤ jstyle ≤ k display 0leq j'leq k. I don't know Note that, given the initial assumption that there are no two x j'displaystyle x_ j'j' are the same, then (when m ≠ j 'displaystyle meq j) x j x m ≠ 0 displaystyle x_ j -x_'m'eq 0 so this expression is always clearly defined. The reason par x i x display style x_i'x_'j'yo y i'≠ y j display style y_i'eq y_ J e is not allowed in that no interpolation function L displaystyle L is such that i and L (x i) displaystyle y_ I L (x_'i) will exist; the feature can only get one value for each argument x i displaystyle x_ i. On the other hand, if also I and y j 'displaystyle y_ 'i'y_'j' then these two points would actually be one point. For all, ≠ j 'displaystyle ieq j', l j (x) (display style (x)) includes the term (x - x i) in the numerator (x-x_), so the whole product will be zero on x x i display style x x_ i: ∀ (j ≠ i) : l (x i ≠ ∏) м.з. (х я х х 0) ( х х х х 0) ⋯ ( х я х х х х х х х х х 0 ) ( х х х х х х я) ( х й х х я ) ⋯ ( х я х к) ( х х х х к) 0. Displaystyle forall jeq i): ell j (x_i)prod meq j'frac (x_x_'m'x_ j'-x_'m'frac (x_'-x_{0}) (x_ (x_{0}) (x_-I-x_) (x_ j'-x_'i) (x_-x_'k) (x_ j x_'k). С другой стороны, l j ( x j ) ∏ м ≠ j x j х x j х м х х х 1 «дисплей»ell j'j'(x_'j)): »prod »meq j'frac »x_ j»-x_'m'x_'j'j x_'m все базисные полиномиалы являются нулевыми на х х х я «displaystyle x»x_'i» , за исключением l j ( x ) »displaystyle »элл йо j»(x)» for which he believes that l j (x j) - 1 x_ (display style) because it lacks (x x j) (x j) (x-x_)) term. It follows that y j l j (x j) - y j 'displaystyle y_ j'ell j'j' (x_ j'j) - y_ j'j' , so at each point x j 'displaystyle x_ j ' , L (x j) j - 0 - 0 - ⋯ - 0 - y j 'displaystyle L (x_'j)) y_j'0'0'dots (0'y_'j' showing that L-displaystyle L' interpolates function accurately. Proof of function L(x) is sought is polynomial in x least that interpolates this data set. that is, it takes the value of yj on the corresponding xj for all data points J: L (x j) and y j y j 0 , ... , k. Displaystyle L (x_ j) y_ j'j'qquad j'0, 'ls,k.) Note this: In l j (x) displaystyle (el yo (x) there are k factors in the product, and each factor contains one x, so that L (x) (which is the sum of these k-degree polynomials) should be a polynomial degree no more than k. l j (x i ≠ ∏) Displaystyle ' ell j (x_) start smallmatrixm'0'meq j'end'smallmatrix-k'frac (x_'i'-x_'m'x_'j'j'x_. , где м й j, если я й J, то все термины, которые появляются являются х j х х х х х х х 1 дисплей стилем frac x_j'-x_'m'x_ jj'-x_'m'1 . Also, if I ≠ j, then one term in the product will be (for m and i), x i x i x i x i x i 0 display style frak x_ x_ I-x_ x_ So L j (x i) - δ j i 1 , x_ if j and i 0 if j ≠ i, cases No1, text if j i'0 , text if jeq iend where δ i j displaystyle delta ij is the Kroneker Delta. So: L (x i) - ∑ j 0 k y j j (x i) - ∑ j 0 k y j δ j i and i i. (L (x_) amount (j'0'k) y_ j'j'ell (x_'i) amount y_ (j'0'k'y_ j'delta So L/x function is polynomial with no more than k and where L (xi) and yi. , polynomial interpolation is unique, as evidenced by the monotheration theorem in the article of polynomial interpolation. It is also true that: ∑ j y 0 k j (x) 1 ∀ x 'display style 'j'0'k'ell j'j'(x) 1qquad 'forall x' it should be a polynomial degree no more than to and passes through all these k 1 data points: ( x 0, 1) , (x j, 1) , ... ( x k, 1) displaystyle (x_{0}.1), ldots,' (x_ j',1), ldots, (x_'k,1) resulting in a lineline. The prospect of linear algebra Solution to the problem of interpolation leads to a problem of linear algebra, equivalent to matrix inversion. Using a standard monomyal base for our polynomial L interpolation (m_ x) - ∑ j 0 k x j j'displaystyle L(x) - we must invert The Vandermond Matrix (x i) j Display style (x_) to address L (x i) - I have an L display (x_ I) y_ i for coefficients m displaystyle m_ j L (x) . Choosing the best foundation Lagrange, L (x) - ∑ j 0 k j (x) y j'displaystyle L(x) , sum i-0'k 'l_ j'j'(x)y_ j'j' , we just get a matrix of identity, δ i j'displaystyle (delta) which is its own reverse side automatically: lagrange base analogue. This structure is similar to the Chinese Theorem of Remainder. Instead of checking the remnants of modulo prime integrators of the prime numbers, we check for the residues of polynomial when divided into linear. Also, when the order is large, Fast Fourier Transformation can be used to solve for interpolated polynomial ratios. Examples Of Example 1 We want to interpolate q (x) x2 in the range of 1 ≤ x ≤ 3, given these three points: x 0 x 1 f ( x 0) - 1 x 1 x 2 f (x 1) 4 x 2 x 3 f (x 2) 9. Display style beginning x_{0} 1f (x_{0})1x_{1} 2 f (x_{1}) 4 x_{2} 3f (x_{2}) 9. Interpolization polynomial: L (x) 1 ⋅ x 2 1 x 2 ⋅ x 3 x 3 x 4 ⋅ x 1 1 ⋅ x 3 x 3 x 9 ⋅ x 1 1 x 1 ⋅ x 2 x 2 x 2 x 2 x 2 x 2 . Displaystyle beginning aligns L(x){1} cdot x-2 over 1-2'cdot x-3 (more than 1-3) {4}kdot x-1 over 2-2-3 x-3 cdot over 2-3{9}cdot x-1 over 3-1 cdot x-2 over 3-2 x-10pt x {2}. Example 2 We want to interpolate q (x) x3 in the 1 ≤ x ≤ 4, Given these four points: x 0 x 1 display style x_{0} f f (x 0) displaystyle f (x_{0}) x 1 x 1 displaystyle x_{1} 2 f (x x 1) 8 displaystyle f (x_{1}) x 2 x 3 (display style x_{2} x 3) f (x 2) 27 display style f (x_{2}) x 27 x 3 x 4 display style x_{3} x 4 f (x 3) display style f (x_{3}) 64 Interpol polynomial: L (x) - 1 ⋅ x 2 1 - 2 ⋅ x 3 x 3 ⋅ x 4 x 4 x 8 ⋅ x 1 2 1 ⋅ x x 3 x 3 x 3 ⋅ x 4 x 27 ⋅ x 1 x 1 ⋅ x 2 2 x 2 ⋅ x 4 3 x 4 64 ⋅ x 1 4 - 1 ⋅ x 2 4 x 2 ⋅ x 3 x 3 x 3 (display {1}) x-1 (more than 2-1) cdot x-3 (more than 2-3) x-4 (more than 2-4) {27}cdot x-1 (more than 3-1)cdot x-2 (more than 3-2) 4'{64} 'cdot x-1 (more than 4-1) cdot x-2 (more than 4-2) cdot x-3 (more than 4-3) 8ptx'x'x'{3}'endaligned Remarks to interpolation discrepancies for the Lagranga polynomials set. The lag-regional form of polynomial interpolation shows the linear nature of polynomial interpolation and the uniqueness of polynomial interpolation. Therefore, it is preferable in evidence and theoretical arguments. The uniqueness can also be seen by the irreversibility of Vandermond's matrix, due to the failure to validate the determinant of Vandermond. But as you can see from the design, every time the xk knot changes, all the polynomials of the Lagrange base must be recalculated. The best form of polynomial interpolation for practical (or computational) purposes is the baricentric form of LaGrange interpolation (see below) or Newton's polynomials. Lagrange and other interpolations at equally placed points, as in the above example, give polynomial vibrations above and below the true function. This behavior tends to increase with the number of points, leading to discrepancies known as the Runge phenomenon; the problem can be fixed by selecting interpolation points in Chebyshev nodes. The polynomials of the Lagrange base can be used in numerical integration to obtain Newton-Cote formulas. Baricentric Form Use l (x x x 0) (x x 1) ⋯ (x x x k) (display (x-x_{0}) (x-x_{1} x_) x x x x x x th ∏ i th 0, I'm ≠ j k (x x x i) style (x_ y j) (x_ y j) mathrm d x_ We can rewrite the x_ x_ polynomials of the lagrange foundations as l j ( x) l q (x j) (x j) displaystyle (ell j'(x) frak (x) (x)'ell' (x_'j) (x-x_'j) by determining baricentric weights (w j' 1 l) (x j) 'displaystyle w_ j'frac {1}'ell' (x_ j) we can simply write l j ( x) l ( l) w j x q'displaystyle (ell j'j) 'ell 'ell (x)-frac (w_ j'x-x_'j'j'j'ell , which is commonly referred to as the first form of baricentric interpolation formula. The advantage of this view is that polynomial interpolation can now be rated as L (x) l (x) ∑ j 0 to W J x x x j j'displaystyle L (x) 0'k'frac (w_ j'j'x-x_'j'j'y_'j which, if the scales w j 'displaystyle w_ j'j' have been pre-calculated, requires only O (k) (display (mathcal (o) (k)) (estimate l ( x) (display (x)) and scales w j /(x j x_ w_) (k'j)) ) {2} to assess the polynomials of the base of Lagrange l The baricentric interpolation formula can also be easily updated to include a new knot x to 1 display style x_K1 by dividing each of the w j 'displaystyle w_ j , j y 0 ... k displaystyle j0'0'0'k dots' by (x q x k 1) Displaystyle (x_ j'-x_'1) and build a new w k 1 displaystyle w_k1 as stated above. We can further simplify the first form by first considering the baricentric interpolation of the constant function g ( x) ≡ 1 displaystyle g(x)equiv 1 (x ) l (x) ∑ j 0 k j x x j j j. (display style g(x)'ell (x) sum (j'0'k'frac (w_ j'j'x-x_'j'.' but gives L (x ∑ ∑) frak (w_)j'j'x-x_'j'j'y_'j'sum, j'0'k'k'frac (w_ j'x-x_'j'j'k), which is called the second form or true form of baritcentric interpolation formula. This second form has the advantage that l (x) displaystyle (x) should not be evaluated for each L (x) displaystyle L (x) rating. Residue in lagrange interpolation formula When interpolating this function f polynomial K in nodes x 0 , . . . x_{0},...,x_ ...... (x)'f(x)-L (x) , который может быть выражен как 5 R ( х ) , f х х 0 , ... , х к , х х ( х х ) х ( х ) f ( к 1 ) ( ξ ) ( к ...... , х 0 < ξ < х к , «дисплей стиль R(x)»f'x_{0}, »ldots,x_'k,x'ell (x)»ell (x) »фрак »f» (к)» (КЗИ) 1)!.., «четверка» x_{0}<-си <x_кз» where f x x x 0, ... x to, x displaystyle fx_{0}, ldots,x_'k,x, is a notation for separated differences. Alternatively, the residue can be expressed as a contour integral in a complex area, as R ( z) l ( z) 2 π i ∫ C f (t) (t q z) (t q 0) ⋯ (t q q q) d q q (z) 2 π i ∫ C f (t) (t) l ( t) «Дисплейстайл R(z)»Фрак (z)»2'pi i'int »C»frac (t-z) (t-z_{0} z_) (к)) Остальная часть может быть связана как R (x) ≤ (х к х х 0 ) к 1 ( к 1 ) ! max x 0 ≤ ξ ≤ x to f (to No 1) (ξ). (Display Style) R (x) Lek (x_-x_{0}) Max x_{0} Si lek x_ Kef (k'1). The conclusion is clear, R (x) displaystyle R(x) is zero in nodes. To find R (x) displaystyle R(x) at the point x p display style x_ p . Identify the new function F (x) - f (x) - L (x) - R (x) ⋅ ∏ (display F(x)-f(x)-L (x)-R (x) and select R (x) and select R (x) style R (x) C'cdot prod i0'k (This provides R (x) ) необходимы для определения для данного х р «дисплей x_». Теперь у F (x) «displaystyle F(x)» есть к 2 нулей (на всех узлах и x p »displaystyle x_ »p») между x 0 «displaystyle x_{0}» и x k «displaystyle x_»k) (включая конечные точки). Предполагая, что f ( x ) «displaystyle f(x)» является k »1 »displaystyle k»1» - раз разной, L ( x ) »displaystyle L (x)» и R ( x ) «displaystyle R(x)» являются полиномиалами, и, следовательно, бесконечно различны. По теореме Ролле, F ( 1 ) ( х ) »displaystyle F »(1)» (x) » имеет k »1 »displaystyle k »1» нулей, F ( 2 ) ( х ) »displaystyle F »(2)» (x) » имеет k »displaystyle k» нули ... F (k й 1 ) (дисплей F) имеет 1 ноль, скажем, ξ , х 0 < ξ < х х к «дисплей стиль »xi ,»,x_{0}<'xi <x_'k. . Явно написание F (k q 1 ) ( ξ ) (displaystyle F )(к'1) (КИ) : F ( к No 1 ) ( ξ ξ ξ ξ ) Стиль Фз (кз 1)» (КЗИ ) (КЗ (КЗХ)» (КЗХ)» (КЗИ )-РЗ (кз 1 ⋅) «Дисплейстайл Лз (к 1)»0,РЗ (к 1) »Кздот »КЗДО (к 1)!» (Потому что самая высокая мощность х «дисплей стиль х» в R ( х ) »displaystyle R(x)» является к 1 «дисплей стиль k » 1 » 0 » f ( q » 1 ) ( ξ ) » C ⋅ ( к ...... «Дисплейстайл 0»f»(к 1)» (Си )-Сёкдо (кз1)!» Уравнение может быть перестроено как C q f ( k q 1 ) ( ξ ) ( k q 1 ) ! «Дисплей стиле C»frac (к)» (КЗИ) »(к)! Derivatives The d {\displaystyle d} th derivatives of the Lagrange polynomial can be written as L ( d ) ( x ) := ∑ j = 0 k y j l j ( d ) ( x ) {\displaystyle L^{(d)}(x):=\sum _{j=0}^{k}y_{j}\ell _{j}^{(d)}(x)} . Для первой производной коэффициенты даются l j ( 1 ) ( x ) : ∑ i 0 i ≠ j k 1 x j х х i ∏ м - 0 м ≠ (i , j ) k x x x x x йо(1)» (x): «Сумма x_ x_ {1} »начать »smallmatrix»i'0'iot (j'end-smallmatrix) «начать»smallmatrix»m'0'mot (i,j)»end'smallmatrix'k'frac (x-x_'m'x_'j'-x_'m'right) и для второго производная l j ( 2 ) ( х ) : ∑ i 0 i ≠ j k 1 x j х х я ∑ м й 0 м ≠ (i , j ) k ( 1 x j х м ∏ л й 0 л ≠ ( i , j , m ) k x x x x x l ) «малыйматрикс»-фрак {1}x_ jj'-x_'i'i'sum (начало)smallmatrix'm'0'meq (i,i,j) Энд-малыйматрикс(Фрак {1})x_ Jj'-x_'m'prod (начало) smallmatrix'l'le q (i ,j,m)'end'smallmatrix'k'k'frac (x-x_'l'l'x_'j'-x_'l'l'right). Через рекурсию можно вычислить формулы для более высоких деривативов. Конечные поля Полинамия Лагранжа также может быть вычислена в конечных полях. Это имеет приложения в криптографии, например, в схеме тайного обмена Шамира. Смотрите также алгоритм Невилла Ньютон форме интерполяции полиномиальной Бернштейн полиномиальной Lebesgue permanent theorem theorem (interpolation) Table of the System Chebfun of the Newton series Frobenius covariant Finite difference factor - Waring, Edward (January 9, 1779). Problems with interpolation. Philosophical deals of the Royal Society. 69: 59–67. doi:10.1098/rstl.1779.0008. Meyering, Eric (2002). Interpolation chronology: from ancient astronomy to modern signal and image processing (PDF). IEEE Procedures. 90 (3): 319–342. doi:10.1109/5.993400. Alfio's quarteroni; Saleri, Fausto (2003). Scientific computing with MATLAB. Texts in computer science and technology. 2. Springer. page 66. ISBN 978-3-540-44363-6. - Berrut, Ian Paul; Trefeten, Lloyd N. (2004). Baricentric Lagrange Interpolation (PDF). SIAM review. 46 (3): 501–517. doi:10.1137/S0036144502417715. Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) (June 1964). Chapter 25, eqn 25.2.3. A guide to mathematical functions with formulas, graphs and mathematical tables. A series of applied mathematics. 55 (Ninth reissue with additional corrections of the tenth original print with corrections (December 1972); first - Washington, D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 878. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. Interpolation (PDF). External Links Implementation of the Wikibook algorithm has a page on the theme: Polynomial Interpolation Formula Interpolation lagrange, Encyclopedia of Mathematics, EMS Press, 2001 (1994) ALGLIB has implementation in C /C /VBA / Pascal. GSL has a polynomial interpolation code in C SO has the example of MATLAB, which demonstrates the algorithm and recreates the first image in this article Lagrange Interpolation Method - Notes, PPT, Mathcad, Mathematica, MATLAB, Maple at the Holistic Institute of Numerical Methods of Lagrange Interpolation Polynomial on the www.math-linux.com Weisstein, Eric W. Lagrange polynomial on ProofWiki Dynamic Lagrange lagrange interpolation with JSXGraph numerical computing with features: Chebfun Project Excel Sheet feature for Bicubic Lagrange Interpolation Lagrange polynomials in Python extracted from lagrange interpolation polynomial calculator. lagrange interpolation polynomial example. lagrange interpolation polynomial formula. lagrange interpolation polynomial matlab. lagrange interpolation polynomial applications. lagrange interpolation polynomial python. lagrange interpolation polynomial calculator with steps. lagrange interpolation polynomial definition

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