Lagrange Interpolation INTERPOLATION
42 Lagrange Interpolation INTERPOLATION
Two-point formula
42.1. px () = fx ()
0 fx x () − x 1
where p (x) is a linear polynomial interpolating two points
( , ( )), ( , ( )), x 0 fx 0 x 1 fx 1 x 0 ≠ x 1
General formula
42.2. px () = fxL () 0 n , 0 () x + fxL () 1 n , 1 () x + ⋯ + fxL () n nn , ( xx )
where
L nk , =
i = ∏ 0 , ik ≠ x k − x i
and where p(x) is an nth-order polynomial interpolating n + 1 points ( , ( )), x k fx k k = 01… ,, ,; n and x i ≠ x j for i ≠ j
Remainder formula
Suppose f x () C ∈ n + 1 [ , ]. ab Then there is a ξ( ) ( , ) x ∈ ab such that:
f n +1 ( ( )) ξ x
42.3. fx () = px () + ( 1 )! ( x − x 0 )( x − x 1 )( ⋯ x − x
Newton’s Interpolation
First-order divided-difference formula
fx () − fx ()
42.4. fxx [,]
Two-point interpolatory formula
42.5. px () = fx () 0 + fxx [ , ]( 0 1 x − x 0 )
where p(x) is a linear polynomial interpolating two points
( , ( )), ( , ( )), x 0 fx 0 xfx 1 1 x 0 ≠ x 1
INTERPOLATION
Second-order divided-difference formula
fxx [,] 1 2 − fxx [,]
42.6. fxxx [,,] 0 1 2 =
Three-point interpolatory formula
42.7. px () = fx () 0 + fxx [ , ]( 0 1 x − x 0 ) + fxxx [ , , ]( 0 1 2 x − x 0 ) (( x − x 1 )
where p(x) is a quadrant polynomial interpolating three points
( , ( )), ( , ( )), ( , ( )) x 0 fx 0 xfx 1 1 x 2 fx 3
General kth-order divided-difference formula
fxx [,,,] 1 2 … x k − fxx [,,, …
k 42.8. − fxx 0 1 x 1
xx k − x 0
General interpolatory formula
42.9. px () = fx () 0 + fxx [ , ]( 0 1 x − x 0 ) + ⋯ + fxx [ , , , ]( 0 1 … x n x −− x 0 )( x − x 1 )( ⋯ x − x n − 1 ) where p(x) is an nth-order polynomial interpolating n + 1 points
( , ( )), x k fx k k = 01… ,,,; n and x i ≠ x j for i ≠ j
Remainder formula
Suppose f x () C ∈ n + 1 [ , ]. ab Then there is a ξ( ) ( , ) x ∈ ab such that
f n +1 ( ( )) x
42.10. fx () = px ()
x − x 0 )( x − x 1 )( ⋯ x x
Newton’s Forward-Difference Formula
First-order forward-difference at x 0
42.11. Δf x () 0 = fx () 1 − fx () 0
Second-order forward difference at x 0
42.12. Δ 2 fx ()
0 = Δ fx () 1 − Δ fx () 0
General kth-order forward difference at x 0
k − 1 k − 42.13. Δ 1 fx ()
0 = Δ fx () 1 − Δ fx () 0
Binomial coefficient
⎛ s ⎞ ss ( − 1 )( ⋯ s −+ k 1 42.14. ) ⎝⎜ k ⎠⎟ =
Newton’s forward-difference formula
42.15. px () = ⎛ ⎞
∑ k Δ fx ()
k = 0 ⎝⎜ k ⎠⎟ where p(x) is an nth-order polynomial interpolating n + 1 equal spaced points
( , ( )), x
k fx k x k = x 0 + kh k = 01… ,, , n
INTERPOLATION
Newton’s Backward-Difference Formula
First-order backward difference at x n
42.16. ∇ fx () n = fx () n − fx ( n − 1 )
Second-order backward difference at x n
42.17. ∇ 2 fx () n =∇ fx () n −∇ fx ( n − 1 )
General kth-order backward difference at x n
42.18. ∇ k fx ()
Newton’s backward-difference formula
where p(x) is an nth-order polynomial interpolating n + 1 equal spaced points
( , ( )), x k fx k x k = x 0 + kh
k = 01… ,, , n
Hermite Interpolation
Two-point basis polynomials
⎞ − x 42.20. 2 H 0 )
Two-point interpolatory formula
42.21. Hx 3 () = fxH () 0 10 , + fxH () 1 11 , +′ fxH ()ˆ 0 10 , + ′ )) ˆ fx ( 1 H 11 ,
where H 3 (x) is a third-order polynomial, agrees with f (x) and its first-order derivatives at two points, i.e., Hx 3 () 0 = fx ( ), 0 Hx 3 ′ () 0 =′ fx ( ), 0 Hx 3 () 1 = fx ( ), 1 H 33 ′ () x 1 =′ fx () 1
General basis polynomials
42.22. H nj , =− 12
L 2 nj , ( ), x H ˆ nj , = ( xx −) x j L nj 2 , L () x x ⎝⎜
nj ′ , () j ⎠⎟ where
L nj , =
i = ∏ 0 , ij ≠ x j − x i
INTERPOLATION
General interpolatory formula
42.23. H 2 n + 1 () x = ∑ fxH () j nj , () x + fxH ′ ()ˆ() j nj , x
j = 00 ∑ j = 0
where H 2 n + 1 () x is a (2n + 1)th-order polynomial, agrees with f(x) and its first order derivatives at n + 1 points, i.e.,
H 2 n + 1 () x k = fx ( ), k H 2 ′ n + 1 () x k =′ fx () k
k = 01 ,, … , n
Remainder formula
Suppose f x ()
∈ n C 2 + 2 [ , ]. ab Then there is a ξ( ) ( , ) x ∈ ab such that
42.24. fx () H = ξ
f 2 n + 2 ( ( )) x
2 n + 1 () x +
x x )( x x ) ⋯x ( x )
( 2 n + 2 )! ( − 0 − 11 − n