1. The existence and uniqueness of the interpolation polynomial
In scientific research and production of a large number of functions encountered in practice, a considerable part obtained by observation or experiment, although the function \ (y = f (x) \) in an interval \ ([a, b ] \) on exist, but we do not know the exact analytic expression, function values observed or only through some of the experiments of discrete points, value of the derivative, it is desirable for such a function with a simple analytical expression for the myopic a description is given on the whole. Some function, although there are clear analytical expression, but due to the analytical expression too complex and inconvenient for its theoretical analysis and numerical computation, and hope to give a function not only reflect the characteristics and functions suitable for simple numerical calculation, instead of the original function is approximated.
Definition 1 : known function \ (y = f (x) \) in the interval \ ([a, b] \ ) a \ (n + 1 \) different points \ (a \ leq x_0 <x_1 <x_2 <\ cdots <x_n \ leq b \) function value \ (F (x_0), F (x_1), \ cdots, F (x_n) \) , if the presence of a simple function \ (P (X) \) , passed through a \ (y = f (x) \) on \ (n + 1 \) known point \ ((x_0, y_0), (x_1, Y_1), \ cdots, (x_n, y_n) \) , \
[ P (x_i) = y_i = f (x_i), \ i = 0,1,2, \ cdots, n \ (1) \]
Establishment, called \ (P (x) \) of \ (f (x) \) of the interpolation function , point \ (x_0, x_1, \ cdots , x_n \) is called interpolation nodes , the node comprising an interpolation section \ ( [a, b] \) is called the interpolation section , find interpolation function \ (P (x) \) method is called interpolation method . If \ (P (x) \) is the number does not exceed \ (n-\) polynomial, i.e.
\ [P_n (x) = a_0 + a_1 x + a_2 x ^ 2 + \ cdots + a_n x ^ n \ (2) \]
Wherein \ (a_i \) are real numbers, called \ (P (x) \) for the polynomial interpolation , an interpolation method referred to as the corresponding polynomial interpolation . If \ (P (x) \) is piecewise polynomial, it is called piecewise interpolation . If \ (P (x) \) is a trigonometric polynomial, it is called trigonometric interpolation .
Looking satisfies (1) interpolation function \ (P (x) \) many ways. \ (P (x) \) may be algebraic polynomials, rational functions like trigonometric polynomial, may be any smooth or piecewise smooth function. Different interpolation function \ (P (X) \) , approximation \ (f (x) \) different effects. Looking interpolation function \ (P (X) \) , the first thought is a polynomial function, not only because the polynomial function is simple expression, and there are many good properties, such as continuous and smooth, differentiable integrable, additionally by a Weierstrass Theory know, any continuous function can be used as the algebraic polynomial approximation of arbitrary precision, while the algebraic polynomial basis or other types of interpolation.
Theorem 1 (existence and uniqueness interpolation polynomial) : Suppose node \ (x_0, x_1, \ cdots , x_n \) mutually different, it does not exceed the number of times \ (n-\) set of polynomials \ (H_n \)In the condition (1) of the interpolating polynomial \ (of P_n (X) \) exists and is unique.
2. Lagrange Interpolation
2.1 Lagrange interpolation polynomial
Seeking interpolation polynomial \ (of P_n (X) \) solution, it can be set by calculating the equation (3) \ (a_0, a_1, \ cdots , a_n \) to give, but this algorithm is computationally intensive, inconvenient for practical use.
\ [\ Left \ {\ begin {aligned} a_0 + a_1 x_0 + \ cdots + a_n x_0 ^ n & = y_0, \\ a_0 + a_1 x_1 + \ cdots + a_n x_1 ^ n & = y_1, \\ \ vdots & \ (3) \\ a_0 + a_1 x_n + \ cdots + a_n x_n ^ n & = y_n, \ end {aligned} \ right. \]
Set \ (\ phi (c_1, c_1 , \ cdots, c_n) \) is the number does not exceed \ (n-\) polynomial space, configured \ (\ phi (c_1, c_1 , \ cdots, c_n) \) of a set of basis function \ (L_0 (X), L_1 (X), \ cdots, L_n (X) \) , so that seeking interpolation polynomial
\ [L_n (x) = \ sum _ {i = 0} ^ {n} a_i l_i ( x) \ (4) \]
The coefficient \ (a_i \) easier.
\ (L_n (x) \) can be written as
\ [L_n (x) = ( l_0 (x), l_1 (x), \ cdots, l_n (x)) (a_0, a_1, \ cdots, a_n) ^ T \ ( 5) \]
and
\ [L_n (x_i) = f (x_i), \ i = 0,1, \ cdots, n. (6) \]
故
\[ \left[ \begin{matrix} l_0(x_0) & l_1(x_0) & \cdots & l_n(x_0) \\ l_0(x_1) & l_1(x_1) & \cdots & l_n(x_1) \\ \vdots & \vdots & & \vdots \\ l_0(x_n) & l_1(x_n) & \cdots & l_n(x_n) \end{matrix} \right] \left[ \begin{matrix} a_0 \\ a_1 \\ \vdots \\ a_n \end{matrix} \right] \left[ \begin{matrix} f(x_0) \\ f(x_1) \\ \vdots \\ f(x_n) \end{matrix} \right] (7) \]
If the coefficients of equations (7) of the matrix is a unit matrix, it is immediately available
\ [a_i = f (x_i)
, \ i = 0,1, \ cdots, n. \ (8) \] For equations (7 ) coefficient matrix is the identity matrix, simply
\ [l_i (x_j) = \ delta _ {ij} = \ left \ {\ begin {aligned} 1, & & i = j, \\ 0, & & i \ neq j, \ end {aligned} \ right. \ i, j = 0,1, \ cdots, n. \ (9) \]
Thus polynomial space () \ \ phi (c_1, c_1, \ cdots, c_n) \ within seek a set of basis functions \ (L_0 (X), L_1 (X), \ cdots, L_n (X) \) , so that coefficients of equations (7) of the matrix is a unit matrix, is converted into configuration satisfying the condition (9) of the basis functions \ (l_i (x) \) , since \ (l_i (x) \) in \ (x = x_j (j = 0,1, \ cdots, i- 1, i + 1, \ cdots, n) \) is at \ (0 \) , so that it can be
\ [l_i (x) = a (x-x_0) ( x-x_1) \ cdots (x -x_ {i-1}) (x-x_ {i + 1}) \ cdots (x-x_n) \ (10) \]
Wherein \ (A \) constant to be determined. In the formula (10) in order \ (X = x_i \) , may be determined \ (A \) to
\ [A = \ frac 1 { (x_i-x_0) (x_i-x_1) \ cdots (x_i-x_ {i -1}) (x_i-x_ { i + 1}) \ cdots (x_i-x_n)} \]
从而
\[ \begin{aligned} l_i(x) & = \frac {(x-x_0)(x-x_1)\cdots(x-x_{i-1})(x-x_{i+1})\cdots(x-x_n)} {(x_i-x_0)(x_i-x_1)\cdots(x_i-x_{i-1})(x_i-x_{i+1})\cdots(x_i-x_n)} \\ & = \prod _{j=0, j\neq i}^{n} \frac {x-x_j}{x_i-x_j} \ (11) \end{aligned} \]
记
\ [\ omega_ {n + 1 } (x) = \ prod _ {i = 0} ^ n (x-x_i) \ (12) \]
则
\[ l_i(x) = \frac {\omega_{n+1}(x)}{(x-x_i)w'_{n+1}(x_i)} \ (13) \]
So as to satisfy the condition may be (1) a \ (n-\) cubic interpolation polynomial
\ [L_n (x) = \ sum _ {i = 0} ^ nf (x_i) l_i (x) \ (14) \]
Called \ (L_n (x) \) is the Lagrange polynomial interpolation , \ (L_i (the X-) \) is the Lagrange interpolation basis functions .
Parabolic interpolation and linear interpolation 2.2
Known function \ (y = f (x) \) at the point \ (x_0, x_1 \) function values are at \ (y_0, Y_1 \) . In the formula (14) in order \ (n-=. 1 \) , LAGRANGE interpolation polynomial
\ [\ begin {aligned} L_1 (x) & = f (x_0) l_0 (x) + f (x_1) l_1 (x) \ \ & = y_0 \ frac {x -x_1} {x_0-x_1} + y_1 \ frac {x-x_0} {x_1-x_0} \\ & = y_0 + \ frac {y_1-y_0} {x_1-x_0} (x -x_0) \ end {aligned} \ (15) \]
其中
\[ l_0(x) = \frac {x-x_1}{x_0-x_1},\ l_1(x) = \frac {x-x_0}{x_1-x_0} \]
It is the result of two \ ((x_0, y_0), (x_1, y_1) \) of a straight line, so this method is generally referred to as linear interpolation .
Known function \ (y = f (x) \) at the point \ (x_0, x_1, x_2 \ ) function values are at \ (y_0, Y_1, Y_2 \) . In order in the formula (14) \ (n-= 2 \) , LAGRANGE interpolation polynomial
\ [\ begin {aligned} L_2 (x) & = f (x_0) l_0 (x) + f (x_1) l_1 (x) + f (x_2) l_2 (x) \\ & = y_0 \ frac {(x-x_1) (x-x_2)} {(x_0-x_1) (x_0-x_2)} + y_1 \ frac {(x-x_0) ( x-x_2)} {(x_1 -x_0) (x_1-x_2)} + y_2 \ frac {(x-x_0) (x-x_1)} {(x_2-x_0) (x_2-x_1)} \\ & = y_0 + \ frac {y_1-y_0} {x_1-x_0} (x-x_0) \ end {aligned} \ (16) \]
其中
\[ l_0(x) = \frac {(x-x_1)(x-x_2)}{(x_0-x_1)(x_0-x_2)}, \ l_1(x) = \frac {(x-x_0)(x-x_2)}{(x_1-x_0)(x_1-x_2)}, \ l_2(x) = \frac {(x-x_0)(x-x_1)}{(x_2-x_0)(x_2-x_1)} \]
Of formula (16) is a quadratic function, is through \ ((x_0, y_0), (x_1, y_1), (x_2, y_2) \) of the parabola, so this method is commonly referred parabolic interpolation .