《计算机数学基础》数值部分第二单元辅导（Computer mathematics foundation numerical part second unit guidance）.docVIP

《计算机数学基础》数值部分第二单元辅导（Computer mathematics foundation numerical part second unit guidance） Computer mathematics foundation numerical part second unit guidance I. key contents 1. function interpolation The function value yk=f (XK) of the known function f (x), k=0,1,2,..., n. Construct a polynomial P (x) that makes P (XK) =yk. P (x) is an interpolation polynomial, f (x) is an interpolating function, and XK is an interpolation node. Error R (x) =f (x) P (x). 2. Lagrange interpolation polynomial With n degree polynomial Pn (x), =y0l0 (x), +y1l1 (x) +, +ynln (x) = The approximate function f (x), that is, f (x), Pn (x), and satisfies Pn (XK), =yk (k=0,1,2,..., n). Base function (i=0,1,2,..., n) When n=1, the linear interpolation P1 (x), =yklk (x), +yk+1lk+1 (x) Base function,. When n=2, two polynomials are obtained, that is, the two interpolation. The remainder of Lagrange interpolation polynomial is Among them, Note: if the n+1 cross node is different, the polynomial of interpolation should be polynomial whose number is not more than n. 3. difference and Newton interpolation polynomial The functional value of difference and difference quotient is the deviation variable, First order difference Two order difference ............ N order difference There are two different common properties: (1) by the linear combination of YK expressed by function value; (2) difference has nothing to do with the interpolation node order (symmetry). The relationship between N and the low order derivative: The mean coefficients of polynomial, is Newton interpolation polynomial, for Nn (x) = f (x0) F (x0, x1) (Xx0) F (x0, x1, x2) (Xx0) (Xx1) +... F (x0, x1, X2,..., xn) (Xx0) (Xx1) (Xx2) (Xxn1)... The remainder of Newton interpolation polynomial is Rn (x) =f (x) Nn (x) =f (x, x0, x1, X2,..., xn) (Xx0) (Xx1) (Xx2) (Xxn1) (Xxn)... = 4. piecewise linear interpolation Divide the interval [a, b] into n cell [xk, xk+1] (k=0,1,..., N - 1) with the point a=x0x1... xn=b. On interval [xk and xk+1], the poly