Given x0 = y0 = −1, x1 = y1 = 0, x2 = y2 = 1, and zi,j = 1 1+x 2 i +y 2 j , where i, j = 0, 1, 2. (a) Approximate the value of z at (x, y) = (1, 1 2 ) by using Lagrange interpolation
时间: 2023-04-06 16:03:46 浏览: 72
I can answer this question. The Lagrange interpolation formula can be used to approximate the value of z at (x, y) = (1, 1/2) using the given values of zi,j. The formula is:
z(x,y) = Σi=0 to 2 Σj=0 to 2 zi,j Li(x) Lj(y)
where Li(x) and Lj(y) are the Lagrange basis polynomials given by:
Li(x) = Πj=0 to 2, j ≠ i (x - xi)/(xi - xj)
Lj(y) = Πi=0 to 2, i ≠ j (y - yj)/(yj - yi)
Substituting the given values, we get:
z(1,1/2) = (1/9) [ (1/2 - 0)(1 - 1)(1 - 0) + (1/2 - 0)(1 - 2)(1 - 0) + (1/2 - 0)(1 - 2)(1 - 1) ] + (4/9) [ (1 - 0)(1/2 - 1)(1/2 - 0) + (1 - 0)(1/2 - 0)(1/2 - 1) + (1 - 1)(1/2 - 0)(1/2 - 1) ] + (1/9) [ (1 - 1)(0 - 1)(1/2 - 0) + (1 - 1)(0 - 0)(1/2 - 1) + (1 - 2)(0 - 0)(1/2 - 1) ]
Simplifying the expression, we get:
z(1,1/2) = 1/9
Therefore, the approximate value of z at (x, y) = (1, 1/2) is 1/9.