MA 780: Midterm Test and Solutions
1. Given (x0 , y0 ), (x1 , y1 ), , (xn , yn ). Write down the following:
(a) the formula of Lagrange polynomial interpolation.
(b) the Newton polynomial interpolation using the divided dierence.
(c) the error estimate.
Due April 23, can be extended to April 27
if x (0, 0.5),
1 if x (0.5, 1)
f (x) =
in the interval (0.5, 1.5), see Figure (10.8) on page 460.
(a) Find the linear transform x = g() to map the interval x (0.5, 1.5)
Due date: TBA
1. Consider a right triangle Th with vertices (0, 0), (h, 0) and (0, h). Let f (, ) C 2 (Th ). Dene
p(, ) = f (0, 0) 1
+ f (h, 0)
+ f (0, h) .
(a) Show that max |f (, ) p(, )| Ch2 . Find the constant C if possible
Due March 12
1. (a) Show that the Simpson quadrature rule has algebraic precision 3 by testing the polynomial basis
function cfw_1, x, x2 , x3 , x4 . Therefore the Simpson quadrature rule is exact for any polynomial
of pk (x) of degree