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MAT 684
Qualifying Exam
Syracuse University
January, Spring 2010
Ex 1. Consider the boundary value problem od determining u(t), 0 < t < 1 which satises
u (t) = 6u (t) tu(t) + u2 (t)
for
0<t<1
u(0) = 20
u(1) = 10
(a) Formulate a nite dierence method to c
MAT 684
Numerical Analysis
August, 2010
NAME:
1. Let f : [0, 1] R be a given continuous function. Consider the following boundary value
problem with Dirichlet boundary conditions
u (x) + u(x) = f (x),
u(0) = 0,
0<x<1
u(1) = 0.
(a) Derive the variational e
MAT 683
Numerical Analysis
January, 2010
NAME:
1. (a) Let S be a linear spline function that interpolates f at a sequence of nodes 0 =
1
x0 < x1 < < xn = 1. Find an expression of 0 S (x) dx in terms of xi and f (xi ),
i = 0, 1, . . . , n.
(b) Can a and b
1
MAT 683
Numerical Analysis
Qualifying Exam Syracuse University
August 24, Fall 2010
Ex 1. Let x0 < x1 be two distinct real numbers. Let
x1 x0 . Find a polynomial p of degree 3 such that
be such that 0 <
<
p ( x0 ) = p ( x0 + ) = 1
p ( x1 ) = p ( x1 + )
MAT 683
Numerical Analysis
January, 2008
NAME:
1. Find a polynomial p(x) of degree 2 that satises
p(x0 ) = a,
p (x0 ) = b,
p (x1 ) = c,
where a, b, c are given constants and x0 , x1 are two dierent points.
2. Let f (0), f (h) and f (2h) be the values of a
MAT 683
Numerical Analysis
August, 2008
NAME:
1. For which values of s [0, 1], will there exist a unique quadratic polynomial p that
satises the following conditions
p(0) = p0 ,
p(1) = p1 ,
p (s) = p2 ?
2. Consider the equation
3x + g (x) = 0
(1)
with g :
MAT 682
Numerical Analysis
August, 2008
NAME:
1. Suppose A be an n n invertible matrix. Let A = U V be the singular
value decomposition of A, where
= diagcfw_1 , 2 , . . . , n and 1 2 n > 0.
(a) Show that 2 (A) = 1 /n , where 2 (A) A
(b) Show that
(n 1)