Math 4363 Numerical Analysis
Test 2
Fall xxxx
Name:
(20)
1 + x + x2 + x3 ,
x [0, 1)
4 + b(x 1) + 4(x 1)2 + (x 1)3 , x [1, 2]
be a clamped cubic spline with f (0) = 1 and f (2) = 17.
1. Let S (x) =
(a) Find b.
(b) Find S (0.5).
(c) Can we nd f (1) from thi
Test 3
(48)
Math 4363 Numerical Analysis
Fall nnnn
1. Numerical Integration
3
2x5 dx using Simpsons rule. You do not need to simplify your result.
(a) Approximate
2
(b) Derive the error estimator for the (Simpsons) adaptive quadrature method.
(c) Starting
Test 3
Math 4363 Numerical Analysis
Fall xxxx
Name:
(30)
1. Numerical Dierentiation
(a) Derive any numerical dierentiation formula you like (including the error term).
(b) Let f (x) = 3x3 . Approximate f (0.2) using the 3-point centered dierence formula
w
Test 3
Math 4363 Numerical Analysis
Fall nnnn
(5)
1. State a theorem regarding the existence and uniqueness of solutions
for (IVP).
(5)
2. Give a short description, in words, of a well-posed IVP.
(5)
3. Let f (0) = 1, f (.25) = 2, f (.5) = 4, f (.75) = 5,
Math 4363 Numerical Analysis
(20)
Fall nnnn
Test 3
1. General
(a) State the initial value problem that we have been calling (IVP).
(b) Give the denition of a well-posed IVP.
(c) State a theorem on the existence and uniqueness of solutions to (IVP).
(60)
2
Math 4363 Numerical Analysis
(40)
Fall nnnn
Test 2
1. Let (x0 , y0 ), (x1 , y1 ), . . . , (xn , yn ) be real ordered pairs with xi = xj for i = j .
(a) Give the general denition of the osculating polynomial for this data. Show how Lagrange
Interpolation,
Math 4363 Numerical Analysis
(40)
Fall nnnn
Test 2
1. Let (x0 , y0 ), (x1 , y1 ), . . . , (xn , yn ) be real ordered pairs with xi = xj for i = j .
(a) Give the general denition of the osculating polynomial for this data. Show how Lagrange
Interpolation,
Math 4363 Numerical Analysis
Fall nnnn
Test 1
Name:
(30)
1. Let f (x) = x2 + x 2. Were looking for a zero of f .
(a) Use the bisection method with a = 6 and b = 0 to nd an interval of length strictly
less than 3 which brackets a zero of f .
(b) Use one it
Math 4363 Numerical Analysis
(30)
Fall nnnn
Test 1
1. Let f (x) = x2 + x 2. Were looking for a zero of f .
(a) Use the bisection method with a = 6 and b = 0 to nd an interval of length strictly
less than 3 which brackets a zero of f .
(b) Use one iteratio
Math 4363 Numerical Analysis
(30)
Fall nnnn
Test 1
1. Let f (x) = x2 + x 2. Were looking for a zero of f .
(a) Use the bisection method with a = 0 and b = 3 to nd an interval of length strictly
less than 2 which brackets a zero of f .
(b) Use one iteratio
Math 4363 Numerical Analysis
(20)
Test 2
Fall nnnn
1 + 2 x + x2 + x3 ,
x [0, 1)
2 + (x 1)3 , x [1, 2]
5 + b(x 1) + 4(x 1)
be a clamped cubic spline for a function f with f (0) = 1.
1. Let S (x) =
(a) Find b.
(b) Find S (0.5).
(c) Find f (2).
(d) Can we nd