Math 315,
Quiz 1
January 22, 2013.
Consider the function
f (x) =
1x
1
.
1 + 2x 1 + x
(a) Explain why evaluating f (x) may give inaccurate results when |x|
1.
(b) Rewrite f (x) so as to avoid the problem in (a).
Solution:
1
(a) When |x|
1 (read: x is much
Math 315,
Quiz 1
January 22, 2013.
Consider the function
f (x) =
1 cos x
.
x
(a) Explain why evaluating f (x) may give inaccurate results when 0 = |x|
1.
(b) Rewrite f (x) so as to avoid the problem in (a). (Hint: Multiply and divide f (x) by the
same fun
Math 315,
Quiz 2
February 7, 2013.
NOTE: You must show all details of your work to receive credit.
0 3 4
1. Let A = 0 4
4 .
20
4
(a) Use Gaussian elimination with partial pivoting to nd the P A = LU factorization. In
particular, be sure to write down the
Math 315,
Quiz 2
February 7, 2013.
NOTE: You must show all details of your work to receive credit.
1. Consider the matrix
2 1
1
A=
0
0
0
2 1
0
.
1
2 1
0 1
2
0
Find the LU factorization of A, where L is unit lower triangular and U is upper triangular. Is
Math 315,
Quiz 3
February 21, 2013.
NOTE: You must show all details of your work to receive credit.
1. Find the Newton form of the interpolation polynomial for the following points: (1, 0), (0, 1)
and (2, 6). After converting the Newton form to the (stand
Math 315,
Quiz 3
February 21, 2013.
NOTE: You must show all details of your work to receive credit.
1. Find the Newton form of the interpolation polynomial for the following points: (1, 1), (0, 3)
and (1, 1). After converting the Newton form to the (stand
Math 315,
Quiz 4
March 26, 2013.
1. Consider the function f (x) = 1 + 10x x2 and let S1,n (x) be its piecewise linear interpolant
at a set of n + 1 equidistant nodes 0 = x0 < x1 < < xn1 < xn = 1 on the interval [0, 1].
Estimate the number of nodes needed
Math 315,
Quiz 4
March 26, 2013.
1. Consider the function f (x) = 2 cos(x) 5x + 1 and let S1,n (x) be its piecewise linear interpolant
at a set of n + 1 equidistant nodes 0 = x0 < x1 < < xn1 < xn = 1 on the interval [0, 1].
Estimate the number of nodes ne
Math 315,
Quiz 5
April 9, 2013.
1. Write down Newtons method for nding the pth root of any positive real number a.
Solution:
We have f (x) = xp a, f (x) = pxp1 , hence Newtons method becomes
xn+1 = xn
xp a
n
p
pxn1
,
n = 0, 1, . . . ,
where x0 is an arbi
Math 315,
Quiz 5
April 9, 2013.
1. Write down Newtons method for nding a root of a dierentiable function f (x).
Solution:
Newtons method is the iteration
xn+1 = xn
f (xn )
,
f (xn )
n = 0, 1, . . . ,
where x0 is an arbitrary initial guess.
2. Starting wi
Math 315,
Quiz 6
April 23, 2013.
Consider the polynomial equation
an xn + an1 xn1 + + a1 x + a0 = 0 .
Answer the following questions:
1. True or false: Multiple roots are typically ill-conditioned.
2. True or false: Well separated roots are always well-co
Math 315
5 March 2013
Test 1
Instructions: For full credit on each problem, you must show all work needed to obtain your
answers. You are expected to abide by all the provisions and spirit of the Emory College Honor
Code in the completion of this exam.
1.
Math 315
5 March 2013
Test 1
Instructions: For full credit on each problem, you must show all work needed to obtain your
answers. You are expected to abide by all the provisions and spirit of the Emory College Honor
Code in the completion of this exam.
1.
Math 315
11 April 2013
Test 2
Instructions: For full credit on each problem, you must show all work needed to obtain your
answers. You are expected to abide by all the provisions and spirit of the Emory College Honor
Code in the completion of this exam.
1