University of Colorado
Department of Computer Science
Numerical Computation
CSCI 3656
Spring 2016
Problem Set 11 Solutions
1. [10 pts] Apply Richardson extrapolation starting from h = 0.1 to get f 0 (1) for
f (x) = ex + 0.5 and using center differences. C
University of Colorado
Department of Computer Science
Numerical Computation
CSCI 3656
Spring 2016
Problem Set 12 Solutions
1.8 x
[e + 0.5]dx with
1. [15 pts] Use your Simpsons 1/3 rule code from PS11 to compute 1.0
h = 0.1 and h = 0.05. Now use one layer
University of Colorado
Department of Computer Science
Numerical Computation
CSCI 3656
Spring 2016
Problem Set 13
Issued:
21 April 2016
Due:
28 April 2016
1. [30 pts] Modify your code from PS12 to solve the Lorenz equations using the trapezoidal
method ins
University of Colorado
Department of Computer Science
Numerical Computation
CSCI 3656
Spring 2016
Problem Set 10 Solutions
Note: these are solutions to the original version of this PS. Were in the process of
updating them to match the actual version.
1. [
Last Assignment
CSCI 3656
due April 28, 2011
1. Consider the initial value problem
y (t) = 1.0/(1.0 + t2 ) 2.0[y (t)]2 .
y (0) = 0
2
The exact solution is y (t) = t/(1.0 + t ). Write a program to solve this problem on the
interval [0,5] using Eulers metho
Assignment 12
CSCI 3656
due April 21, 2011
1. (a) On your computer approximate the derivative of the function f (x) = 2.0/(x2 + 3.2)
exp(1.2x) at the point x = 1.3 using a two point forward dierence formula. Use as
denominators the values h = 2k for k =
Assignment 11
CSCI 3656
due April 14, 2011
1. The following set of simulated data gives the quantity of a decaying substance at various
times. The substance is composed of three isotopes with decay rates 0.0, 0.8, and 0.9. If
the initial quantities are de
Assignment 10
CSCI 3656
due April 7, 2011
1. Given data (xi , yi ) for i = 1, . . . , n, in order to compute the cubic spline coecients
(bi , ci , di ), you must (i) set up the equations (3.30) on page 175, (ii) solve the tridiagonal
system by Gaussian el
Assignment 9
CSCI 3656
due March 31, 2011
1. Suppose you constuct a polynomial approximation to the function f (x) = ln(x) on the
interval [1,3]. If you use n equally spaced points, give an upper bound for the interpolation
error using the error bound for
Assignment 8
CSCI 3656
due March 17, 2011
1. For the data (xi , yi ) = (0, 6.0) (1, 4.0) (2, 3.0), (4, 7.0),
a) Write down the Lagrange form of the interpolating polynomial p(x). (There is no need
to algebraically multiply out the polynomial. Leave it in
Assignment 7
CSCI 3656
due March 3, 2011
1. In Matlab the command A=hilb(10) will produce an 10 10 Hilbert matrix, i.e. A(i, j ) =
1/(i+j 1). Compute the right hand side vector b = Av , where v = [1, 2, 3, 4, 5, 1, 2, 3, 4, 5]T .
(a) Solve the system Ax =
Assignment 6
CSCI 3656
due February 24, 2011
1. (a) By hand use Gaussian elimination with scaled partial pivoting to nd the P, L and U
factors of the matrix A:
2 4
1 2
A=
4
4
12
10 ;
6
Verify that P A = LU
(b) Now use P, L, and U to solve Ax = b, where
2
Assignment 5
CSCI 3656
due February 17, 2011
1. Solve the following system Ax = b by Gaussian elimination with partial pivoting. Use the
partial pivoting rule described in class and in Section 2.4.1 of the text. Show all the steps.
4
A= 2
2
2
6
8
3
3 ; b
Assignment 4
CSCI 3656
due February 10, 2011
1. Given that the distance in meters fallen from rest by a skydiver is
y (t) = ln(cosh(t
g k )/k.
compute the time taken to fall 85 meters using Newtons method. Here, gravitational acceleration
g = 9.8065 m/s/s
Assignment 3
CSCI 3656
due February 3, 2011
1. Write a program to nd a root of a given equation f (x) = 0 by the method of bisection.
Specications
a) It should accept as input two starting points whose function values have opposite signs and a stopping
to
Assignment 2
CSCI 3656
due January 27, 2011
1. Find a more accurate way to calculate each of the following quantities.
a) (x + a)3 a3 ; x close to 0, a near 1.
b) (x y ) 1/(x + 2y ) when x > y > 0
1/
c) 1 + 6h 1; h near 0.
2. The quantity Q may be compute
Assignment 1
CSCI 3656
due January 20, 2011
1. Express the following decimal quantities in binary: 26, 3/8, 91, 39/64.
Now express each as rounded normalized machine numbers with base 2 and 4 bits of precision.
2. Consider a hypothetical computer using ba