# CS3911 Introduction to Numerical Methods with Fortran Final Exam - PDF

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```					CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007   1

CS3911 Introduction to Numerical Methods with Fortran
Final Exam
Fall 2007
200 points { 9 pages

Name:

• Write to the point. If I don't understand what you are
saying, I believe, in most cases, you don't understand the
subject.
is no justification when it is needed, you will receive ZERO
point for that question even though you have provided a
correct answer. I consider a good and correct justification
more important than a right answer. Thus, if you provide
a very vague answer without a convincing argument to
• For computation-based problems, always show the com-
point unless the problem only requires an answer.
• Your calculations should have at least 5 significant digits.
• Do those problems you know how to do first. Otherwise,
you may not be able to complete this exam on time. None
of the following problems require extensive calculations.
CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007                                       2

1. Accuracy and Reliability:

(a) [15 points] Euler proved the following result centuries ago:
n
1
lim    1+           = e = 2.71828 . . .
n→∞         n
A programmer wrote a program to verify this result using single precision (i.e., about 7 signif-
icant digits) and a Fortran 90 statement E = (1.0 + 1.0/n)**n in a DO loop, where n is an
INTEGER and E is a REAL, and obtained the following table with n = 10, 100, . . ., 10000000
and 100000000.

n         Computed (1 + 1/n)n        Absolute Error
10          2.593743               0.12453866
100          2.7048113              0.013470411
1000          2.7170507              1.2309551E-3
10000          2.718597               3.1518936E-4
100000          2.7219622              3.6804676E-3
1000000          2.5952267              0.12305498
10000000          3.2939677              0.575686
100000000          1.0                    1.7182817

The problem is that the computed value (1 + 1/n)n is about 2.718597 when n = 10000;
however, accuracy of further results gets worse if n > 10000, and eventually the computed
result becomes 1, which is obviously incorrect. This programmer is sure that his/her program
logic is correct. So, what is the numerical problem (or problems) in this computation?
You should not make any correction to the Fortran 90 statement. What you have to
do is to point out the numerical problem (or problems) with a clear and to-the-point
such as \should use double precision" will receive zero point.
CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007                                 3

2. Linear Algebra:

(a) [15 points] Matrix A and its LU-decomposition L and U are shown below:
                                            
1 2                        1 0         1 2
                                            
A=               A=L·U =               ·         
2 7                        2 1         0 3

Find A−1 using the given LU-decomposition. Show all calculation steps clearly. Otherwise,

(b) [15 points] Use Jacobi method to find all eigenvalues and their corresponding eigenvectors
of the following symmetric matrix A. You should clearly provide all computation details,
and clearly match each eigenvalue with its corresponding eigenvector. Otherwise, you
method.                                            
1 0 1
           
A= 0 3 0 
1 0 1
CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007                                       4

3. Interpolation: Suppose there are five data points (x0, y0 ) = (−2, 0), (x1, y1) = (−1, 3), (x2, y2) =
(0, 4), (x3 , y3) = (1, 3) and (x4, y4 ) = (2, 0). Do the following problems:

(a) [10 points] Given a polynomial of degree n as follows,

Pn (x) = a0 + a1 x + a2x2 + a3x3 + · · · + an xn

suggest an efficient way to evaluate Pn (x) so that the complexity is O(n). You should provide
an algorithm and its complexity analysis. A method that does not achieve O(n) receives
zero point.

(b) [15 points] Find the Lagrange interpolating polynomial for the given data points. You should
show all computation steps. Only providing an answer and/or using a wrong method
CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007                                  5

(c) [15 points] Find the Newton interpolating polynomial for the given data points using the
divided difference method. You should show all computation steps. Only providing an

4. Numerical Differentiation:

(a) [15 points] Use the 3-point backward difference method to compute the derivative of f (x) =
ex sin(x) at x = 1 with ∆ = 0.01. You should show all computation steps. Only providing
CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007                                 6

(b) [20 points] Use Richardson's extrapolation method to compute the derivative of f (x) =
ex sin(x) at x = 1 with initial ∆ = 0.1. You should carry out the extrapolation steps un-
til the computation for ∆ = 0.025 completes, and clearly indicate the desired answer. You
should show all computation steps. Only providing an answer and/or using a wrong

5. Numerical Integration: The following integral has an exact solution:
1
ex dx = e1 − e0 = e − 1 = 1.71828 . . .
0

(a) [20 points] Use Romberg's method to compute the above integration and use the trapezoid
method for column 0. You should carry out all steps until the computation for ∆ = 0.25
completes, and clearly indicate the desired answer. You should show all computation steps.
Only providing an answer and/or using a wrong method receives zero point.
CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007                                     7

(b) [20 points] Use the iterative 3-point Simpson method to compute the above integration. You
should start with ∆ = 0.5, update the result in each iteration, and stop after the result of
∆ = 0.125 is obtained. You should show all computation steps. Only providing an an-
swer and/or using a wrong method receives zero point.

6. ODE Initial Value Problems: Solve y = xy with initial value y(0) = 1.

(a) [10 points] Use Euler's method to solve the given equation on [0, 4] with ∆ = 1, and fill your
results in the following table.

Iteration     x                        y

Initial Value   0                        1

1          1

2          2

3          3

4          4
CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007                                      8

(b) [15 points] Solve the given equation on [0, 4] using Ralston's second-order Runge-Kutta method
with ∆ = 1, and fill your results in the following table.

Iteration     x                       y

Initial Value   0                       1

1          1

2          2

3          3

7. Random Numbers:

(a) [15 points] Suppose a random vertical cut followed by a random horizontal cut are applied to
a square of edge length 1. This yields four smaller rectangles as shown below. What is the
probability of the area of the largest rectangle being larger than or equal to 0.7?

The following shows a sequence of pre-generated random numbers. Each number has five
digits, and a decimal point is assumed to appear to the left of the first digit. For example,
random number 53479 should be interpreted as 0.53479. Therefore, all random numbers are
in the range of [0,1). Use Monte Carlo method with 10 runs to find an approximation of this
probability. You should show all computation steps. Only providing an answer and/or
using a wrong method receives zero point.

x1 to x8      53479     81115     98036     12217     59526     40238    40577     39351
x9 to x16     43211     69255     97344     70328     58116     91964    26240     44643
x17 to x24    83287     97391     92823     77578     66023     38277    74523     71118
CS3911 Intro. to Numerical Methods with Fortran Final { Fall 2007   9

1 a         15
a        15
2
b        15
a        10
3 b         15
c        15
a        15
4
b        20
a        20
5
b        20
a        10
6
b        15
7 a         15
Total     200
Some Important and Complex Formulas

The following has some important and complex formulas discussed in class. They are provided for
your convenience. If you have anything in doubt, use the version discussed in class or ask questions.
                             
n               n
ai                (x − xj )
i=0           j=0,j=i

1
(3fi − 4fi−1 + fi−2 )
2∆

                  
n                                    i−1
f [x0 ] +           f [x0 , x1, . . . , xi]           (x − xj )
i=1                               j=0

f0 + fn n−1
∆        +     fi
2     i=1

2(n − i) + 1
xi = cos                  π                     where i = 0, 1, . . ., n
2n + 2

1
(−fi+2 + 4fi+1 − 3fi )
2∆

di,j − di−1,j
di,j +
4j+1 − 1

m             m
∆
(f0 + f2m ) + 4     f2i−1 + 2     f2i
3                 i=1           i=1

n
(x − x0 )(x − x1 ) · · · (x − xi−1 )(x − xi+1 ) · · · (x − xn )
yi
i=0
(xi − x0)(xi − x1 ) · · · (xi − xi−1 )(xi − xi+1 ) · · · (xi − xn )

1
c1 = 0           c2 = 1             p2 = a2,1 =
2

1
c1 = c2 =                       p2 = a2,1 = 1
2

1                   2                           3
c1 =               c2 =               p2 = a2,1 =
3                   3                           4

1

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