As a matter of policy, I keep finals for one regular quarter. You may examine your final at any time in my office. If you wish to collect your final, you may do you in June.
Prob 1: (cf. Review page bullet 3,7; Notes 2-4)
Prob 2: (cf. Review page bullet 10; Notes 3-10, 3-11)
Prob 3: (cf. Review page bullet 24; Notes 4-26A)
Prob 4: (cf. Review page bullet 24)
Prob 5: (cf. Notes 6-27)
Prob 6: (cf. Review page bullet 26; Notes 4-26A)
Prob 7: (cf. Review page bullet 23; Notes 4-17)
I remember mentioning in class that "normal" equations are used in
several ways in Numerical Analysis. This way
is used to convert a non-symmetric system into a symmetric system.
Some people had the basic idea of using AT, but both sides
of the original matrix-vector equation must be PRE-multiplied by the
transpose. Some people POST-multiplied one side and PRE-multiplied the
other side, but matrix multiplication is not commutative!
I distinctly remember mentioning in class about the "normal" equations
for a linear system (not for approximating a curve through points) and
mentioning the possibility of putting a question
about "normal equations" on an exam in the past!
Prob 8: (cf. Review page bullet 47; Notes 6-14)
Some had problems doing the simple evaluation of the function
(i.e., the polynomial within the integral) correctly at 2 or
3 points to get the Gauss Quad values.
When n=3, some people evaluated f(0) to be 0, but it is 1.
Both answers are the same as
the analytic evaluation of the integral, which is 10/3 = 3.3333....
Prob 9: (cf. Review page bullet 46)
This problem specified that we wanted an exact formula
for a polynomial of degree 0.
A polynomial of degree 0 is a constant.
Thus f(x)
can be taken
out of the integral (on the left of the given equation) and
"cancelled" with f(3/2) (which equals the same
constant value) on the right. Then the left side of the
initial equation can be evaluated exactly via the
(integration by parts) formula given.
This was not intended so much as a "trick" question but as
a question meant to see whether the reader took into account
all the given information, which is very important in numerical
problems.
A number of people tried to evaluate ln(3/2) which had nothing to do
with the final value.
Some tried to create a quadrature formula with several points, but the
only formula requested was the one with the single point (i.e., 3/2)
and the single corresponding weight.
Prob 10: (cf. Review page bullet 48; Notes 6-18)
This was basically the same procedure that needed to be done for
the Runge-Kutta programming assignment (MP 6).
One introduces (two) new
variables which eliminate some of the derivatives of y and
then one creates several equations with a derivative on the left
side and an expression with no derivatives on the right.
Since you are given a 3rd order DiffEq (i.e., one with
a 3rd derivative in it), you should end up with 3 first order
DiffEqs (with a derivative on the left and an expression without
derivatives on the right) and three (new) initial conditions
related to new variables and based on the original variables and
initial conditions.
Prob 11: (cf. Review bullet 43; Notes 6-8)
One needs to use the error formula to determinine the width
needed and thus also the number of panels.
A few people seemed to confuse the error formula for the Trapezoidal
rule, which only has a 2nd derivative, with the correct formula for
Simpson's rule.
In the error formula
the argument of f(4)(x) is chosen to maximize
the value over the interval of integration. Thus x
should be 3, since that where the 4th derivative is greatest on
the interval given.
One should use the error formula
for the "composite"
version given in the notes.
Since Simpson's rule demands that n be an even number,
the minimum answer was 32 (the next higher even number greater
than 30.006).
Prob 12: (cf. Review page bullet 49; Notes 6-38A, 6-39)
Some had the correct discretization form for y''
but, unfortunately, made various algebraic errors (such as
forgetting to correct for a minus sign inside parentheses)
resulting in an incorrect final linear system.
A few used the discretized formula for the differential equation
given in the notes as if it were a formula that could be used
for any DiffEq.
Also, the question specifically asked for the linear system in "matrix form."
Prob 13: (cf. Review page bullet 44; Notes 6-11)
From information given in class, for the
Trapezoidal rule, the value in the
Richardson formula for the variable j in the
exponent should be 2 since the increase in panels
form the first value (5) to the second value (10).
Thus the denominator of the fraction is
42-1-1 or 3. All one needed to do was
insert the values given into the correct formula to produce
a new value. Several used the correct formula but associated the
given values incorrectly in the formula. When the number of panels
goes from 5 to 10, the width of the panel is cut in half!
Prob 14: (cf. Review page bullet 41; Notes 6-4)
In each case, one was supposed to compute the
approximation to the same integral over the same interval (0,1)
but using various widths for the panels. When the width h
is 1, then there is only one panel (since the distance between
the two limits of the integral is 1) and only the two endpoints are
used. When the width is 1/2, there are two panels and the two
endpoints and the middle point is used. When the width is 1/4,
all 5 points are used.
Some used all 5 points for each
approximation or varied the interval final point depending on the
panel widths. A few used the x value for one of the points
rather than the y value.
final nfinal
187 78
177 72
177 72
171 69
165 66
165 66
160 63
154 60
154 60
153 60
152 59
152 59
145 56
142 54
141 53
138 52
127 46
122 44
122 44
122 44
120 42
120 42
118 41
118 41
108 36
107 36
105 35
83 23
77 20
49 5
MAXIMUM 200 100
x
x x
x x x
x x x x
x x x x
x x x x
x x x x
x x x x x x x x
40- 60- 80- 100- 120- 140- 160- 180-
59 79 99 119 139 159 179 200
(1) (1) (1) (5) (7) (8) (6) (1)
This page is maintained by Dennis C. Smolarski, S.J.
Email: dsmolarski "at" scu.edu
© Copyright 2018 Dennis C. Smolarski, SJ, All rights reserved.
Last changed: 21 March 2018