2.29: Numerical Fluid Mechanics
Solution of Quiz 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
CAMBRIDGE, MASSACHUSETTS 02139
2.29 NUMERICAL FLUID MECHANICS— SPRING 2007
Solution of Quiz 1
Time 1 hour and 15 min, Totally 25 points
Thursday 11 a.m. 03/22/07, Focused on Lecture 1 to 11
Problem 1 (6 points):
State which of the following statements are true and which are false. You do not have to
justify your answer.
1.
The number of significant digits achievable by a specific floating point representation
2.
If
, then the relative error of f is more sensitive to relative error of x,
compared to relative error of y.
3.
Bisection method is capable of predicting the maximum number of iterations required
for a specific error level ahead in time.
4.
If the NewtonRaphson’s method converges for a root finding problem, then the
absolute error in each step will be less than the square of absolute error in previous
step.
5.
The Jacobi iterative method for a linear problem will always converge for a positive
definite matrix.
6.
The numerical stability of Gaussian elimination is guaranteed provided that we do full
pivoting and equilibration.
Solution:
1.
True
. Indeed if the length of mantissa is “t”, then it can distinguish up to
log
10
2
t
=
t
log
10
2
2
t
states and
the number of significant digits will be
in decimal base.
2.
False
. The relative error in multiplication is dependent on absolute value of power.
The sign of power only determines the sign of error.
is not dependent on exponent length.
f
=
ax
2
y
!
2
f
=
ax
2
y
!
2
log
f
=
log
a
+
2log
x
!
2log
y
df
f
=
2
dx
x
!
2
dy
y
"
f
=
2
x
!
2
y
,
: Relative Error
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2.29: Numerical Fluid Mechanics
Solution of Quiz 1
3.
True
. If the first guesses are
x
l
,
x
u
, then the maximum error at the beginning is
x
l
!
x
u
. Also in each step, the maximum error is divided by 2. Consequently for a
specific level of absolute error (e), we can find the maximum number of iterations by
x
l
!
x
u
2
n
"
e
.
4.
False
e
n
+
1
!
a
(
x
r
)
e
n
2
. Quadratic convergence means that the error decays such that the relation
holds for a constant “
a
=
a
(
x
r
)
”. In general “a” is not equal to “1” and
the statement will be false.
5.
False
. However, the GaussSeidel iterative method for a linear problem will always
converge for a positive definite matrix.
6.
False
. Refer to notes and recall that pivoting and equilibration will improve the
solution accuracy. However if we do not have enough significant digits, compared to
matrix condition number the solution will be unstable and inaccurate.
Problem 2 (3 points):
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '07
 HenrikSchmidt
 Mechanical Engineering, Numerical Analysis, Numerical Fluid Mechanics

Click to edit the document details