1 MB = 1 megabyte = 1024 KB
1 KB = 1 kilobyte = 1024 bytes
1 byte = 8 bits
1 bit holds one binary piece of information (0 or 1)
1 double precision variable requires 64 bits
(b)
Repeat this calculation for the Cray-2 supercomputer, which has 512 megawords
(MW) of memory. Assume that 500 MW is available for our use.
1 MW = 1 megaword = 1024
×
1024 words
1 word = 64 bits
(Notice that, on the Cray, the default variable size is double precision.)
(c)
What is the largest system of equations
Ax
=
b
that we could reasonably solve on
a Mac SE/30 if computer speed is our only limitation?
Assume that we will be using Gauss elimination and that “reasonable” means in
less than one hour.
A rough estimate of the speed of a compiled program on a Mac SE/30 is 0.01
MFLOPS (for double precision).
1 MFLOPS = 1 million floating-point operations per second.
Gauss elimination takes approximately
n
3
/
3 operations.
(d)
Repeat this calculation for the Cray-2 supercomputer, if its nominal speed is
assumed to be 100 MFLOPS.
(e)
Based on your answers to (a) through (d), what is the limiting factor for Gauss
elimination on a Mac? On a Cray-2?
2.67
If I want to solve a 200
×
200 matrix with a bandwidth of 5 and
p
=
q
, what is
the ratio of the time required for full Gauss elimination relative to using a banded Gauss
elimination solver?
2.68
Let’s assume we have a computer that can solve a 1000
×
1000 system in 2
seconds using Gauss elimination. If this is in fact a banded system with
p
=
q
=
2 and I
can solve the same problem in five iterations using Jacobi’s method, estimate how long
will it take. Which method would you prefer to use?
2.69
Let’s assume we have a computer that can solve a 100
×
100 system in 1 second
using Gauss elimination. Estimate the time required to solve the following problems on
the same computer. You must indicate the scaling as well as the time.
(a)
1000
×
1000 system using Gauss elimination
(b)
1000
×
1000 system with a band structure
p
=
q
=
2 using banded Gauss
elimination
(c)
The original 100
×
100 system using an iterative method with 10 iterations.

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128
Linear Algebraic Equations
2.70
What are the criteria for the number of iterations such that Jacobi’s method is
faster than Gauss elimination for solving a linear problem that is not banded?
2.71
Determine the value of
y
such that [3,
y
] is an eigenvector of the matrix
A
=
1
3
4
2
for the eigenvalue
λ
=
5.
2.72
What are the magnitudes of the eigenvalues of
A
=
1
−
1
2
1
2.73
Consider the matrix
⎡
⎢
⎢
⎢
⎢
⎢
⎣
1
−
3
π
2
ln 3
0
2
−
50
ln 2
√
3
0
0
3
−
1
2
0
0
0
1
/
2
0
0
0
0
0
1
/
3
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(a)
What is the determinant of the matrix
A
?
(b)
What is the bandwidth of the matrix
A
?
(c)
What is the 1-norm of the matrix
A
?
(d)
How many eigenvalues do you expect to have for
A
? (If you know the eigenvalues,
you can include them too but this is not required.)
(e)
Is matrix
A
diagonally dominant?