1.
A square matrix [A]nxn
i = 1, 2, …, n
i
= 1, 2, …, n and
for
any i = 1, 2, …, n
i
= 1, 2, …, n and
for
any i = 1, 2, …, n
i
= 1, 2, …, n
2.
Using [x 1 x 2 x 3 ]
= [1 3 5] as the initial guess, the value of [x 1
x 2 x 3 ] after three iterations for
[-2.8333
-1.4333 -1.9727]
[1.4959
-0.90464 -0.84914]
[0.90666
-1.0115 -1.0242]
[1.2148
-0.72060 -0.82451]
3.
To ensure that the following system of equations,
converges using Gauss-Siedal Method, one
can rewrite the above equations as follows:
The
equations cannot be rewritten in a form to ensure convergence.
4.
For and
using as
the initial guess, the values of
are
found at the end of each iteration as
Iteration #
x 1
x 2
x 3
1
0.41666
1.1166
0.96818
2
0.93989
1.0183
1.0007
3
0.98908
1.0020
0.99930
4
0.99898
1.0003
1.0000
At
what first iteration number would you trust at least 1 significant digit
in your solution?
1
2
3
4
5.
The algorithm for the
Gauss-Seidal Method to solve [A] [X] = [C] is given as follows for using
nmax iterations. The initial value of [X] is stored in [X].
Sub
Seidal(n, a, x, rhs, nmax)
For k = 1 To nmax
For i = 1 To n
For j = 1 To n
If (i <> j) Then
Sum = Sum + a(i, j) *
x(j)
endif
Next j
x(i) = (rhs(i) - Sum) /
a(i, i)
Next i
Next k
End Sub
Sub
Seidal(n, a, x, rhs, nmax)
For k = 1 To nmax
For i = 1 To n
Sum = 0
For j = 1 To n
If (i <> j) Then
Sum = Sum + a(i, j) * x(j)
endif
Next j
x(i) = (rhs(i) - Sum) / a(i, i)
Next i
Next k
End Sub
Sub Seidal(n, a, x, rhs, nmax)
For k = 1 To nmax
For i = 1 To n
Sum = 0
For j = 1 To n
Sum = Sum + a(i, j) * x(j)
Next j
x(i) = (rhs(i) - Sum) / a(i, i)
Next i
Next k
End Sub
Sub Seidal(n, a, x, rhs, nmax)
For k = 1 To nmax
For i = 1 To n
Sum = 0
For j = 1 To n
If (i <> j) Then
Sum = Sum + a(i, j) * x(j)
endif
Next j
x(i) = rhs(i) / a(i, i)
Next i
Next k
End Sub
6. Thermistors measure temperature, have a
nonlinear output and are valued for a limited range. So when a thermistor is manufactured, the manufacturer supplies a resistance vs.
temperature curve. An accurate representation of the curve is generally
given by
where T is temperature in Kelvin, R is
resistance in ohms, and are
constants of the calibration curve.
Given the following for a thermistor
R
T
ohm
1101.0
911.3
636.0
451.1
25.113
40.120
50.128
the value of temperature in
for
a measured resistance of 900 ohms most nearly is
30.002
30.472
31.272
31.445
Multiple choice questions on other topics
