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Q1.
A square matrix [A]nxn
is diagonally dominant if
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
Q2.
Using [x1 x2 x3]
= [1 3 5] as the initial guess, the value of [x1
x2 x3] after three iterations
of Gauss-Seidal method is
[-2.8333
-1.4333 -1.9727]
[1.4959
-0.90464 -0.84914]
[0.90666
-1.0115 -1.0242]
[1.2148
-0.72060 -0.82451]
Q3.
To ensure that the following system of equations,
converges using the Gauss-Siedal method, one
can rewrite the above equations as follows:
The
equations cannot be rewritten in a form to ensure convergence.
Q4.
For
and
using  as
the initial guess, the values of  are
found at the end of each iteration as
|
Iteration # |
x1 |
x2 |
x3 |
|
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
Q5.
The algorithm for the
Gauss-Seidal method to solve [A] [X] = [C] is given as follows when 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
Q6. 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
|