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 MULTIPLE CHOICE TEST GAUSS-SEIDEL METHOD SIMULTANEOUS LINEAR EQUATIONS

Q1. A square matrix [A]nxn is diagonally dominant if

i = 1, 2, …, n

= 1, 2, …, n  and  for any i  = 1, 2, …, n

i = 1, 2, …, 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 30.131 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

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