Pick the most appropriate answer.

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

 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

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