Holistic Numerical Methods

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MULTIPLE CHOICE TEST

(All Tests)

GAUSS-SEIDEL METHOD

(More on Gauss-Seidel Method)

SIMULTANEOUS LINEAR EQUATIONS

(More on Simultaneous Linear Equations)

Pick the most appropriate answer.


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

 

 

Complete Solution

 

 

 

Multiple choice questions on other topics


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Copyrights: University of South Florida, 4202 E Fowler Ave, Tampa, FL 33620-5350. All Rights Reserved. Questions, suggestions or comments, contact kaw@eng.usf.edu  This material is based upon work supported by the National Science Foundation under Grant# Creative Commons License0126793, 0341468, 0717624,  0836981, 0836916, 0836805, 1322586.  Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.  Other sponsors include Maple, MathCAD, USF, FAMU and MSOE.  Based on a work at http://mathforcollege.com/nm.  Holistic Numerical Methods licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

 

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