Maple Project: Temperature Distributions NameMaple Project: Temperature Distributions NameMaple Project: Temperature Distributions Name
Purpose Given the temperature on a region’s boundary, determine the steady-
state temperature inside.
Prerequisites Section 2.5
Maple commands used IdentityMatrix and MatrixInverse from the LinearAlgebra pack-
age.
1. Suppose a rectangular beam is exposed to boiling water on its top surface and ice water on the
other three surfaces, as shown below.
0 C 0 C
0 C
100 C
0 C 0 C
100 C
0 C
P1 P2 P3
t1=1
4(100 + t2+ 0 + 0)
t2=1
4(100 + t3+ 0 + t1)
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(a) Show that these equations can be written in matrix notation as 4t=b+Ctwhere the
vector of unknown temperatures is t=
t1
t2
t3
, C =
010
101
010
, and b=
100
100
100
.
(b) If the matrix A = 4I−C is invertible the steady-state temperatures are given by t= A−1b.
Without computing the inverse of A, show that A is invertible.
(c) Assuming the matrix C and vector bhave previously been defined in your Maple session,
the following Maple commands can be used to compute the matrices A and A−1and
Record your results in the table below.
A A−1t
(d) Do these temperatures seem reasonable? Why? or Why not?
2. Consider the cross-section and grid shown below. Use the method described in Question 1 to
determine the steady-state temperatures at the grid points.
80 C
10 C
P1 P2 P3 P4
(a) Determine the temperature at the 12 interior points in this problem. Record your results
in the table below.
A A−1t
(b) Do these temperatures seem reasonable? Why? or Why not?
3. Consider the cross-section and grid shown below. Use the method described in Question 1 to
determine the steady-state temperatures at the grid points.
100 C
-10 C
-10 C
P1 P2 P3
(a) Record your results in the table below.
A A−1t
(b) Do these temperatures seem reasonable? Why? or Why not?
Maple Project: Manipulating Matrices in Maple NameMaple Project: Manipulating Matrices in Maple NameMaple Project: Manipulating Matrices in Maple Name
Purpose To gain experience making and justifying conjectures about a family
of structured matrices.
Prerequisites Sections 2.2 and 3.2
1. Find an efficient and painless way to define these matrices — even for large values of n. Record
your way of entering the matrix Mnin the box below.
Hints:
•There are numerous ways to approach this problem.
•One of the most straightforward is to use assignments of the form M[i,j] := max(i,j);
2. Find a formula for the determinant of Mnfor all integers n≥1. Justify your conjecture with
at least 5 examples.
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3. Find the inverse matrices M−1
1, M−1
2, M−1
3, and M−1
4.
Maple Project: Markov Chains and Long-Range Predictions NameMaple Project: Markov Chains and Long-Range Predictions NameMaple Project: Markov Chains and Long-Range Predictions Name
Purpose To use linear algebra to analyze Markov chains and investigate their
steady state vectors.
Read Exercises 2 and 12 in Section 4.9. These problems consider the Markov chain with the
system matrix P =
0.50 0.25 0.25
0.25 0.50 0.25
. Here there are three foods and the (i, j) entry of P is the
1. Load the system matrix with the command markdat( ); (assuming the laylinalg package is
loaded).
(a) Compute the matrix P2. Record the result here.
(b) Suppose an animal chooses food 1 on the first trial. Use P and P2to determine the
probability the animal will:
choose food 2 on the second trial
choose food 2 on the third trial
choose food 3 on the second trial
(c) Use Maple to find the reduced echelon form of P −I. Recall that the 3 ×3 identity matrix
can be defined using Id := IdentityMatrix( 3 );. Then, ref( P – Id ); returns the
P−I P −I (ref) x w
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(d) The steady-state vector, q, for P can be found by dividing the particular solution, w, by
the sum of its components. The Maple commands to compute this vector are:
w := < your data from (c) >; # a particular solution
Is qa (particular) solution to (P −I)x=0? Is qa probability vector? Explain.
2. Read Exercises 4 and 14 in Section 4.9 and solve it as follows.
(b) Compute and record (in the table below) the probability vector for tomorrow’s weather:
Wv0. What is the chance of bad weather tomorrow?
Maple Project page 2 of 4 Markov Chains and Long-Range Predictions
Maple Project page 2 of 4 Markov Chains and Long-Range Predictions
Maple Project page 2 of 4 Markov Chains and Long-Range Predictions
(d) Use the method in Question 1(c) to calculate the steady-state vector, q. In the long run,
what is the probability the weather will be good on a given day?
3. According to Theorem 18, when P is stochastic and regular, and vis any probability vector,
the sequence of vectors v, Pv, P2v, . . . converge, and the limit vector will be the steady-state
vector of P. In other words, when the power kis big enough, Pkvwill look like the unique
steady-state vector. This is not an efficient way to calculate the steady-state vector, but it
is interesting to see the sequence v, Pv, P2v, . . . converge for a few examples. The following
Maple commands can be used to compute the first 10 terms in the sequence. If more terms are
needed, change 10 to a larger number and re-execute.
Estimate kfor both Exercise 2, using matrix P, and Exercise 4, using matrix W. Use each of
the initial vectors shown below and at least one more probability vector vof your own. For each
v, calculate Pkvuntil you find a big enough kso that Pkvlooks like the steady-state vector
for P (compare to the steady-state vectors you found in Questions 1(d) and 2(d)). Repeat this
for each vand W, and record the smallest value of kwhich is big enough in each case. Record
your results, including the initial vector that you chose, in the table below.
Animal Experiment (using matrix P) Weather Forecast (using matrix W)
v0
1
0
0.2
0.6
0.35
0.35
0
1
0.40
0.20
0.33
0.34
4. The matrices P1=
0.7 0.2 0.6
0 0.2 0.4
, P2=
0.7 0.2 0.6
0 0.2 0
, and P3=
010
100
can be
(a) Which of P1, P2, and P3are regular? Explain your answers.
P1
P2
P3
P1P2P3
What steady-
state vector, q
is produced by
the method in
Question 1?
Is qunique?
(Briefly, why?)
If v=
1
0
0
,
does Pkvconverge
as kincreases? If
not, what does
happen?
Maple Project: Real and Complex Eigenvalues NameMaple Project: Real and Complex Eigenvalues NameMaple Project: Real and Complex Eigenvalues Name
Purpose To learn how to use Maple’s eigenvectors command to construct
the matrix of eigenvalues and the matrix of eigenvectors.
Prerequisites Sections 5.1 and 5.3
The following five matrices will be used in this project:
005
054
1. Calculate the eigenvalues of each of the four 2 ×2 matrices by hand. Record your eigenvalues
in the table at the end of Question 2.
2. Repeat Question 1 on four new 2 ×2 matrices created as follows: if the original matrix has real
eigenvalues, change the sign of one entry so that the resulting matrix has complex eigenvalues;
if the original matrix has complex eigenvalues, change one entry in such a way that the resulting