Lesson32 Second Order Difference Equations Slides
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Transcript of Lesson32 Second Order Difference Equations Slides
Lesson 32Second-order Difference Equations
Math 20
May 2, 2007
AnnouncementsI PS 12 due Wednesday, May 2I MT III Friday, May 4 in SC Hall AI Final Exam: Friday, May 25 at 9:15am, Boylston 110 (Fong
Auditorium)I Review Session: Tuesday, May 22I Please do evaluations
Recap
Example: Fibonacci
Another example
Acceleration model of GDP growth
FactThe solution to the homogeneous system of linear differenceequations y(k + 1) = Ay(k) is
y(k) = Aky(0)
FactLet A have a complete system of eigenvalues and eigenvectorsλ1,λ2, . . . ,λn and v1,v2, . . . ,vn. Then the solution to thedifference equation y(k + 1) = Ay(k) is
y(k) = Aky(0) = c1λk1 v1 + c2λ
k2 v2 + · · ·+ cnλ
kn vn
where c1,c2, . . . ,cn are chosen to make
y(0) = c1v1 + c2v2 + · · ·+ cnvn
Applet fun
http://neuron.eng.wayne.edu/bpDynamics/TimeDynSys.html
The big idea
GoalSolve second-order difference equations such as
f (k) = f (k −1) + f (k −2)
MethodTreat a second-order equation as a system of two first-orderequations.
The big idea
GoalSolve second-order difference equations such as
f (k) = f (k −1) + f (k −2)
MethodTreat a second-order equation as a system of two first-orderequations.
Recap
Example: Fibonacci
Another example
Acceleration model of GDP growth
Setting up the Fibonacci sequence
Recall the Fibonacci sequence defined by
f (k + 2) = f (k) + f (k + 1), f (0) = 1, f (0) = 1
Let’s let g(k) = f (k + 1). Then
g(k + 1) = f (k + 2) = f (k) + f (k + 1) = f (k) + g(k).
So if y(k) =
[f (k)g(k)
], we have
y(k + 1) =
[f (k + 1)g(k + 1)
]=
[g(k)
f (k) + g(k)
]=
[1 01 1
]y(k)
Diagram of rabbits
f (0) = 1
f (1) = 1
f (2) = 2
f (3) = 3
f (4) = 5
f (5) = 8
Diagram of rabbits
f (0) = 1
f (1) = 1
f (2) = 2
f (3) = 3
f (4) = 5
f (5) = 8
Diagram of rabbits
f (0) = 1
f (1) = 1
f (2) = 2
f (3) = 3
f (4) = 5
f (5) = 8
Diagram of rabbits
f (0) = 1
f (1) = 1
f (2) = 2
f (3) = 3
f (4) = 5
f (5) = 8
Diagram of rabbits
f (0) = 1
f (1) = 1
f (2) = 2
f (3) = 3
f (4) = 5
f (5) = 8
Diagram of rabbits
f (0) = 1
f (1) = 1
f (2) = 2
f (3) = 3
f (4) = 5
f (5) = 8
Setting up the Fibonacci sequence
Recall the Fibonacci sequence defined by
f (k + 2) = f (k) + f (k + 1), f (0) = 1, f (0) = 1
Let’s let g(k) = f (k + 1). Then
g(k + 1) = f (k + 2) = f (k) + f (k + 1) = f (k) + g(k).
So if y(k) =
[f (k)g(k)
], we have
y(k + 1) =
[f (k + 1)g(k + 1)
]=
[g(k)
f (k) + g(k)
]=
[1 01 1
]y(k)
Setting up the Fibonacci sequence
Recall the Fibonacci sequence defined by
f (k + 2) = f (k) + f (k + 1), f (0) = 1, f (0) = 1
Let’s let g(k) = f (k + 1). Then
g(k + 1) = f (k + 2) = f (k) + f (k + 1) = f (k) + g(k).
So if y(k) =
[f (k)g(k)
], we have
y(k + 1) =
[f (k + 1)g(k + 1)
]=
[g(k)
f (k) + g(k)
]=
[1 01 1
]y(k)
Diagonalize
The eigenvalues of A =
[1 01 1
]are found by solving
0 =
∣∣∣∣−λ 11 1−λ
∣∣∣∣= (−λ )(1−λ )−1
= λ2−λ −1
The roots are
ϕ =1 +√
52
ϕ̄ =1−√
52
Notice thatϕ + ϕ̄ = 1, ϕϕ̄ =−1
(These facts make later calculations simpler.)
Diagonalize
The eigenvalues of A =
[1 01 1
]are found by solving
0 =
∣∣∣∣−λ 11 1−λ
∣∣∣∣= (−λ )(1−λ )−1
= λ2−λ −1
The roots are
ϕ =1 +√
52
ϕ̄ =1−√
52
Notice thatϕ + ϕ̄ = 1, ϕϕ̄ =−1
(These facts make later calculations simpler.)
Diagonalize
The eigenvalues of A =
[1 01 1
]are found by solving
0 =
∣∣∣∣−λ 11 1−λ
∣∣∣∣= (−λ )(1−λ )−1
= λ2−λ −1
The roots are
ϕ =1 +√
52
ϕ̄ =1−√
52
Notice thatϕ + ϕ̄ = 1, ϕϕ̄ =−1
(These facts make later calculations simpler.)
Eigenvectors
We row reduce to find the eigenvectors:
A−ϕI =
[−ϕ 11 1−ϕ
]=
[−ϕ 11 ϕ̄
]←−−ϕ̄
+
[−ϕ 10 0
]
So[
1ϕ
]is an eigenvector for A corresponding to the eigenvalue
ϕ.
Similarly,[
1ϕ̄
]is an eigenvector for A corresponding to the
eigenvalue ϕ̄. So now we know that
y(k) = c1ϕk[
1ϕ
]+ c2ϕ̄
k[
1ϕ̄
]
Eigenvectors
We row reduce to find the eigenvectors:
A−ϕI =
[−ϕ 11 1−ϕ
]=
[−ϕ 11 ϕ̄
]←−−ϕ̄
+
[−ϕ 10 0
]
So[
1ϕ
]is an eigenvector for A corresponding to the eigenvalue
ϕ.
Similarly,[
1ϕ̄
]is an eigenvector for A corresponding to the
eigenvalue ϕ̄.
So now we know that
y(k) = c1ϕk[
1ϕ
]+ c2ϕ̄
k[
1ϕ̄
]
Eigenvectors
We row reduce to find the eigenvectors:
A−ϕI =
[−ϕ 11 1−ϕ
]=
[−ϕ 11 ϕ̄
]←−−ϕ̄
+
[−ϕ 10 0
]
So[
1ϕ
]is an eigenvector for A corresponding to the eigenvalue
ϕ.
Similarly,[
1ϕ̄
]is an eigenvector for A corresponding to the
eigenvalue ϕ̄. So now we know that
y(k) = c1ϕk[
1ϕ
]+ c2ϕ̄
k[
1ϕ̄
]
What are the constants?
To find c1 and c2, we solve[11
]= c1
[1ϕ
]+ c2
[1ϕ̄
]=
[1 1ϕ ϕ̄
][c1c2
]=⇒
[c1c2
]=
[1 1ϕ ϕ̄
]−1[11
]=
1ϕ̄−ϕ
[ϕ̄ −1−ϕ 1
]=
1ϕ̄−ϕ
[ϕ̄−1ϕ + 1
][11
]=
1√5
[ϕ
−ϕ̄
]
Finally
Putting this all together we have
y(k) =ϕ√5
ϕk[
1ϕ
]− ϕ̄√
5ϕ̄
k[
1ϕ̄
][
f (k)g(k)
]=
1√5
[ϕk+1− ϕ̄k+1
ϕk+2− ϕ̄k+2
]
So
f (k) =1√5
(1 +√
52
)k+1
−
(1−√
52
)k+1
Finally
Putting this all together we have
y(k) =ϕ√5
ϕk[
1ϕ
]− ϕ̄√
5ϕ̄
k[
1ϕ̄
][
f (k)g(k)
]=
1√5
[ϕk+1− ϕ̄k+1
ϕk+2− ϕ̄k+2
]So
f (k) =1√5
(1 +√
52
)k+1
−
(1−√
52
)k+1
Recap
Example: Fibonacci
Another example
Acceleration model of GDP growth
QuestionSuppose that we have a large number of blocks of three colors:yellow, red, and gree. Each yellow block fills one space, andeach red or green block fills two spaces. How many differentways are there to arrange the block in a line so that they fill kspaces?
Guessing the first few values
k Arrangements r(k)
0
∅ 1
1 Y 1
2 YY, R, G 33 YYY, RY, YR, GY, YG 5
Guessing the first few values
k Arrangements r(k)
0
∅ 1
1 Y 12 YY, R, G 3
3 YYY, RY, YR, GY, YG 5
Guessing the first few values
k Arrangements r(k)
0
∅ 1
1 Y 12 YY, R, G 33 YYY, RY, YR, GY, YG 5
Guessing the first few values
k Arrangements r(k)
0 ∅ 11 Y 12 YY, R, G 33 YYY, RY, YR, GY, YG 5
Deriving a difference equationThink about a row of k blocks. Let r(k) be the number ofarrangements.
I Suppose we put a yellow block in the last spot. We can fillk spaces by filling the first k −1 spaces anyway we want,then stick on a yellow block. This accounts for r(k −1)ways to fill k spaces.
I Suppose the last two spots are taken up by a red block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a red block. This accounts forr(k −2) ways to fill k spaces.
I Suppose the last two spots are taken up by a green block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a green block. This accounts forr(k −2) ways to fill k spaces.
Thus
r(k) = r(k −1) + 2r(k −2) ⇐⇒ r(k + 2) = r(k + 1) + 2r(k)
Deriving a difference equationThink about a row of k blocks. Let r(k) be the number ofarrangements.I Suppose we put a yellow block in the last spot. We can fill
k spaces by filling the first k −1 spaces anyway we want,then stick on a yellow block. This accounts for r(k −1)ways to fill k spaces.
I Suppose the last two spots are taken up by a red block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a red block. This accounts forr(k −2) ways to fill k spaces.
I Suppose the last two spots are taken up by a green block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a green block. This accounts forr(k −2) ways to fill k spaces.
Thus
r(k) = r(k −1) + 2r(k −2) ⇐⇒ r(k + 2) = r(k + 1) + 2r(k)
Deriving a difference equationThink about a row of k blocks. Let r(k) be the number ofarrangements.I Suppose we put a yellow block in the last spot. We can fill
k spaces by filling the first k −1 spaces anyway we want,then stick on a yellow block. This accounts for r(k −1)ways to fill k spaces.
I Suppose the last two spots are taken up by a red block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a red block. This accounts forr(k −2) ways to fill k spaces.
I Suppose the last two spots are taken up by a green block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a green block. This accounts forr(k −2) ways to fill k spaces.
Thus
r(k) = r(k −1) + 2r(k −2) ⇐⇒ r(k + 2) = r(k + 1) + 2r(k)
Deriving a difference equationThink about a row of k blocks. Let r(k) be the number ofarrangements.I Suppose we put a yellow block in the last spot. We can fill
k spaces by filling the first k −1 spaces anyway we want,then stick on a yellow block. This accounts for r(k −1)ways to fill k spaces.
I Suppose the last two spots are taken up by a red block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a red block. This accounts forr(k −2) ways to fill k spaces.
I Suppose the last two spots are taken up by a green block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a green block. This accounts forr(k −2) ways to fill k spaces.
Thus
r(k) = r(k −1) + 2r(k −2) ⇐⇒ r(k + 2) = r(k + 1) + 2r(k)
Deriving a difference equationThink about a row of k blocks. Let r(k) be the number ofarrangements.I Suppose we put a yellow block in the last spot. We can fill
k spaces by filling the first k −1 spaces anyway we want,then stick on a yellow block. This accounts for r(k −1)ways to fill k spaces.
I Suppose the last two spots are taken up by a red block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a red block. This accounts forr(k −2) ways to fill k spaces.
I Suppose the last two spots are taken up by a green block.We can fill k spaces by filling the first k −2 spaces anywaywe want, then stick on a green block. This accounts forr(k −2) ways to fill k spaces.
Thus
r(k) = r(k −1) + 2r(k −2) ⇐⇒ r(k + 2) = r(k + 1) + 2r(k)
So let y(k) =
[r(k)
r(k + 1)
]. Then
y(k + 1) =
[r(k + 1)r(k + 2)
]=
[r(k + 1)
r(k + 1) + 2r(k)
]=
[0 12 1
]y(k)
The eigenvalues of[0 12 1
]are 2 and −1 with eigenvectors
[12
]and
[−11
]. We have r(0) = 1 and r(1) = 1, so
[c1c2
]=
[1 −12 1
]−1[11
]=
13
[1 1−2 1
][11
]=
13
[2−1
]
So let y(k) =
[r(k)
r(k + 1)
]. Then
y(k + 1) =
[r(k + 1)r(k + 2)
]=
[r(k + 1)
r(k + 1) + 2r(k)
]=
[0 12 1
]y(k)
The eigenvalues of[0 12 1
]are 2 and −1 with eigenvectors
[12
]and
[−11
]. We have r(0) = 1 and r(1) = 1, so
[c1c2
]=
[1 −12 1
]−1[11
]=
13
[1 1−2 1
][11
]=
13
[2−1
]
Solution
So [r(k)
r(k + 1)
]=
23
(2)k[12
]− 1
3(−1)k
[−11
]
meaning
r(k) =13
(2k+1 + (−1)k+2
)=
13
(2k+1 + (−1)k
)You can check that this does match up with the sequence.
Solution
So [r(k)
r(k + 1)
]=
23
(2)k[12
]− 1
3(−1)k
[−11
]meaning
r(k) =13
(2k+1 + (−1)k+2
)=
13
(2k+1 + (−1)k
)
You can check that this does match up with the sequence.
Solution
So [r(k)
r(k + 1)
]=
23
(2)k[12
]− 1
3(−1)k
[−11
]meaning
r(k) =13
(2k+1 + (−1)k+2
)=
13
(2k+1 + (−1)k
)You can check that this does match up with the sequence.
Recap
Example: Fibonacci
Another example
Acceleration model of GDP growth
Model of national income
Y (k) = C(k) + I(k) + G(k)
whereI Y (k) is national incomeI C(k) is consumer expenditureI I(k) is induced investmentI G(k) is government expenditure
all at time step k .
Assumptions and Equation
I Consumer expenditure is proportional to national incomeat each time step
I Induced investment proportional to the increase in thenational income at the previous time step
I Government expenditure remains constant.
Putting this together gives an inhomogeneous second-orderequation
Y (k) + aY (k −1) + bY (k −2) = c
which we can solve using these same techniques.
Assumptions and Equation
I Consumer expenditure is proportional to national incomeat each time step
I Induced investment proportional to the increase in thenational income at the previous time step
I Government expenditure remains constant.
Putting this together gives an inhomogeneous second-orderequation
Y (k) + aY (k −1) + bY (k −2) = c
which we can solve using these same techniques.