Understanding Amendments 11-27 Mrs. Vajgrt’s 9 th Grade Social Studies Class.
Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and...
Transcript of Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and...
![Page 1: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/1.jpg)
Understanding Mrs. Divitz
Metaphor and Computational Mathematics
Charlie Van Loan
Department of Computer Science
Cornell University
Householder XVIII: Tahoe City
![Page 2: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/2.jpg)
What Recent Math Test Scores Show...
4th Grade 8th Grade 12th Grade
1. Singapore 1. Singapore 1. Netherlands2. Korea 2. Korea 2. Sweden3. Japan 3. Japan 3. Denmark4. Hong Kong 4. Hong Kong 4. Switzerland5. Netherlands 5. Belgium 5. Iceland6. Czech Republic 6. Czech Republic 6. Norway
... ... ...12. United States 28. United States 19. United States
A theory to cover the facts: Word Problems
![Page 3: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/3.jpg)
Romy and Michele’s High School Reunion
Romy
(Mira Sorvino)
Michele
(Lisa Kudrow)
![Page 4: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/4.jpg)
THE Quote
Hey Romy, remember Mrs. Divitz’s class?
There was like always a word problem.
![Page 5: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/5.jpg)
THE Quote
Like there’s a guy in a rowboat going X miles,
and the current is going like, you know,
some other miles, and how long does it take
him to get to town?
![Page 6: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/6.jpg)
THE Quote
It’s like, who cares? Who wants to go to town
with a guy who drives a rowboat?
![Page 7: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/7.jpg)
Understanding Mrs. Divitz
How important is it to teach computation by usingexciting applications?
Can a rowboat problem ever be interesting?
Was Mrs. Divitz using the rowboat as a metaphor?
![Page 8: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/8.jpg)
Rowboat Problems Around the World
Different Strokes for Different Folks...
![Page 9: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/9.jpg)
The Scandinavian Approach
Who wants to plunder the north coast of Europewith somebody to drives a rowboat?
![Page 10: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/10.jpg)
The British Approach
Who wants to read Maths at Oxford or Cambridgewith somebody who crews on a rowboat?
![Page 11: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/11.jpg)
The East Coast Approach
Who wants to join a revolution with somebody whocrosses the Delaware in a rowboat?
![Page 12: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/12.jpg)
The West Coast Approach
Who heads to Berkeley in the ’60’s with somebodywho drives a Volkswagen with a rowboat “option”?
![Page 13: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/13.jpg)
The Householder Approach
Who sets out for Tahoe City with a guy in a Kayak?
![Page 14: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/14.jpg)
Are Rowboat Problems Too Abstract?
It depends...
![Page 15: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/15.jpg)
Matlab Has a Role To Play...
function T = RowBoat(X,Y,Distance)
% X is some miles per hour
% Y is like you know some other miles.
% T is when whats-his-name gets to town.
T = Distance/(X-Y);
Note how easy it is to vectorize this!
![Page 16: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/16.jpg)
Metaphors Also Have a Role to Play
The Worst Lack-of-Rowboat Problem of All Time
![Page 17: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/17.jpg)
Example 1. The Xeno Problem
The Wrong Approach
Prove that if |r| < 1 and
Sn = r + r2 + r3 + · · · + rn,
then
limn→∞Sn =
r
1 − r.
![Page 18: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/18.jpg)
Example 1. The Xeno Problem
The Divitz “Metaphor” Approach
Bob is one meter away from the head of the line at McDonalds.Every minute his distance to the counter is halved. When doesBob get his Big Mac? Show work.
Solution.
Bob never gets his Big Mac...
1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, ...
Van Loan, C.F. (2011). The Xeno Diet, SIAM Publications, Philadelphia.
![Page 19: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/19.jpg)
Example 2. The Missing Data Problem
The Wrong Approach
What is 3 × 4? What is 3 × 5? What is 4 × 4? What is 4 × 5?
![Page 20: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/20.jpg)
Example 2. The Missing Data Problem
The Divitz “Metaphor” Approach
Complete the following matrix so that it has minimum rank:
![Page 21: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/21.jpg)
Example 3. The Sudoku Problem
The Wrong Approach
For the following matrix, verify by hand that every row, column,and block is made up of the integers 1 through 9.
2 7 3 9 5 4 1 6 89 6 4 1 2 8 3 7 55 1 8 3 7 6 2 9 4
7 2 5 4 8 1 9 3 68 3 1 5 6 9 7 4 24 9 6 2 3 7 5 8 1
1 4 7 8 9 5 6 2 33 5 9 6 4 2 8 1 76 8 2 7 1 3 4 5 9
![Page 22: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/22.jpg)
Example 3. The Sudoku Problem
The Divitz “Metaphor” Approach
UT
2 7 3 9 5 4 1 6 89 6 4 1 2 8 3 7 55 1 8 3 7 6 2 9 4
7 2 5 4 8 1 9 3 68 3 1 5 6 9 7 4 24 9 6 2 3 7 5 8 1
1 4 7 8 9 5 6 2 33 5 9 6 4 2 8 1 76 8 2 7 1 3 4 5 9
V =
![Page 23: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/23.jpg)
On to the Feature Film...
Slides have
Crude Matrix/Vector Humor
![Page 24: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/24.jpg)
A Freshman Programming Problem
Given a polygon, generate a new polygon
by connecting the midpoints of its sides.
Repeat the process many times and show
what happens graphically.
Turns out to be an excellent exampleof the Divitz “Metaphor Approach”.
![Page 25: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/25.jpg)
A Polygon..
A
B
C
D
E
![Page 26: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/26.jpg)
The Midpoints...
A
B
C
D
E
a
b
c
d
e
![Page 27: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/27.jpg)
The New Polygon...
A
B
C
D
E
a
b
c
d
e
![Page 28: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/28.jpg)
The Average of Two Polygons
x = [ 0.0 5.0 3.0 -2.0 5.0 ];
xShift = [ 5.0 3.0 -2.0 5.0 0.0 ];
xNew = [ 2.5 4.0 0.5 1.5 2.5 ];
y = [ 1.0 4.0 0.0 5.0 6.0 ];
yShift = [ 4.0 0.0 5.0 6.0 1.0 ];
yNew = [ 2.5 2.0 2.5 5.5 3.5 ];
PNew =(P + PShift
)/2
![Page 29: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/29.jpg)
Solution 1
−1.5 −1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
2
A Random Polygon (n = 20 )
![Page 30: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/30.jpg)
Solution 1
−1.5 −1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
2
n = 20 Averagings = 8
![Page 31: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/31.jpg)
Solution 1
−1.5 −1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
2
n = 20 Averagings = 50
![Page 32: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/32.jpg)
Solution 1
−1.5 −1 −0.5 0 0.5 1 1.5 2−1.5
−1
−0.5
0
0.5
1
1.5
2
n = 20 Averagings = 213
![Page 33: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/33.jpg)
A Freshman Programming Problem(Revised)
Given a polygon, generate a new polygon
by connecting the midpoints of its sides.
Repeat the process many times and show
what happens graphically. Use autoscaling.
![Page 34: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/34.jpg)
Solution 2
−2 −1.5 −1 −0.5 0 0.5 1 1.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
A Random Polygon (n = 20 )
![Page 35: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/35.jpg)
Solution 2
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
n = 20 Averagings = 65
![Page 36: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/36.jpg)
Solution 2
−0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
n = 20 Averagings = 65
![Page 37: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/37.jpg)
Solution 2
−0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02−0.2
−0.15
−0.1
−0.05
0
0.05
n = 20 Averagings = 130
![Page 38: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/38.jpg)
The Scandinavian Analysis
”...a strange feeling like we have just beengoing in circles.”
![Page 39: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/39.jpg)
A Senior-Level ScientificComputing Problem
Given a polygon with centroid (0,0), generate a
a new polygon by connecting the midpoints of its
sides. The x and y vertex vectors should always
have unit 2-norm.
Repeat the process many times, show what
happens graphically, and EXPLAIN WHAT
YOU SEE.
![Page 40: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/40.jpg)
The Iteration
% Random vectors with mean zero...
x = rand(n,1); x = x - mean(x); x = x/norm(x);
y = rand(n,1); y = y - mean(y); y = y/norm(y);
for i=1:nRepeat
% Connect midpoints, scale, and plot...
x = (x + [x(2:n);x(1)])/2; x = x/norm(x);
y = (y + [y(2:n);y(1)])/2; y = y/norm(y);
plot(x,y)
end
![Page 41: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/41.jpg)
Solution 3
A Random Polygon (n = 30 )
![Page 42: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/42.jpg)
Solution 3
n = 30 Averagings = 20
![Page 43: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/43.jpg)
Solution 3
n = 30 Averagings = 100
![Page 44: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/44.jpg)
Solution 3
n = 30 Averagings = 225
![Page 45: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/45.jpg)
Matrix Description
Midpoint generation is a matrix-vector product computation...
x1x2x3x4x5
=
1
2
x1 + x2x2 + x3x3 + x4x4 + x5x5 + x1
=
1
2
1 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 11 0 0 0 1
x1x2x3x4x5
⇑M5⇓
y1y2y3y4y5
=
1
2
y1 + y2y2 + y3y3 + y4y4 + y5y5 + y1
=
1
2
1 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 11 0 0 0 1
y1y2y3y4y5
![Page 46: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/46.jpg)
The Iteration
% Random vectors with mean zero...
x = rand(n,1); x = x - mean(x); x = x/norm(x);
y = rand(n,1); y = y - mean(y); y = y/norm(y);
for i=1:nRepeat
% Connect midpoints, scale, and plot...
x = M(n)*x; x = x/norm(x);
y = M(n)*y; y = y/norm(y);
plot(x,y)
end
Side-by-side instances of the power method.
![Page 47: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/47.jpg)
The Eigenvalues of M5
−0.2 0 0.2 0.4 0.6 0.8 1 1.2
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
λ1
λ2
λ3
λ4
λ5
![Page 48: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/48.jpg)
The Eigenspaces of M5
1
1
1
1
1
,
cos(0)
cos(2π/5)
cos(4π/5)
cos(6π/5)
cos(8π/5)
,
sin(0)
sin(2π/5)
sin(4π/5)
sin(6π/5)
sin(8π/5)
,
cos(0)
cos(4π/5)
cos(8π/5)
cos(12π/5)
cos(16π/5)
,
sin(0)
sin(4π/5)
sin(8π/5)
sin(12π/5)
sin(16π/5)
λ1 = 1 |λ2| = |λ2| = cos2(π
5
)|λ4| = |λ5| = cos2
(2π
5
)
![Page 49: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/49.jpg)
The Damping Factor
|λ4||λ2|
=
cos
(2πn
)
cos
(πn
) = 1 − 3
2
( π
n
)2+ O
(1
n4
)
After k averagings:
dist( Vertices , Limiting Ellipse ) ≈(|λ4||λ2|
)k
![Page 50: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/50.jpg)
At the Start...
x(0) = α1
cos(
0π5
)
cos(
2π5
)
cos(
4π5
)
cos(
6π5
)
cos(
8π5
)
+ α2
sin(
0π5
)
sin(
2π5
)
sin(
4π5
)
sin(
6π5
)
sin(
8π5
)
+ α3
cos(
0π5
)
cos(
4π5
)
cos(
8π5
)
cos(
12π5
)
cos(
16π5
)
+ α4
sin(
0π5
)
sin(
4π5
)
sin(
8π5
)
sin(
12π5
)
sin(
16π5
)
y(0) = β1
ditto
+β2
ditto
+β3
ditto
+β4
ditto
![Page 51: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/51.jpg)
In the Limit...
After k averagings, vertex vectors x(k) and y(k) are unit vectorsin the direction of
Mk5 x(0) ≈ α1M
k5
cos(
0π5
)
cos(
2π5
)
cos(
4π5
)
cos(
6π5
)
cos(
8π5
)
+ α2Mk5
sin(
0π5
)
sin(
2π5
)
sin(
4π5
)
sin(
6π5
)
sin(
8π5
)
Mk5 y(0) ≈ β1M
k5
[ditto
]+ β2M
k5
[ditto
]
![Page 52: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/52.jpg)
What Those Unit Vectors Are.....
x(k) ≈ cos(θx)
cos(
(0+k)π
5
)
cos(
(2+k)π
5
)
cos(
(4+k)π
5
)
cos(
(6+k)π
5
)
cos(
(8+k)π
5
)
+ sin(θx)
sin(
(0+k)π
5
)
sin(
(2+k)π
5
)
sin(
(4+k)π
5
)
sin(
(6+k)π
5
)
sin(
(8+k)π
5
)
y(k) ≈ cos(θy)
ditto
+ sin(θy)
ditto
![Page 53: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/53.jpg)
The Four Magic Numbers.....
cos(θx) =cTx(0)
√(cTx(0))2 + (sTx(0))2
cT =[
cos(
0π5
)cos
(2π5
)cos
(4π5
)cos
(6π5
)cos
(8π5
) ]
sT =[
sin(
0π5
)sin
(2π5
)sin
(4π5
)sin
(6π5
)sin
(8π5
) ]
The recipes for sin(θx), cos(θy), and sin(θy) are similar.
![Page 54: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/54.jpg)
Where the Vertices Sit.....
x(t) ≈ cos(θx) cos(t) + sin(θx) sin(t)
y(t) ≈ cos(θy) cos(t) + sin(θy) sin(t)
This describes an ellipse!
![Page 55: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/55.jpg)
What Are Its Semiaxes and Tilt?
[x(t)
y(t)
]
≈[
cos(θx) sin(θx)
cos(θy) sin(θy)
] [cos(t)
sin(t)
]
= U
[σ1 0
0 σ2
]V T
[cos(t)
sin(t)
]
The SVD Tells All...
![Page 56: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/56.jpg)
The Limiting Ellipse
[x(t)
y(t)
]
≈[
cos(45) − sin(45)
cos(45) sin(45)
] [σ1 0
0 σ2
][cos(t)
sin(t)
]
σ1 =2√n· cos
(θy − θx
2
)σ2 =
2√n· sin
(θy − θx
2
)
![Page 57: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/57.jpg)
An Initial “Random” Polygon
![Page 58: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/58.jpg)
After 32 Averagings...
![Page 59: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/59.jpg)
After 80 Averagings...
![Page 60: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/60.jpg)
After 140 Averagings...
![Page 61: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/61.jpg)
After 278 Averagings...
![Page 62: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/62.jpg)
An Undergraduate Research Problem
Explain why the limiting ellipse has a 45-degree tilt? What arethe lengths of its semiaxes?
A. Elmachtaub and C. Van Loan (2010). “From Random Polygonto Ellipse,” SIAM Review, 52, 151-170.
![Page 63: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/63.jpg)
When the Initial Vertex VectorsAre Deficient...
x orthoginal to D2n = 50 Averagings = 2328
![Page 64: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/64.jpg)
More Interesting Behavior...
1. What happens if we scale by other norms? E.g.,
x = x/‖ x ‖p
2. What happens if we use alternative “midpoints”? E.g.,
xi = λ · xi + (1 − λ) · xi+1
![Page 65: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/65.jpg)
The Divitz Metaphor Analysis...
Who would want to go out
with somebody who averages polygons?
![Page 66: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/66.jpg)
The Problem is a Metaphor forComputational Science and Engineering
First:
We experimented with a simple iteration
and observed that it transforms something that
is chaotic and rough into something that is
organized and smooth.
![Page 67: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/67.jpg)
The Problem is a Metaphor forComputational Science and Engineering
Second:
As a step towards explaining the limiting behavior
of the polygon sequence, we described the averaging
process using matrix-vector notation.
![Page 68: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/68.jpg)
The Problem is a Metaphor forComputational Science and Engineering
Third:
This led to an eigenanalysis, the identification of a
crucial invariant subspace, and a vertex-vector
convergence analysis.
![Page 69: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/69.jpg)
The Problem is a Metaphor forComputational Science and Engineering
Fourth:
We then used the singular value decomposition to
connect our algebraic manipulations to a simple
underlying geometry.
![Page 70: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/70.jpg)
High-Level Summary
A rowboat is a gondola...
![Page 71: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/71.jpg)
Voyage of Life: Childhood
Lucky to be here.
![Page 72: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/72.jpg)
Voyage of Life: Youth
The view beats the boat. Any boat.
![Page 73: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/73.jpg)
Voyage of Life: Middle Age
Research w/o Rudder? Who cares because...
![Page 74: Understanding Mrs. Divitz - Cornell University · Understanding Mrs. Divitz Metaphor and Computational Mathematics Charlie Van Loan Department of Computer Science Cornell University](https://reader033.fdocuments.in/reader033/viewer/2022060422/5f18b17f738f0e2a32702179/html5/thumbnails/74.jpg)
Voyage of Life: Youth
You can always start over.