When Computers Don't Compute and Other Fun with Numbers

When Computers Don't Compute
And Other

pde1d.blogspot.com

Have you ever gotten a wrong
answer from a computer ?
Finite precision numbers
● Chaos
● Limited number of numbers
●

–

Insufficient resolution / memory

Software errors
● Branch cuts
● User errors / poor documentation
●

pde1d.blogspot.com

There are 10 types of people,
those that understand binary
numbers, and those that don't.
4s / sign bit

2s bit

1s bit

unsigned

signed

0
0
0
0
1
1
1
1

0
0
1
1
0
0
1
1

0
1
0
1
0
1
0
1

0
1
2
3
4
5
6
7

0
1
2
3
-4
-3
-2
-1

3 + 1 → -4

7+1→0

5 = -3

2/3 → 0
pde1d.blogspot.com

Factorial !!!

n

n! =

∏k

= 1∗2∗3∗4∗...∗(n−1)∗n

k=1

n ! = n∗(n−1)!
8 Byte
signed
integers

20! = 2432902008176640000
21! = -4249290049419214848
64! = -9223372036854775808
65! = -9223372036854775808
66! = 0
pde1d.blogspot.com

70! = Googol +
– C,

C++, Bash, Csh, Fortran, Java → 0

– Hand

Calculator → Error

– LibraOffice,

Calligra Sheets, Gnumeric →
1.19785716699699E+100

– Python,

Lisp, dc, bc, GMP, BigInteger →
119785716699698917960727837216890
987364589381425464258575553628646
280095827898453196800000000000000
00
pde1d.blogspot.com

Spreadsheets Googol +
–

LibraOffice →
1197857166996990000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000

–

Calligra Sheets →
1197857166996989221259474668628791114472893603304820
9921351258923353917776673784237595759172044980224

–

Gnumeric →
1197857166996989026992585446055884022526702920952930
3278944419871214396524861374498691473966836482048

–

70! =
119785716699698917960727837216890987364589
381425464258575553628646280095827898453196
80000000000000000
pde1d.blogspot.com

dc & bc
●

dc – Desktop calculator
–
–
–
–

●

Stack based calculator for Unixes
dc was bc ( Before C ) and before bc
Arbitrary Precision
Integer arithmetic with decimal point shift

bc – basic/bench calculator
–
–
–
–

C like syntax
Originally piped through dc
'bc -l' adds math library and scale=20 digits
Sine, Cosine, Arctan, ln, exp, Bessel fcn
pde1d.blogspot.com

High Precision Libraries
GMP – Gnu Multiple Precision
● MPFR – Gnu Multiple Precision
floating-point with rounding
●

–

MPC – Complex numbers

DD, QD – 32 and 64 digit
accuracy
● PARI/GP – Algebra C library +
shell
●

pde1d.blogspot.com

Floating Point Numbers
●

EPSILON
–

Smallest number such that

1.0 + ϵ != 1.0
or, if δ<ϵ 1.0 + δ = 1.0
●

Floating Point numbers
–

IEEE 754 - 2008
●
●

–
–

FLT_EPSILON = 1.19209e-07
DBL_EPSILON = 2.22045e-16

CPU registers may use higher precision
Detailed definitions vary by factor of ~2
pde1d.blogspot.com

Precision Loss
●

Rapidly varying functions
o

arccos( 0.99999999984769129011 ) = 0.001
o
arccos( 0.99999999939076516049 ) = 0.002
th

Change in 10 place doubles result
● Subtraction of nearly equal numbers
3.14159292035xxx
3.14159265359xxx
0.00000026676xxx
12 place accuracy reduced to 5 places
pde1d.blogspot.com

Quadratic Formula
−b± √ b −4 a c
X=
2a
2

2

a X +b X +c=0
●

2

What if

b ≈ 4ac

√

or

2

2

b ≫ ∣4 a c∣ or b ≪ ∣4 a c∣

1
EPSILON ⇒ Loss of digits
2
pde1d.blogspot.com

Quadratic Equation Examples
Traditional and alternate solution formulations*

●

2

Enter a,b,c: 1 -2 0.9999999999999996
X = 1.00000002107342 , 0.999999978926576
Err: 0 , -1.11022e-16
Alt. sol. X = 1.00000002107342 , 0.999999978926576
Err: 0 , 0

2

Enter a,b,c: 1 -2000000 1
X = 1999999.9999995 , 5.00003807246685e-07
Err: -0.000240445 , -7.61449e-06
Alt. sol. X = 1999999.9999995 , 5.00000000000125e-07
Err: -0.000240445 , 0

b ≈ 4ac
b ≫ 4ac

2

b ≪ 4ac

Enter a,b,c: 10000 1 -10000
X = 0.99995000125 , -1.00005000125
Err: -1.81899e-12 , 0
Alt. sol. X = -1.00005000125 , 0.99995000125
Err: 0 , 0

*Acton, Forman S., “REAL Computing Made Real”,pp 5-6, Dover, 2005/ Princeton Press 1996

pde1d.blogspot.com

It's About Time
Not the 1966 TV show
● The value of time is inversely
proportional to the amount remaining
● Example, 50 times slower
●

–
–
●

1 millisecond computation
4 day computation

Some times for 500X500 grid
–

Python ~30 min, numpy ~30sec, f2py ~2 sec
“Numerical Computing with SAGE”, pg 18, Aug 15, 2013
pde1d.blogspot.com

Time Cost of Precision
Double(15) – baseline gotoBLAS
● DD(32) – 11 to 312 X
● QD(64) – 109 to 3060 X
● GMP(77) – 132 to 3700 X
● GMP(154) – 197 to 5900 X
● MPFR(154) – 405 to 11500 X
●

Times/speeds from MPACK

http://www.slideshare.net/NakataMaho/the-mpack-multiple-precision-version-of-blas-and-lapack

pde1d.blogspot.com

Pick 3 Digits Calculation
●
●
●
●
●
●

Pick 3 ordered digits ( 1 2 3 )
Enter twice ( 123123 )
Divide by 11
Divide by 7
Divide by original 3 digit number
- Result >bc -l
123123
last/11
last/7
last/123

>dc
123123
11 /
7/
123 /
p
pde1d.blogspot.com

Paraskevidekatriaphobia
th

Fear of Friday the 13
● Today is the first Friday the
Thirteenth of 2013
● In Thirteen weeks the next Friday
th
the 13
● Triskaidekaphobia
●

–

Fear of the number 13

pde1d.blogspot.com

13
A prime number
● Is the smallest emirp
●

–
●

A happy number
–

●

12 + 32 = 10 → 12 + 02 = 1

Wilson prime ( 3 known – 5, 13, 563 )
–

●

31 and 13 are both prime

( (13-1)! + 1 ) % 132 = 0

A Fibonacci number
–

1 , 1, 2, 3, 5, 8, 13, 21, 34, …

pde1d.blogspot.com

What is the smartest
mammal ?
●

Rabbits
–

●

*

they can multiply very fast

Fibonacci
–
–
–
–

Leonardo Pisano Bigollo of Pisa
Illustrated rabbit breeding in 1202 “Liber Abaci”
F(n+1) = Fn + F(n-1) , F1 = 1 , F2 = 1
F(n+1) = number of possible arrangements of long
and short items
n

* http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
pde1d.blogspot.com

Fibonacci Spiral

8
Chris 73 / Wikimedia Commons
13
1

2

1
5

Appears in Nature
- Sunflower seed spirals
Spiral with equal packing
- Sea Shells ( not csh )

3

pde1d.blogspot.com

Fibonacci &
Golden Ratio
a

F n+1
→ ϕ
F n n→∞

b

b
a+b
ϕ =
=
a
b
●

b
a

Beautiful
Proportion

φ, Phi symbol after Phidias
– Greek sculpture used and studied

pde1d.blogspot.com

Golden Ratio Uses
●

Penrose Tiles
–
–

Non-repeating pattern
Phi ratio of tiles
1

●

φ

1

Φ

2011 Nobel Prize
–
–

Quasicrystals
3D Non-repeating

Sir Roger Penrose
Solorflare100
Wikimedia Commons

1
2π
o
ϕ = 2 cos( 5 ) = 2 cos(72 ) = Φ
pde1d.blogspot.com

Calculating Golden Ratio
1
ϕ = 1+
1
1+
1
1+
1+. . .
ϕ =

Golde

n Mea
n
Divin
e P ro
portio
n
Gold
en Se
c ti o n

√1+ √1+ √ 1+ √ 1+ √1+...

1+ √ 5
ϕ =
2

Do you like stand-up comedians?
pde1d.blogspot.com

One Liners for Gloden Ratio
●

Continued fraction
–
–

●

Continued squareroot
–
–

●

echo “67k 1[dsb1r/1+pdlb!=c]dscxq” | dc
163 iterations
dc -e “67k1[dsb1+vpdlb!=c]dscxq”
133 iterations

Fibonacci
–
–

echo "67k1sa1d[sclardsa+dla/pdlc!=b]dsbx" | dc
163 iterations

pde1d.blogspot.com

Golden Ratio by dc
●

Equation - 67k5v1+2/p
Actions

67 Push '67'

Stack
67

k

Pop → set precision

5

Push '5'

5

v

Pop → Sqrt → Push

2.236...

1

Push '1'

1 ; 2.236...

+

Pop 2 → add → Push

3.236...

2

Push '2'

2 ; 3.236...

/

Pop 2 → divide → Push

1.618...

p

Pop → print → Push

1.618...
pde1d.blogspot.com

More dc
echo 67k5v1+2/dd*rd1r/[0]Ppsapsapq | dc
0.6180339887498948482045868343656381177203091798057628621354486227052
1.6180339887498948482045868343656381177203091798057628621354486227052
2.6180339887498948482045868343656381177203091798057628621354486227050

0.816033...6227052
1.816033...6227052
2.816033...6227050
1
ϕ = Φ = ϕ−1

2

ϕ = ϕ+1
pde1d.blogspot.com

Python Decimal Numbers
> python
>>> 1.1+2.2
3.3000000000000003
>>> from decimal import *
>>> getcontext().prec=67
>>> phi = ( Decimal('5').sqrt() + 1 ) / 2
>>> print 1/phi, 'n', phi, 'n', phi*phi
0.618033988749894848...21354486227050
1.618033988749894848...2135448622706
2.618033988749894848...2135448622708
pde1d.blogspot.com

Irrational Number Power
●

Can an irrational number to an
irrational power give a rational result ?

√(2) √(2)

(( √2) )

= 2

> python
>>> from decimal import *
>>> getcontext().prec=67
>>> s = Decimal(2).sqrt()
>>> print (s**s), "**", s
1.6325269194381528447734... ** 1.4142135623730950488016...
>>> print (s**s)**s
2.000000000000000000000000000000000000000000000000000
000000000000002
pde1d.blogspot.com

Chaos
●

Maxwell Smart's arch nemesis
–

●

●

Lorenz's Strange Attractor
Wikimedia commons

Butterfly effect
–

●

KAOS

Small initial difference yield a large impact

Non-periodic behavior
Example recurrence relations
–

Logistics equation

x n+1 = r x n (1−x n )
–

, 3.82843 < r ≤ 4 , 0 < x < 1

or
2
n

x n+1=x − 2

, −2 < x < 2
pde1d.blogspot.com

2
n

X0 = 0.5 vs 0.49...

x n+1=x − 2

2
1.5
1
0.5
Xn

0
-0.5
-1
-1.5
-2
-2.5
0

2

4

6

8

10

12

14

16

18

20

Iteration n

X_0=0.5

2.5

Twelve Nines

X_0=0.4999999999999

Rain

2
1.5
1

Xn

0.5
0
-0.5
-1
-1.5
-2
-2.5
80

82

84

86

88

90

92

94

96

98

100

Iteration n

X_0=0.5

X_0=0.4999999999999

pde1d.blogspot.com

2
n

Recurrence Relation x n+1 = x − 2
●

An infinite number of initial values
end with x = 2
–

x 0 = √ 2 , x 1=0 , x 2 =−2 , x 3=2 , x 4 =2 , . . .
2
n

x n+1 = x − 2
x=0
for( i=1; i<30; i++) {
x= sqrt(x+2)
print i, " ", x, "n"
x= - sqrt(x+2)
print i, " ", x, "n"
}

→

x n−1 = ± √ x n + 2
27 .6180339887498950...
27 -1.6180339887498949...
28 .6180339887498947...
28 -1.6180339887498948...
29 .6180339887498948...
29 -1.6180339887498948520...

±ϕ → Φ → −ϕ → Φ → −ϕ → . . .
pde1d.blogspot.com

●

I know I am not
good at math,
but if I can get
95% of
problems right,
who cares
about the other
10%
pde1d.blogspot.com

Founding Fathers & Mathematics
●

Thomas Jefferson
–

Kept good numerical records
●

–
–

U.S. Currency
Mathematics to U.S. Universities
●
●

–

University of Virginia
West Point

Statistics to guide laws
●

–

Corrected Newton's Math

Limited duration laws ( 20 Yrs )

Concerning distribution of Representatives by state
●

Any law should “reduces the apportionment always
to an arithmetic operation, about which no two men
can ever possibly differ.”
Cohen, I.B.,”The Triumph of Numbers, How Counting Shaped Modern Life”,
W.W.Norton & Company, New York, 2005

pde1d.blogspot.com

Eliminate Soldier's Helmets ?
Britain almost stopped issuing steal
helmets in WW I
● generals began to call for their
removal as they increased incidences
of headwounds twelvefold and
doubled total casualties
● A fatality was not a casualty
●

http://tvtropes.org/pmwiki/pmwiki.php/Main/LiesDamnedLiesAndStatistics
pde1d.blogspot.com

Flue Shot ?
●

●

●

Based on three studies, getting
the flu shot
reduces chance of getting the flu by
5%
73% effective at preventing the flu

http://blog.minitab.com/blog/adventures-in-statistics/how-effective-are-flu-shots
pde1d.blogspot.com

Founding Fathers & Numbers
●

Benjamin Franklin
–

Inventor
●
●
●

–

Founder
●
●
●

–
–
–

Lightening Rod
Franklin Stove
Bifocals
Public Library
Fire Company
Police Service

Post Office
Electricity
Magic Squares
http://www.math.wichita.edu/~richardson/
pde1d.blogspot.com

Magic Squares
Sum of numbers in rows, columns
8 1 6
and diagonals are equal
3 5 7
● Unique number per cell
4 9 2
● Pattern for odd number squares
●

B

C

D

2

=H3+1

=B3+1

=C3+1

3

=H4+1

=B4+1

=C4+1

4

=H5+1

=B5+1

5

=H6+1

6

E
1

F

G

H

=E3+1

=F3+1

=G3+1

=D4+1

=E4+1

=F4+1

=H2+1

=D3+1

=D5+1

=E5+1

=F5+1

=G5+1

=B6+1

=C6+1

=D6+1

=E6+1

=G4+1

=G6+1

=H7+1

=C5+1

=C7+1

=D7+1

=E7+1

=F7+1

=G7+1

7

=H8+1

=B8+1

=C8+1

=D8+1

=F6+1

=F8+1

=G8+1

8

=B7+1

=B2+1

=C2+1

=D2+1

=E2+1

=F2+1

=G2+1
pde1d.blogspot.com

11 X 11 Magic Square – by Spreadsheet

pde1d.blogspot.com

4x4
Half Rows = Half Sum
–
–

Ben's Bent Diagonals
●

2056

http://www.math.wichita.edu/~richardson/
pde1d.blogspot.com

Music is About Fractions
●

Not just quarter
note, eight note
but tuning scales

1
Step =
2

12

√2

frequency ( A# )
12
= √ 2 ∗ frequency ( A )
●

Independent of
base note

1
Fraction Error
1.0595
1.1225 ≈ 9/8
-0.0023
1.1892 ≈ 6/5
-0.0091
1.2599 ≈ 5/4
0.0079
1.3348 ≈ 4/3
0.0011
1.4142 ≈ 7/5
0.0101
1.4983 ≈ 3/2
-0.0011
1.5874 ≈ 8/5
-0.0079
1.6818 ≈ 5/3
0.0090
1.7818 ≈ 7/4
0.0178
1.8877 ≈ 15/8
0.0068
2
pde1d.blogspot.com

A

C

A#

C#

B

D

C

D#

C#

E

D

F

D#

F#

E

G

F

G#

F#

A

G

A#

G#

B

A

C

1

Fraction
1.0595
1.1225
1.1892
1.2599
1.3348
1.4142
1.4983
1.5874
1.6818
1.7818
1.8877
2

Error

≈ 9/8
≈ 6/5

-0.0023
-0.0091

≈
≈
≈
≈
≈
≈
≈
≈

0.0079
0.0011
0.0101
-0.0011
-0.0079
0.0090
0.0178
0.0068

5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8

pde1d.blogspot.com

A Golden Ratio Scale
FREQUENCY

STEP

1
8*Φ4

1
1.16718

1.167184

2*Φ

1.23607

1.059017

φ2/2

1.30902

1.059017

4*Φ2

1.52786

1.167184

φ

1.61803

1.059017

8*Φ3

1.88854

1.167184

2

1.059017

2

Fractions
within 1%
7
6
13
8
15
8

pde1d.blogspot.com

Golden Ratio Intervals
3

1.059017... = φ / 4
● 12
√2 = 1.059463... → 0.04%
● 4 each octave
●

4

1.167184... = 8*Φ
● Can be factored into 3 'half' steps
●

–
–

1.059017... * 1.059017... * 1.040719...
3 each octave

How many total intervals(steps) per octave ?
pde1d.blogspot.com

I hope you
had fun
pde1d.blogspot.com

When Computers Don't Compute and Other Fun with Numbers

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