Download Computing π

CLEVE’S CORNER
Computing π
By Cleve Moler
Computing hundreds, or trillions, of digits of π has long been used to stress hardware, validate software, and establish
bragging rights. MATLAB® implementations of the most widely used algorithms for computing π illustrate two different styles
of arithmetic available in Symbolic Math Toolbox™: exact rational arithmetic and variable-precision floating-point arithmetic.
L
ast August, two computer hobbyists, Alexander Yee and Shigeru
Kondo, announced that they had set a world record by computing
5 trillion digits of π (Figure 1). Their computation took 90 days on a
“homebrew” PC. Yee is now a graduate student at the University of
Illinois. His computing software, “y-cruncher,” began as a class project at Palo Alto High School. Kondo is a systems engineer who assembles personal computers at his home in Japan. The machine that
Cleve's Corner: Computing π
he built for this project (Figure 2) has two Intel® Xeon® processors
Computing hundreds, or trillions, of digits of π has long been used to stress hardware, validate software,
and
establish
braggingof
rights.
of the most
used algorithms
with
a total
12 MATLAB
cores,implementations
96 gigabytes
ofwidely
RAM,
and 20for external hard
computing π illustrate two different styles of arithmetic available in Symbolic Math Toolbox: exact
rational
variable-precisioncapacity
floating-point of
arithmetic.
disksarithmetic
with aandcombined
32 terabytes.
Last August, two computer hobbyists, Alexander Yee and Shigeru Kondo, announced that they had
The computation of 5 trillion digits is a huge processing task
set a world record by computing five trillion digits of π. Their computation took 90 days on a homebrew
PC. Yee is now a graduate student at the University of Illinois. His computing software, “y-cruncher”,
where
the time required for the data movement is just as important
began as a class project at Palo Alto High School. Kondo is a systems engineer who assembles personal
computers at his home in Japan. The machine that he built for this project (Figure 1) has two Intel Xeons
as
the
time
required
for
the
arithmetic.
with a total of 12 cores,
96 gigabytes of
RAM,
and 20
external hard disks with a combined capacity of 32
terabytes.
y-cruncher uses several different formulas for computation and
The computation of five trillion digits is a huge processing task where the time required for the data
verification.
The primary
toolforisthea arithmetic.
formidable formula discovered by
movement
is just as important
as the time required
y-cruncher
usesGregory
several different
formulas for computation
and verification. The primary tool is a
David
and
Chudnovsky
in 1987:
formidable formula discovered by David and Gregory Chudnovsky in 1987.
The Chudnovsky brothers, who were the subject of two fascinating articles in the New Yorker and a
NOVA documentary, have also built “home brew” computers in their apartment in Manhattan. The
massively parallel machine that they assembled from commodity parts in the 1990s was considered a
supercomputer at the time.
The Chudnovsky brothers, who were the subject of two fascinating articles in The New Yorker and a NOVA documentary, have also
High-Precision Arithmetic in Symbolic Math Toolbox
built homebrew computers in their apartment in Manhattan. The
Symbolic Math Toolbox provides two styles of high-precision arithmetic, sym and vpa.
massively parallel machine that they assembled from commodity
The sym function initiates exact rational arithmetic: Arithmetic quantities are represented by
quotients
large integers.
length increasesaassupercomputer
necessary, limited only by computer
parts and
in roots
theof1990s
wasInteger
considered
at the time.
time and storage requirements. Quotients and roots are not computed unless the result is an exact
integer.
High-Precision Arithmetic in Symbolic Math Toolbox
Symbolic
Math
Toolbox
provides
tworoundoff
styles
ofat the
high-precision
arithArithmetic
operations,
including
divisions and
roots, can involve
errors
level of the
specified accuracy.For example, the MATLAB statement
metic: sym and vpa.
p = rat(pi)
The sym function initiates exact rational arithmetic: Arithmetic
quantities are represented by quotients and roots of large integers.
Integer length increases as necessary, limited only by computer
time and storage requirements. Quotients and roots are not computed unless the result is an exact integer.
Figure 1. A log plot of the data from the Wikipedia article “Chronology of
computation of pi” showing how the world record for number of digits has
increased over the last 60 years. A linear fit to the logarithm reveals a Moore’s
Law phenomenon. The number of digits doubles about every 22.5 months.
The vpa function initiates variable-precision floating-point
arithmetic: Arithmetic quantities are represented by decimal fractions of a specified length together with a power-of-10 exponent.
Arithmetic operations, including divisions and roots, can involve
roundoff errors at the level of the specified accuracy. For example,
the MATLAB statement
The vpa function initiates variable-precision floating-point arithmetic: Arithmetic quantities are
represented by decimal fractions of a specified length together with a power-of-ten exponent.
1
p = rat(pi)
produces three terms in the continued fraction approximation of π
p = 3 + 1/(7 + 1/16)
Then the statement
sym(p)
yields the rational expression
355/113
which is an attractive alternative to the familiar 22/7. On the other hand,
the statement
vpa(p,8)
produces the 8-digit floating-point value
produces three terms in the continued fraction approximation of π
p =
3 + 1/(7 + 1/16)
3.1415929
Then the statement
sym(p)
The Chudnovsky Algorithm
Our
MATLAB programs implement three of the hundreds of possible
355/113
algorithms
for computing
π. OnThe
Chudnovsky
which
is an attractive alternative
to the familiar 22/7.
the other
hand, the statementalgorithm (Figure 3)
vpa(p,8)
uses
exact rational arithmetic until the last step. Each term in the
produces
the 8-digit floating-point
value
Chudnovsky
series
provides about 14 decimal digits, so to obtain d
3.1415929
digits
requires [d/14] terms. For example, the statement
yields the rational expression
Figure 2. Shigeru Kondo’s homebrew PC, which currently holds the world
record for computing digits of π.
function P = chud _ pi(d)
% CHUD _ PI Chudnovsky algorithm for pi.
The Chudnovsky Algorithm
% chud _ pi(d) produces d decimal digits.
Our MATLAB programs implement three of the hundreds of possible algorithms for computing π.
The Chudnovsky
algorithm (Figure 2) uses exact rational arithmetic until the last step. Each term in the
chud_pi(42)
Chudnovsky series provides about 14 decimal digits, so to obtain d digits requires
terms. For
example, the statement
k = sym(0);
chud_pi(42)
uses
exact rational computation to produce
uses exact rational computation to produce
s = sym(0);
sig = sym(1);
n = ceil(d/14);
for j = 1:n
s = s + sig * prod(3*k+1:6*k)/prod(1:k)^3 * ...
and then the final step uses variable-precision floating-point arithmetic with 42 digits to produce
and
then the final step uses variable-precision floating-point arithmetic
3.14159265358979323846264338327950288419717
with
42 digits
to the
produce
One interesting
fact about
digits of π is revealed by
chud_pi(775)
or
3.14159265358979323846264338327950288419717
vpa(pi,775)
One interesting fact about the digits of π is revealed by
chud_pi(775)
(13591409+545140134*k) / 640320^(3*k+3/2);
k = k+1;
sig = -sig;
end
S = 1/(12*s);
P = vpa(S,d);
Figure 3. MATLAB implementation of the Chudnovsky algorithm, using
sym to initiate exact rational arithmetic.
or
vpa(pi,775)
These both produce 775 digits, beginning and ending with
3.141592653589793238 ...... 707211349999998372978
We see not only that chud_pi is working correctly but also that
there are six 9s in a row around position 765 in the decimal expansion of π (Figure 4).
Computing 5000 digits with chud_pi requires 358 terms and
takes about 20 seconds on my laptop. The final symbolic expression
contains 12,425 characters.
2
=
=
bn+1 = �𝑎𝑎𝑛𝑛 𝑏𝑏𝑛𝑛
=
until a==
n and bn agree to a desired accuracy. For
example,
arithmetic
mean ofaccuracy.
1 and 9 isFor
5, the
until
andthe
bnagree
agree
to aa desired
desired
example,
the
arithm
until
aannand
and
bmean
to
accuracy.
Forexample,
example,
the
arithm
nagree
geometric
is
3,
and
the
agm
is
3.9362.
until
a
b
to
a
desired
accuracy.
For
the
arithme
n
n
geometric
mean
is
3,
and
the
agm
is
3.9362.
geometric
mean
is
3,
and
the
agm
is
3.9362.
geometric
mean isproperties
3, and theof
agm
3.9362.
The powerful
theisagm
lie not just in
The
powerful
properties
ofresults
the agm
lie not just in the final value
theThe
final
value
but
also
in
the
generated
powerfulproperties
propertiesofofthe
theagm
agmlielienot
notjust
justininthe
the
final
valu
The
powerful
final
value
along
the
way.
Computing
isisjust
one
compu
along
theway.
way.Computing
Computingπππis
justone
oneexample.
example.ItItItinvolves
along
the
just
example.
involves
comp
along the way. Computing π is just one example. It involves comput
involves computing
(
)
((
))
M = agm(1,1/√2)
During the iteration, keep track of the changes,
During
the
iteration,
keep
track
ofthe
thechanges,
changes,
During
the
iteration,
keep
track
During
the
iteration,
keep
track
of track
theof
changes,
During
the
iteration,
keep
of the
changes,
Duringthe
theiteration,
iteration,keep
keeptrack
trackofofthe
thechanges,
changes,
During
cn = an+1 - an
an+1 - an
c
ncn==an+1
Finally,
letlet- an
Finally,
Finally,
let
Finally,
let
Finally,
Finally,1let
let
Finally,
let
𝑛𝑛 2
S = 1 – ∑∞
𝑛𝑛=0 2 𝑐𝑐𝑛𝑛
14
∞ 𝑛𝑛 𝑛𝑛 2 2
∞
∑
S
=
–
𝑐𝑐𝑛𝑛
2
S = 4– ∑𝑛𝑛=0
𝑛𝑛=02 𝑐𝑐𝑛𝑛
Then4
Then
Then
Then
Then
Then
Then
𝑀𝑀2
2
𝜋𝜋
=
𝑀𝑀𝑀𝑀2
These both produce 775 digits, beginning and ending with
𝜋𝜋𝜋𝜋== 𝑆𝑆𝑆𝑆
Our MATLAB 𝑆𝑆
function for the agm algorithm (Figure 3) uses variable- precision floating-point
arithmetic from the very beginning. If we used symbolic rational arithmetic, we would end up with a
Our
MATLAB
function
the
agm
algorithm
(Figure
uses
variableprecision
floating-point
3.141592653589793238 ...... 707211349999998372978
Our
MATLAB
function
forfor
the
agm
algorithm
(Figure
3) 3)
uses
precision
floating-point
nested
sequence
of square
roots.
After
just two
iterations
wevariablewould have
arithmetic
from
very
beginning.
If we
used
symbolic
rational
arithmetic,
would
end
with
arithmetic
from
thethe
very
beginning.
If we
used
symbolic
rational
arithmetic,
wewe
would
end
upup
with
a a
nested
sequence
square
roots.
After
two
iterations
would
have
nested
sequence
of of
square
roots.
After
justjust
two
iterations
wewe
would
have
2
Our MATLAB function for the agm algorithm (Figure 5) uses
variable-precision
floating-point
arithmetic
from the very beginning.
We see not only that chud_pi is working correctly but also that there are six
9s�in
around
√2a row
22
1
If wefunction
usedforsymbolic
rational
we would
end up with a nest√2
2 3)arithmetic,
Our MATLAB
the agm algorithm
(Figure
uses√2
variable- precision
floating-point
√2
position 765 in the decimal expansion of π (Figure 4). arithmetic
⎛
��
+
+1 ⎞
from the very beginning. If we used symbolic
end up with a
Figure 4. 10,000 digits of π, visualized in MATLAB. Can you see the six
√2
8arithmetic,
4⎞we would
1⎞
√2
222+rational
⎛roots.
ed sequence
of After
square
After
just
two
iterations
we would have
⎛two
+
+
nested sequence
of square roots.
just
iterations
we would
have
+
22
consecutive 9s in the eighth row?
88 44
⎝ 20 seconds on
⎠ my
Computing five thousand digits with chud_pi requires 358 terms and takes about
⎝⎝
⎠⎠
laptop. The final symbolic expression contains 12,425 characters.
2
�√2
The Algebraic-Geometric Mean Algorithm
TheAlgebraic-Geometric
Chudnovsky formula is a power
series:
Each new term in the partial
The
Mean
Algorithm
2
22
1
√2
√2 ��
√2 1
2 + 1⎞
√2
− �√2 −112�2 − 2 ⎛√2 −
11
4⎞
4− �
√28 − 222 +11
√24 − 2� − 2⎛⎛
⎞
− � 4− 2
� −2
8− 22 + 44
8
44
4
2
⎝
⎠
⎝⎝
⎠⎠ to 𝜋𝜋. The
The decimal value of this monstrosity is 3.14168, so
it is not yet a very good approximation
These both
produce
775 digits,
beginning
endingof
withaccuracy. The algebraic-geometric
sum
adds
a fixed
number
ofand
digits
3.141592653589793238
......
707211349999998372978
mean
algorithm
is
completely
different;
it is quadratically
convergent:
The Chudnovsky formula is a power
series: Each
new term
in the
partial
sumis 3.14168,
adds ait isfixed
number
of to 𝜋𝜋. The
complexity
doubles
with each
iteration.
The
decimal
value
monstrosity
a very
good
approximation
The
decimal
value
of of
thisthis
monstrosity
is 3.14168, so so
it is notnot
yetyet
a very
good
approximation
to 𝜋𝜋. The
We
see
not
only
that
chud_pi
is
working
correctly
but
also
that
there
are
six
9s
in
a
row
around
complexity
doubles
with
each
iteration.
Each of
new
iteration doubles
the number of correct digits.
Thealgorithm
agm al- complexity
digits
accuracy.
The
algebraic-geometric
mean
is
completely
different;
it
is
quadratically
doubles
with
each
iteration.
Such exact symbolic expressions are computationally unwieldy and inefficient. With vpa, however, the
position 765 in the decimal expansion of π (Figure 4).
quadratic
convergence
can be seen
in
the successive unwieldy
iterates: and inefficient. With vpa, however, the
Such
exact
symbolic
expressions
computationally
gorithm has a Each
long history
dating backdoubles
to Gauss and
Such
exact
symbolic
expressions
areare
computationally
unwieldy
and inefficient.
convergent:
new iteration
theLegendre.
numberItsofabilcorrect
digits.
The
agm
algorithm
has
a longWith vpa, however, the
Computing five thousand digits with chud_pi requires 358 terms and takes about 20 seconds on my
quadratic
convergence
seen
in the
successive
iterates:
quadratic
convergence
cancan
bebe
seen
in the
successive
iterates:
3.2
laptop.
The
final symbolic
expression
contains 12,425independently
characters.
ity
to
compute
π
was
discovered
by
Richard
Brent
and
history dating back to Gauss and Legendre. Its ability toThecompute
π was discovered independently by
3.142
3.2
decimal3.2
value
of this monstrosity is 3.14168, so it is not yet a very good approximation to . The
The Algebraic-Geometric
Mean
Algorithm
3.142
Eugene
Salamin
in
1975.
complexity doubles
each iteration.
Thewith
decimal
value of this monstrosity is 3.14168, so it is not yet a very
3.142
Richard
Brent and
Eugene Salamin in 1975.
The Chudnovsky
formula is
a power of
series:
Eachnumbers,
new term in the(a+b)/2,
partial sum adds
fixed number
of Such exactgood
symbolicapproximation
expressions are computationally
unwieldy
and inefficient.doubles
With vpa,with
however,
the iteration.
The
arithmetic
mean
two
is aalways
greater
to π. The
complexity
each
digits of accuracy. The algebraic-geometric mean algorithm is completely different; it is quadratically quadratic convergence can be seen in the successive iterates:
convergent:
Eachgeometric
new iteration doubles
the number
ofTheir
correct digits.
The agm algorithm has a long mean,
than
their
mean,
√ab.
arithmetic-geometric
Such
exact
symbolic
expressions
are
computationally
unwieldy
The
arithmetic
meanItsof
two
numbers,
(a+b)/2,
greater than their geometric mean,
.
3.2
history
dating
back to Gauss and Legendre.
ability
to compute
π was discovered
independentlyis
byalways
or
agm(a,b),
is
computed
3.142 and inefficient. With vpa, however, the quadratic convergence can
Richard
Brent and Eugene
Salamin in 1975.by repeatedly taking arithmetic and geoTheir arithmetic-geometric mean, or agm(a,b), is computed
by repeatedly taking arithmetic and
3.1415927
metric
means.
with
a
=
a
and
b
=
b,
iterate
be seen in the successive iterates:
3.141592653589793
The arithmetic
meanStarting
of two numbers,
(a+b)/2,
is
always
greater
than
their
geometric mean, √𝑎𝑎𝑎𝑎.
0
0
geometric means. Starting with a = a and b = b, iterate
0
Their arithmetic-geometric mean, or agm(a,b), is computed by 0repeatedly taking arithmetic
and
geometric means. Starting with a0 = a and b0 = b, iterate
3.1415926535897932384626433832795
3.141592653589793238462643383279502884197169399375105820974944592
3.2
3.142
The quadratic
convergence makes agm very fast. To compute 100,000 digits requires only 17 steps
2
and about 2.5 seconds on my laptop.
3.1415927
bn+1 == �𝑎𝑎𝑛𝑛 𝑏𝑏𝑛𝑛
The BBP Formula
3.141592653589793
Yee spot-checked his computation with the BBP formula. This formula, named after David Bailey,
3.1415926535897932384626433832795
Jonathan Borwein,
and Simon Plouffe and discovered by Plouffe in 1995, is a power series involving
until an and bn agree to a desired accuracy. For
inverse
powers
of 16.
until
a
b
agree
to
a
desired
accuracy.
For
example,
the
arithmetic
mean of 1 and 9 is 5, the
until
a
and
b
agree
to
a
desired
accuracy.
For
example,
the
arith3.1415926535897932…79502884197169399375105820974944592
n nand
n
n arithmetic mean of 1 and 9 is 5, the
example, the
geometric
mean
is
3,
and
the
agm
is
3.9362.
metic
mean of
1 andis9 3,
is 5,
thethe
geometric
is 3, and the agm
geometric
mean
and
agm ismean
3.9362.
is 3.9362.
The sixth entry in this output has 64 correct digits, and the tenth
Thepowerful
properties
ofagm
thelieagm
agm
lie not
not
just
invalue
powerful
properties
the
lie
just
in
the finalhasvalue
1024.but also in the results generated
The
properties
of theof
not just
in the
final
thealso
final
value
butgenerated
also inπthe
results
generated
along
theinway.
Computing
is just
one
example.
but
the results
along
the
way.
ComputingItπinvolves
is just computing
The quadratic convergence makes agm very fast. To compute 100,000
along
the way.
Computing
π is just one example. It
digits requires only 17 steps and about 2.5 seconds on my laptop.
one
example.
It involves
computing
an+1 ==
𝑎𝑎𝑛𝑛 + 𝑏𝑏𝑛𝑛
involves computing
(
)
M =the
agm(1,1/√2)
During
iteration, keep track of the changes,
3
Finally, let
The tenth entry is this output would have 1024 correct digits.
function P = agm _ pi(d)
% AGM _ PI Arithmetic-geometric mean for pi.
% agm _ pi(d) produces d decimal digits.
Our MATLAB function for the agm algorithm (Figure 3) uses variable- precision floating-point
arithmetic
from the very beginning. If we used symbolic rational arithmetic, we would end up with a
digits(d)
nested sequence of square roots. After just two iterations we would have
a = vpa(1,d);
b = 1/sqrt(vpa(2,d));
s = 1/vpa(4,d);
p = 1;
n = ceil(log2(d));
Richard Preston, “The Mountains of Pi,” The New Yorker,
March 2, 1992.
www.newyorker.com/archive/1992/03/02/1992_03_02_036_TNY_
CARDS_000362534
Richard Preston, “Capturing the Unicorn,” The New Yorker,
April 11, 2005.
www.newyorker.com/archive/2005/04/11/050411fa_fact
for k = 1:n
c = (a+b)/2;
b = sqrt(a*b);
s = s - p*(c-a)^2;
p = 2*p;
The decimal
of this monstrosity is 3.14168, so it is not yet a very good approximation to . The
a value
= c;
complexity doubles with each iteration.
end
Such exact symbolic expressions are computationally unwieldy and inefficient. With vpa, however, the
P = a^2/s;
quadratic convergence can be seen in the successive iterates:
3.2
3.142 5. MATLAB implementation of the arithmetic-geometric mean
Figure
3.1415927
algorithm,
using digits and vpa to initiate variable-precision floating3.141592653589793
3.1415926535897932384626433832795
point
arithmetic.
3.141592653589793238462643383279502884197169399375105820974944592
The tenth entry is this output would have 1024 correct digits.
The
BBP Formula
The quadratic
convergence makes agm very fast. To compute 100,000 digits requires only 17 steps
and
about
2.5 seconds on myhis
laptop.
Yee spot-checked
computation with the BBP formula. This forThe
BBP Formula
mula,
named
after David Bailey, Jonathan Borwein, and Simon
Plouffe
and discovered
bywith
Plouffe
1995,This
is aformula,
power
series
Yee spot-checked
his computation
the BBPin
formula.
named
afterinvolving
David Bailey,
Jonathan Borwein, and Simon Plouffe and discovered by Plouffe in 1995, is a power series involving
inverse
powers
of
16:
inverse powers of 16.
The remarkable property of the BBP formula is that it allows the
computation of several consecutive hexadecimal digits in the base
16 expansion of π without computing the earlier digits and without
using extra-precision arithmetic. A formula for the hex digits starting at position d is obtained by simply multiplying this formula by
16d and then taking the fractional part.
I can’t resist concluding that BBP can be used to compute just a
piece of π.
Our BBP program is too long to be included here, but all three
programs, chudovsky_pi, agm_pi, and bbp_pi, are available for
download on MATLAB Central. ■
Published 2011
91954v00
References
Alexander J. Yee and Shigeru Kondo, “5 Trillion Digits of Pi,”
August 2, 2010.
www.numberworld.org/misc_runs/pi-5t/details.html
“Brothers Chudnovsky,” PBS, NOVA, July 26, 2005.
www.pbs.org/wgbh/nova/physics/chudnovsky-math.html
David Bailey, Jonathan Borwein, Peter Borwein, and Simon Plouffe,
“The Quest for Pi,” Mathematical Intelligencer 19, pp. 50-57, 1997.
http://crd.lbl.gov/~dhbailey/dhbpapers/pi-quest.pdf
Jonathan M. Borwein and David H. Bailey, Mathematics by
Experiment: Plausible Reasoning in the 21st Century, AK Peters,
Natick, MA, 2008.
“Chronology of computation of pi,” Wikipedia, March 9, 2011.
http://en.wikipedia.org/wiki/Chronology_of_computation_of_π
Learn More
Download: MATLAB Implementations of
the Chudnovsky, agm, and BBP Algorithms
mathworks.com/computing-pi
Cleve's Corner Collection
mathworks.com/clevescorner
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