Exact real arithmetic

Keith Briggs

Keith.Briggs@bt.com

http://more.btexact.com/people/briggsk2/xr-kent-talk-pp.pdf

University of Kent Computing Laboratory 2004 Feb 10 1400

typeset 2004 February 9 12:18 in pdfLATEX on a linux system

Exact real arithmetic 1 of 32

Contents. Outline

. IEEE floating point

. Another floating point disaster

. Irrationals

. Digit expansions

. What's the problem?

. Continued fraction arithmetic

. Scaled-integer representation

. Data flow in XR system

. Some easy operations

. Addition and multiplication

. The big software challenge

. Algebraic number construction

. Transcendental functions

Exact real arithmetic 2 of 32

. exp

. Computable bounds

. My python implementation

. Lambdas

. Comparison

. Python implementation in action

. C++ implementation in action

. Features of my implementations

. Computational geometry application

. Simultaneous Diophantine approximation application

. Limitations

. Maths ⇐⇒ software

. Conclusion

. Bibliography

Exact real arithmetic 3 of 32

Outline

☞ what's the problem?

☞ representation and processing of reals

☞ my implementation of the Boehm scheme

☞ applications

☞ notation. integers: Z = { . . . ,−2,−1, 0, 1, 2, . . . }. rationals: Q = { p/q; p, q ∈ Z }. R\Q = { algebraics ∪ transcendentals }. bxe is the nearest integer to x

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IEEE floating point

☞ finite set of rationals and approximate operations:

F = { m 2e : |m| < 253, |e| < 1024, NaN, Inf }O = { +,−, ∗, / }, round to even or nearest

☞ problems!

x0 = F(0.9)xk+1 = F(3.999)∗xk∗(1−xk) k = 0, 1, 2, . . .

☞ in F, x53 > 0.5, but the exact result is

x53 = 0.130235874811773733039643730080570 . . .

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Another floating point disaster

u0 = e−1uk = k uk−1−1 k = 1, . . . , 25

0 5 10 15 20 25 30−50

−40

−30

−20

−10

0

10

20

30

decimal digits

log(

rela

tive

erro

r)

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Irrationals

☞ construction conventionally involves a limit of a sequence ofrationals

. rate of convergence is not specified

. traditional digit expansions converge from below

. what about negative digits?

☞ other possibilities:. symbolic dynamics of an expanding map. continued fractions, Möbius maps, LFTs. nested intervals with rational endpoints. non-integer base - e.g. golden ratio. does something even better exist?

We want a representation that allows thecomputation of more significant digits first

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Digit expansions

x 7→ f(x)digit=branch number

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What's the problem?

☞ example: find sign of one root x = −b+√

b2−4ac2a of quadratic

ax2+bx+c = 0

a

** b

sqr

-

c

*

2 4

output

/

-

sqrt

. input nodes don't know how much precision to send

. all input nodes send data, even if it eventually may not be needed

. to recalculate requires the whole tree to be re-evaluated

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Continued fraction arithmetic

x = [x0;x1, x2, x3, . . .] = x0+1

x1+1

x2+· · ·

☞ studied by Gosper, Vuillemin, Liardet, Stambul, Ménissier-Morain, Potts

☞ only easy operation: ax+bcx+d

☞ for +, ∗ etc., inefficient

☞ for exp, sin etc., very inefficient!

☞ problem:

How do we convert inputs to cf form?

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Scaled-integer representation (Boehm)

☞ for x ∈ R, consider a function x : Z+ → Z such that

| Bn x−x(n) | < 1

for fixed B = 2, 3, 4, . . .

☞ x : Z+ → Z ⇔ x ∈ XR

☞ in other words: x(n)−1Bn

< x <x(n)+1

Bn

☞ example: x ∈ Q, x(n) ≡ bBn xc

All computation with XRs is reduced tolarge integer operations

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Data flow in XR system

a

** b

sqr

-

c

*

2 4

output

/

-

sqrt

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Some easy operations

☞ |x|(n) ≡ |x(n)|

☞ [−x](n) ≡ −x(n)

☞ [sqrt(x)](n) ≡√

x(2n)

☞ for m ∈ Z, [m x](n) ≡ bm x(n+p)/2p e, p = 1+log2 |m|

☞ for m ∈ Z∗, [x/m](n) ≡ bx(n)/m e

☞ comparison: if for some n, x(n) and y(n) differ by morethan 1, they are unequal

☞ caching: x(n) can be evaluated as bx(m)/2m−n e if x(m) form > n is available

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Addition and multiplication (for B = 2)

[x+y](n) ≡ b (x(n+2)+y(n+2)+2) /4 c

| [x+y](n)−2n(x+y) |= | b (x(n+2)+y(n+2)+2 ) /4c−2n(x+y) |6 1/2+| (x(n+2)+y(n+2))/4−2n(x+y) |= 1/2+| x(n+2)+y(n+2)−2n+2(x+y) | / 4

6 1/2+| x(n+2)−2n+2 x | / 4+| y(n+2)−2n+2 y | / 4< 1

[x∗y](n) ≡ b(x(q+r+1)y(p+s+1)+2p+q+3

)/2p+q+4 c

where

r = b (n+2)/2 c, s = n+2−r, p = b log2 |x(r)| c, q = b log2 |x(s)| c

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Algebraic number construction

☞ given a polynomial p with integer coefficients, and integersa, k > 0, we can compute the sign of p at a/Bk with onlyinteger operations

def signat(p,a,k): # return the sign of polynomial p at a/B^kn,w=len(p),p[0]for j in 1. .n:w*=aw+=p[j]*B^(k*j)

return w>0

☞ thus, given a bracketed root:

sign p(a/Bk) < 0, sign p(b/Bk) > 0

we may refine it by bisection to any desired accuracy

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Transcendental functions

☞ f = exp, arctan, sin etc. can be computed if we can imple-ment an approximating function f : Q×Z+ → Z such that∣∣∣ Bn f(q)−f(q, n)

∣∣∣ < 1

☞ we need approximating functions which give us a priori errorbounds

☞ or, at least, error bounds must be computable in rationalarithmetic

☞ for π, use

π =∞∑

n=0

[4

8n+1− 2

8n+4− 1

8n+5− 1

8n+6

]1

16n

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exp

☞ easy to get exp(x) with bounded absolute error if we know|x| < 1: use Padé approximants:

max|x|<1

| ex−Rn/n(x) | 6 8(n!)2

(2n)!(2n+1)!

. we can't test |x| < 1!. . .

. but we can safely find k such that |x/2k| < 1

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The big software challenge

Produce efficient software which hides thebottom-up data flow and can be used bya programmer as if it were conventionaltop-down code

☞ needed features:. λ anonymous function constructor. operator overloading

☞ not easily possible in Fortran, C, java, . . .

☞ easy in python, Haskell, clean, ML, ocaml

☞ possible in C++! (with tricks)

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My python implementation

☞ www.python.org

☞ `perl done right'

☞ much cleaner syntax than java

☞ portable (Linux, Unix, Mac, . . . , Windows)

☞ functional features - lambda, map, filter, reduce

☞ operator overloading

☞ free

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Lambdas

def +(x,y):# compute the sum of x and y for B=2return lambda n: (x(n+2)+y(n+2)+2)/4

def sqrt(x):# compute the square root of x for B=2

return lambda n: sqrt(x(2*n))

☞ possibility of distributed computation: a lambda may beexecuted remotely

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Comparison

This is a system for proving inequalities

. . . but, equality is undecidable

def <(x,y):# return true if x<y, false if x>y, else don’t returnn=1while 1:if x(n)<y(n)−1: return trueif x(n)>y(n)+1: return false

n+=1

def max(x,y):# compute the maximum of x and y without comparison

return (x+y+abs(x−y))/2

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Python implementation in action

> import XR> e=exp(XR(1))> print contfrac(e)[2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, 1, 1, 12, 1, 1, 14]> print cos(2*pi()/7).dec(50)0.62348980185873353052500488400423981063227473> print polyroot([8,4,−4,−1],0,1,0).dec(50)0.62348980185873353052500488400423981063227473> x=XR(Q(1,3))> for i in range(10):print 2*x>1x=4*x*(1−x)

0 1 0 1 0 1 1 0 0 1

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C++ implementation in action

#include "XR.h"

int main() {XR e=exp(XR(1));cout<<contfrac(e);cout<<cos(2*pi()/7).dec(50);ZZ coeffs[ ]={8,4,−4,−1};cout<<polyroot(coeffs,0,1,0).dec(50);x=XR(Q(1,3));for (int i=0; i<10; i++) {cout<<2*x>1;x=4*x*(1−x);}}

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My C implementation

☞ more.btexact.com/people/briggsk2/xrc.html

☞ portable ISO C

☞ Uses pointers to construct DAG representing compositions oflambdas

☞ Function call ust be written for every arithmetic operation

☞ C++ wrapper

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Features of my implementations

☞ python, C++, C

☞ all integer arithmetic, using gmp

☞ automatic caching of all intermediate results

☞ easily integrated with existing code

☞ C, C++ versions are fully compiled

☞ possible application areas:. computational number theory. computational geometry. computer-assisted proofs in analysis. computer algebra

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Computational geometry application

☞ Simple planar example: line through (x1, y1), (x2, y2); is thepoint (x0, y0) to the left or right of the line?

��

��

��

��

��

��

��

��

��

b (x1, y1)

b (x2, y2)

b (x0, y0)

☞ This is determined by sign ([(y1−y0)(x2−x1)−(x1−x0)(y2−y1)])

With XR arithmetic, the sign is alwaysdetermined correctly and with the minimalnecessary computation

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Simultaneous Diophantine approximation application

☞ typical subproblem: given. x1, x2 ∈ R. p1, p2, q ∈ Z. e1 = |qx1−p1|, e2 = |qx2−p2|

☞ determine whether e1 > e2. very efficiently solved in XR with trivial coding by user

☞ is exp(π√

163) an integer?

x=exp(pi*sqrt(XR(163)))

print x>floor(x) # prints ’True’

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Limitations

☞ memory demands

☞ non-incrementality

☞ the floor function does not terminate on integer inputs

☞ transcendental functions still incomplete

☞ no verified decimal output

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Mathematics?⇐⇒ software

☞ mathematics =⇒ mathematical software. Numerical analysis. Statistics. Computer algebra. Computational number theory. Combinatorics and graph theory. Finite groups. Theorem proving. . . .

☞ what about the other direction?

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Conclusion

☞ it's worth rethinking how we represent numbers and doarithmetic

☞ lots of future work possible:. granularity - automatic choice for B?. optimize caching strategy. log, sin, cos etc.. non-functional - graph method. parallel and multi-threaded techniques. characterize the set XR. complexity analysis (not assuming each ∗, + etc. has the same cost)

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Bibliography 1

☞ H-J Boehm, R Cartwright, M Riggle, and M J O'Donnell: Exactreal arithmetic: A case study in higher order programmingdev.acm.org/pubs/citations/proceedings/lfp/319838/p162-boehm/

☞ V Ménissier-Morain: Arithmétique exacte, conception, algo-rithmique et performances d'une implémentation informatiqueen précision arbitraire Paris thesis 1994 calfor.lip6.fr/∼vmm/

☞ J R Harrison: Theorem proving with the real numbers Cam-bridge thesis 1996 www.ftp.cl.cam.ac.uk/ftp/papers/reports/

☞ P Potts: Exact Real Arithmetic using Möbius Transformationswww.purplefinder.com/∼potts/

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Bibliography 2

☞ Computability and Complexity in Analysis Networkwww.informatik.fernuni-hagen.de/import/cca/

☞ K Weihrauch: Computable Analysiswww.informatik.fernuni-hagen.de/import/thi1/klaus.weihrauch/book.html

☞ P Liardet and P Stambul: Algebraic Computations with Con-tinued Fractions www.idealibrary.com/links/doi/10.1006/jnth.1998.2274

☞ K M Briggs: XR homepage more.btexact.com/people/briggsk2/XR.html;This talk: more.btexact.com/people/briggsk2/xr-kent-talk-pp.pdf

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