IEEE-1788 standardization of interval arithmetic

117 Pages·2014·0.74 MB·English

Intervalarithmetic:anintroduction
Standardizationofintervalarithmetic:IEEEP1788
Conclusion
IEEE-1788 standardization
of interval arithmetic:
introduction - link with MPFI
Nathalie Revol
INRIA - Universit´e de Lyon
LIP (UMR 5668 CNRS - ENS Lyon - INRIA - UCBL)
Third MPFR-MPC Developers Meeting
Nancy, 20-22 January 2014
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction
Standardizationofintervalarithmetic:IEEEP1788
Conclusion
Agenda
Interval arithmetic: an introduction
Introduction
Cons and pros
Standardization
Standardization of interval arithmetic: IEEE P1788
Facts about the working group
Overview of the IEEE-1788 standard
Intervals
Operations
Predicates
Exceptions and decorations
Flavours
Conclusion
Exceptions and decorations
Interval Newton iteration
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction Introduction
Standardizationofintervalarithmetic:IEEEP1788 Consandpros
Conclusion Standardization
Agenda
Interval arithmetic: an introduction
Introduction
Cons and pros
Standardization
Standardization of interval arithmetic: IEEE P1788
Facts about the working group
Overview of the IEEE-1788 standard
Intervals
Operations
Predicates
Exceptions and decorations
Flavours
Conclusion
Exceptions and decorations
Interval Newton iteration
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction Introduction
Standardizationofintervalarithmetic:IEEEP1788 Consandpros
Conclusion Standardization
Agenda
Interval arithmetic: an introduction
Introduction
Cons and pros
Standardization
Standardization of interval arithmetic: IEEE P1788
Facts about the working group
Overview of the IEEE-1788 standard
Intervals
Operations
Predicates
Exceptions and decorations
Flavours
Conclusion
Exceptions and decorations
Interval Newton iteration
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction Introduction
Standardizationofintervalarithmetic:IEEEP1788 Consandpros
Conclusion Standardization
A brief introduction
Interval arithmetic:
instead of numbers, use intervals and compute.
Fundamental theorem of interval arithmetic:
(or “Thou shalt not lie”) (or “Inclusion property”):
the exact result (number or set) is contained in the computed
interval.
No result is lost, the computed interval is guaranteed to contain
every possible result.
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction Introduction
Standardizationofintervalarithmetic:IEEEP1788 Consandpros
Conclusion Standardization
Definitions: intervals
Objects:
(cid:73) intervals of real numbers = closed connected sets of R
(cid:73) interval for π: [3.14159,3.14160]
(cid:73) data d measured with an absolute error less than ±ε:
[d −ε,d +ε]
(cid:73) interval vector: components = intervals; also called box
[0 ; 2] [0;2]
[0 ; 2]
[4 ; 4.5]
[4 ; 5]
[−6 ; −5]
5
4.5
4 4 −5
0 2
−6
0 2 0 2
(cid:73) interval matrix: components = intervals.
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction Introduction
Standardizationofintervalarithmetic:IEEEP1788 Consandpros
Conclusion Standardization
Definitions: operations
x(cid:5)y = Hull{x (cid:5)y : x ∈ x,y ∈ y}
Arithmetic and algebraic operations: use the monotonicity
(cid:2) (cid:3) (cid:2) (cid:3)
[x, x]+ y, y = x +y, x + y
(cid:2) (cid:3) (cid:2) (cid:3)
[x, x]− y, y = x − y, x −y
(cid:2) (cid:3) (cid:2) (cid:3)
[x, x]× y, y = min(x ×y,x × y, x ×y, x × y),max(ibid.)
[x, x]2 = (cid:2)min(x2, x2),max(x2, x2)(cid:3) if 0 (cid:54)∈ [x, x]
(cid:2)0,max(x2, x2)(cid:3) otherwise
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction Introduction
Standardizationofintervalarithmetic:IEEEP1788 Consandpros
Conclusion Standardization
Definitions: functions
Definition:
an interval extension f of a function f satisfies
∀x, f(x) ⊂ f(x), and ∀x, f({x}) = f({x}).
Elementary functions: again, use the monotonicity.
expx = [expx,exp x]
logx = [logx,log x] if x ≥ 0,[−∞,log x] if x > 0
sin[π/6,2π/3] = [1/2,1]
...
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction Introduction
Standardizationofintervalarithmetic:IEEEP1788 Consandpros
Conclusion Standardization
Interval arithmetic:
implementation using floating-point arithmetic
Implementation using floating-point arithmetic:
use directed rounding modes (cf. IEEE 754 standard)
√ √
(cid:112)
[2,3] = [(cid:53) 2,(cid:52) 3]
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic
Intervalarithmetic:anintroduction Introduction
Standardizationofintervalarithmetic:IEEEP1788 Consandpros
Conclusion Standardization
Definitions: function extension
f(x) = x2−x +1 = x(x −1)+1 = (x −1/2)2+3/4 on [−2,1].
Using x2−x +1, one gets
[−2,1]2−[−2,1]+1=[0,4]+[−1,2]+1= [0,7].
Using x(x −1)+1, one gets
[−2,1]·([−2,1]−1)+1=[−2,1]·[−3,0]+1=[−3,6]+1= [−2,7].
Using (x −1/2)2+3/4, one gets
([−2,1]−1/2)2+3/4=[−5/2,1/2]2+3/4=[0,25/4]+3/4=
[3/4,7] = f([−2,1]).
Problem with this definition: infinitely many interval extensions,
syntactic use (instead of semantic).
How to choose the best extension? A good one?
NathalieRevol-INRIA-Universit´edeLyon-LIP IEEE-1788standardizationofintervalarithmetic