Interval arithmetic: an introduction Standardization of interval arithmetic: IEEE P1788 Conclusion IEEE-1788 standardization of interval arithmetic: introduction - link with MPFI Nathalie Revol INRIA - Universit´ede Lyon LIP (UMR 5668 CNRS - ENS Lyon - INRIA - UCBL) Third MPFR-MPC Developers Meeting Nancy, 20-22 January 2014 Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Standardization of interval arithmetic: IEEE P1788 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 NathalieInterval Revol - INRIANewton - Universit´ede iteration Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros 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 NathalieInterval Revol - INRIANewton - Universit´ede iteration Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros 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 NathalieInterval Revol - INRIANewton - Universit´ede iteration Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros 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. Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Definitions: intervals Objects: I intervals of real numbers = closed connected sets of R I interval for π: [3:14159; 3:14160] I data d measured with an absolute error less than ±": [d − "; d + "] I interval vector: components = intervals; also called box [0 ; 2] [0;2] [0 ; 2] [4 ; 5] [4 ; 4.5] [−6 ; −5] 5 4.5 4 4 −5 0 2 −6 0 2 0 2 I interval matrix: components = intervals. Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Definitions: operations x y = Hullfx y : x 2 x; y 2 yg Arithmetic and algebraic operations: use the monotonicity [x; x] + y; y = x + y; x + y [x; x] − y; y = x − y; x − y [x; x] × y; y = min(x × y; x × y; x × y; x × y); max(ibid.) [x; x]2 = min(x2; x2); max(x2; x2) if 0 62 [x; x] 0; max(x2; x2) otherwise Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Definitions: functions Definition: an interval extension f of a function f satisfies 8x; f (x) ⊂ f(x); and 8x; f (fxg) = f(fxg): Elementary functions: again, use the monotonicity. exp x = [exp x; exp x] log x = [log x; log x] if x ≥ 0; [−∞; log x] if x > 0 sin[π=6; 2π=3] = [1=2; 1] ::: Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Interval arithmetic: implementation using floating-point arithmetic Implementation using floating-point arithmetic: use directed rounding modes (cf. IEEE 754 standard) p p p[2; 3] = [5 2; 4 3] Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros 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? Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros 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 NathalieInterval Revol - INRIANewton - Universit´ede iteration Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Cons: overestimation (1/2) The result encloses the true result, but it is too large: overestimation phenomenon. Two main sources: variable dependency and wrapping effect. (Loss of) Variable dependency: x − x = fx − y : x 2 x; y 2 xg= 6 fx − x : x 2 xg = f0g: Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Cons: overestimation (2/2) Wrapping effect f(X) F(X) image of f (x) 2 successive rotations of π=4 with f : R2 ! R2 of the little central square Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Cons: complexity and efficiency Complexity: most problems are NP-hard (Gaganov, Rohn, Kreinovich. ) I evaluate a function on a box. even up to " 4 I solve a linear system. even up to 1=4n I determine if the solution of a linear system is bounded Efficiency Implementation using floating-point arithmetic: use directed roundings, towards ±∞. Overhead in execution time: in theory, at most 4, or 8, cf. [x; x] × y; y = [ min(RD(x × y); RD(x × y); RD( x × y); RD( x × y)); max(RU(x × y); RU(x × y); RU( x × y); RU( x × y)) in practice, around 20: changing the rounding modes implies flushing the pipelines (on most architectures and implementations). Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Pros: set computing Computing with whole sets or with sets enclosing uncertainties. Behaviour safe? On x, are the extrema of the function f 1 controllable? dangerous? > f , < f2? f 1 f f(x) f f 2 x 1 always controllable. No if f (x) = [f ; f ] ⊂ [f2; f ]. Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros Conclusion Standardization Pros: Brouwer-Schauder theorem A function f which is continuous on the unit ball B and which satisfies f (B) ⊂ B has a fixed point on B. Furthermore, if f (B) ⊂ intB then f has a unique fixed point on B. f(K) K The theorem remains valid if B is replaced by a compact K and in particular an interval. Nathalie Revol - INRIA - Universit´ede Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction Standardization of interval arithmetic: IEEE P1788 Cons and pros 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 NathalieInterval Revol - INRIANewton - Universit´ede iteration Lyon - LIP IEEE-1788 standardization of interval arithmetic Interval arithmetic: an introduction Introduction