Difference between revisions of "Interval package"
(Split introduction into simple and advanced topics, added syntax highlighting, use decorated intervals per default) 
m (→Related work: Fixed libieeep1788 name) 

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== Related work ==  == Related work ==  
−  For C++ there is an open source interval library [https://github.com/nehmeier/libieeep1788/  +  For C++ there is an open source interval library [https://github.com/nehmeier/libieeep1788/ libieeep1788] by Marco Nehmeier (member of IEEE P1788). It aims to be standard compliant with IEEE 1788 and is designed in a modular way, supporting several interval data types and different flavors of interval arithmetic [http://www.youtube.com/watch?v=GOa9aWAZO_Q]. The GNU Octave interval package shares several unit tests with libieeep1788. 
For C++, Pascal and Fortran there is a free interval library [http://www2.math.uniwuppertal.de/~xsc/xscsprachen.html XSC]. It is not standard compliant with IEEE 1788. Some parts of the GNU Octave interval package have been derived from CXSC.  For C++, Pascal and Fortran there is a free interval library [http://www2.math.uniwuppertal.de/~xsc/xscsprachen.html XSC]. It is not standard compliant with IEEE 1788. Some parts of the GNU Octave interval package have been derived from CXSC. 
Revision as of 07:08, 22 March 2015
The GNU Octave interval package provides data types and fundamental operations for real valued interval arithmetic based on the common floatingpoint format “binary64” a. k. a. doubleprecision. It aims to be standard compliant with the (upcoming) IEEE 1788 and therefore implements the setbased interval arithmetic flavor. Interval arithmetic produces mathematically proven numerical results.
Motivation
Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. … An interval computation yields a pair of numbers, an upper and a lower bound, which are guaranteed to enclose the exact answer. Maybe you still don’t know the truth, but at least you know how much you don’t know.—Brian Hayes, DOI: 10.1511/2003.6.484
Standard floating point arithmetic  Interval arithmetic 

octave:1> 19 * 0.1  2 + 0.1
ans = 1.3878e16

octave:1> x = infsup ("0.1"); octave:2> 19 * x  2 + x ans ⊂ [3.1918911957973251e16, +1.3877787807814457e16] 
Floatingpoint arithmetic, as specified by IEEE 754, is available in almost every computer system today. It is widespread, implemented in common hardware and integral part in programming languages. For example, the doubleprecision format is the default numeric data type in GNU Octave. Benefits are obvious: The results of arithmetic operations are welldefined and comparable between different systems and computation is highly efficient.
However, there are some downsides of floatingpoint arithmetic in practice, which will eventually produce errors in computations.
 Floatingpoint arithmetic is often used mindlessly by developers. [1] [2] [3]
 The binary data types categorically are not suitable for doing financial computations. Very often representational errors are introduced when using “real world” decimal numbers. [4]
 Even if the developer would be proficient, most developing environments / technologies limit floatingpoint arithmetic capabilities to a very limited subset of IEEE 754: Only one or two data types, no rounding modes, missing functions … [5]
 Results are hardly predictable. [6] All operations produce the best possible accuracy at runtime, this is how a floating point works. Contrariwise, financial computer systems typically use a fixedpoint arithmetic (COBOL, PL/I, …), where overflow and rounding can be precisely predicted at compiletime.
 Results are system dependent. All but the most basic floatingpoint operations are not guaranteed to be accurate and produce different results depending on low level libraries and hardware. [7] [8]
 If you do not know the technical details (cf. first bullet) you ignore the fact that the computer lies to you in many situations. For example, when looking at numerical output and the computer says “
ans = 0.1
,” this is not absolutely correct. In fact, the value is only close enough to the value 0.1. Additionally, many functions produce limit values (∞ × −∞ = −∞, ∞ ÷ 0 = ∞, ∞ ÷ −0 = −∞, log (0) = −∞), which is sometimes (but not always!) useful when overflow and underflow occur.
Interval arithmetic addresses above problems in its very special way and introduces new possibilities for algorithms. For example, the interval newton method is able to find all zeros of a particular function.
Theory
Online Introductions
Interval analysis in MATLAB Note: The INTLAB toolbox for Matlab is not entirely compatible with this interval package for GNU Octave, cf. #Compatibility. However, basic operations can be compared and should be compatible for common intervals.
Moore's fundamental theroem of interval arithmetic
Let y = f(x) be the result of intervalevaluation of f over a box x = (x_{1}, … , x_{n}) using any interval versions of its component library functions. Then
 In all cases, y contains the range of f over x, that is, the set of f(x) at points of x where it is defined: y ⊇ Rge(f  x) = {f(x)  x ∈ x ∩ Dom(f) }
 If also each library operation in f is everywhere defined on its inputs, while evaluating y, then f is everywhere defined on x, that is Dom(f) ⊇ x.
 If in addition, each library operation in f is everywhere continuous on its inputs, while evaluating y, then f is everywhere continuous on x.
 If some library operation in f is nowhere defined on its inputs, while evaluating y, then f is nowhere defined on x, that is Dom(f) ∩ x = Ø.
What to expect
The interval arithmetic provided by this interval package is slow, but accurate.
Why is the interval package slow? All arithmetic interval operations are simulated in highlevel octave language using C99 or multiprecision floatingpoint routines, which is a lot slower than a hardware implementation [9]. Building interval arithmetic operations from floatingpoint routines is easy for simple monotonic functions, e. g., addition and subtraction, but is complex for others, e. g., interval power function, atan2, or reverse functions. For some interval operations it is not even possible to rely on floatingpoint routines, since not all required routines are available in C99 or BLAS.
For example, multiplication of matrices with many elements becomes unfeasible as it takes a lot of time.
Interval matrix size  plus

times

log

pow

mtimes

mtimes

inv


tightest accuracy 
tightest accuracy 
tightest accuracy 
tightest accuracy 
valid accuracy 
tightest accuracy 
valid accuracy  
10 × 10  < 0.001  < 0.001  0.001  0.008  0.001  0.002  0.025 
100 × 100  0.003  0.010  0.055  0.61  0.012  0.53  0.30 
500 × 500  0.060  0.24  1.3  15  0.30  63  4.2 
Why is the interval package accurate? The GNU Octave builtin floatingpoint routines are not useful for interval arithmetic: Their results depend on hardware, system libraries and compilation options. The interval package handles all arithmetic functions with the help of the GNU MPFR library. With MPFR it is possible to compute systemindependent, valid and tight enclosures of the correct results for most functions. However, it should be noted that some reverse operations and matrix operations do not exists in GNU MPFR and therefore cannot be computed with the same accuracy.
Installation
The interval package is available at Octave Forge [10] and can be installed from within GNU Octave (version ≥ 3.8.2). During installation parts of the package are compiled for the target system, which requires the GNU MPFR development libraries (version ≥ 3.1.0) to be installed.
octave:1> pkg install forge interval octave:2> pkg load interval
The development version may be obtained from its mercurial repository. For convenience a Makefile target allows running the package from source.
hg clone http://hg.code.sf.net/p/octave/interval octaveinterval cd octaveinterval; make run
Quick start introduction
Input and output
Before exercising interval arithmetic, interval objects must be created from noninterval data. There are interval constants empty
and entire
and the interval constructors infsupdec
(create an interval from boundaries), midrad
(create an interval from midpoint and radius) and hull
(create an interval enclosure for a list of mixed arguments: numbers, intervals or interval literals). The constructors are very sophisticated and can be used with several kinds of parameters: Interval boundaries can be given by numeric values or string values with decimal numbers.
Code: Create intervals for performing interval arithmetic 
infsupdec (1) # [1]_com
infsupdec (1, 2) # [1, 2]_com
infsupdec ("3", "4") # [3, 4]_com
infsupdec ("1.1") # [1.0999999999999998, 1.1000000000000001]_com
infsupdec ("5.8e17")
# [5.799999999999999e17, 5.800000000000001e17]_com
midrad (12, 3) # [9, 15]_com
midrad ("4.2", "1e7") # [4.199999899999999, 4.2000001000000005]_com
hull (3, 42, "19.3", "2.3") # [2.3000000000000003, +42]_trv
hull ("pi", "e") # [2.718281828459045, 3.1415926535897936]_trv

The default text representation of intervals is not guaranteed to be exact, because this would massively spam console output. For example, the exact text representation of realmin
would be over 700 decimal places long! However, the default text representation is correct as it guarantees to contain the actual boundaries.
Code: Beware of the conversion pitfall 
## The numeric constant “0.2” is an approximation of the
## decimal number 0.2. An interval around this approximation
## will not contain the decimal number 0.2.
infsupdec (0.2) # [.20000000000000001, .20000000000000002]_com
## However, passing the decimal number 0.2 as a string
## to the interval constructor will create an interval which
## actually encloses the decimal number.
infsupdec ("0.2") # [.19999999999999998, .20000000000000002]_com

Interval vectors and matrices
Vectors and matrices of intervals can be created by passing numerical matrices, char vectors or cell arrays to the interval constructors. With cell arrays it is also possible to mix several types of boundaries.
Interval matrices behave like normal matrices in GNU Octave and can be used for broadcasting and vectorized function evaluation.
Code: Create interval matrices 
M = infsup (magic (3))
# [8] [1] [6]
# [3] [5] [7]
# [4] [9] [2]
infsup (magic (3), magic (3) + 1)
# [8, 9] [1, 2] [6, 7]
# [3, 4] [5, 6] [7, 8]
# [4, 5] [9, 10] [2, 3]
infsupdec (["0.1"; "0.2"; "0.3"; "0.4"; "0.5"])
# [.09999999999999999, .10000000000000001]_com
# [.19999999999999998, .20000000000000002]_com
# [.29999999999999998, .30000000000000005]_com
# [.39999999999999996, .40000000000000003]_com
# [.5]_com
infsup ({1, eps; "4/7", "pi"}, {2, 1; "e", "0xff"})
# [1, 2] [2.220446049250313e16, 1]
# [.5714285714285713, 2.7182818284590456] [3.1415926535897931, 255]

Arithmetic operations
The interval packages comprises many interval arithmetic operations. A complete list can be found in its function reference. Function names match GNU Octave standard functions where applicable and follow recommendations by IEEE 1788 otherwise, cf. IEEE 1788 index.
Arithmetic functions in a setbased interval arithmetic follow these rules: Intervals are sets. They are subsets of the set of real numbers. The interval version of an elementary function such as sin(x) is essentially the natural extension to sets of the corresponding pointwise function on real numbers. That is, the function is evaluated for each number in the interval where the function is defined and the result must be an enclosure of all possible values that may occur.
By default arithmetic functions are computed with best possible accuracy (which is more than what is guaranteed by GNU Octave core functions). The result will therefore be a tight and very accurate enclosure of the true mathematical value in most cases. Details on each function's accuracy can be found in its documentation, which is accessible with GNU Octave's help command.
Code: Examples of using interval arithmetic functions 
sin (infsupdec (0.5)) # [.47942553860420294, .47942553860420301]_com
pow (infsupdec (2), infsupdec (3, 4)) # [8, 16]_com
atan2 (infsupdec (1), infsupdec (1))
# [.7853981633974482, .7853981633974484]_com
midrad (magic (3), 0.5) * pascal (3)
# [13.5, 16.5]_com [25, 31]_com [42, 52]_com
# [13.5, 16.5]_com [31, 37]_com [55, 65]_com
# [13.5, 16.5]_com [25, 31]_com [38, 48]_com

Numerical operations
Some operations on intervals do not return an interval enclosure, but a single number (in doubleprecision). Most important are inf
and sup
, which return the lower and upper interval boundaries.
More such operations are mid
(approximation of the interval's midpoint), wid
(approximation of the interval's width), rad
(approximation of the interval's radius), mag
(interval's magnitude) and mig
(interval's mignitude).
Boolean operations
Interval comparison operations produce boolean results. While some comparisons are especially for intervals (subset, interior, ismember, isempty, disjoint, …) others are extensions of simple numerical comparison. For example, the lessorequal comparison is mathematically defined as ∀_{a} ∃_{b} a ≤ b ∧ ∀_{b} ∃_{a} a ≤ b.
octave:1> infsup (1, 3) <= infsup (2, 4)
ans = 1
Matrix operations
Above mentioned operations can also be applied elementwise to interval vectors and matrices. Many operations use vectorization techniques.
In addition, there are matrix operations on interval matrices. These operations comprise: dot product, matrix multiplication, vector sums (all with tightest accuracy), matrix inversion, matrix powers, and solving linear systems (the latter are less accurate). As a result of missing hardware / lowlevel library support and missing optimizations, these operations are relatively slow compared to familiar operations in floatingpoint arithmetic.
Code: Examples of using interval matrix functions 
A = infsup ([1, 2, 3; 4, 0, 0; 0, 0, 1]); A (2, 3) = "[0, 6]"
# [1] [2] [3]
# [4] [0] [0, 6]
# [0] [0] [1]
B = inv (A)
# [0] [.25] [1.5, 0]
# [.5] [.125] [1.5, .75]
# [0] [0] [1]
A * B
# [1] [0] [1.5, +1.5]
# [0] [1] [6, +6]
# [0] [0] [1]
A = infsup (magic (3))
# [8] [1] [6]
# [3] [5] [7]
# [4] [9] [2]
c = A \ [3; 4; 5]
# [.18333333333333326, .18333333333333349]
# [.43333333333333329, .43333333333333341]
# [.18333333333333315, .18333333333333338]
A * c
# [2.9999999999999982, 3.0000000000000018]
# [3.9999999999999982, 4.0000000000000018]
# [4.9999999999999982, 5.0000000000000018]

Notes on linear systems
A system of linear equations in the form Ax = b with intervals can be seen as a range of classical linear systems, which can be solved simultaneously. Whereas classical algorithms compute an approximation for a single solution of a single linear system, interval algorithms compute an enclosure for all possible solutions of (possibly several) linear systems. Some characteristics should definitely be known when linear interval systems are solved:
 If the linear system is underdetermined and has infinitely many solutions, the interval solution will be unbound in at least one of its coordinates. Contrariwise, from an unbound result it can not be concluded whether the linear system is underdetermined or has solutions.
 If the interval result is empty in at least one of its coordinates, the linear system is guaranteed to be underdetermined and has no solutions. Contrariwise, from a nonempty result it can not be concluded whether all or some of the systems have solutions or not.
 Wide intervals within the matrix A can easily lead to a superposition of cases, where the rank of A is no longer unique. If the linear interval system contains cases of linear independent equations as well as linear dependent equations, the resulting enclosure of solutions will inevitably be very broad.
However, solving linear systems with interval arithmetic can produce useful results in many cases and automatically carries a guaranty for error boundaries. Additionally, it can give better information than the floatingpoint variants for some cases.
Code: Standard floating point arithmetic versus interval arithmetic on illconditioned linear systems 
A = [1, 0; 2, 0];
## This linear system has no solutions
A \ [3; 0]
# warning: matrix singular to machine precision, rcond = 0
# 0.60000
# 0.00000
## This linear system has many solutions
A \ [4; 8]
# 4
# 0
## The empty interval vector proves that there is no solution
infsup (A) \ [3; 0]
# [Empty]
# [Empty]
## The unbound interval vector indicates that there may be many solutions
infsup (A) \ [4; 8]
# [4]
# [Entire]

Advanced topics
Error handling
Due to the nature of setbased interval arithmetic, one should not observe errors (in the sense of raised GNU Octave error messages) during computation unless operations are evaluated for incompatible data types. Arithmetic operations which are not defined for (parts of) their input, simply ignore anything that is outside of their domain.
However, the interval constructors can produce errors depending on the input. The infsup
constructor will fail if the interval boundaries are invalid. Contrariwise, the (preferred) infsupdec
, midrad
and hull
constructors will only issue a warning and return a [NaI] object, which will propagate and survive through computations. NaI stands for “not an interval”.
Code: Effects of setbased interval arithmetic on partial functions and the NaI object 
## Evaluation of a function outside of its domain returns an empty interval
infsupdec (2) / 0 # [Empty]_trv
infsupdec (0) ^ infsupdec (0) # [Empty]_trv
## Illegal interval construction creates a NaI
infsupdec (3, 2) # [NaI]
## NaI even survives through computations
ans + 1 # [NaI]

Decorations
The interval package provides a powerful decoration system for intervals, as specified by the IEEE standard for interval arithmetic. By default any interval carries a decoration, which collects additional information about the course of function evaluation on the interval data.
Only the (unfavored) infsup
constructor creates bare, undecorated intervals and the intervalpart
operation may be used to demote decorated intervals into bare, undecorated ones. It is highly recommended to always use the decorated interval arithmetic, which gives additional information about an interval result in exchange for a tiny overhead.
The following decorations are available:
Decoration  Bounded  Continuous  Defined  Definition 

com (common) 
✓  ✓  ✓  x is a bounded, nonempty subset of Dom(f); f is continuous at each point of x; and the computed interval f(x) is bounded 
dac (defined & continuous) 
✓  ✓  x is a nonempty subset of Dom(f); and the restriction of f to x is continuous  
def (defined) 
✓  x is a nonempty subset of Dom(f)  
trv (trivial) 
always true (so gives no information)  
ill (illformed) 
Not an interval, at least one interval constructor failed during the course of computation 
The decoration information is especially useful after a very long and complicated function evaluation. For example, when the “def” decoration survives until the final result, it is proven that the overall function is actually defined for all values covered by the input intervals.
Code: Examples of using the decoration system 
x = infsupdec (3, 4) # [3, 4]_com
y = x  3.5 # [.5, +.5]_com
## The square root function ignores any negative part of the input,
## but the decoration indicates whether this has or has not happened.
sqrt (x) # [1.7320508075688771, 2]_com
sqrt (y) # [0, .7071067811865476]_trv

Specialized interval constructors
Above mentioned interval construction with decimal numbers or numeric data is straightforward. Beyond that, there are more ways to define intervals or interval boundaries.
 Hexadecimalfloatingconstant form: Each interval boundary may be defined by a hexadecimal number (optionally containing a point) and an exponent field with an integral power of two as defined by the C99 standard (ISO/IEC9899, N1256, §6.4.4.2). This can be used as a convenient way to define interval boundaries in doubleprecision, because the hexadecimal form is much shorter than the decimal representation of many numbers.
 Rational literals: Each interval boundary may be defined as a fraction of two decimal numbers. This is especially useful if interval boundaries shall be tightest enclosures of fractions, that would be hard to write down as a decimal number.
 Uncertain form: The interval as a whole can be defined by a midpoint or upper/lower boundary and an integral number of “units in last place” (ULPs) as an uncertainty. The format is
m?ruE
, wherem
is a mantissa in decimal,r
is either empty (which means ½ ULP) or is a nonnegative decimal integral ULP count or is the?
character (for unbounded intervals),u
is either empty (symmetrical uncertainty of r ULPs in both directions) or is eitheru
(up) ord
(down),E
is either empty or an exponent field comprising the charactere
followed by a decimal integer exponent (base 10).
Code: Examples of different formats during interval construction 
infsup ("0x1.999999999999Ap4") # hexform
# [.1, .10000000000000001]
infsup ("1/3", "7/9") # rational form
# [.33333333333333331, .7777777777777778]
infsup ("121.2?") # uncertain form
# [121.14999999999999, 121.25]
infsup ("5?32e2") # uncertain form with ulp count
# [2700, +3700]
infsup ("42??u") # unbound uncertain form
# [42, +Inf]

Reverse arithmetic operations
Some arithmetic functions also provide reverse mode operations. That is inverse functions with interval constraints. For example the sqrrev
can compute the inverse of the sqr
function on intervals. The syntax is sqrrev (C, X)
and will compute the enclosure of all numbers x ∈ X that fulfill the constraint x² ∈ C.
In the following example, we compute the constraints for base and exponent of the power function pow
as shown in the figure.
octave:1> x = powrev1 (infsup ("[1.1, 1.45]"), infsup (2, 3)) x ⊂ [1.6128979635153646, 2.7148547265657915] octave:2> y = powrev2 (infsup ("[2.14, 2.5]"), infsup (2, 3)) y ⊂ [.7564707973660299, 1.4440113978403284]
Tips & Tricks
For convenience it is possible to implicitly call the interval constructor during all interval operations if at least one input already is an interval object.
octave:1> infsupdec ("17.7") + 1 ans ⊂ [18.699999999999999, 18.700000000000003]_com octave:2> ans + "[0, 2]" ans ⊂ [18.699999999999999, 20.700000000000003]_com
Interval functions with only one argument can be called by using property syntax, e. g. x.inf
, x.sup
or even x.sqr
.
When matrices are resized using subscripted assignment, any implicit new matrix elements will carry an empty interval.
octave:1> M = infsup (magic (3)); M (4, 4) = 42
M = 4×4 interval matrix
[8] [1] [6] [Empty]
[3] [5] [7] [Empty]
[4] [9] [2] [Empty]
[Empty] [Empty] [Empty] [42]
Whilst most functions (size
, isvector
, ismatrix
, …) work as expected on interval data types, the function isempty
is evaluated elementwise and checks if an interval equals the empty set.
octave:1> builtin ("isempty", empty ()), isempty (empty ())
ans = 0
ans = 1
IEEE 1788 index
IEEE 1788  GNU Octave 

newDec  infsupdec [11] 
setDec  infsupdec 
numsToInterval  infsup [12] 
textToInterval  infsup or infsupdec 
exp10  pow10 [13] 
exp2  pow2 [14] 
recip  inv [15] 
sqrt  realsqrt [16] 
rootn  nthroot [17] 
logp1  log1p [18] 
roundTiesToAway  round [19] 
roundTiesToEven  roundb [20] 
trunc  fix [21] 
sum  on intervals: sum [22] on numbers: mpfr_vector_sum_d [23] 
dot  on intervals: dot [24] on numbers: mpfr_vector_dot_d [25] 
sumAbs  on intervals: sumabs [26] on numbers: use mpfr_vector_sum_d (roundingMode, abs (x)) 
sumSquare  on intervals: sumsq [27] on numbers: use mpfr_vector_dot_d (roundingMode, abs (x), abs (x)) 
intersection  and (& ) [28]

convexHull  or ( ) [29]

mulRevToPair  mulrev [30] with two output parameters 
Compatibility
The interval package's main goal is to be compliant with IEEE 1788, so it is compatible with other standardconforming implementations (on the set of operations described by the standard document).
This interval package is not meant to be a replacement for INTLAB and any compatibility with it is pure coincidence. Since both are compatible with GNU Octave, they happen to agree on many function names and programs written for INTLAB may possibly run with this interval package as well. Some fundamental differences that I am currently aware of:
 INTLAB is nonfree software, it grants none of the four essential freedoms of free software
 INTLAB is not conforming to IEEE 1788 and the parsing of intervals from strings uses a different format—especially for the uncertain form
 INTLAB supports intervals with complex numbers and sparse interval matrices, but no empty intervals
 INTLAB uses inferior accuracy for most arithmetic operations, because it focuses on speed
 Basic operations can be found in both packages, but the availability of special functions depends
Code: In GNU Octave the interval package can also be run alongside INTLAB. 
# INTLAB intervals
A1 = infsup (2, 3);
B1 = hull (4, A1);
C1 = midrad (0, 2);
# Interval package intervals
pkg load interval
A2 = infsup (2, 3);
B2 = hull (4, A2);
C2 = midrad (0, 2);
pkg unload interval
# Computation with INTLAB
A1 + B1 * C1
# Computation without INTLAB
A2 + B2 * C2

Related work
For C++ there is an open source interval library libieeep1788 by Marco Nehmeier (member of IEEE P1788). It aims to be standard compliant with IEEE 1788 and is designed in a modular way, supporting several interval data types and different flavors of interval arithmetic [31]. The GNU Octave interval package shares several unit tests with libieeep1788.
For C++, Pascal and Fortran there is a free interval library XSC. It is not standard compliant with IEEE 1788. Some parts of the GNU Octave interval package have been derived from CXSC.
For Java there is a library jinterval by Dmitry Nadezhin (member of IEEE P1788). It aims to be standard compliant with IEEE 1788, but is not complete yet.
For MATLAB there is a popular, nonfree interval arithmetic toolbox INTLAB by Siegfried Rump. It had been free of charge for academic use in the past, but no longer is. Its origin dates back to 1999, so it is well tested and comprises a lot of functionality, especially for vector / matrix operations. INTLAB is compatible with GNU Octave since Version 9 [32].