Interval package

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—Brian Hayes, DOI: 10.1511/2003.6.484
Standard floating point arithmetic Interval arithmetic
octave:1> 19 * 0.1 - 2 + 0.1
ans =  1.3878e-16
octave:2> 19 * x - 2 + x
ans ⊂ [-3.191891195797326e-16, +1.3877787807814457e-16]

Floating-point arithmetic, as specified by IEEE 754, is available in almost every computer system today. It is wide-spread, implemented in common hardware and integral part in programming languages. For example, the double-precision format is the default numeric data type in GNU Octave. Benefits are obvious: The results of arithmetic operations are well-defined and comparable between different systems and computation is highly efficient.

However, there are some downsides of floating-point arithmetic in practice, which will eventually produce errors in computations.

  • Floating-point arithmetic is often used mindlessly by developers. [1] [2] [3]
  • Even if the developer would be proficient, most developing environments / technologies limit floating-point 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 fixed-point arithmetic (COBOL, PL/I, …), where overflow and rounding can be precisely predicted at compile-time.
  • Results are system dependent. All but the most basic floating-point operations are not guaranteed to be accurate and produce different results depending on low level libraries and hardware. [7] [8]



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



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 high-level octave language using C99 or multi-precision floating-point routines, which is a lot slower than a hardware implementation [9]. Building interval arithmetic operations from floating-point 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 floating-point 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.

Approximate runtime (wall clock time in seconds) — Results have been produced with GNU Octave 3.8.2 and Interval package 0.1.4 on an Intel® Core™ i5-4340M CPU (2.9–3.6 GHz)
Interval matrix size plus times log pow mtimes mtimes inv
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 built-in floating-point 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 system-independent, 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.


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 octave-interval
cd octave-interval; make run

Quick start introduction

Input and output

Before exercising interval arithmetic, interval objects must be created from non-interval 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, 1.100000000000001]_com
infsupdec ("5.8e-17")
   # [5.799999999999999e-17, 5.800000000000001e-17]_com
midrad (12, 3) # [9, 15]_com
midrad ("4.2", "1e-7") # [4.199999899999999, 4.200000100000001]_com
hull (3, 42, "19.3", "-2.3") # [-2.300000000000001, +42]_com
hull ("pi", "e") # [2.718281828459045, 3.1415926535897936]_com

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.

Warning icon.svg
Code: Beware of the conversion pitfall
## The numeric constant “0.3” is an approximation of the
## decimal number 0.3.  An interval around this approximation
## will not contain the decimal number 0.3.
infsupdec (0.3) # [0.29999999999999998, 0.29999999999999999]_com
## However, passing the decimal number 0.3 as a string
## to the interval constructor will create an interval which
## actually encloses the decimal number.
infsupdec ("0.3") # [0.29999999999999998, 0.3000000000000001]_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"])
   # [0.09999999999999999, 0.1000000000000001]_com
   # [0.19999999999999998, 0.2000000000000001]_com
   # [0.29999999999999998, 0.3000000000000001]_com
   # [0.39999999999999996, 0.4000000000000001]_com
   #                                     [0.5]_com
infsup ({1, eps; "4/7", "pi"}, {2, 1; "e", "0xff"})
   #                                   [1, 2]   [2.220446049250313e-16, 1]
   # [0.5714285714285713, 2.7182818284590456]     [3.141592653589793, 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.


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)) # [.4794255386042029, .4794255386042031]_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 double-precision). 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 less-or-equal 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 element-wise 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 / low-level library support and missing optimizations, these operations are relatively slow compared to familiar operations in floating-point 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]     [0.25]       [-1.5, 0]
   # [0.5]   [-0.125]   [-1.5, -0.75]
   #   [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]
   # [0.18333333333333315, 0.18333333333333358]
   # [0.43333333333333318, 0.43333333333333346]
   # [0.18333333333333307, 0.18333333333333355]
A * c
   # [2.9999999999999964, 3.0000000000000036]
   #  [3.9999999999999964, 4.000000000000004]
   #   [4.999999999999997, 5.000000000000003]

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 non-empty 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 floating-point variants for some cases.

Code: Standard floating point arithmetic versus interval arithmetic on ill-conditioned 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 set-based 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 set-based 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]

There are some situations where the interval package cannot decide whether an error occurred or not and issues a warning. The user may choose to ignore these warnings or handle them as errors, see help warning for instructions.

Warning ID Reason Possible consequences
interval:PossiblyUndefined Interval construction with boundaries in decimal format, and the constructor can't decide whether the lower boundary is smaller than the upper boundary. Both boundaries are very close and lie between subsequent binary64 numbers. The constructed interval is a valid and tight enclosure of both numbers. If the lower boundary was actually greater than the upper boundary, this illegal interval is not considered an error.
interval:ImplicitPromote An interval operation has been evaluated on both, a bare and a decorated interval. The bare interval has been converted into a decorated interval in order to produce a decorated result. Note: This warning does not occur if a bare interval literal string gets promoted into a decorated interval, e. g., infsupdec (1, 2) + "[3, 4]" does not produce this warning whereas infsupdec (1, 2) + infsup (3, 4) does. A bare interval can be explicitly promoted with the newdec [11] function. The implicit conversion applies the best possible decoration for the bare interval. If the bare interval has been produced from an interval arithmetic computation, this branch of computation is not covered by the decoration information and the final decoration could be considered wrong. For example, infsupdec (1, 2) + infsup (0, 1) ^ 0 would ignore that 00 is undefined.
interval:FixedDecoration During interval construction the desired decoration of the interval has been changed. An unbound interval cannot carry the common decoration. An empty interval must carry the trivial decoration. The interval's decoration is degraded as necessary. The input for the interval constructor could be considered illegal, e. g. [0, Inf]_com is no valid decorated interval literal.
interval:NaI An error has occured during interval construction and the NaI object has been produced. The warning text contains further details. A NaI can be explicitly created with the nai [12] function. Nothing bad is going to happen, because the semantics of NaI are well defined by IEEE 1788. However, the user might choose to cancel the algorithm immediately when the NaI is encountered for the first time.


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
(defined & continuous)
x is a nonempty subset of Dom(f)
always true (so gives no information)

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 # [-0.5, +0.5]_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) # [1.732050807568877, 2]_com
sqrt (y) # [0, 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.

  • Hexadecimal-floating-constant 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, § This can be used as a convenient way to define interval boundaries in double-precision, 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, where
    • m is a mantissa in decimal,
    • r is either empty (which means ½ ULP) or is a non-negative 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 either u (up) or d (down),
    • E is either empty or an exponent field comprising the character e followed by a decimal integer exponent (base 10).
Code: Examples of different formats during interval construction
infsup ("0x1.999999999999Ap-4") # hex-form
   # [0.1, 0.1000000000000001]
infsup ("1/3", "7/9") # rational form
   # [0.3333333333333333, 0.7777777777777778]
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

Reverse power operations. A relevant subset of the function's domain is outlined and hatched. In this example we use xy ∈ [2, 3].

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.


x ⊂ [1.6128979635153646, 2.714854726565793]
y ⊂ [0.7564707973660299, 1.4440113978403289]

Tips & Tricks


octave:1> infsupdec ("17.7") + 1
ans ⊂ [18.699999999999999, 18.70000000000001]_com
octave:2> ans + "[0, 2]"
ans ⊂ [18.699999999999999, 20.70000000000001]_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 element-wise and checks if an interval equals the empty set.

octave:1> builtin ("isempty", empty ()), isempty (empty ())
ans = 0
ans =  1

IEEE 1788 index

In terms of a better integration into the GNU Octave language, several operations use a function name which is different from the name proposed in the standard document.
IEEE 1788 GNU Octave
setDec infsupdec
numsToInterval infsup [13]
textToInterval infsup or
exp10 pow10 [14]
exp2 pow2 [15]
recip inv [16]
sqrt realsqrt [17]
rootn nthroot [18]
logp1 log1p [19]
roundTiesToAway round [20]
roundTiesToEven roundb [21]
trunc fix [22]
sum on intervals: sum [23]
on numbers: mpfr_vector_sum_d [24]
dot on intervals: dot [25]
on numbers: mpfr_vector_dot_d [26]
sumAbs on intervals: sumabs [27]
on numbers: use mpfr_vector_sum_d (roundingMode, abs (x))
sumSquare on intervals: sumsq [28]
on numbers: use mpfr_vector_dot_d (roundingMode, abs (x), abs (x))
intersection intersect [29]
convexHull union [30]
mulRevToPair mulrev [31] with two output parameters


The interval package's main goal is to be compliant with IEEE 1788, so it is compatible with other standard-conforming implementations (on the set of operations described by the standard document).


  • INTLAB is not conforming to IEEE 1788 and the parsing of intervals from strings uses a different format—especially for the uncertain form
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 [32]. 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 C-XSC.


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 [33].