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− | {{ | + | The GNU Octave [http://octave.sourceforge.net/interval/ interval package] provides data types and fundamental operations for real valued interval arithmetic based on the common floating-point format “binary64” a. k. a. double-precision. It aims to be standard compliant with the (upcoming) [http://standards.ieee.org/develop/project/1788.html IEEE 1788] and therefore implements the ''set-based'' interval arithmetic flavor. '''Interval arithmetic''' produces mathematically proven numerical results. |

− | | | + | |

− | | | + | == Motivation == |

− | | | + | |

− | + | {{quote|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|[http://dx.doi.org/10.1511/2003.6.484 DOI: 10.1511/2003.6.484]}} | |

− | + | ||

− | | | + | {| class="wikitable" style="margin: auto" |

− | + | !Standard floating point arithmetic | |

− | + | !Interval arithmetic | |

− | + | |- | |

− | + | | style = "vertical-align: top" | | |

− | | | + | <span style="opacity:.5">octave:1> </span>19 * 0.1 - 2 + 0.1 |

− | + | ans = 1.3878e-16 | |

− | + | | style = "vertical-align: top" | | |

− | + | <span style="opacity:.5">octave:1> </span>x = infsup ("0.1"); | |

− | + | <span style="opacity:.5">octave:2> </span>19 * x - 2 + x | |

− | | | + | ans ⊂ [-3.191891195797326e-16, +1.3877787807814457e-16] |

− | + | |} | |

− | | | + | |

+ | Floating-point arithmetic, as specified by [http://en.wikipedia.org/wiki/IEEE_floating_point 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. [http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html] [http://www.cs.berkeley.edu/~wkahan/Mindless.pdf] [http://www.ima.umn.edu/~arnold/disasters/] | ||

+ | * The binary data types categorically are not suitable for doing financial computations. Very often representational errors are introduced when using “real world” decimal numbers. [http://en.wikipedia.org/wiki/Decimal_computer] | ||

+ | * 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 … [http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf] | ||

+ | * Results are hardly predictable. [https://hal.archives-ouvertes.fr/hal-00128124/en/] All operations produce the best possible accuracy ''at runtime'', this is how a floating point works. Contrariwise, financial computer systems typically use a [http://en.wikipedia.org/wiki/Fixed-point_arithmetic 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. [http://www.gnu.org/software/libc/manual/html_node/Errors-in-Math-Functions.html#Errors-in-Math-Functions] [http://developer.amd.com/tools-and-sdks/cpu-development/libm/] | ||

+ | * 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 “<code>ans = 0.1</code>,” 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 [http://en.wikipedia.org/wiki/Interval_arithmetic#Interval_Newton_method interval newton method] is able to find ''all'' zeros of a particular function. | ||

+ | |||

+ | == Theory == | ||

+ | |||

+ | === Online Introductions === | ||

+ | |||

+ | [http://www.maths.manchester.ac.uk/~higham/narep/narep416.pdf 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 | ||

+ | interval-evaluation of ''f'' over a box '''''x''''' = (''x''<sub>1</sub>, … , ''x''<sub>''n''</sub>) | ||

+ | 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 high-level octave language using C99 or multi-precision floating-point routines, which is a lot slower than a hardware implementation [https://books.google.de/books?id=JTc4XdXFnQIC&pg=PA61]. 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., [http://exp.ln0.de/heimlich-power-2011.htm interval power function], atan2, or [[#Reverse_arithmetic_operations|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. | ||

+ | |||

+ | {| class="wikitable" style="margin: auto; align:right" | ||

+ | |+ Approximate runtime (wall clock time in seconds) <span style="font-weight: normal">— 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)</span> | ||

+ | !rowspan="2"|Interval matrix size | ||

+ | !valign="top" style="min-width:6em"|<code>plus</code> | ||

+ | !valign="top" style="min-width:6em"|<code>times</code> | ||

+ | !valign="top" style="min-width:6em"|<code>log</code> | ||

+ | !valign="top" style="min-width:6em"|<code>pow</code> | ||

+ | !valign="top" style="min-width:6em"|<code>mtimes</code> | ||

+ | !valign="top" style="min-width:6em"|<code>mtimes</code> | ||

+ | !valign="top" style="min-width:6em"|<code>inv</code> | ||

+ | |- | ||

+ | !style="font-weight:normal"|tightest<br/>accuracy | ||

+ | !style="font-weight:normal"|tightest<br/>accuracy | ||

+ | !style="font-weight:normal"|tightest<br/>accuracy | ||

+ | !style="font-weight:normal"|tightest<br/>accuracy | ||

+ | !style="font-weight:normal"|valid<br/>accuracy | ||

+ | !style="font-weight:normal"|tightest<br/>accuracy | ||

+ | !style="font-weight:normal"|valid<br/>accuracy | ||

+ | |- | ||

+ | | 10 × 10 | ||

+ | |align="right"| < 0.001 | ||

+ | |align="right"| < 0.001 | ||

+ | |align="right"| 0.001 | ||

+ | |align="right"| 0.008 | ||

+ | |align="right"| 0.001 | ||

+ | |align="right"| 0.002 | ||

+ | |align="right"| 0.025 | ||

+ | |- | ||

+ | | 100 × 100 | ||

+ | |align="right"| 0.003 | ||

+ | |align="right"| 0.010 | ||

+ | |align="right"| 0.055 | ||

+ | |align="right"| 0.61 | ||

+ | |align="right"| 0.012 | ||

+ | |align="right"| 0.53 | ||

+ | |align="right"| 0.30 | ||

+ | |- | ||

+ | | 500 × 500 | ||

+ | |align="right"| 0.060 | ||

+ | |align="right"| 0.24 | ||

+ | |align="right"| 1.3 | ||

+ | |align="right"| 15 | ||

+ | |align="right"| 0.30 | ||

+ | |align="right"| 63 | ||

+ | |align="right"| 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 [http://www.mpfr.org/ 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. | ||

+ | |||

+ | == Installation == | ||

+ | The interval package is available at Octave Forge [http://octave.sourceforge.net/interval/] 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. | ||

+ | <span style="opacity:.5">octave:1> </span>pkg install -forge interval | ||

+ | <span style="opacity:.5">octave:2> </span>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 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 <code>empty</code> and <code>entire</code> and the interval constructors <code>infsupdec</code> (create an interval from boundaries), <code>midrad</code> (create an interval from midpoint and radius) and <code>hull</code> (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|<syntaxhighlight lang="octave"> | ||

+ | 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 | ||

+ | </syntaxhighlight> | ||

}} | }} | ||

− | The | + | 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 <code>realmin</code> would be over 700 decimal places long! However, the default text representation is correct as it guarantees to contain the actual boundaries. |

− | + | ||

− | + | {{Warning|Decimal fractions as well as numbers of high magnitude (> 2<sup>53</sup>) should always be passed as a string to the constructor. Otherwise it is possible, that GNU Octave introduces conversion errors when the numeric literal is converted into floating-point format '''before''' it is passed to the constructor.}} | |

− | + | ||

+ | {{Code|Beware of the conversion pitfall|<syntaxhighlight lang="octave"> | ||

+ | ## 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 | ||

+ | </syntaxhighlight> | ||

+ | }} | ||

− | + | ==== 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|<syntaxhighlight lang="octave"> |

− | + | 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] | ||

+ | </syntaxhighlight> | ||

+ | }} | ||

− | == | + | === Arithmetic operations === |

− | + | The interval packages comprises many interval arithmetic operations. A complete list can be found in its [http://octave.sourceforge.net/interval/overview.html function reference]. Function names match GNU Octave standard functions where applicable and follow recommendations by IEEE 1788 otherwise, cf. [[#IEEE_1788_index|IEEE 1788 index]]. | |

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− | + | Arithmetic functions in a set-based 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 point-wise 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. | |

− | The interval | + | |

+ | 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|<syntaxhighlight lang="octave"> | ||

+ | 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 | ||

+ | </syntaxhighlight> | ||

+ | }} | ||

+ | |||

+ | === Numerical operations === | ||

+ | Some operations on intervals do not return an interval enclosure, but a single number (in double-precision). Most important are <code>inf</code> and <code>sup</code>, which return the lower and upper interval boundaries. | ||

+ | |||

+ | More such operations are <code>mid</code> (approximation of the interval's midpoint), <code>wid</code> (approximation of the interval's width), <code>rad</code> (approximation of the interval's radius), <code>mag</code> (interval's magnitude) and <code>mig</code> (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 ∀<sub>''a''</sub> ∃<sub>''b''</sub> ''a'' ≤ ''b'' ∧ ∀<sub>''b''</sub> ∃<sub>''a''</sub> ''a'' ≤ ''b''. | ||

− | + | <span style="opacity:.5">octave:1> </span>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 [http://www.gnu.org/software/octave/doc/interpreter/Vectorization-and-Faster-Code-Execution.html#Vectorization-and-Faster-Code-Execution 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|<syntaxhighlight lang="octave"> | |

+ | 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] | ||

+ | </syntaxhighlight> | ||

+ | }} | ||

+ | |||

+ | ==== Notes on linear systems ==== | ||

+ | A system of linear equations in the form A''x'' = 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|<syntaxhighlight lang="octave"> | |

− | + | A = [1, 0; 2, 0]; | |

− | {{Code| | + | ## 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] | ||

</syntaxhighlight> | </syntaxhighlight> | ||

}} | }} | ||

− | |||

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− | |||

− | + | == 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 <code>infsup</code> constructor will fail if the interval boundaries are invalid. Contrariwise, the (preferred) <code>infsupdec</code>, <code>midrad</code> and <code>hull</code> 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|<syntaxhighlight lang="octave"> | ||

+ | ## 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] | ||

</syntaxhighlight> | </syntaxhighlight> | ||

}} | }} | ||

− | |||

− | |||

− | + | 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 <code>help warning</code> for instructions. | |

− | + | ||

− | {| | + | {| class="wikitable" style="margin: auto" |

− | ! | + | !Warning ID |

− | ! | + | !Reason |

+ | !Possible consequences | ||

|- | |- | ||

− | | | + | | interval:PossiblyUndefined |

− | | | + | |style="vertical-align:top" | 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. |

+ | |style="vertical-align:top" | 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 |

− | | | + | |style="vertical-align:top" | 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., <code>infsupdec (1, 2) + "[3, 4]"</code> does not produce this warning whereas <code>infsupdec (1, 2) + infsup (3, 4)</code> does. A bare interval can be explicitly promoted with the newdec [http://octave.sourceforge.net/interval/function/@infsup/newdec.html] function. |

+ | |style="vertical-align:top" | 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, <code>infsupdec (1, 2) + infsup (0, 1) ^ 0</code> would ignore that 0<sup>0</sup> is undefined. | ||

|- | |- | ||

− | | | + | | interval:NaI |

− | | | + | |style="vertical-align:top" | 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 [http://octave.sourceforge.net/interval/function/nai.html] function. |

− | |- | + | |style="vertical-align:top" | 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. |

− | | | + | |} |

− | + | ||

− | + | === 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) <code>infsup</code> constructor creates bare, undecorated intervals and the <code>intervalpart</code> 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: | ||

+ | |||

+ | {| class="wikitable" style="margin: auto" | ||

+ | !Decoration | ||

+ | !Bounded | ||

+ | !Continuous | ||

+ | !Defined | ||

+ | !Definition | ||

|- | |- | ||

− | | | + | | com<br/>(common) |

− | | | + | | style="text-align: center" | ✓ |

+ | | style="text-align: center" | ✓ | ||

+ | | style="text-align: center" | ✓ | ||

+ | | '''''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<br/>(defined & continuous) |

− | | | + | | |

+ | | style="text-align: center" | ✓ | ||

+ | | style="text-align: center" | ✓ | ||

+ | | '''''x''''' is a nonempty subset of Dom(''f''); and the restriction of ''f'' to '''''x''''' is continuous | ||

|- | |- | ||

− | | | + | | def<br/>(defined) |

− | | | + | | |

+ | | | ||

+ | | style="text-align: center" | ✓ | ||

+ | | '''''x''''' is a nonempty subset of Dom(''f'') | ||

|- | |- | ||

− | | | + | | trv<br/>(trivial) |

− | | | + | | |

+ | | | ||

+ | | | ||

+ | | always true (so gives no information) | ||

|- | |- | ||

− | | | + | | ill<br/>(ill-formed) |

− | | | + | | |

+ | | | ||

+ | | | ||

+ | | 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|<syntaxhighlight lang="octave"> |

− | + | 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 | ||

+ | </syntaxhighlight> | ||

+ | }} | ||

− | === | + | === 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. | |

− | * <code> | + | * 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 ([http://www.open-std.org/jtc1/sc22/WG14/www/docs/n1256.pdf ISO/IEC9899, N1256, §6.4.4.2]). 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. |

− | * <code> | + | * 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. |

− | * <code> | + | * Uncertain form: The interval as a whole can be defined by a midpoint or upper/lower boundary and an integral number of [http://en.wikipedia.org/wiki/Unit_in_the_last_place “units in last place” (ULPs)] as an uncertainty. The format is <code>''m''?''ruE''</code>, where |

+ | ** <code>''m ''</code> is a mantissa in decimal, | ||

+ | ** <code>''r ''</code> is either empty (which means ½ ULP) or is a non-negative decimal integral ULP count or is the <code>?</code> character (for unbounded intervals), | ||

+ | ** <code>''u ''</code> is either empty (symmetrical uncertainty of ''r'' ULPs in both directions) or is either <code>u</code> (up) or <code>d</code> (down), | ||

+ | ** <code>''E ''</code> is either empty or an exponent field comprising the character <code>e</code> followed by a decimal integer exponent (base 10). | ||

− | + | {{Code|Examples of different formats during interval construction|<syntaxhighlight lang="octave"> | |

− | < | + | 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] | ||

+ | </syntaxhighlight> | ||

+ | }} | ||

− | === | + | === Reverse arithmetic operations === |

+ | [[File:Reverse-power-functions.png|400px|thumb|right|Reverse power operations. A relevant subset of the function's domain is outlined and hatched. In this example we use ''x''<sup>''y''</sup> ∈ [2, 3].]] | ||

− | + | Some arithmetic functions also provide reverse mode operations. That is inverse functions with interval constraints. For example the <code>sqrrev</code> can compute the inverse of the <code>sqr</code> function on intervals. The syntax is <code>sqrrev (C, X)</code> and will compute the enclosure of all numbers ''x'' ∈ X that fulfill the constraint ''x''² ∈ C. | |

− | |||

− | |||

− | + | In the following example, we compute the constraints for base and exponent of the power function <code>pow</code> as shown in the figure. | |

+ | <span style="opacity:.5">octave:1> </span>x = powrev1 (infsup ("[1.1, 1.45]"), infsup (2, 3)) | ||

+ | x ⊂ [1.6128979635153646, 2.714854726565793] | ||

+ | <span style="opacity:.5">octave:2> </span>y = powrev2 (infsup ("[2.14, 2.5]"), infsup (2, 3)) | ||

+ | y ⊂ [0.7564707973660299, 1.4440113978403289] | ||

− | + | === 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. | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | = | + | <span style="opacity:.5">octave:1> </span>infsupdec ("17.7") + 1 |

− | + | ans ⊂ [18.699999999999999, 18.70000000000001]_com | |

− | + | <span style="opacity:.5">octave:2> </span>ans + "[0, 2]" | |

− | + | ans ⊂ [18.699999999999999, 20.70000000000001]_com | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | Interval functions with only one argument can be called by using property syntax, e. g. <code>x.inf</code>, <code>x.sup</code> or even <code>x.sqr</code>. | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | + | When matrices are resized using subscripted assignment, any implicit new matrix elements will carry an empty interval. | |

− | + | <span style="opacity:.5">octave:1> </span>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 (<code>size</code>, <code>isvector</code>, <code>ismatrix</code>, …) work as expected on interval data types, the function <code>isempty</code> is evaluated element-wise and checks if an interval equals the empty set. | |

− | + | <span style="opacity:.5">octave:1> </span>builtin ("isempty", empty ()), isempty (empty ()) | |

− | + | ans = 0 | |

− | + | ans = 1 | |

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | |||

− | == | + | == IEEE 1788 index == |

− | |||

− | {| | + | {| class="wikitable" style="margin: auto; min-width: 50%" |

− | + | |+ 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 | |

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|- | |- | ||

− | | | + | | numsToInterval |

− | | | + | | bare version: infsup (''l'', ''u'') [http://octave.sourceforge.net/interval/function/@infsup/infsup.html]<br/>decorated version: infsupdec (''l'', ''u'') [http://octave.sourceforge.net/interval/function/@infsupdec/infsupdec.html] |

− | |||

|- | |- | ||

− | | | + | | textToInterval |

− | | | + | | bare version: infsup (''s'')<br/>decorated version: infsupdec (''s'') |

− | |||

|- | |- | ||

− | | | + | | setDec |

− | | | + | | infsupdec (''x'', ''dx'') |

− | |||

|- | |- | ||

− | | | + | | exp10 |

− | | | + | | pow10 [http://octave.sourceforge.net/interval/function/@infsup/pow10.html] |

− | |||

|- | |- | ||

− | | | + | | exp2 |

− | | | + | | pow2 [http://octave.sourceforge.net/interval/function/@infsup/pow2.html] |

− | |||

|- | |- | ||

− | | | + | | recip |

− | | | + | | inv [http://octave.sourceforge.net/interval/function/@infsup/inv.html] |

− | |||

|- | |- | ||

− | | | + | | sqrt |

− | | | + | | realsqrt [http://octave.sourceforge.net/interval/function/@infsup/realsqrt.html] |

− | |||

|- | |- | ||

− | | | + | | rootn |

+ | | nthroot [http://octave.sourceforge.net/interval/function/@infsup/nthroot.html] | ||

|- | |- | ||

− | | | + | | logp1 |

− | | | + | | log1p [http://octave.sourceforge.net/interval/function/@infsup/log1p.html] |

− | |||

|- | |- | ||

− | | | + | | roundTiesToAway |

− | | | + | | round [http://octave.sourceforge.net/interval/function/@infsup/round.html] |

− | |||

|- | |- | ||

− | | | + | | roundTiesToEven |

− | | | + | | roundb [http://octave.sourceforge.net/interval/function/@infsup/roundb.html] |

− | |||

|- | |- | ||

− | | | + | | trunc |

+ | | fix [http://octave.sourceforge.net/interval/function/@infsup/fix.html] | ||

|- | |- | ||

− | | | + | | sum |

− | | | + | | on intervals: sum [http://octave.sourceforge.net/interval/function/@infsup/sum.html]<br/>on numbers: mpfr_vector_sum_d [http://octave.sourceforge.net/interval/function/mpfr_vector_sum_d.html] |

− | |||

|- | |- | ||

− | | | + | | dot |

− | | | + | | on intervals: dot [http://octave.sourceforge.net/interval/function/@infsup/dot.html]<br/>on numbers: mpfr_vector_dot_d [http://octave.sourceforge.net/interval/function/mpfr_vector_dot_d.html] |

− | |||

|- | |- | ||

− | | | + | | sumAbs |

− | | | + | | on intervals: sumabs [http://octave.sourceforge.net/interval/function/@infsup/sumabs.html]<br/>on numbers: mpfr_vector_sum_d (''rounding mode'', abs (''x'')) |

− | |||

|- | |- | ||

− | | | + | | sumSquare |

− | | | + | | on intervals: sumsq [http://octave.sourceforge.net/interval/function/@infsup/sumsq.html]<br/>on numbers: mpfr_vector_dot_d (''rounding mode'', abs (''x''), abs (''x'')) |

− | |||

|- | |- | ||

− | | | + | | intersection |

+ | | intersect [http://octave.sourceforge.net/interval/function/@infsup/intersect.html] | ||

|- | |- | ||

− | | | + | | convexHull |

− | | | + | | union [http://octave.sourceforge.net/interval/function/@infsup/union.html] |

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+ | | mulRevToPair | ||

+ | | mulrev [http://octave.sourceforge.net/interval/function/@infsup/mulrev.html] with two output parameters | ||

|} | |} | ||

+ | == Compatibility == | ||

+ | |||

+ | 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). | ||

+ | |||

+ | 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 non-free software, it grants none of the [http://www.gnu.org/philosophy/free-sw.html 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.|<syntaxhighlight lang="octave"> | ||

+ | # 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 | ||

+ | </syntaxhighlight> | ||

+ | }} | ||

+ | |||

+ | == Related work == | ||

+ | |||

+ | 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.uni-wuppertal.de/~xsc/xsc-sprachen.html XSC]. It is not standard compliant with IEEE 1788. Some parts of the GNU Octave interval package have been derived from C-XSC. | ||

+ | |||

+ | For Java there is a library [https://java.net/projects/jinterval/ 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 [http://www.ti3.tu-harburg.de/rump/intlab/ 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 [http://www.ti3.tuhh.de/rump/intlab/Octave/octave.html]. | ||

− | [[Category:Octave Forge]] | + | [[Category:Octave-Forge]] |