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##
Interval package
(edit)

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Revision as of 06:01, 22 March 2015

1,662 bytes added
, 06:01, 22 March 2015

Split introduction into simple and advanced topics, added syntax highlighting, use decorated intervals per default

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~~ ~~octave~~:1~~> ~~infsup ~~(1)+~~ ans = ~~[1]+~~ octave:2> infsup ~~(1, 2)+~~ ans = ~~[1, 2]+~~ octave:3> infsup ~~("3", "4")+~~ ans = ~~[3, 4]+~~ octave:4> infsup ~~("1.1")+~~ ans ⊂ ~~[1.0999999999999998, 1.1000000000000001]+~~ octave:~~5~~> infsup (~~"[5, ~~6~~.~~5~~]")+~~ ans = ~~[~~5~~, ~~6~~.~~5~~]+~~ octave:6> infsup ~~("[~~5~~.~~8e-17~~]")+~~ ans ⊂ ~~[~~5~~.~~799999999999999e-17~~, ~~5~~.~~800000000000001e-17~~]+~~It is possible to access the exact numeric interval boundaries with the functions <code>inf</code> and <code>sup</code>. ~~The ~~shown ~~text representation of intervals ~~can be created with <code>intervaltotext</code>. The default text representation ~~is not guaranteed to be exact ~~(see function <code>intervaltoexact</code> for that purpose)~~, 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 and is accurate enough to separate different ~~boundaries.+~~ octave:7> infsup ~~(~~1, 1 + eps)~~+~~ ans ⊂ [1, 1.0000000000000003]~~
−~~ octave:8~~> ~~infsup (1, 1 + ~~2 ~~* eps~~)
−~~ ans ⊂ [1~~, ~~1~~.~~0000000000000005]~~
−~~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~~. ~~In simple cases it may help to construct intervals with the ~~[~~https://www~~.~~gnu~~.~~org~~/~~software/octave/doc/interpreter/Commands.html#Commands command syntax].~~+~~ octave:9> infsup (~~<~~span style ~~= "~~color:red~~">~~0.~~2~~</span>~~)+~~ ans ⊂ ~~[~~.20000000000000001~~, ~~.20000000000000002~~]+~~ octave:~~10~~> infsup ~~(~~<span style = ~~"~~color:green~~"~~>~~"0.2"~~</span>~~)+
−~~ ans ⊂ ~~[.19999999999999998, .20000000000000002]+~~ octave:11> ~~infsup ~~<span style = ~~"~~color:green~~"~~>0~~.2</~~span~~>+
−~~ ans ⊂ ~~[.~~19999999999999998~~, .~~20000000000000002~~]+
+~~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:12> infsup ~~(~~"17.7"~~) ~~+ 1~~+~~ ans ⊂ [18.699999999999999, 18~~.~~700000000000003]~~
−~~ octave:13> ans + "[0, 2]"~~
−~~ ans ⊂ [18~~.~~699999999999999~~, ~~20~~.~~700000000000003]~~
−~~==== 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 ~~([~~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~~.+~~* 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 ~~[~~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).~~+~~ octave:14> infsup ~~(~~"0x1.999999999999Ap~~-~~4"~~)+~~ ans ⊂ [~~.~~1, .10000000000000001]~~+~~ octave:15~~> ~~infsup ("1~~/~~3", "7~~/~~9")~~
−~~ ans ⊂ [.33333333333333331, .7777777777777778]~~
−~~ octave:16~~> ~~infsup ("121.2?")~~
−~~ ans ⊂ [121.14999999999999~~, ~~121~~.~~25]~~
−~~ octave:17> infsup ("5?32e2")~~
−~~ ans = [-2700, +3700]~~
−~~ octave:18> infsup ("-42??u")~~
−~~ ans = [-42, +Inf]~~
−~~==== Interval vectors and matrices ====~~+~~Vectors and matrices ~~of ~~intervals can be created by passing numerical matrices~~, ~~char vectors or cell arrays to the ~~<code>~~infsup~~</code> ~~constructor. With cell arrays it is also possible to mix several types ~~of ~~boundaries.~~
−~~ octave:19> M = infsup (magic (3))~~
−~~ M = 3×3 ~~interval ~~matrix~~
−~~ ~~
−~~ [8] [1] [6]~~
−~~ [3] [5] [7]~~
−~~ [4] [9] [2]~~
−~~ ~~
−~~ octave:20> infsup (magic (3~~), ~~magic ~~(~~3~~) ~~+ 1)~~
−~~ ans = 3×3 interval matrix~~
−~~ ~~
−~~ [8~~, ~~9] [1, 2] [6, 7]~~
−~~ [3, 4] [5, 6] [7, 8]~~
−~~ [4, 5] [9, 10] [2, 3]~~
−~~ ~~
−~~ octave:21~~> ~~infsup ~~(~~["0.1"; "0.2"; "0.3"; "0.4"; "0.5"]~~)
−~~ ans ⊂ 5×1 interval vector~~
−~~ ~~
−~~ [.09999999999999999, .10000000000000001]~~
−~~ [.19999999999999998, .20000000000000002]~~
−~~ [.29999999999999998, .30000000000000005]~~
−~~ [.39999999999999996, .40000000000000003]~~
−~~ [.5]~~
−~~ ~~
−~~ octave:22~~> ~~infsup ~~(~~{1, eps; "4/7", "pi"}, {2, 1; "e", "0xff"}~~)
−~~ ans ⊂ 2×2 interval matrix~~
−~~ ~~
−~~ [1, 2] [2~~.~~220446049250313e-16, 1]~~
−~~ [.5714285714285713, 2.7182818284590456] [3.1415926535897931, 255]~~
−~~When matrices ~~are ~~resized using subscripted assignment~~, ~~any implicit new matrix elements will carry an empty interval~~.+~~ octave:23~~> ~~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]~~
−~~Note~~: ~~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.~~+~~ octave:24> builtin ~~(~~"isempty"~~, ~~empty ()~~)~~, isempty (empty ())~~
−~~ ans = 0~~
~~With the subclass <code>infsupdec</code> it is possible to extend ~~interval ~~arithmetic with ~~a decoration system. ~~Every ~~interval ~~and intermediate result will additionally carry ~~a decoration, which ~~may provide ~~additional information about the ~~final result~~.+~~Although ~~the ~~examples in this wiki are mostly presented with ~~bare, undecorated intervals ~~(for simplicity)~~, ~~it ~~is highly recommended to use the decorated interval arithmetic ~~by default~~.+~~In the following example, all ~~decoration information is ~~lost ~~when the ~~interval ~~is ~~possibly divided by zero, i. e., ~~the overall function is ~~not guaranteed to be ~~defined for all ~~possible inputs~~.+~~ ~~octave~~:1~~> infsupdec (3, 4)+~~ ans = ~~[3, 4]_com+~~ octave:2> ans + 12~~+~~ ans ~~= [~~15~~, ~~16~~]_com+~~ octave:3> ans / "~~[~~0~~, 2]~~"~~+~~ ans = ~~[~~7~~.~~5, Inf~~]_trv+~~The ~~interval ~~packages comprises many ~~interval ~~arithmetic operations~~. ~~Function names match GNU Octave ~~standard ~~functions where applicable~~, ~~and follow recommendations by IEEE 1788 otherwise~~, ~~cf~~. ~~[[#IEEE_1788_index|IEEE 1788 index]~~].+ +~~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~~.+~~ ~~octave~~:1~~> ~~sin (~~infsup (~~0~~.~~5)~~)+~~ ans ⊂ ~~[.~~47942553860420294~~, .~~47942553860420301~~]+~~ octave:2> pow (~~infsup (~~2~~), infsup (~~3, 4)~~)+~~ ans = ~~[~~8~~, ~~16~~]+~~ octave:3> atan2 (~~infsup (~~1~~), infsup (~~1)~~)+~~ ans ⊂ ~~[~~.7853981633974482~~, ~~.7853981633974484~~]+~~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> and <code>mig</code>~~.+~~=== Boolean operations ===~~+~~ 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 [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.~~
−
−~~ ~~octave:1> ~~A ~~= infsup (~~[1, 2, ~~3~~; 4, 0, 0; 0, 0, 1]~~); ~~A ~~(~~2, 3) = "[0~~, ~~6]"~~
−~~ A = 3×3 interval matrix~~
−~~ ~~
−~~ [1] [2] [3]~~
−~~ [~~4~~] [0] [0, 6]~~
−~~ [0] [0] [1]~~
−~~ ~~
−~~ octave:2> B = inv (A~~)
−~~ B ~~= ~~3×3 interval matrix~~
−~~ ~~
−~~ [0] [.25] [-1.5, 0]~~
−~~ [.5] [-.125] [-1.5, -.75]~~
−~~ [0] [0] [1]~~
−
−~~ octave:3> A * B~~
−~~ ans ~~= ~~3×3 ~~interval matrix
−~~ ~~
−~~ [1] [0] [-1.5, +1.5]~~
−~~ [0] [1] [-6, +6]~~
−~~ [0] [0] [1]~~
+~~ octave:4> A = infsup ~~(~~magic (3))~~+~~ A = 3×3 interval matrix~~+~~ ~~+~~ [8] [1] [6]~~+~~ [3] [5] [7]~~
−~~ [4] [9] [2]~~
−~~ ~~
−~~ octave:5> c = A \ [3; 4; 5]~~
−~~ c ⊂ 3×1 interval vector~~
−~~ ~~
−~~ [.18333333333333326, .18333333333333349]~~
−~~ [.43333333333333329, .43333333333333341]~~
−~~ [.18333333333333315, .18333333333333338]~~
−~~ ~~
−~~ octave:6> A * c~~
−~~ ans ⊂ 3×1 interval vector~~
−~~ ~~
−~~ [2.9999999999999982, 3.0000000000000018]~~
−~~ [3.9999999999999982, 4.0000000000000018]~~
−~~ [4.9999999999999982, 5.0000000000000018]~~
−
−~~==== 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.~~
−
−~~{| class="wikitable" style="margin: auto"~~
−~~!Standard floating point arithmetic~~
−~~!Interval arithmetic~~
−~~|-~~
−~~| style = "vertical-align: top" |~~
−~~ ~~<~~span style="opacity:.5">octave:1~~> </~~span~~>~~A = [1, 0; 2~~, ~~0];~~
−~~ ~~<~~span style="opacity:.5">octave:2~~> </~~span~~>~~A \ [3; 0] # no solution~~
−~~ warning: matrix singular to machine precision~~, ~~rcond = 0~~
−~~ ans =~~
−~~ ~~
−~~ 0.60000~~
−~~ 0.00000~~
−~~ ~~
−~~ ~~<~~span style="opacity:.5">octave:3~~> </~~span~~>~~A \ [4; 8] # many solutions~~
−~~ ans =~~
−~~ ~~
−~~ 4~~
−~~ 0~~
−~~| style = "vertical-align: top" |~~
−~~ <span style="opacity:.5">octave:4> </span>A = infsup (A~~)~~;~~
−~~ <span style="opacity:.5">octave:5> </span>A \ [3; 0] # no solution~~
−~~ ans = 2×1 ~~interval ~~vector~~
−~~ ~~
−~~ [Empty]~~
−~~ [Empty]~~
−~~ ~~
−~~ ~~<~~span style="opacity:.5">octave:6~~> </~~span~~>~~A \ [4; 8] # many solutions~~
−~~ ans = 2×1 ~~interval ~~vector~~
−~~ ~~
−~~ [4]~~
−~~ [Entire]~~
−~~|}~~
−
−~~=== Error handling ===~~
−~~Due to ~~the ~~nature of ~~set~~-based interval arithmetic, one should never observe errors (in the sense of raised GNU Octave error messages) during computation. If you do, there either is a bug in the code or there are unsupported data types. Arithmetic operations which are not defined for (parts of) their input, simply ignore anything that is outside of their domain~~.
−
−~~ ans = [Empty]~~
−~~ <span style=~~"~~opacity:.5~~"~~>octave:2> </span>infsup ~~(~~0~~) ~~^ infsup (0~~)
−~~ ans = [Empty]~~
−
−~~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 <code>infsupdec</code> constructor will only issue a warning and return a [NaI], which will propagate and survive through computations.~~
−
−~~ <span style="opacity:.5">octave:3> </span>infsup ~~(~~3, 2~~) ~~+ 1~~
−~~ error: illegal interval boundaries: infimum greater than supremum~~
−~~ ''… (call stack~~) ~~…''~~
− ~~<span style~~=~~"opacity:.5">octave:3> </span>infsupdec (3, 2) + 1~~
−~~ warning: illegal interval boundaries: infimum greater than supremum~~
−

=== Input and output ===

=== 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 ~~class ~~constructors <code>~~infsup~~</code> ~~for bare intervals ~~and <code>~~infsupdec~~</code> for ~~decorated ~~intervals. The ~~class ~~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~~. Also it is possible to use so called interval literals with square brackets~~.

+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, 1.1000000000000001]_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, 4.2000001000000005]_com

−hull (3, 42, "19.3", "-2.3") # [-2.3000000000000003, +42]_trv

−hull ("pi", "e") # [2.718281828459045, 3.1415926535897936]_trv

−</syntaxhighlight>

+}}

−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 “0.2” 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

+</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"])

+ # [.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] [2.220446049250313e-16, 1]

+ # [.5714285714285713, 2.7182818284590456] [3.1415926535897931, 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]].

−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.

−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)) # [.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

+</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

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] [.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]

+</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];

+## 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>

+}}

+
+== 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>

+}}

=== Decorations ===

=== 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:

The following decorations are available:

Line 266:
Line 335:
|}

|}

−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 # [-.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

+</syntaxhighlight>

+}}

−=== ~~Arithmetic operations ~~===

+=== 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 ([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.

−* 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 [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

− # [.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]

+</syntaxhighlight>

+}}

=== Reverse arithmetic operations ===

=== 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].]]

+[[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~~.

+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.

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

− octave:1> x = powrev1 (infsup ("[1.1, 1.45]"), infsup (2, 3))

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

x ⊂ [1.6128979635153646, 2.7148547265657915]

x ⊂ [1.6128979635153646, 2.7148547265657915]

− octave:2> y = powrev2 (infsup ("[2.14, 2.5]"), infsup (2, 3))

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

y ⊂ [.7564707973660299, 1.4440113978403284]

y ⊂ [.7564707973660299, 1.4440113978403284]

−=== ~~Numerical operations ~~===

+=== 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, 18.700000000000003]_com

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

+ ans ⊂ [18.699999999999999, 20.700000000000003]_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>.

−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''~~.

−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

− <span style="opacity:.5">octave:1> </span>~~infsup ~~(~~2, 3) / 0~~

− ans = ~~[NaI]~~

== IEEE 1788 index ==

== IEEE 1788 index ==

240

edits

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