RRi -- the class of all real intervals

Description

A real interval is entered as a pair of real numbers to the interval function. It is stored internally as an arbitrary precision interval using the MPFI library.

i1 : interval(3.1415,3.1416)

o1 = [3.1415,3.1416]

o1 : RRi (of precision 53)

The precision is measured in bits, is visible in the ring displayed on the second of each pair of output lines, and can be recovered using precision.

i2 : precision interval(3.1415,3.1416)

o2 = 53

For real intervals, the functions class and ring yield different results. That allows numbers of various precisions to be used without creating a new ring for each precision.

i3 : class interval(3.1,3.5)

o3 = RRi

o3 : InexactFieldFamily

i4 : ring interval(3.1,3.5)

o4 = RRi
        53

o4 : RealIntervalField

The precision can be specified on input by specifying the precision of both input RR numbers.Alternatively, the precision can be specified by including the option Precision.

i5 : interval(2.5p100,3.2p1000)

o5 = [2.5,3.2]

o5 : RRi (of precision 100)

i6 : interval(2.5,3.2,Precision=>200)

o6 = [2.5,3.20000000000000017763568394002504646778106689453125]

o6 : RRi (of precision 200)

Intervals can also be created using span(Sequence) to create the smallest interval containing the inputs.

i7 : span(2,Precision=>100)

o7 = [2,2]

o7 : RRi (of precision 100)

i8 : span(2,3,interval(-1.5,-0.5),73)

o8 = [-1.5,73]

o8 : RRi (of precision 53)

Operations using intervals are computed as sets so that the resulting intervals contain all possible outputs from pairs of points in input intervals.

i9 : interval(1,3)+interval(2,4)

o9 = [3,7]

o9 : RRi (of precision 53)

i10 : interval(-1,1)*interval(2,3)

o10 = [-3,3]

o10 : RRi (of precision 53)

i11 : interval(0,1)-interval(0,1)

o11 = [-1,1]

o11 : RRi (of precision 53)

i12 : interval(1,2)/interval(1,2)

o12 = [.5,2]

o12 : RRi (of precision 53)

The notion of equality tested by == amounts to checking the equality of the endpoints of intervals.The notion of equality tested by === takes into account the precision of the inputs as well.

i13 : interval(1,3) == interval(1,3,Precision=>100)

o13 = true

i14 : interval(1,3) === interval(1,3,Precision=>100)

o14 = false

i15 : interval(1/3,1,Precision=>100)==interval(1/3,1,Precision=>1000)

o15 = false

The notion of inequalities for intervals amounts to testing the inequality for all points in the intervals. In particular, <= is not the same as < or ==.

i16 : interval(1,2)<=interval(2,3)

o16 = true

i17 : interval(1,2)<=interval(1,2)

o17 = false

i18 : interval(1,2)<interval(2,3)

o18 = false

i19 : interval(1,2)<interval(3,4)

o19 = true

Transcendental functions on intervals produce intervals containing the image of the function on the interval.

i20 : exp(interval(2,4))

o20 = [7.38905609893065,54.5981500331442]

o20 : RRi (of precision 53)

i21 : cos(interval(1,1.3))

o21 = [.267498828624587,.54030230586814]

o21 : RRi (of precision 53)

i22 : sqrt(interval(2))

o22 = [1.41421356237309,1.4142135623731]

o22 : RRi (of precision 53)

Transcendental functions are available to high precision, with numericInterval.

i23 : numericInterval(100,pi)

o23 = [3.14159265358979323846264338328,3.14159265358979323846264338328]

o23 : RRi (of precision 100)

i24 : numericInterval_200 EulerConstant

o24 = [.577215664901532860606512090082402431042159335939923598805767
      ,.577215664901532860606512090082402431042159335939923598805768]

o24 : RRi (of precision 200)

Functions and methods returning a real interval :

QQ * RRi -- see * -- a binary operator, usually used for multiplication
RR * RRi -- see * -- a binary operator, usually used for multiplication
RRi * QQ -- see * -- a binary operator, usually used for multiplication
RRi * RR -- see * -- a binary operator, usually used for multiplication
RRi * RRi -- see * -- a binary operator, usually used for multiplication
RRi * ZZ -- see * -- a binary operator, usually used for multiplication
ZZ * RRi -- see * -- a binary operator, usually used for multiplication
+ RRi -- see + -- a unary or binary operator, usually used for addition
QQ + RRi -- see + -- a unary or binary operator, usually used for addition
RR + RRi -- see + -- a unary or binary operator, usually used for addition
RRi + QQ -- see + -- a unary or binary operator, usually used for addition
RRi + RR -- see + -- a unary or binary operator, usually used for addition
RRi + RRi -- see + -- a unary or binary operator, usually used for addition
RRi + ZZ -- see + -- a unary or binary operator, usually used for addition
ZZ + RRi -- see + -- a unary or binary operator, usually used for addition
- RRi -- see - -- a unary or binary operator, usually used for negation or subtraction
QQ - RRi -- see - -- a unary or binary operator, usually used for negation or subtraction
RR - RRi -- see - -- a unary or binary operator, usually used for negation or subtraction
RRi - QQ -- see - -- a unary or binary operator, usually used for negation or subtraction
RRi - RR -- see - -- a unary or binary operator, usually used for negation or subtraction
RRi - RRi -- see - -- a unary or binary operator, usually used for negation or subtraction
RRi - ZZ -- see - -- a unary or binary operator, usually used for negation or subtraction
ZZ - RRi -- see - -- a unary or binary operator, usually used for negation or subtraction
QQ / RRi -- see / -- a binary operator, usually used for division
RR / RRi -- see / -- a binary operator, usually used for division
RRi / QQ -- see / -- a binary operator, usually used for division
RRi / RR -- see / -- a binary operator, usually used for division
RRi / RRi -- see / -- a binary operator, usually used for division
RRi / ZZ -- see / -- a binary operator, usually used for division
ZZ / RRi -- see / -- a binary operator, usually used for division
acos(RRi) -- see acos -- arccosine
asin(RRi) -- see asin -- arcsine
atan(RRi) -- see atan -- compute the arctangent of a number
atan2(RRi,RRi) -- see atan2 -- compute an angle of a certain triangle
atan2(RR,RRi) (missing documentation)
atan2(RRi,RR) (missing documentation)
cos(RRi) -- see cos -- compute the cosine
cosh(RRi) -- see cosh -- compute the hyperbolic cosine
cot(RRi) -- see cot -- cotangent
coth(RRi) -- see coth -- hyperbolic cotangent
csc(RRi) -- see csc -- cosecant
csch(RRi) -- see csch -- hyperbolic cosecant
exp(RRi) -- see exp -- exponential function
expm1(RRi) -- see expm1 -- exponential minus 1
intersect(RRi) -- computes intersection of input intervals
intersect(RRi,RRi) (missing documentation)
log(RRi) -- see log -- logarithm function
log(RRi,RRi) -- see log -- logarithm function
log(RR,RRi) (missing documentation)
log(RRi,RR) (missing documentation)
log1p(RRi) -- see log1p -- logarithm of 1+x
sec(RRi) -- see sec -- secant
sech(RRi) -- see sech -- hyperbolic secant
sin(RRi) -- see sin -- compute the sine
sinh(RRi) -- see sinh -- compute the hyperbolic sine
sqrt(RRi) -- see sqrt -- square root function
tan(RRi) -- see tan -- compute the tangent
tanh(RRi) -- see tanh -- compute the hyperbolic tangent

Methods that use a real interval :

QQ == RRi -- see == -- equality
RR == RRi -- see == -- equality
RRi == QQ -- see == -- equality
RRi == RR -- see == -- equality
RRi == RRi -- see == -- equality
RRi == ZZ -- see == -- equality
ZZ == RRi -- see == -- equality
abs(RRi) -- see abs -- absolute value function
diameter(RRi) -- see diameter -- diameter of an interval
intersection(RRi,RRi) -- computes intersection of input intervals
isEmpty(RRi) -- empty test for an interval
isMember(QQ,RRi) -- membership test in an interval
isMember(RR,RRi) -- see isMember(QQ,RRi) -- membership test in an interval
isMember(ZZ,RRi) -- see isMember(QQ,RRi) -- membership test in an interval
isReal(RRi) -- see isReal -- whether a number is real
isSubset(RRi,RRi) -- subset test for intervals
RRi << ZZ -- see left shift
lift(RRi,type of QQ) -- see lift -- lift to another ring
lift(RRi,type of RR_*) -- see lift -- lift to another ring
lift(RRi,type of ZZ) -- see lift -- lift to another ring
numeric(RRi) -- see numeric -- convert to floating point
numeric(ZZ,RRi) -- see numeric -- convert to floating point
RRi >> ZZ -- see right shift
ring(RRi) -- see ring -- get the associated ring of an object
size2(RRi) -- see size2 -- number of binary digits to the left of the point

For the programmer

The object RRi is an inexact field family, with ancestor classes InexactNumber < Number < Thing.

RRi -- the class of all real intervals

Description

See also

Functions and methods returning a real interval :

Methods that use a real interval :

For the programmer