Module Interval_intel

The interval type. Be careful however when creating intervals. For example, the following code: let a = {low=1./.3.; high=1./.3.} creates an interval which does NOT contain the mathematical object 1/3.

If you want to create an interval representing 1/3, you have to write let a = I.(inv(v 3. 3.)) because rounding will then be properly handled by I.inv and the resulting interval will indeed contain the exact value of 1/3.

Interval operations. Locally open this module — using e.g. I.(...) — to redefine classical arithmetic operators for interval arithmetic.

Access to low level floating point functions. THIS LIBRARY ONLY WORKS FOR INTEL PROCESSORS.

Neutral element for addition

Neutral element for multiplication

pi with bounds properly rounded

e with bounds properly rounded

Prints an interval with the same format applied to both endpoints. Formats follow the same specification than the one used for the regular printf function

Prints an interval into an out_channel with the same format applied to both endpoints

Returns a string holding the interval printed with the same format applied to both endpoints

Returns the interval containing the float conversion of an integer

compare_I_f a x returns 1 if a.high<x, 0 if a.low<=x<=a.high and -1 if x<a.low

size_I a returns a.high-a.low

sgn a returns {low=float (compare a.low 0.);high=float (compare a.high 0.)}

truncate_I a returns {low=floor a.low;high=ceil a.high}

abs_I a returns {low=a.low;high=a.high} if a.low>=0., {low=-a.high;high=-a.low} if a.high<=0., and {low=0.;high=max -a.low a.high} otherwise

union_I_I a b returns {low=min a.low b.low; high=max a.high b.high}

max_I_I a b returns {low=max a.low b.low; high=max a.high b.high}

min_I_I a b returns {low=min a.low b.low; high=min a.high b.high}

a +$ b returns {low=a.low+.b.low;high=a.high+.b.high}

a +$. x returns {low=a.low+.x;high=a.high+.x}

x +.$ a returns {low=a.low+.x;high=a.high+.x}

a -$ b returns {low=a.low-.b.high;high=a.high-.b.low}

a -$. x returns {low=a.low-.x;high=a.high-.x}

x -.$ a returns {low=x-.a.high;high=x-.a.low}

~-$ a returns {low=-a.high;high=-a.low}

a *$. x multiplies a by x according to interval arithmetic and returns the proper result. If x=0. then zero_I is returned

x *$. a multiplies a by x according to interval arithmetic and returns the proper result. If x=0. then zero_I is returned.

a *$ b multiplies a by b according to interval arithmetic and returns the proper result. If a=zero_I or b=zero_I then zero_I is returned

a /$. x divides a by x according to interval arithmetic and returns the proper result. Raise Failure "/$." if x=0.

x /.$ a divides x by a according to interval arithmetic and returns the result. Raise Failure "/.$" if a=zero_I

a /$ b divides the first interval by the second according to interval arithmetic and returns the proper result. Raise Failure "/$" if b=zero_I

mod_I_f a f returns a mod f according to interval arithmetic et OCaml mod_float definition. Raise Failure "mod_I_f" if f=0.

inv_I a returns 1. /.$ a. Raise Failure "inv_I" if a=zero_I

sqrt_I a returns {low=sqrt a;high=sqrt b} if a>=0., {low=0.;high=sqrt b} if a<0.<=b. Raise Failure "sqrt_I" if b<0.

Pow_I_i a n with n integer returns interval a raised to nth power according to interval arithmetic. If n=0 then {low=1.;high=1.} is returned. Raise Failure "pow_I_f" if n<=0 and a=zero_I. Computed with exp-log in base2.

a **$. f returns interval a raised to f power according to interval arithmetic. If f=0. then {low=1.;high=1.} is returned. Raise Failure "**$." if f<=0. and a=zero_I or if f is not an integer value and a.high<0.. Computed with exp-log in base2.

a **$ b returns interval a raised to b power according to interval arithmetic, considering the restriction of x power y to x >= 0. Raise Failure "**$" if a.high < 0 or (a.high=0. and b.high<=0.)

x **.$ a returns float x raised to interval a power according to interval arithmetic, considering the restiction of x power y to x >= 0. Raise Failure "**.$" if x < 0 and a.high <= 0

log_I a returns {low=log a.low; high=log a.high} if a.low>0., {low=neg_infinity; high=log a.high} if a.low<0<=a.high. Raise Failure "log_I" if a.high<=0.

exp_I a returns {low=exp a.high;high=exp b.high}

cos_I a returns the proper extension of cos to arithmetic interval Returns [-1,1] if one of the bounds is greater or lower than +/-2**53

sin_I a returns the proper extension of sin to arithmetic interval Returns [-1,1] if one of the bounds is greater or lower than +/-2**53

tan_I a returns the proper extension of tan to arithmetic interval Returns [-Inf,Inf] if one of the bounds is greater or lower than +/-2**53

acos_I a raise Failure "acos_I" if a.low>1. or a.high<-1., else returns {low=if a.high<1. then acos a.high else 0; high=if a.low>-1. then acos a.low else pi}. All values are in [0,pi].

asin_I a raise Failure "asin_I" if a.low>1. or a.high<-1. else returns {low=if a.low>-1. then asin a.low else -pi/2; high=if a.low<1. then asin a.high else pi/2}. All values are in [-pi/2,pi/2].

atan_I a returns {low=atan a.low;high=atan a.high}

atan2mod_I_I y x returns the proper extension of interval arithmetic to atan2 but with values in [-pi,2 pi] instead of [-pi,pi]. This can happen when y.low<0 and y.high>0 and x.high<0: then the returned interval is {low=atan2 y.high x.high;high=(atan2 y.low x.high)+2 pi}. This preserves the best inclusion function possible but is not compatible with the standard definition of atan2

Same function as above but when y.low<0 and y.high>0 and x.high<0 the returned interval is [-pi,pi]. This does not preserve the best inclusion function but is compatible with the atan2 regular definition

cosh_I is the proper extension of interval arithmetic to cosh

sinh_I is the proper extension of interval arithmetic to sinh

tanh_I is the proper extension of interval arithmetic to tanh

Computes the size of the largest interval of the interval vector

Computes the mean of the size of intervals of the interval vector

Prints an interval vector with the same format applied to all endpoints.

Prints an interval vector into an out_channel with the same format applied to all endpoints

Returns a string holding the interval vector printed with the same format applied to all endpoints

Deprecated

Deprecated

Deprecated

Deprecated

Deprecated

Deprecated

Deprecated