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Upper and lower bounds of a real valued function over an interval or a rectangular
multivariate domain are computed by the operator bounds
. The algorithmic basis is the
computation with inequalities: starting from the interval(s) of the variables, the bounds
are propagated in the expression using the rules for inequality computation. Some
knowledge about the behavior of special functions like ABS, SIN, COS, EXP, LOG,
fractional exponentials etc. is integrated and can be evaluated if the operator bounds
is
called with rounded mode on (otherwise only algebraic evaluation rules are
available).
If bounds
finds a singularity within an interval, the evaluation is stopped with an error
message indicating the problem part of the expression.
Syntax:
\((exp,var_1=(l_1 .. u_1) [,var_2=(l_2 .. u_2) \ldots ])\)
\((exp,\{var_1=(l_1 .. u_1) [,var_2=(l_2 .. u_2)\ldots ]\})\)
where \(exp\) is the function to be investigated,
\(var_1, var_2 , \ldots \) are the variables of exp,
\(l_1, l_2 , \ldots \) and \(u_1, u_2 , \ldots \) specify the area (intervals).
bounds
computes upper and lower bounds for the expression in the given
area. An interval is returned.
Example:
bounds(sin x,x=(1 .. 2)); - 1 .. 1 on rounded; bounds(sin x,x=(1 .. 2)); 0.841470984808 .. 1 bounds(x**2+x,x=(-0.5 .. 0.5)); - 0.25 .. 0.75
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