17.6 Bounds of a Function

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:

bounds

\((exp,var_1=(l_1 .. u_1) [,var_2=(l_2 .. u_2) \ldots ])\)

bounds

\((exp,\{var_1=(l_1 .. u_1) [,var_2=(l_2 .. u_2)\ldots ]\})\)

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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