Sollya User's Manual

Name:

checkinfnorm checks whether the infinity norm of a function is bounded by a value

Usage:

checkinfnorm(function,range,constant) : (function, range, constant) -> boolean

Parameters:

function represents the function whose infinity norm is to be checked
range represents the infinity norm is to be considered on
constant represents the upper bound the infinity norm is to be checked to

Description:

The command checkinfnorm checks whether the infinity norm of the given function function in the range range can be proven (by Sollya) to be less than the given bound bound. This means, if checkinfnorm evaluates to true, the infinity norm has been proven (by Sollya's interval arithmetic) to be less than the bound. If checkinfnorm evaluates to false, there are two possibilities: either the bound is less than or equal to the infinity norm of the function or the bound is greater than the infinity norm but Sollya could not conclude using its internal interval arithmetic.

checkinfnorm is sensitive to the global variable diam. The smaller diam, the more time Sollya will spend on the evaluation of checkinfnorm in order to prove the bound before returning false although the infinity norm is bounded by the bound. If diam is equal to 0, Sollya will eventually spend infinite time on instances where the given bound bound is less or equal to the infinity norm of the function function in range range. In contrast, with diam being zero, checkinfnorm evaluates to true iff the infinity norm of the function in the range is bounded by the given bound.

Example 1:

   > checkinfnorm(sin(x),[0;1.75], 1);
   true
   > checkinfnorm(sin(x),[0;1.75], 1/2); checkinfnorm(sin(x),[0;20/39],1/2);
   false
   true

Example 2:

   > p = remez(exp(x), 5, [-1;1]);
   > b = dirtyinfnorm(p - exp(x), [-1;1]);
   > checkinfnorm(p - exp(x), [-1;1], b);
   false
   > b1 = round(b, 15, RU);
   > checkinfnorm(p - exp(x), [-1;1], b1);
   true
   > b2 = round(b, 25, RU);
   > checkinfnorm(p - exp(x), [-1;1], b2);
   false
   > diam = 1b-20!;
   > checkinfnorm(p - exp(x), [-1;1], b2);
   true

See also: infnorm, dirtyinfnorm, supnorm, accurateinfnorm, remez, diam