Sollya User's Manual

Name:

dirtyinfnorm computes a numerical approximation of the infinity norm of a function on an interval.

Usage:

dirtyinfnorm(f,I) : (function, range) -> constant

Parameters:

f is a function.
I is an interval.

Description:

dirtyinfnorm(f,I) computes an approximation of the infinity norm of the given function f on the interval I, e.g. max {|f(x)|, x in I}.
The interval must be bound. If the interval contains one of -Inf or +Inf, the result of dirtyinfnorm is NaN.
The result of this command depends on the global variables prec and points. Therefore, the returned result is generally a good approximation of the exact infinity norm, with precision prec. However, the result is generally underestimated and should not be used when safety is critical. Use infnorm instead.
The following algorithm is used: let n be the value of variable points and t be the value of variable prec.
- Evaluate |f| at n evenly distributed points in the interval I. The evaluation are faithful roundings of the exact results at precision t.
- Whenever the derivative of f changes its sign for two consecutive points, find an approximation x of its zero with precision t. Then compute a faithful rounding of |f(x)| at precision t.
- Return the maximum of all computed values.

Example 1:

> dirtyinfnorm(sin(x),[-10;10]);
1

Example 2:

   > prec=15!;
   > dirtyinfnorm(exp(cos(x))*sin(x),[0;5]);
   1.45856
   > prec=40!;
   > dirtyinfnorm(exp(cos(x))*sin(x),[0;5]);
   1.458528537135
   > prec=100!;
   > dirtyinfnorm(exp(cos(x))*sin(x),[0;5]);
   1.458528537136237644438147455024
   > prec=200!;
   > dirtyinfnorm(exp(cos(x))*sin(x),[0;5]);
   1.458528537136237644438147455023841718299214087993682374094153

Example 3:

> dirtyinfnorm(x^2, [log(0);log(1)]);
@NaN@

See also: prec, points, infnorm, checkinfnorm, supnorm

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