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Taylor functions are the most important class in the the Kepler package in many ways. They should be thought of as a rigorously implemented math library function. To evaluate a function is to supply it with bounds on the lower and upper bounds on each of the variables, and an approximate center point where the taylor series approximation is to be computed.
When evaluated, instead of returning a mere double, a taylorFunction returns a taylorInterval, which constains information about the upper and lower bounds of the function on the given domain, and also about the upper and lower bounds of the derivatives of the function on the given domain.
Taylor functions can be added together and multiplied by scalars. To get an upper bound on the sum of two taylorFunctions it is better to add them and then evaluate the sum, rather than evaluate the individual functions and then add the results. The reason for this is that there can be cancellation of terms, so that the bound by adding first is, roughly speaking |x-x|, rather than |x|+|-x|.
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Thomas C. Hales
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taylorFunction operator+(const taylorFunction&) const;
Add a taylorFunctions to a given one.
taylorFunction operator+(const taylorFunction&) const;
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taylorFunction operator*(const interval&) const;
Scale a taylorFunction by a interval multiple.
taylorFunction operator*(const interval&) const;
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taylorFunction(int capacity );
Constructor. For advanced users:
the capacity refers to the number of distinct
primitive functions occuring in the linear combination represented
by the taylorFunction. If in doubt,
use the default argument!
taylorFunction(int capacity =0);
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taylorFunction(primitive&);
taylorFunctions are built up from certain primitive functions.
This is the constructor that converts a primitive function to
a taylorFunction.
The class primitive and this class is only used in the implementation
details. End-users can safely ignore this constructor.
taylorFunction(primitive&);
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taylorFunction(const taylorFunction&);
create a bitwise copy of a taylorFunction
taylorFunction(const taylorFunction&);
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taylorFunction& operator=(const taylorFunction& f);
assignment of a taylorFunction
taylorFunction& operator=(const taylorFunction& f);
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~taylorFunction();
Deallocate memory:
~taylorFunction();
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taylorInterval evalf(const domain& x,const domain& z) const;
Evaluate a taylorFunction
There are two arguments, x = lower bounds on variables,
z = upper bounds on variables,
taylorInterval evalf(const domain& x,const domain& z) const;
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lineInterval evalAt(const domain&) const;
Evaluate a taylorFunction at a single point x
lineInterval evalAt(const domain&) const;
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int getReducibleState() const;
int getReducibleState() const;
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void setReducibleState(int);
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void selfTest();
Check the correctness of Taylor routines.
static void selfTest();
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