Chebyshev approximation (GSL)
Approximate a function on a finite interval using a Chebyshev series:
\[
f(x) = \sum_n c_n T_n(x)
\]
where \( T_n(x)=\cos(n \arccos x) \)
See also the Chebyshev approximation example.
Initialization methods
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template<class func_t>
inline void init(func_t &func, size_t ord, fp_t a1, fp_t b1)
Initialize a Chebyshev approximation of the function func over the interval from a1 to b1.
The interval must be specified so that \( a < b \) , so a and b are swapped if this is not the case.
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template<class vec_t>
inline void init(fp_t a1, fp_t b1, size_t ord, vec_t &v)
Create an approximation from a vector of coefficients.
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template<class vec_t>
inline void init_func_values(fp_t a1, fp_t b1, size_t ord, vec_t &fval)
Create an approximation from a vector of function values.
Evaulation methods
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inline fp_t eval(fp_t x) const
Evaluate the approximation.
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inline fp_t operator()(fp_t x) const
Evaluate the approximation.
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inline fp_t eval_n(size_t n, fp_t x) const
Evaluate the approximation to a specified order.
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inline void eval_err(fp_t x, fp_t &result, fp_t &abserr)
Evaluate the approximation and give the uncertainty.
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inline void eval_n_err(size_t n, fp_t x, fp_t &result, fp_t &abserr)
Evaluate the approximation to a specified order and give the uncertainty.
Maniupulating coefficients and endpoints
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inline fp_t get_coefficient(size_t ix) const
Get a coefficient.
Legal values of the argument are 0 to order (inclusive)
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inline void set_coefficient(size_t ix, fp_t co)
Set a coefficient.
Legal values of the argument are 0 to order (inclusive)
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inline void get_endpoints(fp_t &la, fp_t &lb)
Return the endpoints of the approximation.
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template<class vec_t>
inline void get_coefficients(size_t n, vec_t &v) const
Get the coefficients.
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template<class vec_t>
inline void set_coefficients(size_t n, const vec_t &v)
Set the coefficients.
Derivatives and integrals
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inline void deriv(cheb_approx_tl &gc) const
Make gc an approximation to the derivative.
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inline void integ(cheb_approx_tl &gc) const
Make gc an approximation to the integral.
Public Types
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typedef boost::numeric::ublas::vector<fp_t> ubvector
Protected Attributes
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ubvector c
Coefficients.
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size_t order
Order of the approximation.
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fp_t a
Lower end of the interval.
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fp_t b
Upper end of the interval.
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size_t order_sp
Single precision order.
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ubvector f
Function evaluated at Chebyshev points.
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bool init_called
True if init has been called.
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fp_t pi
The constant \( \pi \).