special Namespace Reference

Namespaces
	constant
	integral

Functions
int	coulomb_zeros (double eta, int L, int nzeros, double *zeros, double epsrel=1e-8)
	Get zeros of the Coulomb wave function \( F_L(-1/k,kr) \). More...
std::vector< std::vector< int > >	FdB_partition (int n)
	Faa di Bruno partitioning. More...
template<class T , class OuterFunctionDerivative , class InnerFunctionDerivative >
T	chain_rule (OuterFunctionDerivative Df, InnerFunctionDerivative Dg, int n, double x)
	Chain rule for n-th derivative. More...

Function Documentation

template<class T , class OuterFunctionDerivative , class InnerFunctionDerivative >

T special::chain_rule	(	OuterFunctionDerivative	Df,
		InnerFunctionDerivative	Dg,
		int	n,
		double	x
	)

This function implements the Faa di Bruno's formula for n-th derivative of a nested function,

\[ \frac{\mathrm{d}^n}{\mathrm{d}^n x} f(g(x)) = \sum \frac{n!}{m_1! 1!^{m_1} m_2! 2!^{m_2} \dots m_n! n!^{m_n}} f^{(m_1+m_2+\dots+m_n)}(g(x)) \prod_{j=1}^n \left(g^{(j)}(x)\right)^{m_j} \ . \]

The sum runs over all n-tuples \( (m_1, \dots, m_n) \) that satisfy the Faa di Bruno's sum condition

\[ 1 m_1 + 2 m_2 + \dots + n m_n = n \ . \]

Those n-tuples are retrieved from the function FdB_partition.

int special::coulomb_zeros	(	double	eta,
		int	L,
		int	nzeros,
		double *	zeros,
		double	epsrel = 1e-8
	)

Calculates given number of leading zeros of the Coulomb wave function \( F_L(\eta,\rho) \), where \( \eta = -1/k \) and \( \rho = kr \).

The method used comes from Ikebe Y.: The zeros of regular Coulomb wave functions and of their derivatives, Math. Comp. 29, 131 (1975) 878-887. It uses eigenvalues of a special tridiagonal matrix. The eigenvalues are computed using the standard Lapack function DSTEV .

std::vector<std::vector<int> > special::FdB_partition

(

int

n

)

The Faa di Bruno partitioning is computed by looping over possible n-tuples

\[ (m_1, m_2, \dots, m_n) \]

of integers and by picking only such that satisfy the Faa di Bruno's sum condition

\[ 1 m_1 + 2 m_2 + \dots + n m_n = n \ . \]

The initial trial partitioning is a zero tuple

\[ (0, 0, \dots, 0) \]

and the further tuples are constructed by incrementing a corresponding multidigit number, that has a number system varying with the position. The number system base for the left-most (least significant) position is \( n + 1 \), for the next position it is \( \lceil (n + 1)/2 \rceil \), for the next it is \( \lceil (n + 1)/3 \rceil \), etc. For example, if \( n = 4 \), the increments are

\[ (0, 0, 0, 0), (1, 0, 0, 0), (2, 0, 0, 0), (3, 0, 0, 0), \mathbf{(4, 0, 0, 0)}, \]

\[ (0, 1, 0, 0), (1, 1, 0, 0), (2, 1, 0, 0), \mathbf{(3, 1, 0, 0)}, (4, 1, 0, 0), \]

\[ (0, 2, 0, 0), (1, 2, 0, 0), \mathbf{(2, 2, 0, 0)}, (3, 2, 0, 0), (4, 2, 0, 0), \]

\[ (0, 0, 1, 0), \mathbf{(1, 0, 1, 0)}, (2, 0, 1, 0), (3, 0, 1, 0), (4, 0, 1, 0), \]

\[ (0, 1, 1, 0), (1, 1, 1, 0), (2, 1, 1, 0), (3, 1, 1, 0), (4, 1, 1, 0), \]

\[ (0, 2, 1, 0), (1, 2, 1, 0), (2, 2, 1, 0), (3, 2, 1, 0), (4, 2, 1, 0), \]

\[ \mathbf{(0, 0, 0, 1)}, \ \mathrm{etc.} \]

Here, the number system are 5, 3, 2 and 2. Only those tuples in bold satisfy the sum condition and will be returned.

This function is needed by the generalized chain_rule.

Todo:

Cache results.