basic_letter Class Reference

A basic_letter is an element of an alphabet. More...

#include <basic_letter.h>

Inheritance diagram for basic_letter:


Public Member Functions
	basic_letter (const GiNaC::ex &l, const GiNaC::ex &d, const GiNaC::ex &o)
	basic_letter (const GiNaC::ex &l, const GiNaC::ex &d, const GiNaC::ex &o, const GiNaC::ex &i)
void	archive (GiNaC::archive_node &node) const
void	read_archive (const GiNaC::archive_node &node, GiNaC::lst &sym_lst)
unsigned	return_type (void) const
void	print (const GiNaC::print_context &c, unsigned level=0) const
unsigned	precedence (void) const
GiNaC::ex	eval (int level=0) const
GiNaC::ex	subs (const GiNaC::exmap &m, unsigned options=0) const
virtual GiNaC::ex	concat (const GiNaC::ex &l) const
virtual GiNaC::ex	eval_explicit (int level=0) const
virtual GiNaC::ex	expand_members (int level=0) const
GiNaC::ex	get_scale (void) const
GiNaC::ex	get_degree (void) const
GiNaC::ex	get_offset (void) const
GiNaC::ex	get_index (void) const
GiNaC::ex	set_index (const GiNaC::ex &i) const
GiNaC::ex	forget_index (void) const
GiNaC::ex	shift_index (const GiNaC::ex &new_index) const
GiNaC::ex	index_eq_one (void) const
GiNaC::ex	shift_plus_one (void) const
GiNaC::ex	shift_minus_one (void) const
GiNaC::ex	degree_minus_one (void) const
GiNaC::ex	degree_plus_one (void) const
Protected Member Functions
GiNaC::ex	eval_ncmul (const GiNaC::exvector &v) const
GiNaC::ex	derivative (const GiNaC::symbol &s) const
unsigned	calchash (void) const
virtual GiNaC::ex	concat_speedy (const GiNaC::ex &l) const
Protected Attributes
GiNaC::ex	scale
GiNaC::ex	degree
GiNaC::ex	offset
GiNaC::ex	index
Friends
GiNaC::ex	concat (const basic_letter &l1, const basic_letter &l2)

Detailed Description

A basic_letter is an element of an alphabet.

The model for a basic_letter is

$\frac{x^i}{(i+c)^m}$

Here $x$ is the scale, $m$ the degree, $c$ the offset and $i$ a parameter (usually a summation index). $i$ does not need to be specified. In this case the index is set to the default index.

Two basic_letters can be multiplied ("concatenated") to form a new basic_letter.

Member Function Documentation

ex concat ( const GiNaC::ex & l ) const [virtual]

Concats two letters. The routine checks first if the argument is of type basic_letter.
It then checks, if the indices are compatible (e.g., equal indices, or at least one index equal to the default one).
If this is the case, the method concat_speedy is called, otherwise the routine returns unevaluated.

ex concat_speedy ( const GiNaC::ex & l ) const [protected, virtual]

Same as concat, but does not perform any checking. Assumes that the indices are the same or equal to the default one.
The multiplication is performed if the two basic_letters have

the same offset
$\frac{x^i}{(i+c)^{m_1}} \frac{y^i}{(i+c)^{m_2}} = \frac{(xy)^i}{(i+c)^{m_1+m_2}}$

unequal offsets, but both degrees are integers (and not symbols).

If one of the degrees is zero, we have
$\frac{x^i}{(i+c)^{m}} y^i = \frac{(xy)^i}{(i+c)^{m}}$

If both degrees are positive, we have
$\frac{x^i}{(i+c_1)^{m_1}} \frac{y^i}{(i+c_2)^{m_2}} = \frac{1}{c_2-c_1} \left[ \frac{x^i}{(i+c_1)^{m_1}} \frac{y^i}{(i+c_2)^{m_2-1}} \mbox{} - \frac{x^i}{(i+c_1)^{m_1-1}} \frac{y^i}{(i+c_2)^{m_2}} \right]$

If one degree is positive ( $m_1$ ), the other negative ( $m_2$ ), we have
$\frac{x^i}{(i+c_1)^{m_1}} \frac{y^i}{(i+c_2)^{m_2}} = \frac{x^i}{(i+c_1)^{m_1-1}} \frac{y^i}{(i+c_2)^{m_2+1}} + (c_2-c_1) \frac{x^i}{(i+c_1)^{m_1}} \frac{y^i}{(i+c_2)^{m_2+1}}$

Finally, if both degrees are negative, we have
$\frac{x^i}{(i+c_1)^{m_1}} \frac{y^i}{(i+c_2)^{m_2}} = \frac{x^i}{(i+c_1)^{m_1-1}} \frac{y^i}{(i+c_2)^{m_2+1}} + (c_2-c_1) \frac{x^i}{(i+c_1)^{m_1}} \frac{y^i}{(i+c_2)^{m_2+1}}$
To avoid an infinite recursion in the last case, the algorithm does not commute two basic_letters.
The multiplication does not alter the offset, e.g. if we have $i+c_1 > 0$ and $i+c_2 > 0$ for all $i$ , then the evaluated product of the two basic_letters has only the offsets $c_1$ or $c_2$ . Multiplying two basic_letters does not introduce poles.
The result gets the index from the first letter, if this one differs from _default_index, otherwise the result gets the index from the second letter.
The routine returns unevaluated if

the offsets are not equal and at least one degree is not an integer.

Reimplemented in letter, and unit_letter.

ex degree_minus_one ( void ) const

Returns a basic_letter with degree-1

ex degree_plus_one ( void ) const

Returns a basic_letter with degree+1

ex eval ( int level = 0 ) const

If the offset is zero, evaluation returns a letter .
If the index is an integer, the object is evaluated explicitly.
Reimplemented in letter, and unit_letter.

ex eval_explicit ( int level = 0 ) const [virtual]

Explicit evaluation.

ex eval_ncmul ( const GiNaC::exvector & v ) const [protected]

A product of two basic_letter's is simplified, if

the two indices are equal, or at least one of the indices is equal to the default index;
and both basic_letter's are actually of type letter;
or the types are only basic_letter's, but both degrees are integer.

A product is also simplified if the difference between the two indices is an integer, according to
$\frac{x^{i+l}}{((i+l)+c)^m} \frac{y^i}{(i+c')^{m'}} = x^l \frac{x^{i}}{(i+(c+l))^m} \frac{y^i}{(i+c')^{m'}}$
where $ l > 0 $ . Note that the algorithm returns a product, where both factors have the "smaller" index.