Zsum Class Reference

Zsums form a Hopf algebra. More...

#include <Zsum.h>

Inheritance diagram for Zsum:


Public Member Functions
	Zsum (const GiNaC::ex &nc)
	Zsum (const GiNaC::ex &nc, const GiNaC::ex &llc)
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	convert_to_Ssum_exvector (const GiNaC::exvector &Z0, const GiNaC::exvector &Z1) const
virtual GiNaC::ex	shuffle_exvector (const GiNaC::exvector &Z0, const GiNaC::exvector &Z1, const GiNaC::exvector &Z2) const
virtual GiNaC::ex	set_index (const GiNaC::ex &i) const
virtual GiNaC::ex	shift_plus_one (void) const
virtual GiNaC::ex	shift_minus_one (void) const
virtual GiNaC::ex	adjust_upper_limit_downwards (const GiNaC::ex &i) const
virtual GiNaC::ex	adjust_upper_limit_upwards (const GiNaC::ex &i) const
virtual GiNaC::ex	adjust_upper_limit_plus_one (void) const
virtual GiNaC::ex	index_eq_one (void) const
virtual GiNaC::ex	get_head (int k) const
virtual GiNaC::ex	get_tail (int k) const
virtual GiNaC::ex	antipode (void) const
virtual GiNaC::ex	expand_members (int level=0) const
virtual GiNaC::ex	eval_explicit (int level=0) const
virtual GiNaC::ex	get_first_letter (void) const
virtual GiNaC::ex	remove_first_letter (void) const
virtual GiNaC::ex	remove_first_letter (const GiNaC::ex &nc) const
virtual GiNaC::ex	prepend_letter (const GiNaC::ex &lc) const
virtual GiNaC::ex	prepend_letter (const GiNaC::ex &nc, const GiNaC::ex &lc) const
virtual GiNaC::ex	append_letter (const GiNaC::ex &lc) const
virtual GiNaC::ex	append_letter_list (const GiNaC::ex &lc) const
GiNaC::ex	get_index (void) const
GiNaC::ex	get_letter_list (void) const
unsigned	get_depth (void) const
GiNaC::ex	get_weight (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	cast_to_Ssum (void) const
virtual GiNaC::ex	multiply_letter_with_last_letter (const GiNaC::ex &lc) const
virtual GiNaC::ex	multiply_letter_with_first_letter (const GiNaC::ex &lc) const
virtual GiNaC::ex	concat_two_sums (const GiNaC::ex &Z1, const GiNaC::ex &Z2) const
Protected Attributes
GiNaC::ex	n
GiNaC::ex	letter_list
Friends
GiNaC::ex	shuffle_Zsum (const GiNaC::ex &Z1, const GiNaC::ex &Z2)
GiNaC::ex	convert_Zsum_to_Ssum (const GiNaC::ex &Z1)
GiNaC::ex	remove_trivial_Zsum (const GiNaC::ex &expr)

Detailed Description

Zsums form a Hopf algebra.

A Zsum is recursively defined by

$Z(n;m_1,...,m_k;x_1,...,x_k) = \sum\limits_{i=1}^n \frac{x_1^i}{i^{m_1}} Z(i-1;m_2,...,m_k;x_2,...,x_k)$

with

$Z(n) = 1$

for $n \ge 0$ and

$Z(n) = 0$

for $ n < 0 $

All letters should have their index set to _default_index.

Member Function Documentation

ex adjust_upper_limit_downwards ( const GiNaC::ex & i ) const [virtual]

Adjusts the upper summation limit $n$ down to $i$ , e.g.
$Z(n;m_1,...;x_1,...) = Z(i;m_1,...;x_1,...) + \sum\limits_{j=1}^{n-i} x_1^j \frac{x_1^i}{(i+j)^{m_1}} Z(i-1+j;m_2,...;x_2,...)$
with $n > i $ .
For the empty sum we have
$Z(n) = Z(i)$

This routine assumes $i \ge 0$ .
Reimplemented in Euler_Zagier_sum.

ex adjust_upper_limit_plus_one ( void ) const [virtual]

Adjusts the upper summation limit $n$ to $n+1$ , e.g.
$Z(n;m_1,...;x_1,...) = Z(n+1;m_1,...;x_1,...) \mbox{} - \frac{x_1^{n+1}}{(n+1)^{m_1}} Z(n;m_2,...;x_2,...).$

Reimplemented in Euler_Zagier_sum.

ex adjust_upper_limit_upwards ( const GiNaC::ex & i ) const [virtual]

Adjusts the upper summation limit $n$ up to $i$ , e.g.
$Z(n;m_1,...;x_1,...) = Z(i;m_1,...;x_1,...) \mbox{} - \sum\limits_{j=0}^{i-n-1} x_1^{-j} \frac{x_1^i}{(i-j)^{m_1}} Z(i-j-1;m_2,...;x_2,...)$
with $n < i $ .
For the empty sum we have
$Z(n) = Z(i)$

This routine assumes $n \ge 0$ .
This routine might give rise to poles for some values of $i$ and should be used with care.
Reimplemented in Euler_Zagier_sum.

ex antipode ( void ) const [virtual]

Calculates the antipode.
Only for Zsums.

ex append_letter ( const GiNaC::ex & lc ) const [virtual]

Appends the letter lc to the letter_list.

ex append_letter_list ( const GiNaC::ex & lc ) const [virtual]

Appends the list lc to the letter_list.

ex cast_to_Ssum ( void ) const [protected, virtual]

Creates a Ssum with the same upper summation index and the same letter_list.
This is a crude cast, something like reinterpret_cast.

ex concat_two_sums ( const GiNaC::ex & Z1,

const GiNaC::ex & Z2

) const [protected, virtual]

Takes Z1 as head and Z2 as tail and forms a new sum as the concatenation of the two.

ex convert_to_Ssum_exvector ( const GiNaC::exvector & Z0,

const GiNaC::exvector & Z1

) const [virtual]

Method to convert a Zsum to a Ssum.
An efficient version for the conversion to Ssums, based on exvector (e.g. std::vector<GiNaC::ex> ). Z0 contains the result. Z1 is reversed order, so that we can use pop_back.
The basic formula is
$Z(n;m_1,...;x_1,...) = \sum\limits_{i=1}^n \frac{x_1^i}{i^{m_1}} Z(i;m_2,...;x_2,...) \mbox{} - Z(n;m_1+m_2,...;x_1 x_2, ...)$

Reimplemented in Euler_Zagier_sum.

ex eval ( int level = 0 ) const

The simplifications are done in the following order:

If the upper summation limit is equal to infinity, we have a multiple polylog.

If all $x_j$ 's are equal to 1, we have a Euler-Zagier sum.

If the upper summation index is an integer, perform the sum explicitly.

Reimplemented in classical_polylog, Euler_Zagier_sum, harmonic_polylog, multiple_polylog, multiple_zeta_value, and nielsen_polylog.

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

Explicit evaluation

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

A product of Zsums is simplified, if the last two have the same upper summation index. The function shuffle_Zsum is called for the multiplication.
A product is also simplified if the difference between the two upper summation indices is an integer. The factor with the "larger" upper summation index is converted to the "smaller" index with the help of the function Zsum::adjust_upper_limit_downwards. Note that the algorithm returns a product, where both factors have the "smaller" index.

unsigned get_depth ( void ) const [inline]

Returns the depth.

ex get_first_letter ( void ) const [virtual]

Returns the first letter from the letter_list.

ex get_head ( int k ) const [virtual]

Returns from the Z-sum $ Z(n;m_1,...,m_l;x_1,...,x_l) $ the head $Z(n;m_1,...,m_{k-1};x_1,...,x_{k-1})$ .

GiNaC::ex get_index ( void ) const [inline]

Returns the upper summation limit.

GiNaC::ex get_letter_list ( void ) const [inline]

Returns the letter_list.

ex get_tail ( int k ) const [virtual]

Returns from the Z-sum $ Z(n;m_1,...,m_l;x_1,...,x_l) $ the tail $ Z(n;m_k,...,m_l;x_k,...,x_l) $ .

ex get_weight ( void ) const

Returns the weight.

ex index_eq_one ( void ) const [virtual]

Returns
$Z(1;m_1,...,m_k;x_1,...,x_k) = 0$
for $k>1$ . For $k=1$ it returns
$x_1$
and for $k=0$ we have
$1$

ex multiply_letter_with_first_letter ( const GiNaC::ex & lc ) const [protected, virtual]

Returns a Zsum, where the letter lc is multiplied with the first letter in the letter list of the original Zsum.
If the sum is empty, e.g. $Z(n)$ , the result is $\frac{x^n}{n^m} Z(n)$ .

ex multiply_letter_with_last_letter ( const GiNaC::ex & lc ) const [protected, virtual]

Returns a Zsum, where the letter lc is multiplied with the last letter in the letter list of the original Zsum.
If the sum is empty, e.g. $Z(n)$ , the result is $\frac{x^n}{n^m} Z(n)$ .

ex prepend_letter ( const GiNaC::ex & nc,

const GiNaC::ex & lc

) const [virtual]

Prepends the letter lc to the letter_list and sets the upper summation index to nc.

ex prepend_letter ( const GiNaC::ex & lc ) const [virtual]

Prepends the letter lc to the letter_list.

ex remove_first_letter ( const GiNaC::ex & nc ) const [virtual]

Returns a Zsum with the first letter removed from the letter_list and the upper summation index set to nc.
Reimplemented in Euler_Zagier_sum.

ex remove_first_letter ( void ) const [virtual]

Returns a Zsum with the first letter removed from the letter_list.
Reimplemented in Euler_Zagier_sum.

ex set_index ( const GiNaC::ex & i ) const [virtual]

Sets the upper summation index to $i$
Reimplemented in Euler_Zagier_sum.

ex shift_minus_one ( void ) const [virtual]

Returns $Z(n-1,m_1,...,x_1,...)$
Reimplemented in Euler_Zagier_sum.

ex shift_plus_one ( void ) const [virtual]

Returns $Z(n+1,m_1,...,x_1,...)$
Reimplemented in Euler_Zagier_sum.

ex shuffle_exvector ( const GiNaC::exvector & Z0,

const GiNaC::exvector & Z1,

const GiNaC::exvector & Z2

) const [virtual]

Method to multiply two Zsums.
An efficient version for the multiplication of Zsums, based on exvector (e.g. std::vector<GiNaC::ex> ). Z0 contains the result. Z1 and Z2 are in reversed order, so that we can use pop_back.
The basic formula is
$Z(n;m_1,...;x_1,...) Z(n;m_1',...;x_1',...) = \sum\limits_{i=1}^n \frac{x_1^i}{i^{m_1}} Z(i-1;m_2,...;x_2,...) Z(i-1;m_1',...;x_1',...)$

$\mbox{} + \sum\limits_{i=1}^n \frac{{x_1'}^i}{i^{m_1'}} Z(i-1;m_1,...;x_1,...) Z(i-1;m_2',...;x_2',...) \mbox{} + \sum\limits_{i=1}^n \frac{(x_1 {x_1'})^i}{i^{m_1+m_1'}} Z(i-1;m_2,...;x_2,...) Z(i-1;m_2',...;x_2',...)$

Reimplemented in Euler_Zagier_sum.

The documentation for this class was generated from the following files:

Generated on Wed Jun 10 22:59:11 2009 for Nestedsums library by

1.3.7

Zsum Class Reference

Public Member Functions

Protected Member Functions

Protected Attributes

Friends

Detailed Description

Member Function Documentation