Ssum Class Reference

Ssums form an algebra. More...

#include <Ssum.h>

Inheritance diagram for Ssum:


Public Member Functions
	Ssum (const GiNaC::ex &nc)
	Ssum (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_Zsum_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	refine_exvector (unsigned q, const GiNaC::exvector &Z0, const GiNaC::exvector &Z1) const
virtual GiNaC::ex	refine (unsigned q) 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 int	flag_remove_negative_degrees (void) const
virtual GiNaC::ex	remove_negative_degrees (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_Zsum (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_Ssum (const GiNaC::ex &Z1, const GiNaC::ex &Z2)
GiNaC::ex	convert_Ssum_to_Zsum (const GiNaC::ex &Z1)
GiNaC::ex	remove_trivial_Ssum (const GiNaC::ex &expr)

Detailed Description

Ssums form an algebra.

A Ssum is recursively defined by

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

with

$S(n) = 1$

for $ n > 0 $ and

$S(n) = 0$

for $n \le 0$ .

Ssums can be converted to Zsums (and vice versa).

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.
$S(n;m_1,...;x_1,...) = S(i;m_1,...;x_1,...) + \sum\limits_{j=0}^{n-i-1} x_1^{j+1} \frac{x_1^i}{(i+j+1)^{m_1}} S(i+j+1;m_2,...;x_2,...)$
with $n > i $ .
For the empty sum we have
$S(n) = S(i)$

This routine assumes $i \ge 1$ .
Reimplemented in harmonic_sum.

ex adjust_upper_limit_plus_one ( void ) const [virtual]

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

Reimplemented in harmonic_sum.

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

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

This routine assumes $n \ge 1$ .
This routine might give rise to poles for some values of $i$ and should be used with care.
Reimplemented in harmonic_sum.

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_Zsum ( void ) const [protected, virtual]

Creates a Zsum 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_Zsum_exvector ( const GiNaC::exvector & Z0,

const GiNaC::exvector & Z1

) const [virtual]

Method to convert a Ssum to a Zsum.
An efficient version for the conversion to Zsums, 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
$S(n;m_1,...;x_1,...) = \sum\limits_{i=1}^n \frac{x_1^i}{i^{m_1}} S(i-1;m_2,...;x_2,...) \mbox{} + S(n;m_1+m_2,...;x_1 x_2, ...)$

Reimplemented in harmonic_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 Ssum_to_Infinity

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

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

Reimplemented in harmonic_sum, harmonic_sum_to_Infinity, and Ssum_to_Infinity.

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

Explicit evaluation

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

A product of Ssums is simplified, if the last two have the same upper summation index.
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.

int flag_remove_negative_degrees ( void ) const [virtual]

Signals if there are letters with negative or zero degrees.
Only for Ssums.

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

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 S-sum $ S(n;m_1,...,m_l;x_1,...,x_l) $ the tail $ S(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
$S(1;m_1,...,m_k;x_1,...,x_k) = x_1 \cdot ... \cdot x_k$

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

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

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

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

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 refine ( unsigned q ) const [virtual]

Method to convert a Ssum with upper limit $n$ to Ssums with upper limit $q \cdot n$ .

ex refine_exvector ( unsigned q,

const GiNaC::exvector & Z0,

const GiNaC::exvector & Z1

) const [virtual]

Helper routine to convert a Ssum with upper limit $n$ to Ssums with upper limit $q \cdot n$ .
Z0 contains the result in reversed order. Z1 the letter list, which has to be converted.
The basic formula is
$\sum\limits_{i=1}^n \frac{x^i}{i^m} S( q \cdot i;...) = q^{m-1} \sum\limits_{l=0}^{q-1} \sum\limits_{i=1}^{q \cdot n} \frac{1}{i^m} \left( r_q^l x^{1/q} \right)^i S(i;...)$
Here $r_q^l$ are roots of unity.

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

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

ex remove_first_letter ( void ) const [virtual]

Returns a Ssum with the first letter removed from the letter_list.
Reimplemented in harmonic_sum.

ex remove_negative_degrees ( void ) const [virtual]

Removes letters with non-positive degrees.
The basic formula is
$S(n;-m_1,m_2,...;x_1,x_2,...) = \sum\limits_{i2=1}^n \frac{x_2^{i_2}}{i_2^{m_2}} S(i_2;m_3,...;x_3,...) \sum\limits_{i_1=i_2}^n i_1^{m_1} x_1^{i_1}$
For $x_1 \neq 1$ we have
$\sum\limits_{i_1=i_2}^n i_1^{m_1} x_1^{i_1} = \left( x_1^- \right)^{m_1} \left( \frac{1}{1-x_1} x_1^{i_2} - \frac{x_1}{1-x_1} x_1^n \right)$
For $x_1 = 1$ we have
$\sum\limits_{i_1=i_2}^n i_1^{m_1} = \sum\limits_{i_1=1}^n i_1^{m_1} - \sum\limits_{i_1=1}^{i_2-1} i_1^{m_1}$
and the sums can be done analytically with the help of the Bernoulli numbers:
$\sum\limits_{i=1}^{n} i^m = n^m + \sum\limits_{k=0}^m \frac{B_k}{k!} \frac{m!}{(m+1-k)!} n^{m+1-k}$

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

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

ex shift_minus_one ( void ) const [virtual]

Returns $S(n-1,m_1,...,x_1,...)$
Reimplemented in harmonic_sum.

ex shift_plus_one ( void ) const [virtual]

Returns $S(n+1,m_1,...,x_1,...)$
Reimplemented in harmonic_sum.

ex shuffle_exvector ( const GiNaC::exvector & Z0,

const GiNaC::exvector & Z1,

const GiNaC::exvector & Z2

) const [virtual]

Method to multiply two Ssums.
An efficient version for the multiplication of Ssums, 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
$S(n;m_1,...;x_1,...) S(n;m_1',...;x_1',...) = \sum\limits_{i=1}^n \frac{x_1^i}{i^{m_1}} S(i;m_2,...;x_2,...) S(i;m_1',...;x_1',...)$

$\mbox{} + \sum\limits_{i=1}^n \frac{{x_1'}^i}{i^{m_1'}} S(i;m_1,...;x_1,...) S(i;m_2',...;x_2',...) \mbox{} - \sum\limits_{i=1}^n \frac{(x_1 {x_1'})^i}{i^{m_1+m_1'}} S(i;m_2,...;x_2,...) S(i;m_2',...;x_2',...)$
Note the minus sign in front of the last term.
Reimplemented in harmonic_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

Ssum Class Reference

Public Member Functions

Protected Member Functions

Protected Attributes

Friends

Detailed Description

Member Function Documentation