nestedsums Namespace Reference

Classes

class basic_letter

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

class Bsum

Bsums arise from Hoelder convolutions. More...

class classical_polylog

A classical polylog is a special case of a Nielsen polylog. More...

class Csum

Csums involve a conjugation. More...

class Euler_Zagier_sum

A Euler-Zagier sum is a special case of a Zsum. More...

class expand_status

Flags for the status of the expansion. More...

class expand_request

Flags for the nestedsum_helper_expand function. More...

class print_format

Flags for the print format for polylogs. More...

class nestedsums_status_flags

Additional status flags. More...

struct map_remove_Zsums

struct map_insert_Zsums

class harmonic_polylog

A harmonic polylog is a special case of a multiple polylog. More...

class harmonic_sum

A harmonic sum is a special case of a Ssum. More...

class harmonic_sum_to_Infinity

class hash_entry

This class provides a container for expressions to be stored in hash tables. More...

class nestedsums_helper_less

class letter

A letter is a basic_letter with offset=0. More...

class list_of_tgamma

The class list_of_tgamma is a container for the class ratio_of_tgamma. More...

class multiple_polylog

A multiple polylog sum is a special case of a Zsum. More...

class multiple_zeta_value

class nielsen_polylog

A Nielsen polylog is a special case of a harmonic polylog. More...

class ratio_of_tgamma

The class ratio_of_tgamma contains the ratio of two Gamma functions. More...

class root_of_unity

class Ssum

Ssums form an algebra. More...

class Ssum_to_Infinity

A Ssum_to_Infinity is a special case of a Ssum. More...

class transcendental_fct_type_A

class transcendental_fct_type_B

class transcendental_fct_type_C

class transcendental_fct_type_D

class transcendental_sum_type_A

class transcendental_sum_type_B

The class transcendental_sum_type_B may contain $i$ and $(n-i)$ . More...

class transcendental_sum_type_C

The class transcendental_sum_type_C involves a conjugation. More...

class transcendental_sum_type_D

class unit_letter

A unit_letter is a letter whose letter content equals 1. More...

class Zsum

Zsums form a Hopf algebra. More...

Typedefs

typedef unsigned int p_int

Functions

GINAC_BIND_UNARCHIVER (basic_letter)

GiNaC::ex concat (const basic_letter &l1, const basic_letter &l2)

GINAC_DECLARE_UNARCHIVER (basic_letter)

GiNaC::ex create_basic_letter (const GiNaC::ex &l, const GiNaC::ex &d, const GiNaC::ex &o)

GiNaC::ex create_basic_letter (const GiNaC::ex &l, const GiNaC::ex &d, const GiNaC::ex &o, const GiNaC::ex &i)

GINAC_BIND_UNARCHIVER (Bsum)

ex convert_Bsum_to_Ssum (const ex &C)

ex convert_Bsum_to_Zsum (const ex &C)

ex create_Bsum_from_exvector (const ex &nc, const exvector &v)

GINAC_DECLARE_UNARCHIVER (Bsum)

GiNaC::ex create_Bsum (const GiNaC::ex &nc, const GiNaC::ex &llc)

GiNaC::ex create_Bsum_from_exvector (const GiNaC::ex &nc, const GiNaC::exvector &v)

GiNaC::ex convert_Bsum_to_Ssum (const GiNaC::ex &C)

GiNaC::ex convert_Bsum_to_Zsum (const GiNaC::ex &C)

GINAC_BIND_UNARCHIVER (classical_polylog)

GINAC_DECLARE_UNARCHIVER (classical_polylog)

GiNaC::ex create_classical_polylog (const GiNaC::ex &llc)

const symbol Infinity ("Infinity","\\infty")

const symbol _default ("default")

const symbol _default_index ("dummy")

GINAC_BIND_UNARCHIVER (Csum)

ex convert_Csum_to_Ssum (const ex &C)

ex convert_Csum_to_Zsum (const ex &C)

ex create_Csum_from_exvector (const ex &nc, const exvector &v)

GINAC_DECLARE_UNARCHIVER (Csum)

GiNaC::ex create_Csum (const GiNaC::ex &nc, const GiNaC::ex &llc)

GiNaC::ex create_Csum_from_exvector (const GiNaC::ex &nc, const GiNaC::exvector &v)

GiNaC::ex convert_Csum_to_Ssum (const GiNaC::ex &C)

GiNaC::ex convert_Csum_to_Zsum (const GiNaC::ex &C)

GINAC_BIND_UNARCHIVER (Euler_Zagier_sum)

ex create_Euler_Zagier_sum_with_ones (const ex &n, const int &k)

ex create_Euler_Zagier_sum_from_exvector (const ex &nc, const exvector &v)

GINAC_DECLARE_UNARCHIVER (Euler_Zagier_sum)

GiNaC::ex create_Euler_Zagier_sum (const GiNaC::ex &nc, const GiNaC::ex &llc)

GiNaC::ex create_Euler_Zagier_sum_with_ones (const GiNaC::ex &n, const int &k)

GiNaC::ex create_Euler_Zagier_sum_from_exvector (const GiNaC::ex &nc, const GiNaC::exvector &v)

ex Harmonic_sum_eval (const ex &x1, const ex &x2)

REGISTER_FUNCTION (Harmonic_sum, eval_func(Harmonic_sum_eval))

ex EulerZagier_sum_eval (const ex &x1, const ex &x2)

REGISTER_FUNCTION (EulerZagier_sum, eval_func(EulerZagier_sum_eval))

ex S_sum_infinity_eval (const ex &x1, const ex &x2)

REGISTER_FUNCTION (S_sum_infinity, eval_func(S_sum_infinity_eval))

ex S_sum_eval (const ex &x1, const ex &x2, const ex &x3)

REGISTER_FUNCTION (S_sum, eval_func(S_sum_eval))

ex Z_sum_eval (const ex &x1, const ex &x2, const ex &x3)

REGISTER_FUNCTION (Z_sum, eval_func(Z_sum_eval))

ex convert_to_nestedsums (const ex &e)

ex convert_to_ginac_functions (const ex &e)

GiNaC::ex convert_to_nestedsums (const GiNaC::ex &e)

GiNaC::ex convert_to_ginac_functions (const GiNaC::ex &e)

GINAC_BIND_UNARCHIVER (harmonic_polylog)

GINAC_DECLARE_UNARCHIVER (harmonic_polylog)

GiNaC::ex create_harmonic_polylog (const GiNaC::ex &llc)

GINAC_BIND_UNARCHIVER (harmonic_sum)

ex create_harmonic_sum_with_ones (const ex &n, const int &k)

ex create_harmonic_sum_from_exvector (const ex &nc, const exvector &v)

GINAC_DECLARE_UNARCHIVER (harmonic_sum)

GiNaC::ex create_harmonic_sum (const GiNaC::ex &nc, const GiNaC::ex &llc)

GiNaC::ex create_harmonic_sum_with_ones (const GiNaC::ex &n, const int &k)

GiNaC::ex create_harmonic_sum_from_exvector (const GiNaC::ex &nc, const GiNaC::exvector &v)

GiNaC::ex create_harmonic_sum_to_Infinity (const GiNaC::ex &llc)

GINAC_BIND_UNARCHIVER (hash_entry)

GINAC_DECLARE_UNARCHIVER (hash_entry)

GiNaC::ex create_hash_entry (const GiNaC::ex &cd, const GiNaC::ex &sd, const GiNaC::ex &rd, unsigned rkey)

ex nestedsums_helper_expand (const ex &expr, int level)

ex nestedsums_helper_multiply_via_triangle (const ex &A, const ex &B, const ex &expansion_parameter, int order)

ex nestedsums_helper_multiply_via_triangle_speedy (const ex &A, const ex &B, const ex &expansion_parameter, int order)

ex nestedsums_helper_multiply_laurent (const ex &A, const ex &B, const ex &expansion_parameter, int order)

ex nestedsums_helper_multiply_laurent_speedy (const ex &A, const ex &B, const ex &expansion_parameter, int order)

ex nestedsums_helper_expand_tgamma_fct (const ex &a1, const ex &b1, const ex &a2, const ex &b2, const ex &expansion_parameter, int order)

ex nestedsums_helper_expand_lst_tgamma_fct (const ex &ll1, const ex &ll2, const ex &expansion_parameter, int order)

ex nestedsums_helper_series (const ex &f, const ex &expansion_parameter, int order)

exvector exvector_from_lst (const ex &l)

exvector reverse_exvector_from_lst (const ex &l)

exvector exvector_append_lst (const exvector &v, const ex &l)

exvector exvector_multiply_last_letter (const exvector &v, const ex &x)

exvector exvector_increase_last_degree (const exvector &v)

ex nestedsums_helper_bernoulli (int n)

ex nestedsums_helper_lowering_op_geo_0 (const ex &x, int m)

ex nestedsums_helper_lowering_op_geo_1 (const ex &x, int m)

ex nestedsums_helper_arithmetic_sum_type_Z (const ex &n, const ex &m)

ex nestedsums_helper_arithmetic_sum_type_S (const ex &n, const ex &m)

ex nestedsums_helper_expand_power (const ex &x, const ex &m, const ex &expansion_parameter, int order)

int nestedsums_helper_get_degree (const ex &expr, const ex &eps)

int nestedsums_helper_ldegree (const ex &expr, const ex &eps)

ex nestedsums_helper_coeff (const ex &expr, const ex &eps, int n)

bool is_class_in_nestedsums (const ex &expr)

bool is_expr_in_sym_lst (const ex &expr, const lst &sym_lst)

ex nestedsums_helper_normalize (const ex &expr, const lst &sym_lst)

ex nestedsums_helper_sort (const ex &expr, const lst &sym_lst)

ex convert_Zsums_to_standard_form (const ex &expr, const lst &sym_lst)

void nestedsums_helper_clear_hashes (void)

GiNaC::ex nestedsums_helper_expand (const GiNaC::ex &expr, int level=0)

GiNaC::ex nestedsums_helper_multiply_via_triangle (const GiNaC::ex &A, const GiNaC::ex &B, const GiNaC::ex &expansion_parameter, int order)

GiNaC::ex nestedsums_helper_multiply_via_triangle_speedy (const GiNaC::ex &A, const GiNaC::ex &B, const GiNaC::ex &expansion_parameter, int order)

GiNaC::ex nestedsums_helper_multiply_laurent (const GiNaC::ex &A, const GiNaC::ex &B, const GiNaC::ex &expansion_parameter, int order)

GiNaC::ex nestedsums_helper_multiply_laurent_speedy (const GiNaC::ex &A, const GiNaC::ex &B, const GiNaC::ex &expansion_parameter, int order)

GiNaC::ex nestedsums_helper_expand_tgamma_fct (const GiNaC::ex &a1, const GiNaC::ex &b1, const GiNaC::ex &a2, const GiNaC::ex &b2, const GiNaC::ex &expansion_parameter, int order)

GiNaC::ex nestedsums_helper_expand_lst_tgamma_fct (const GiNaC::ex &ll1, const GiNaC::ex &ll2, const GiNaC::ex &expansion_parameter, int order)

GiNaC::ex nestedsums_helper_expand_power (const GiNaC::ex &x, const GiNaC::ex &m, const GiNaC::ex &expansion_parameter, int order)

GiNaC::ex nestedsums_helper_series (const GiNaC::ex &f, const GiNaC::ex &expansion_parameter, int order)

GiNaC::ex nestedsums_helper_lowering_op_geo_0 (const GiNaC::ex &x, int m)

GiNaC::ex nestedsums_helper_lowering_op_geo_1 (const GiNaC::ex &x, int m)

GiNaC::ex nestedsums_helper_arithmetic_sum_type_Z (const GiNaC::ex &n, const GiNaC::ex &m)

GiNaC::ex nestedsums_helper_arithmetic_sum_type_S (const GiNaC::ex &n, const GiNaC::ex &m)

int nestedsums_helper_get_degree (const GiNaC::ex &expr, const GiNaC::ex &eps)

int nestedsums_helper_ldegree (const GiNaC::ex &expr, const GiNaC::ex &eps)

GiNaC::ex nestedsums_helper_coeff (const GiNaC::ex &expr, const GiNaC::ex &eps, int n=1)

GiNaC::exvector exvector_from_lst (const GiNaC::ex &l)

GiNaC::exvector reverse_exvector_from_lst (const GiNaC::ex &l)

GiNaC::exvector exvector_append_lst (const GiNaC::exvector &v, const GiNaC::ex &l)

GiNaC::exvector exvector_multiply_last_letter (const GiNaC::exvector &v, const GiNaC::ex &x)

GiNaC::exvector exvector_increase_last_degree (const GiNaC::exvector &v)

bool is_class_in_nestedsums (const GiNaC::ex &expr)

bool is_expr_in_sym_lst (const GiNaC::ex &expr, const GiNaC::lst &sym_lst)

GiNaC::ex nestedsums_helper_normalize (const GiNaC::ex &expr, const GiNaC::lst &sym_lst=_empty_list)

GiNaC::ex nestedsums_helper_sort (const GiNaC::ex &expr, const GiNaC::lst &sym_lst=_empty_list)

GiNaC::ex convert_Zsums_to_standard_form (const GiNaC::ex &expr, const GiNaC::lst &sym_lst=_empty_list)

GINAC_BIND_UNARCHIVER (letter)

GINAC_DECLARE_UNARCHIVER (letter)

GiNaC::ex create_letter (const GiNaC::ex &l, const GiNaC::ex &d)

GiNaC::ex create_letter (const GiNaC::ex &l, const GiNaC::ex &d, const GiNaC::ex &i)

GINAC_BIND_UNARCHIVER (list_of_tgamma)

GINAC_DECLARE_UNARCHIVER (list_of_tgamma)

GiNaC::ex create_list_of_tgamma (const GiNaC::ex &l)

GiNaC::ex create_list_of_tgamma (const GiNaC::ex &l, const GiNaC::ex &i, const GiNaC::ex &eps, int o)

GiNaC::ex create_list_of_tgamma_and_set_gammas (const GiNaC::ex &l, const GiNaC::ex &i, const GiNaC::ex &eps, int o)

GINAC_BIND_UNARCHIVER (multiple_polylog)

ex eval_multiple_polylog_approx (const ex &expr)

GINAC_DECLARE_UNARCHIVER (multiple_polylog)

GiNaC::ex create_multiple_polylog (const GiNaC::ex &llc)

GiNaC::ex eval_multiple_polylog_approx (const GiNaC::ex &expr)

GINAC_BIND_UNARCHIVER (multiple_zeta_value)

GINAC_DECLARE_UNARCHIVER (multiple_zeta_value)

GiNaC::ex create_multiple_zeta_value (const GiNaC::ex &llc)

GINAC_BIND_UNARCHIVER (nielsen_polylog)

GINAC_DECLARE_UNARCHIVER (nielsen_polylog)

GiNaC::ex create_nielsen_polylog (const GiNaC::ex &llc)

GINAC_BIND_UNARCHIVER (ratio_of_tgamma)

ex eval_ratio_of_tgamma_to_scalar (const ex &expr)

GINAC_DECLARE_UNARCHIVER (ratio_of_tgamma)

GiNaC::ex create_ratio_of_tgamma (const GiNaC::ex &a1, const GiNaC::ex &b1, const GiNaC::ex &a2, const GiNaC::ex &b2)

GiNaC::ex create_ratio_of_tgamma (const GiNaC::ex &a1, const GiNaC::ex &b1, const GiNaC::ex &a2, const GiNaC::ex &b2, const GiNaC::ex &i, const GiNaC::ex &eps, int o, int f)

GiNaC::ex eval_ratio_of_tgamma_to_scalar (const GiNaC::ex &expr)

GINAC_BIND_UNARCHIVER (root_of_unity)

GINAC_DECLARE_UNARCHIVER (root_of_unity)

GiNaC::ex create_root_of_unity (unsigned l, unsigned k)

GINAC_BIND_UNARCHIVER (Ssum)

ex shuffle_Ssum (const ex &Z1, const ex &Z2)

ex convert_Ssum_to_Zsum (const ex &Z1)

ex create_Ssum_from_exvector (const ex &nc, const exvector &v)

ex Ssum_to_Zsum (const ex &expr)

ex shift_upper_limit_plus_one_for_Ssum (const ex &expr)

ex remove_negative_degrees_from_Ssum (const ex &expr)

ex remove_trivial_Ssum (const ex &expr)

ex refine_Ssum (const ex &expr, unsigned q)

GINAC_DECLARE_UNARCHIVER (Ssum)

GiNaC::ex create_Ssum (const GiNaC::ex &nc, const GiNaC::ex &llc)

GiNaC::ex shuffle_Ssum (const GiNaC::ex &Z1, const GiNaC::ex &Z2)

GiNaC::ex convert_Ssum_to_Zsum (const GiNaC::ex &Z1)

GiNaC::ex create_Ssum_from_exvector (const GiNaC::ex &nc, const GiNaC::exvector &v)

GiNaC::ex Ssum_to_Zsum (const GiNaC::ex &expr)

GiNaC::ex shift_upper_limit_plus_one_for_Ssum (const GiNaC::ex &expr)

GiNaC::ex remove_negative_degrees_from_Ssum (const GiNaC::ex &expr)

GiNaC::ex remove_trivial_Ssum (const GiNaC::ex &expr)

GiNaC::ex refine_Ssum (const GiNaC::ex &expr, unsigned q)

GiNaC::ex create_Ssum_to_Infinity (const GiNaC::ex &llc)

const symbol symbol_factory (const std::string &symbol_name, const std::string &latex_name)

const symbol next_index (void)

const symbol next_x (void)

GINAC_BIND_UNARCHIVER (transcendental_fct_type_A)

GINAC_DECLARE_UNARCHIVER (transcendental_fct_type_A)

GiNaC::ex create_transcendental_fct_type_A (const GiNaC::ex &xx, const GiNaC::ex &ii_num, const GiNaC::ex &ii_denom, const GiNaC::ex &pp_num, const GiNaC::ex &pp_denom)

GiNaC::ex create_transcendental_fct_type_A (const GiNaC::ex &xx, const GiNaC::ex &ii_num, const GiNaC::ex &ii_denom, const GiNaC::ex &pp_num, const GiNaC::ex &pp_denom, const GiNaC::ex &eps, int o, int f)

GINAC_BIND_UNARCHIVER (transcendental_fct_type_B)

GINAC_DECLARE_UNARCHIVER (transcendental_fct_type_B)

GiNaC::ex create_transcendental_fct_type_B (const GiNaC::ex &xx1, const GiNaC::ex &xx2, const GiNaC::ex &ii_num, const GiNaC::ex &ii_denom, const GiNaC::ex &jj_num, const GiNaC::ex &jj_denom, const GiNaC::ex &iijj_num, const GiNaC::ex &iijj_denom, const GiNaC::ex &pp_num, const GiNaC::ex &pp_denom)

GiNaC::ex create_transcendental_fct_type_B (const GiNaC::ex &xx1, const GiNaC::ex &xx2, const GiNaC::ex &ii_num, const GiNaC::ex &ii_denom, const GiNaC::ex &jj_num, const GiNaC::ex &jj_denom, const GiNaC::ex &iijj_num, const GiNaC::ex &iijj_denom, const GiNaC::ex &pp_num, const GiNaC::ex &pp_denom, const GiNaC::ex &eps, int o, int f)

GINAC_BIND_UNARCHIVER (transcendental_fct_type_C)

GINAC_DECLARE_UNARCHIVER (transcendental_fct_type_C)

GiNaC::ex create_transcendental_fct_type_C (const GiNaC::ex &xx1, const GiNaC::ex &xx2, const GiNaC::ex &ii_num, const GiNaC::ex &ii_denom, const GiNaC::ex &iijj_num, const GiNaC::ex &iijj_denom, const GiNaC::ex &pp_num, const GiNaC::ex &pp_denom)

GiNaC::ex create_transcendental_fct_type_C (const GiNaC::ex &xx1, const GiNaC::ex &xx2, const GiNaC::ex &ii_num, const GiNaC::ex &ii_denom, const GiNaC::ex &iijj_num, const GiNaC::ex &iijj_denom, const GiNaC::ex &pp_num, const GiNaC::ex &pp_denom, const GiNaC::ex &eps, int o, int f)

GINAC_BIND_UNARCHIVER (transcendental_fct_type_D)

GINAC_DECLARE_UNARCHIVER (transcendental_fct_type_D)

GiNaC::ex create_transcendental_fct_type_D (const GiNaC::ex &xx1, const GiNaC::ex &xx2, const GiNaC::ex &ii_num, const GiNaC::ex &ii_denom, const GiNaC::ex &jj_num, const GiNaC::ex &jj_denom, const GiNaC::ex &iijj_num, const GiNaC::ex &iijj_denom, const GiNaC::ex &pp_num, const GiNaC::ex &pp_denom)

GiNaC::ex create_transcendental_fct_type_D (const GiNaC::ex &xx1, const GiNaC::ex &xx2, const GiNaC::ex &ii_num, const GiNaC::ex &ii_denom, const GiNaC::ex &jj_num, const GiNaC::ex &jj_denom, const GiNaC::ex &iijj_num, const GiNaC::ex &iijj_denom, const GiNaC::ex &pp_num, const GiNaC::ex &pp_denom, const GiNaC::ex &eps, int o, int f)

GINAC_BIND_UNARCHIVER (transcendental_sum_type_A)

ex create_transcendental_sum_type_A_and_expand (const ex &nn, const ex &i, const ex &l, const ex &v, const ex &ss, const ex &eps, int o)

GINAC_DECLARE_UNARCHIVER (transcendental_sum_type_A)

GiNaC::ex create_transcendental_sum_type_A (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &v, const GiNaC::ex &ss, const GiNaC::ex &eps, int o, int f)

GiNaC::ex create_transcendental_sum_type_A_and_set_gammas (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &v, const GiNaC::ex &ss, const GiNaC::ex &eps, int o, int f)

GiNaC::ex create_transcendental_sum_type_A_and_expand (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &v, const GiNaC::ex &ss, const GiNaC::ex &eps, int o)

GINAC_BIND_UNARCHIVER (transcendental_sum_type_B)

ex create_transcendental_sum_type_B_and_expand (const ex &nn, const ex &i, const ex &l, const ex &lr, const ex &v, const ex &vr, const ex &ss, const ex &ssr, const ex &eps, int o)

GINAC_DECLARE_UNARCHIVER (transcendental_sum_type_B)

GiNaC::ex create_transcendental_sum_type_B (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &lr, const GiNaC::ex &v, const GiNaC::ex &vr, const GiNaC::ex &ss, const GiNaC::ex &ssr, const GiNaC::ex &eps, int o, int f)

GiNaC::ex create_transcendental_sum_type_B_and_set_gammas (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &lr, const GiNaC::ex &v, const GiNaC::ex &vr, const GiNaC::ex &ss, const GiNaC::ex &ssr, const GiNaC::ex &eps, int o, int f)

GiNaC::ex create_transcendental_sum_type_B_and_expand (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &lr, const GiNaC::ex &v, const GiNaC::ex &vr, const GiNaC::ex &ss, const GiNaC::ex &ssr, const GiNaC::ex &eps, int o)

GINAC_BIND_UNARCHIVER (transcendental_sum_type_C)

ex create_transcendental_sum_type_C_and_expand (const ex &nn, const ex &i, const ex &l, const ex &v, const ex &ss, const ex &eps, int o)

GINAC_DECLARE_UNARCHIVER (transcendental_sum_type_C)

GiNaC::ex create_transcendental_sum_type_C (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &v, const GiNaC::ex &ss, const GiNaC::ex &eps, int o, int f)

GiNaC::ex create_transcendental_sum_type_C_and_set_gammas (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &v, const GiNaC::ex &ss, const GiNaC::ex &eps, int o, int f)

GiNaC::ex create_transcendental_sum_type_C_and_expand (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &v, const GiNaC::ex &ss, const GiNaC::ex &eps, int o)

GINAC_BIND_UNARCHIVER (transcendental_sum_type_D)

ex create_transcendental_sum_type_D_and_expand (const ex &nn, const ex &i, const ex &l, const ex &lr, const ex &v, const ex &vr, const ex &ss, const ex &ssr, const ex &eps, int o)

GINAC_DECLARE_UNARCHIVER (transcendental_sum_type_D)

GiNaC::ex create_transcendental_sum_type_D (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &lr, const GiNaC::ex &v, const GiNaC::ex &vr, const GiNaC::ex &ss, const GiNaC::ex &ssr, const GiNaC::ex &eps, int o, int f)

GiNaC::ex create_transcendental_sum_type_D_and_set_gammas (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &lr, const GiNaC::ex &v, const GiNaC::ex &vr, const GiNaC::ex &ss, const GiNaC::ex &ssr, const GiNaC::ex &eps, int o, int f)

GiNaC::ex create_transcendental_sum_type_D_and_expand (const GiNaC::ex &nn, const GiNaC::ex &i, const GiNaC::ex &l, const GiNaC::ex &lr, const GiNaC::ex &v, const GiNaC::ex &vr, const GiNaC::ex &ss, const GiNaC::ex &ssr, const GiNaC::ex &eps, int o)

GINAC_BIND_UNARCHIVER (unit_letter)

GINAC_DECLARE_UNARCHIVER (unit_letter)

GiNaC::ex create_unit_letter (const GiNaC::ex &d)

GiNaC::ex create_unit_letter (const GiNaC::ex &d, const GiNaC::ex &i)

unsigned rotate_left_31 (unsigned n)

unsigned golden_ratio_hash (unsigned n)

unsigned make_hash_seed (const std::type_info &tinfo)

GINAC_BIND_UNARCHIVER (Zsum)

ex shuffle_Zsum (const ex &Z1, const ex &Z2)

ex convert_Zsum_to_Ssum (const ex &Z1)

ex create_Zsum_from_exvector (const ex &nc, const exvector &v)

ex Zsum_to_Ssum (const ex &expr)

ex shift_upper_limit_plus_one_for_Zsum (const ex &expr)

ex remove_negative_degrees_from_Zsum (const ex &expr)

ex remove_trivial_Zsum (const ex &expr)

GINAC_DECLARE_UNARCHIVER (Zsum)

GiNaC::ex create_Zsum (const GiNaC::ex &nc, const GiNaC::ex &llc)

GiNaC::ex shuffle_Zsum (const GiNaC::ex &Z1, const GiNaC::ex &Z2)

GiNaC::ex convert_Zsum_to_Ssum (const GiNaC::ex &Z1)

GiNaC::ex create_Zsum_from_exvector (const GiNaC::ex &nc, const GiNaC::exvector &v)

GiNaC::ex Zsum_to_Ssum (const GiNaC::ex &expr)

GiNaC::ex shift_upper_limit_plus_one_for_Zsum (const GiNaC::ex &expr)

GiNaC::ex remove_negative_degrees_from_Zsum (const GiNaC::ex &expr)

GiNaC::ex remove_trivial_Zsum (const GiNaC::ex &expr)

Variables

const int nestedsums_version_major = NESTEDSUMS_MAJOR

const int nestedsums_version_minor = NESTEDSUMS_MINOR

const int nestedsums_version_micro = NESTEDSUMS_MICRO

int _debug_level = 0

int _nestedsums_evaluation_veto = expand_status::zombie

int _nestedsums_evaluation_veto_type_A = expand_status::zombie

int _nestedsums_evaluation_veto_type_B = expand_status::zombie

int _nestedsums_evaluation_veto_type_C = expand_status::zombie

int _nestedsums_evaluation_veto_type_D = expand_status::zombie

int _NMAX = 10

int _print_format = 0

const lst _empty_list = lst()

const unit_letter _unit_zero_letter = unit_letter((ex) 0)

const unit_letter _unit_one_letter = unit_letter((ex) 1)

const list_of_tgamma _empty_list_of_tgamma = list_of_tgamma()

const basic_letter _default_basic_letter = basic_letter()

const Zsum _default_Zsum = Zsum()

const multiple_polylog _default_multiple_polylog = multiple_polylog()

const Ssum _default_Ssum = Ssum()

const Ssum_to_Infinity _default_Ssum_to_Infinity = Ssum_to_Infinity()

std::map< unsigned, ex > _table_list_of_tgamma

int _flag_table_list_of_tgamma = 1

int _count_table_list_of_tgamma = 0

std::map< unsigned, ex > _table_transcendental_sum_type_A

int _flag_table_transcendental_sum_type_A = 1

int _count_table_transcendental_sum_type_A = 0

std::map< unsigned, ex > _table_transcendental_sum_type_C

int _flag_table_transcendental_sum_type_C = 1

int _count_table_transcendental_sum_type_C = 0

const GiNaC::symbol Infinity

const int add_precedence = 40

const int mul_precedence = 50

const int pow_precedence = 60

const GiNaC::symbol _default

const GiNaC::symbol _default_index

const int version_major = NESTEDSUMS_MAJOR

const int version_minor = NESTEDSUMS_MINOR

const int version_micro = NESTEDSUMS_MICRO

Detailed Description

All classes and functions of the nestedsums package are defined in the nestedsums namespace.

Function Documentation

const symbol _default ( "default" )

A global constant to represent an unspecified symbol, used in default ctors.

const symbol _default_index ( "dummy" )

A global constant to represent an unspecified summation index

GiNaC::ex concat ( const basic_letter & l1,

const basic_letter & l2

)

Concat two letters.

ex convert_Bsum_to_Ssum ( const ex & C )

Functional form for the conversion to a Ssum.

ex convert_Bsum_to_Zsum ( const ex & C )

Functional form for the conversion to a Zsum.

ex convert_Csum_to_Ssum ( const ex & C )

Functional form for the conversion to a Ssum.

ex convert_Csum_to_Zsum ( const ex & C )

Functional form for the conversion to a Zsum.

ex convert_Ssum_to_Zsum ( const ex & Z1 )

Functional form for the conversion to a Ssum.

ex convert_Zsum_to_Ssum ( const ex & Z1 )

Functional form for the conversion to a Ssum.

ex convert_Zsums_to_standard_form ( const ex & expr,

const lst & sym_lst

)

Converts an expression involving Zsums to a standard form by first removing sums with non-positive degrees, expanding the expression and bringing the coefficients to normal form.
The result is in "normal" form.

GiNaC::ex create_basic_letter ( const GiNaC::ex & l,

const GiNaC::ex & d,

const GiNaC::ex & o,

const GiNaC::ex & i

) [inline]

Named ctor on the heap.

GiNaC::ex create_basic_letter ( const GiNaC::ex & l,

const GiNaC::ex & d,

const GiNaC::ex & o

) [inline]

Named ctor on the heap.

GiNaC::ex create_Bsum ( const GiNaC::ex & nc,

const GiNaC::ex & llc

) [inline]

Named ctor on the heap.

ex create_Bsum_from_exvector ( const ex & nc,

const exvector & v

)

Construct a Bsum on the heap from an exvector.

GiNaC::ex create_classical_polylog ( const GiNaC::ex & llc ) [inline]

Named ctor on the heap.

GiNaC::ex create_Csum ( const GiNaC::ex & nc,

const GiNaC::ex & llc

) [inline]

Named ctor on the heap.

ex create_Csum_from_exvector ( const ex & nc,

const exvector & v

)

Construct a Csum on the heap from an exvector.

GiNaC::ex create_Euler_Zagier_sum ( const GiNaC::ex & nc,

const GiNaC::ex & llc

) [inline]

Named ctor on the heap.

ex create_Euler_Zagier_sum_from_exvector ( const ex & nc,

const exvector & v

)

Construct a Euler_Zagier_sum on the heap from an exvector.

ex create_Euler_Zagier_sum_with_ones ( const ex & n,

const int & k

)

Named ctor on the heap. Creates Z_{11...1}(n) with $k$ unit indices.

GiNaC::ex create_harmonic_polylog ( const GiNaC::ex & llc ) [inline]

Named ctor on the heap.

GiNaC::ex create_harmonic_sum ( const GiNaC::ex & nc,

const GiNaC::ex & llc

) [inline]

Named ctor on the heap.

ex create_harmonic_sum_from_exvector ( const ex & nc,

const exvector & v

)

Construct a harmonic_sum on the heap from an exvector.

GiNaC::ex create_harmonic_sum_to_Infinity ( const GiNaC::ex & llc ) [inline]

Named ctor on the heap.

ex create_harmonic_sum_with_ones ( const ex & n,

const int & k

)

Named ctor on the heap. Creates S_{11...1}(n) with $k$ unit indices.

GiNaC::ex create_hash_entry ( const GiNaC::ex & cd,

const GiNaC::ex & sd,

const GiNaC::ex & rd,

unsigned rkey

) [inline]

Named ctor on the heap.

GiNaC::ex create_letter ( const GiNaC::ex & l,

const GiNaC::ex & d,

const GiNaC::ex & i

) [inline]

Named ctor on the heap.

GiNaC::ex create_letter ( const GiNaC::ex & l,

const GiNaC::ex & d

) [inline]

Named ctor on the heap.

GiNaC::ex create_list_of_tgamma ( const GiNaC::ex & l,

const GiNaC::ex & i,

const GiNaC::ex & eps,

int o

) [inline]

Named ctor on the heap.

GiNaC::ex create_list_of_tgamma ( const GiNaC::ex & l ) [inline]

Named ctor on the heap.

GiNaC::ex create_list_of_tgamma_and_set_gammas ( const GiNaC::ex & l,

const GiNaC::ex & i,

const GiNaC::ex & eps,

int o

) [inline]

Named ctor on the heap.

GiNaC::ex create_multiple_polylog ( const GiNaC::ex & llc ) [inline]

Named ctor on the heap.

GiNaC::ex create_multiple_zeta_value ( const GiNaC::ex & llc ) [inline]

Named ctor on the heap.

GiNaC::ex create_nielsen_polylog ( const GiNaC::ex & llc ) [inline]

Named ctor on the heap.

GiNaC::ex create_ratio_of_tgamma ( const GiNaC::ex & a1,

const GiNaC::ex & b1,

const GiNaC::ex & a2,

const GiNaC::ex & b2,

const GiNaC::ex & i,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_ratio_of_tgamma ( const GiNaC::ex & a1,

const GiNaC::ex & b1,

const GiNaC::ex & a2,

const GiNaC::ex & b2

) [inline]

Named ctor on the heap.

GiNaC::ex create_root_of_unity ( unsigned l,

unsigned k

) [inline]

Named ctor on the heap.

GiNaC::ex create_Ssum ( const GiNaC::ex & nc,

const GiNaC::ex & llc

) [inline]

Named ctor on the heap.

ex create_Ssum_from_exvector ( const ex & nc,

const exvector & v

)

Construct a Ssum on the heap from an exvector.

GiNaC::ex create_Ssum_to_Infinity ( const GiNaC::ex & llc ) [inline]

Named ctor on the heap.

GiNaC::ex create_transcendental_fct_type_A ( const GiNaC::ex & xx,

const GiNaC::ex & ii_num,

const GiNaC::ex & ii_denom,

const GiNaC::ex & pp_num,

const GiNaC::ex & pp_denom,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_transcendental_fct_type_B ( const GiNaC::ex & xx1,

const GiNaC::ex & xx2,

const GiNaC::ex & ii_num,

const GiNaC::ex & ii_denom,

const GiNaC::ex & jj_num,

const GiNaC::ex & jj_denom,

const GiNaC::ex & iijj_num,

const GiNaC::ex & iijj_denom,

const GiNaC::ex & pp_num,

const GiNaC::ex & pp_denom,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_transcendental_fct_type_C ( const GiNaC::ex & xx1,

const GiNaC::ex & xx2,

const GiNaC::ex & ii_num,

const GiNaC::ex & ii_denom,

const GiNaC::ex & iijj_num,

const GiNaC::ex & iijj_denom,

const GiNaC::ex & pp_num,

const GiNaC::ex & pp_denom,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_transcendental_fct_type_D ( const GiNaC::ex & xx1,

const GiNaC::ex & xx2,

const GiNaC::ex & ii_num,

const GiNaC::ex & ii_denom,

const GiNaC::ex & jj_num,

const GiNaC::ex & jj_denom,

const GiNaC::ex & iijj_num,

const GiNaC::ex & iijj_denom,

const GiNaC::ex & pp_num,

const GiNaC::ex & pp_denom,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_transcendental_sum_type_A ( const GiNaC::ex & nn,

const GiNaC::ex & i,

const GiNaC::ex & l,

const GiNaC::ex & v,

const GiNaC::ex & ss,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

ex create_transcendental_sum_type_A_and_expand ( const ex & nn,

const ex & i,

const ex & l,

const ex & v,

const ex & ss,

const ex & eps,

int o

)

Stop and go

GiNaC::ex create_transcendental_sum_type_A_and_set_gammas ( const GiNaC::ex & nn,

const GiNaC::ex & i,

const GiNaC::ex & l,

const GiNaC::ex & v,

const GiNaC::ex & ss,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_transcendental_sum_type_B ( const GiNaC::ex & nn,

const GiNaC::ex & i,

const GiNaC::ex & l,

const GiNaC::ex & lr,

const GiNaC::ex & v,

const GiNaC::ex & vr,

const GiNaC::ex & ss,

const GiNaC::ex & ssr,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

ex create_transcendental_sum_type_B_and_expand ( const ex & nn,

const ex & i,

const ex & l,

const ex & lr,

const ex & v,

const ex & vr,

const ex & ss,

const ex & ssr,

const ex & eps,

int o

)

Stop and go

GiNaC::ex create_transcendental_sum_type_B_and_set_gammas ( const GiNaC::ex & nn,

const GiNaC::ex & i,

const GiNaC::ex & l,

const GiNaC::ex & lr,

const GiNaC::ex & v,

const GiNaC::ex & vr,

const GiNaC::ex & ss,

const GiNaC::ex & ssr,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_transcendental_sum_type_C ( const GiNaC::ex & nn,

const GiNaC::ex & i,

const GiNaC::ex & l,

const GiNaC::ex & v,

const GiNaC::ex & ss,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

ex create_transcendental_sum_type_C_and_expand ( const ex & nn,

const ex & i,

const ex & l,

const ex & v,

const ex & ss,

const ex & eps,

int o

)

Stop and go

GiNaC::ex create_transcendental_sum_type_C_and_set_gammas ( const GiNaC::ex & nn,

const GiNaC::ex & i,

const GiNaC::ex & l,

const GiNaC::ex & v,

const GiNaC::ex & ss,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_transcendental_sum_type_D ( const GiNaC::ex & nn,

const GiNaC::ex & i,

const GiNaC::ex & l,

const GiNaC::ex & lr,

const GiNaC::ex & v,

const GiNaC::ex & vr,

const GiNaC::ex & ss,

const GiNaC::ex & ssr,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

ex create_transcendental_sum_type_D_and_expand ( const ex & nn,

const ex & i,

const ex & l,

const ex & lr,

const ex & v,

const ex & vr,

const ex & ss,

const ex & ssr,

const ex & eps,

int o

)

Stop and go

GiNaC::ex create_transcendental_sum_type_D_and_set_gammas ( const GiNaC::ex & nn,

const GiNaC::ex & i,

const GiNaC::ex & l,

const GiNaC::ex & lr,

const GiNaC::ex & v,

const GiNaC::ex & vr,

const GiNaC::ex & ss,

const GiNaC::ex & ssr,

const GiNaC::ex & eps,

int o,

int f

) [inline]

Named ctor on the heap.

GiNaC::ex create_unit_letter ( const GiNaC::ex & d,

const GiNaC::ex & i

) [inline]

Named ctor on the heap.

GiNaC::ex create_unit_letter ( const GiNaC::ex & d ) [inline]

Named ctor on the heap.

GiNaC::ex create_Zsum ( const GiNaC::ex & nc,

const GiNaC::ex & llc

) [inline]

Named ctor on the heap.

ex create_Zsum_from_exvector ( const ex & nc,

const exvector & v

)

Construct a Zsum on the heap from an exvector.

ex eval_multiple_polylog_approx ( const ex & expr )

Evaluates an expression involving multiple polylog with a rough and brute force evaluation routine.
This is not a routine designed for performance. It only provides a simple way to check a result for a few selected points.
The method eval_approx is called for the rough numerical approximation.

ex eval_ratio_of_tgamma_to_scalar ( const ex & expr )

Sets the status flag of ratio_of_tgamma objects to eval_to_scalar. Ratio_of_tgamma objects with integer index are then evaluated explicitly.

exvector exvector_append_lst ( const exvector & v,

const ex & l

)

Returns an exvector, where the elements of the list l have been appended to the exvector v.

exvector exvector_from_lst ( const ex & l )

Takes a list lst(a1,a2,...,an) and constructs an exvector from it, e.g.
v[0] = a1
...
v[n-1] = an

exvector exvector_increase_last_degree ( const exvector & v )

If v is an exvector of letters, then this function returns an exvector, where the degree of the last letter has been raised by one.

exvector exvector_multiply_last_letter ( const exvector & v,

const ex & x

)

If v is an exvector of letters, then this function returns an exvector, where the scale of the last letter has been multiplied by x. The degree of the last letter is not changed.

unsigned golden_ratio_hash ( unsigned n ) [inline]

Golden ratio hash function for the 31 least significant bits.

const symbol Infinity ( "Infinity" ,

"\\infty"

)

A global constant to represent infinity

bool is_class_in_nestedsums ( const ex & expr )

Returns true if expr is of type Zsum, Ssum, basic_letter, transcendental_sum_type_A to transcendental_sum_type_D, list_of_tgamma, ratio_of_tgamma, Bsum or Csum.
Returns false otherwise.

bool is_expr_in_sym_lst ( const ex & expr,

const lst & sym_lst

)

Returns true if expr is a symbol (or a symbol raised to some power) contained in sym_lst.
Returns false otherwise.

ex nestedsums_helper_arithmetic_sum_type_S ( const ex & n,

const ex & m

)

Returns
$\sum\limits_{i=1}^{n} i^m = n^m + \sum\limits_{k=0}^m \frac{B_k}{k!} \frac{m!}{(m+1-k)!} n^{m+1-k}$

ex nestedsums_helper_arithmetic_sum_type_Z ( const ex & n,

const ex & m

)

Returns
$\sum\limits_{i=1}^{n-1} i^m = \sum\limits_{k=0}^m \frac{B_k}{k!} \frac{m!}{(m+1-k)!} n^{m+1-k}$

GiNaC::ex nestedsums_helper_bernoulli ( int n )

Returns the n-th Bernoulli number. They are calculated as $B_0 = 1$ , $B_1 = -1/2$ , $B_{2k+1} = 0$ for $k>0$ and recursively through
$B_n = - \frac{1}{n+1} \sum\limits_{i=0}^{n-1} \left( \begin{array}{c} n+1 \\ i \\ \end{array} \right) B_i$

void nestedsums_helper_clear_hashes ( void )

Clear the hash tables

ex nestedsums_helper_coeff ( const ex & expr,

const ex & eps,

int n

)

Returns the coefficient of the order $n$ term in a Laurent series.
The expression is assumed to be in "expanded" or "normal" form.

ex nestedsums_helper_expand ( const ex & expr,

int level

)

A helper routine to expand expressions. GiNaC's proper expand method is buggy, therefore we provide a method for the things we need. This helper routine knows about the classes "add", "mul" and "ncmul".
If (level & expand_request::member_variables) is true, the routine expands also recursively the arguments of the classes basic_letter, Zsum, Ssum, Csum and Bsum.
This routine is not a "complete" expand-function, for example, it does not know about the class "power" and uses in this case the native GiNaC method, if (level & expand_request::power) is set.

ex nestedsums_helper_expand_lst_tgamma_fct ( const ex & ll1,

const ex & ll2,

const ex & expansion_parameter,

int order

)

Expands ratios of tgamma functions in eps. the first argument ll1 is a list, containing the arguments in the numerator, for example
lst( eps, 1-eps, 3+eps ).
The second argument ll2 is a list containing the arguments of the denominator. The neglected terms are of order O(eps^order).
The result is in "collected" form.

ex nestedsums_helper_expand_power ( const ex & x,

const ex & m,

const ex & expansion_parameter,

int order

)

Expands $x^m$ with $m=a+b\varepsilon$ in $\varepsilon$ up to the desired order.

$x^{a+b\varepsilon} = x^a \sum\limits_{i=0}^\infty \varepsilon^i \frac{(-b)^i}{i!} \left( \mbox{Li}_1(1-x) \right)^i$

The result is in expanded form.

ex nestedsums_helper_expand_tgamma_fct ( const ex & a1,

const ex & b1,

const ex & a2,

const ex & b2,

const ex & expansion_parameter,

int order

)

Expands
$\frac{\Gamma(a_1+b_1 \varepsilon)}{\Gamma(a_2+b_2 \varepsilon)}$
in $\varepsilon$ up to $O(\varepsilon^{order})$ .
The neglected term is exactly of order $O(\varepsilon^{order})$ .
The result is in "expanded" form.
The inversion of a power series 1/( a0 + a1 eps + a2 eps^2 + ... ) sucks time and memory in GiNaC.
It is therefore more efficient to calculate the series as
$\frac{1}{\Gamma(a_2+b_2 \varepsilon)} = \frac{1}{\pi} \sin \left( \pi \left( 1 - a_2 - b_2 \varepsilon \right) \right) \Gamma(1-a_2-b_2 \varepsilon)$

The build-in "series"-method of GiNaC does not guarantee that the neglected term is exactly of order $O(\varepsilon^{order})$ .
For example
expr.series(eps==0,order) works for
tgamma(1+eps),
tgamma(eps),
(1+a*eps ...)^{-1},
but does not work for
(1 + a*eps + ... + a_n eps^n),
(1/eps + a0 + a1 eps ... )^{-1}.

int nestedsums_helper_get_degree ( const ex & expr,

const ex & eps

)

Returns the power to which a variable appears in an expression.
On sums this function return zero, e.g. sums are not allowed to contain powers of eps.

int nestedsums_helper_ldegree ( const ex & expr,

const ex & eps

)

Returns the degree of the lowest order term in a Laurent series.
The expression is assumed to be in "expanded" or "normal" form.

ex nestedsums_helper_lowering_op_geo_0 ( const ex & x,

int m

)

Returns
$\left( x \frac{d}{dx} \right)^m \frac{1}{1-x}$

ex nestedsums_helper_lowering_op_geo_1 ( const ex & x,

int m

)

Returns
$\left( x \frac{d}{dx} \right)^m \frac{x}{1-x}$

ex nestedsums_helper_multiply_laurent ( const ex & A,

const ex & B,

const ex & expansion_parameter,

int order

)

A routine to multiply Laurent series in eps.
A and B are Laurent series in $\varepsilon$ . Multiplies to Laurent series and neglects terms of $O(\varepsilon^{order})$ .
The result is in "expanded" form.

ex nestedsums_helper_multiply_laurent_speedy ( const ex & A,

const ex & B,

const ex & expansion_parameter,

int order

)

As nestedsums_helper_multiply_laurent but assumes that the input is already in expanded form.
The result is in "expanded" form.

ex nestedsums_helper_multiply_via_triangle ( const ex & A,

const ex & B,

const ex & expansion_parameter,

int order

)

A routine to multiply power series in eps.
A and B are power series in eps :

$A = a_0 + a_1 \varepsilon + a_{n-1} \varepsilon^{n-1} + O(\varepsilon^n),$

$B = b_0 + b_1 \varepsilon + b_{n-1} \varepsilon^{n-1} + O(\varepsilon^n)$

The function calculates the power series $A \cdot B$ . It only calculates terms of order < n.
It does not handle the case of Laurent series, e.g. with terms like $1/\varepsilon$ . In that case one should use the routine nestedsums_helper_multiply_laurent.
The result is in "expanded" form.

ex nestedsums_helper_multiply_via_triangle_speedy ( const ex & A,

const ex & B,

const ex & expansion_parameter,

int order

)

As nestedsums_helper_multiply_via_triangle but assumes that the input is already in expanded form.
The result is in "expanded" form.

ex nestedsums_helper_normalize ( const ex & expr,

const lst & sym_lst

)

This function deals with expressions of the form a1 Term1 + b1 Term2 + a2 Term1 + c1 Term3 + ..., where a1, b1, c1 are scalars and Term1, Term2 are non-commutative objects defined in this package (Zsums etc.) or products thereof.
The function combines all terms with the same Term and applies the method "normal" to them.
This function is by far more efficient than a brute force application of "normal" on the complete expression.
The result is in "normal" form.

ex nestedsums_helper_series ( const ex & f,

const ex & expansion_parameter,

int order

)

Wrapper for series expansion around expansion parameter == 0. Handles order<1 correctly.

ex nestedsums_helper_sort ( const ex & expr,

const lst & sym_lst

)

Expands an expression first and collects and normalizes terms of the same type.
The result is in "normal" form.

ex refine_Ssum ( const ex & expr,

unsigned q

)

Refines all Ssums from $S(n)$ to $S(q \cdot n )$ .

ex remove_negative_degrees_from_Ssum ( const ex & expr )

Removes non-positive degrees in Ssums.

ex remove_negative_degrees_from_Zsum ( const ex & expr )

Removes non-positive degrees in Zsums.

ex remove_trivial_Ssum ( const ex & expr )

Does some trivial simplifications.

$S(\infty) \rightarrow 1$

This function has to be used with care, since $S(\infty)$ is of type non-commutative, whereas $1$ is a commuting object.

ex remove_trivial_Zsum ( const ex & expr )

Does some trivial simplifications.

$Z(\infty) \rightarrow 1$

This function has to be used with care, since $Z(\infty)$ is of type non-commutative, whereas $1$ is a commuting object.

exvector reverse_exvector_from_lst ( const ex & l )

Takes a list lst(a1,a2,...,an) and constructs an exvector in the reverse order from it, e.g.
v[0] = an
...
v[n-1] = a1

unsigned rotate_left_31 ( unsigned n ) [inline]

Rotate lower 31 bits of unsigned value by one bit to the left (upper bit gets cleared).

ex shift_upper_limit_plus_one_for_Ssum ( const ex & expr )

Convert all S(n,...) appearing in the expression to S(n+1,...).

ex shift_upper_limit_plus_one_for_Zsum ( const ex & expr )

Convert all Z(n,...) appearing in the expression to Z(n+1,...).

ex shuffle_Ssum ( const ex & Z1,

const ex & Z2

)

Functional form for the multiplication of two Ssums.
The method shuffle_exvector is called for the multiplication. If both sums are harmonic sums, the reimplementation of shuffle_exvector in the class harmonic_sum is used directly.
If the upper summation limits are not equal, the function returns unevaluated.

ex shuffle_Zsum ( const ex & Z1,

const ex & Z2

)

Functional form for the multiplication of two Zsums.
The method shuffle_exvector is called for the multiplication. If both sums are Euler-Zagier sums, the reimplementation of shuffle_exvector in the class Euler_Zagier_sum is used directly.
If the upper summation limits are not equal, the function returns unevaluated.

ex Ssum_to_Zsum ( const ex & expr )

Convert all Ssums occuring in expr into Zsums.

const GiNaC::symbol symbol_factory ( const std::string & symbol_name,

const std::string & latex_name

)

The symbol factory

ex Zsum_to_Ssum ( const ex & expr )

Convert all Zsums occuring in expr into Ssums.

Variable Documentation

int _debug_level = 0

A global variable which allows additional information to be printed out for debugging. If bit 0 is set, additional information during the evaluation of sums of type A is printed. If bit 1 is set, additional information during the evaluation of sums of type B is printed. If bit 2 is set, additional information during the evaluation of sums of type C is printed. If bit 3 is set, additional information during the evaluation of sums of type D is printed.
If bit 4 is set, additional information for Csum::convert_to_Ssum is printed. If bit 5 is set, additional information for Bsum::convert_to_Ssum is printed.

const GiNaC::lst _empty_list = lst()

A global variable for an empty list.

const list_of_tgamma _empty_list_of_tgamma = list_of_tgamma()

A global variable for an empty list of ratio_of_tgamma's.

int _nestedsums_evaluation_veto = expand_status::zombie

A global variable to stop evaluation at a certian stage

int _nestedsums_evaluation_veto_type_A = expand_status::zombie

A global variable to stop evaluation for sums of type A at a certian stage

int _nestedsums_evaluation_veto_type_B = expand_status::zombie

A global variable to stop evaluation for sums of type B at a certian stage

int _nestedsums_evaluation_veto_type_C = expand_status::zombie

A global variable to stop evaluation for sums of type C at a certian stage

int _nestedsums_evaluation_veto_type_D = expand_status::zombie

A global variable to stop evaluation for sums of type D at a certian stage

const unit_letter _unit_one_letter = unit_letter((ex) 1)

The unit letter of weight 1, e.g.
$\frac{1}{i^1}$

const unit_letter _unit_zero_letter = unit_letter((ex) 0)

ONE in the algebra of letters, e.g.
$\frac{1}{i^0}$

const int nestedsums_version_major = NESTEDSUMS_MAJOR

Version information

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

1.3.7