"Fossies" - the Fresh Open Source Software Archive  

Source code changes of the file "doc/tutorial/ginac.info" between
ginac-1.7.11.tar.bz2 and ginac-1.8.0.tar.bz2

About: GiNaC (GiNaC is Not a CAS (Computer Algebra System)) is a C++ library for symbolic calculations.

ginac.info  (ginac-1.7.11.tar.bz2):ginac.info  (ginac-1.8.0.tar.bz2)
File: ginac.info, Node: Top, Next: Introduction, Prev: (dir), Up: (dir) File: ginac.info, Node: Top, Next: Introduction, Prev: (dir), Up: (dir)
GiNaC GiNaC
***** *****
This is a tutorial that documents GiNaC 1.7.11, an open framework for This is a tutorial that documents GiNaC 1.8.0, an open framework for
symbolic computation within the C++ programming language. symbolic computation within the C++ programming language.
* Menu: * Menu:
* Introduction:: GiNaC's purpose. * Introduction:: GiNaC's purpose.
* A tour of GiNaC:: A quick tour of the library. * A tour of GiNaC:: A quick tour of the library.
* Installation:: How to install the package. * Installation:: How to install the package.
* Basic concepts:: Description of fundamental classes. * Basic concepts:: Description of fundamental classes.
* Methods and functions:: Algorithms for symbolic manipulations. * Methods and functions:: Algorithms for symbolic manipulations.
* Extending GiNaC:: How to extend the library. * Extending GiNaC:: How to extend the library.
skipping to change at line 3039 skipping to change at line 3039
* Pattern matching and advanced substitutions:: * Pattern matching and advanced substitutions::
* Applying a function on subexpressions:: * Applying a function on subexpressions::
* Visitors and tree traversal:: * Visitors and tree traversal::
* Polynomial arithmetic:: Working with polynomials. * Polynomial arithmetic:: Working with polynomials.
* Rational expressions:: Working with rational functions. * Rational expressions:: Working with rational functions.
* Symbolic differentiation:: * Symbolic differentiation::
* Series expansion:: Taylor and Laurent expansion. * Series expansion:: Taylor and Laurent expansion.
* Symmetrization:: * Symmetrization::
* Built-in functions:: List of predefined mathematical functions. * Built-in functions:: List of predefined mathematical functions.
* Multiple polylogarithms:: * Multiple polylogarithms::
* Iterated integrals::
* Complex expressions:: * Complex expressions::
* Solving linear systems of equations:: * Solving linear systems of equations::
* Input/output:: Input and output of expressions. * Input/output:: Input and output of expressions.
File: ginac.info, Node: Information about expressions, Next: Numerical evaluat ion, Prev: Methods and functions, Up: Methods and functions File: ginac.info, Node: Information about expressions, Next: Numerical evaluat ion, Prev: Methods and functions, Up: Methods and functions
5.1 Getting information about expressions 5.1 Getting information about expressions
========================================= =========================================
5.1.1 Checking expression types 5.1.1 Checking expression types
skipping to change at line 4694 skipping to change at line 4695
} }
File: ginac.info, Node: Built-in functions, Next: Multiple polylogarithms, Pr ev: Symmetrization, Up: Methods and functions File: ginac.info, Node: Built-in functions, Next: Multiple polylogarithms, Pr ev: Symmetrization, Up: Methods and functions
5.12 Predefined mathematical functions 5.12 Predefined mathematical functions
====================================== ======================================
* Menu: * Menu:
* Multiple polylogarithms:: * Multiple polylogarithms::
* Iterated integrals::
5.12.1 Overview 5.12.1 Overview
--------------- ---------------
GiNaC contains the following predefined mathematical functions: GiNaC contains the following predefined mathematical functions:
*Name* *Function* *Name* *Function*
'abs(x)' absolute value 'abs(x)' absolute value
'step(x)' step function 'step(x)' step function
'csgn(x)' complex sign 'csgn(x)' complex sign
skipping to change at line 4738 skipping to change at line 4740
polylogarithm polylogarithm
'G(a, y)' multiple polylogarithm 'G(a, y)' multiple polylogarithm
'G(a, s, y)' multiple polylogarithm with explicit signs for 'G(a, s, y)' multiple polylogarithm with explicit signs for
the imaginary parts the imaginary parts
'S(n, p, x)' Nielsen's generalized polylogarithm 'S(n, p, x)' Nielsen's generalized polylogarithm
'H(m, x)' harmonic polylogarithm 'H(m, x)' harmonic polylogarithm
'zeta(m)' Riemann's zeta function as well as multiple zeta 'zeta(m)' Riemann's zeta function as well as multiple zeta
value value
'zeta(m, s)' alternating Euler sum 'zeta(m, s)' alternating Euler sum
'zetaderiv(n, x)' derivatives of Riemann's zeta function 'zetaderiv(n, x)' derivatives of Riemann's zeta function
'iterated_integral(a, iterated integral
y)'
'iterated_integral(a, iterated integral with explicit truncation
y, N)' parameter
'tgamma(x)' gamma function 'tgamma(x)' gamma function
'lgamma(x)' logarithm of gamma function 'lgamma(x)' logarithm of gamma function
'beta(x, y)' beta function 'beta(x, y)' beta function
('tgamma(x)*tgamma(y)/tgamma(x+y)') ('tgamma(x)*tgamma(y)/tgamma(x+y)')
'psi(x)' psi (digamma) function 'psi(x)' psi (digamma) function
'psi(n, x)' derivatives of psi function (polygamma 'psi(n, x)' derivatives of psi function (polygamma
functions) functions)
'EllipticK(x)' complete elliptic integral of the first kind
'EllipticE(x)' complete elliptic integral of the second kind
'factorial(n)' factorial function n! 'factorial(n)' factorial function n!
'binomial(n, k)' binomial coefficients 'binomial(n, k)' binomial coefficients
'Order(x)' order term function in truncated power series 'Order(x)' order term function in truncated power series
For functions that have a branch cut in the complex plane, GiNaC follows For functions that have a branch cut in the complex plane, GiNaC follows
the conventions of C/C++ for systems that do not support a signed zero. the conventions of C/C++ for systems that do not support a signed zero.
In particular: the natural logarithm ('log') and the square root In particular: the natural logarithm ('log') and the square root
('sqrt') both have their branch cuts running along the negative real ('sqrt') both have their branch cuts running along the negative real
axis. The 'asin', 'acos', and 'atanh' functions all have two branch axis. The 'asin', 'acos', and 'atanh' functions all have two branch
cuts starting at +/-1 and running away towards infinity along the real cuts starting at +/-1 and running away towards infinity along the real
skipping to change at line 4792 skipping to change at line 4800
| expand_options::expand_transcendental) << endl; | expand_options::expand_transcendental) << endl;
// -> exp(y^2)*exp(2*x*y)*exp(x^2) // -> exp(y^2)*exp(2*x*y)*exp(x^2)
} }
If both flags are set (as in the last call), then GiNaC tries to get the If both flags are set (as in the last call), then GiNaC tries to get the
maximal expansion. For example, for the exponent GiNaC firstly expands maximal expansion. For example, for the exponent GiNaC firstly expands
the argument and then the function. For the logarithm and absolute the argument and then the function. For the logarithm and absolute
value, GiNaC uses the opposite order: firstly expands the function and value, GiNaC uses the opposite order: firstly expands the function and
then its argument. Of course, a user can fine-tune this behavior by then its argument. Of course, a user can fine-tune this behavior by
sequential calls of several 'expand()' methods with desired flags. sequential calls of several 'expand()' methods with desired flags.
File: ginac.info, Node: Multiple polylogarithms, Next: Complex expressions, P rev: Built-in functions, Up: Methods and functions File: ginac.info, Node: Multiple polylogarithms, Next: Iterated integrals, Pr ev: Built-in functions, Up: Methods and functions
5.12.3 Multiple polylogarithms 5.12.3 Multiple polylogarithms
------------------------------ ------------------------------
The multiple polylogarithm is the most generic member of a family of The multiple polylogarithm is the most generic member of a family of
functions, to which others like the harmonic polylogarithm, Nielsen's functions, to which others like the harmonic polylogarithm, Nielsen's
generalized polylogarithm and the multiple zeta value belong. Each of generalized polylogarithm and the multiple zeta value belong. Each of
these functions can also be written as a multiple polylogarithm with these functions can also be written as a multiple polylogarithm with
specific parameters. This whole family of functions is therefore often specific parameters. This whole family of functions is therefore often
referred to simply as multiple polylogarithms, containing 'Li', 'G', referred to simply as multiple polylogarithms, containing 'Li', 'G',
skipping to change at line 4922 skipping to change at line 4930
'Harmonic Polylogarithms', E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys. 'Harmonic Polylogarithms', E.Remiddi, J.A.M.Vermaseren, Int.J.Mod.Phys.
A15 (2000), pp. 725-754 A15 (2000), pp. 725-754
'Special Values of Multiple Polylogarithms', J.Borwein, D.Bradley, 'Special Values of Multiple Polylogarithms', J.Borwein, D.Bradley,
D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp. D.Broadhurst, P.Lisonek, Trans.Amer.Math.Soc. 353/3 (2001), pp.
907-941 907-941
'Numerical Evaluation of Multiple Polylogarithms', J.Vollinga, 'Numerical Evaluation of Multiple Polylogarithms', J.Vollinga,
S.Weinzierl, hep-ph/0410259 S.Weinzierl, hep-ph/0410259
File: ginac.info, Node: Complex expressions, Next: Solving linear systems of e File: ginac.info, Node: Iterated integrals, Next: Complex expressions, Prev:
quations, Prev: Multiple polylogarithms, Up: Methods and functions Multiple polylogarithms, Up: Methods and functions
5.12.4 Iterated integrals
-------------------------
Multiple polylogarithms are a particular example of iterated integrals.
An iterated integral is defined by the function
'iterated_integral(a,y)'. The variable 'y' gives the upper integration
limit for the outermost integration, by convention the lower integration
limit is always set to zero. The variable 'a' must be a GiNaC 'lst'
containing sub-classes of 'integration_kernel' as elements. The depth
of the iterated integral corresponds to the number of elements of 'a'.
The available integrands for iterated integrals are (for a more detailed
description the user is referred to the publications listed at the end
of this section)
*Class* *Description*
'integration_kernel()' Base class, represents the one-form dy
'basic_log_kernel()' Logarithmic one-form dy/y
'multiple_polylog_kernel(z_j)'The one-form dy/(y-z_j)
'ELi_kernel(n, m, x, y)' The one form ELi_{n;m}(x;y;q) dq/q
'Ebar_kernel(n, m, x, y)' The one form \overline{E}_{n;m}(x;y;q)
dq/q
'Kronecker_dtau_kernel(k, The one form C_k K (k-1)/(2 \pi i)^k
z_j, K, C_k)' g^{(k)}(z_j,K \tau) dq/q
'Kronecker_dz_kernel(k, The one form C_k (2 \pi i)^{2-k}
z_j, tau, K, C_k)' g^{(k-1)}(z-z_j,K \tau) dz
'Eisenstein_kernel(k, N, a, The one form C_k E_{k,N,a,b,K}(\tau) dq/q
b, K, C_k)'
'Eisenstein_h_kernel(k, N, The one form C_k h_{k,N,r,s}(\tau) dq/q
r, s, C_k)'
'modular_form_kernel(k, P, The one form C_k P dq/q
C_k)'
'user_defined_kernel(f, y)' The one form f(y) dy
All parameters are assumed to be such that all integration kernels have
a convergent Laurent expansion around zero with at most a simple pole at
zero. The iterated integral may also be called with an optional third
parameter 'iterated_integral(a,y,N_trunc)', in which case the numerical
evaluation will truncate the series expansion at order 'N_trunc'.
The classes 'Eisenstein_kernel()', 'Eisenstein_h_kernel()' and
'modular_form_kernel()' provide a method 'q_expansion_modular_form(q,
order)', which can used to obtain the q-expansion of
E_{k,N,a,b,K}(\tau), h_{k,N,r,s}(\tau) or P to the specified order.
Useful publications:
'Numerical evaluation of iterated integrals related to elliptic Feynman
integrals', M.Walden, S.Weinzierl, arXiv:2010.05271
File: ginac.info, Node: Complex expressions, Next: Solving linear systems of e
quations, Prev: Iterated integrals, Up: Methods and functions
5.13 Complex expressions 5.13 Complex expressions
======================== ========================
For dealing with complex expressions there are the methods For dealing with complex expressions there are the methods
ex ex::conjugate(); ex ex::conjugate();
ex ex::real_part(); ex ex::real_part();
ex ex::imag_part(); ex ex::imag_part();
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- 'The Role of gamma5 in Dimensional Regularization', Dirk Kreimer, - 'The Role of gamma5 in Dimensional Regularization', Dirk Kreimer,
hep-ph/9401354 hep-ph/9401354
File: ginac.info, Node: Concept index, Prev: Bibliography, Up: Top File: ginac.info, Node: Concept index, Prev: Bibliography, Up: Top
Concept index Concept index
************* *************
* Menu: * Menu:
* abs(): Built-in functions. (line 16) * abs(): Built-in functions. (line 17)
* accept(): Visitors and tree traversal. * accept(): Visitors and tree traversal.
(line 6) (line 6)
* accuracy: Numbers. (line 61) * accuracy: Numbers. (line 61)
* acos(): Built-in functions. (line 28) * acos(): Built-in functions. (line 29)
* acosh(): Built-in functions. (line 35) * acosh(): Built-in functions. (line 36)
* add: Fundamental containers. * add: Fundamental containers.
(line 6) (line 6)
* add <1>: Internal representation of products and sums. * add <1>: Internal representation of products and sums.
(line 6) (line 6)
* advocacy: A comparison with other CAS. * advocacy: A comparison with other CAS.
(line 6) (line 6)
* alternating Euler sum: Multiple polylogarithms. * alternating Euler sum: Multiple polylogarithms.
(line 6) (line 6)
* antisymmetrize(): Symmetrization. (line 6) * antisymmetrize(): Symmetrization. (line 6)
* append(): Lists. (line 6) * append(): Lists. (line 6)
* archive (class): Input/output. (line 391) * archive (class): Input/output. (line 391)
* archiving: Input/output. (line 391) * archiving: Input/output. (line 391)
* asin(): Built-in functions. (line 27) * asin(): Built-in functions. (line 28)
* asinh(): Built-in functions. (line 34) * asinh(): Built-in functions. (line 35)
* atan(): Built-in functions. (line 29) * atan(): Built-in functions. (line 30)
* atanh(): Built-in functions. (line 36) * atanh(): Built-in functions. (line 37)
* atom: The class hierarchy. (line 12) * atom: The class hierarchy. (line 12)
* atom <1>: Symbols. (line 6) * atom <1>: Symbols. (line 6)
* Autoconf: Configuration. (line 6) * Autoconf: Configuration. (line 6)
* basic_log_kernel(): Iterated integrals. (line 18)
* bernoulli(): Numbers. (line 224) * bernoulli(): Numbers. (line 224)
* beta(): Built-in functions. (line 56) * beta(): Built-in functions. (line 61)
* binomial(): Built-in functions. (line 61) * binomial(): Built-in functions. (line 68)
* branch cut: Built-in functions. (line 64) * branch cut: Built-in functions. (line 71)
* building GiNaC: Building GiNaC. (line 6) * building GiNaC: Building GiNaC. (line 6)
* calchash(): Adding classes. (line 308) * calchash(): Adding classes. (line 308)
* canonicalize_clifford(): Non-commutative objects. * canonicalize_clifford(): Non-commutative objects.
(line 407) (line 407)
* Catalan: Constants. (line 6) * Catalan: Constants. (line 6)
* chain rule: Symbolic differentiation. * chain rule: Symbolic differentiation.
(line 6) (line 6)
* charpoly(): Matrices. (line 200) * charpoly(): Matrices. (line 200)
* clifford (class): Non-commutative objects. * clifford (class): Non-commutative objects.
(line 103) (line 103)
skipping to change at line 7454 skipping to change at line 7512
(line 452) (line 452)
* color_trace(): Non-commutative objects. * color_trace(): Non-commutative objects.
(line 532) (line 532)
* compare(): Information about expressions. * compare(): Information about expressions.
(line 231) (line 231)
* compare_same_type(): Adding classes. (line 103) * compare_same_type(): Adding classes. (line 103)
* compile_ex: Input/output. (line 326) * compile_ex: Input/output. (line 326)
* compiling expressions: Input/output. (line 276) * compiling expressions: Input/output. (line 276)
* complex numbers: Numbers. (line 46) * complex numbers: Numbers. (line 46)
* configuration: Configuration. (line 6) * configuration: Configuration. (line 6)
* conjugate(): Built-in functions. (line 18) * conjugate(): Built-in functions. (line 19)
* conjugate() <1>: Complex expressions. (line 6) * conjugate() <1>: Complex expressions. (line 6)
* constant (class): Constants. (line 6) * constant (class): Constants. (line 6)
* const_iterator: Information about expressions. * const_iterator: Information about expressions.
(line 124) (line 124)
* const_postorder_iterator: Information about expressions. * const_postorder_iterator: Information about expressions.
(line 154) (line 154)
* const_preorder_iterator: Information about expressions. * const_preorder_iterator: Information about expressions.
(line 154) (line 154)
* container: The class hierarchy. (line 12) * container: The class hierarchy. (line 12)
* container <1>: Information about expressions. * container <1>: Information about expressions.
(line 107) (line 107)
* content(): Polynomial arithmetic. * content(): Polynomial arithmetic.
(line 193) (line 193)
* contravariant: Indexed objects. (line 23) * contravariant: Indexed objects. (line 23)
* Converting ex to other classes: Information about expressions. * Converting ex to other classes: Information about expressions.
(line 9) (line 9)
* copy-on-write: Expressions are reference counted. * copy-on-write: Expressions are reference counted.
(line 6) (line 6)
* cos(): Built-in functions. (line 25) * cos(): Built-in functions. (line 26)
* cosh(): Built-in functions. (line 32) * cosh(): Built-in functions. (line 33)
* covariant: Indexed objects. (line 23) * covariant: Indexed objects. (line 23)
* csrc: Input/output. (line 60) * csrc: Input/output. (line 60)
* csrc_cl_N: Input/output. (line 60) * csrc_cl_N: Input/output. (line 60)
* csrc_double: Input/output. (line 60) * csrc_double: Input/output. (line 60)
* csrc_float: Input/output. (line 60) * csrc_float: Input/output. (line 60)
* CUBA library: Input/output. (line 321) * CUBA library: Input/output. (line 321)
* DECLARE_FUNCTION: Symbolic functions. (line 10) * DECLARE_FUNCTION: Symbolic functions. (line 10)
* degree(): Polynomial arithmetic. * degree(): Polynomial arithmetic.
(line 92) (line 92)
* delta_tensor(): Indexed objects. (line 467) * delta_tensor(): Indexed objects. (line 467)
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* dirac_ONE(): Non-commutative objects. * dirac_ONE(): Non-commutative objects.
(line 128) (line 128)
* dirac_slash(): Non-commutative objects. * dirac_slash(): Non-commutative objects.
(line 153) (line 153)
* dirac_trace(): Non-commutative objects. * dirac_trace(): Non-commutative objects.
(line 185) (line 185)
* divide(): Polynomial arithmetic. * divide(): Polynomial arithmetic.
(line 167) (line 167)
* doublefactorial(): Numbers. (line 222) * doublefactorial(): Numbers. (line 222)
* dummy index: Indexed objects. (line 339) * dummy index: Indexed objects. (line 339)
* Ebar_kernel(): Iterated integrals. (line 22)
* Eisenstein_h_kernel(): Iterated integrals. (line 29)
* Eisenstein_kernel(): Iterated integrals. (line 28)
* ELi_kernel(): Iterated integrals. (line 20)
* EllipticE(): Built-in functions. (line 66)
* EllipticK(): Built-in functions. (line 65)
* epsilon_tensor(): Indexed objects. (line 586) * epsilon_tensor(): Indexed objects. (line 586)
* eta(): Built-in functions. (line 40) * eta(): Built-in functions. (line 41)
* Euler: Constants. (line 6) * Euler: Constants. (line 6)
* Euler numbers: Symbolic differentiation. * Euler numbers: Symbolic differentiation.
(line 36) (line 36)
* eval(): Automatic evaluation. * eval(): Automatic evaluation.
(line 44) (line 44)
* eval() <1>: Adding classes. (line 248) * eval() <1>: Adding classes. (line 248)
* evalf(): Constants. (line 6) * evalf(): Constants. (line 6)
* evalf() <1>: Numerical evaluation. * evalf() <1>: Numerical evaluation.
(line 6) (line 6)
* evalm(): Matrices. (line 153) * evalm(): Matrices. (line 153)
* evaluation: Automatic evaluation. * evaluation: Automatic evaluation.
(line 6) (line 6)
* evaluation <1>: Symbolic functions. (line 86) * evaluation <1>: Symbolic functions. (line 86)
* evaluation <2>: Adding classes. (line 248) * evaluation <2>: Adding classes. (line 248)
* exceptions: Error handling. (line 6) * exceptions: Error handling. (line 6)
* exp(): Built-in functions. (line 37) * exp(): Built-in functions. (line 38)
* expand trancedent functions: Built-in functions. (line 80) * expand trancedent functions: Built-in functions. (line 87)
* expand(): Indexed objects. (line 455) * expand(): Indexed objects. (line 455)
* expand() <1>: Polynomial arithmetic. * expand() <1>: Polynomial arithmetic.
(line 21) (line 21)
* expand_dummy_sum(): Indexed objects. (line 390) * expand_dummy_sum(): Indexed objects. (line 390)
* expand_options::expand_function_args: Built-in functions. (line 80) * expand_options::expand_function_args: Built-in functions. (line 87)
* expand_options::expand_transcendental: Built-in functions. (line 80) * expand_options::expand_transcendental: Built-in functions. (line 87)
* expression (class ex): Expressions. (line 6) * expression (class ex): Expressions. (line 6)
* ex_is_equal (class): Information about expressions. * ex_is_equal (class): Information about expressions.
(line 231) (line 231)
* ex_is_less (class): Information about expressions. * ex_is_less (class): Information about expressions.
(line 231) (line 231)
* ex_to<...>(): Information about expressions. * ex_to<...>(): Information about expressions.
(line 9) (line 9)
* factor(): Polynomial arithmetic. * factor(): Polynomial arithmetic.
(line 300) (line 300)
* factorial(): Built-in functions. (line 60) * factorial(): Built-in functions. (line 67)
* factorization: Polynomial arithmetic. * factorization: Polynomial arithmetic.
(line 276) (line 276)
* factorization <1>: Polynomial arithmetic. * factorization <1>: Polynomial arithmetic.
(line 300) (line 300)
* fibonacci(): Numbers. (line 225) * fibonacci(): Numbers. (line 225)
* find(): Pattern matching and advanced substitut ions. * find(): Pattern matching and advanced substitut ions.
(line 186) (line 186)
* fraction: Numbers. (line 6) * fraction: Numbers. (line 6)
* fsolve: What it can do for you. * fsolve: What it can do for you.
(line 148) (line 148)
* FUNCP_1P: Input/output. (line 314) * FUNCP_1P: Input/output. (line 314)
* FUNCP_2P: Input/output. (line 314) * FUNCP_2P: Input/output. (line 314)
* FUNCP_CUBA: Input/output. (line 314) * FUNCP_CUBA: Input/output. (line 314)
* function (class): Mathematical functions. * function (class): Mathematical functions.
(line 6) (line 6)
* G(): Built-in functions. (line 44) * G(): Built-in functions. (line 45)
* G() <1>: Built-in functions. (line 46) * G() <1>: Built-in functions. (line 47)
* Gamma function: Mathematical functions. * Gamma function: Mathematical functions.
(line 17) (line 17)
* gamma function: Built-in functions. (line 53) * gamma function: Built-in functions. (line 58)
* garbage collection: Expressions are reference counted. * garbage collection: Expressions are reference counted.
(line 6) (line 6)
* GCD: Polynomial arithmetic. * GCD: Polynomial arithmetic.
(line 220) (line 220)
* gcd(): Polynomial arithmetic. * gcd(): Polynomial arithmetic.
(line 220) (line 220)
* get_dim(): Indexed objects. (line 111) * get_dim(): Indexed objects. (line 111)
* get_free_indices(): Indexed objects. (line 340) * get_free_indices(): Indexed objects. (line 340)
* get_metric(): Non-commutative objects. * get_metric(): Non-commutative objects.
(line 288) (line 288)
skipping to change at line 7593 skipping to change at line 7657
* get_TeX_name(): Symbols. (line 148) * get_TeX_name(): Symbols. (line 148)
* get_value(): Indexed objects. (line 111) * get_value(): Indexed objects. (line 111)
* ginac-excompiler: Input/output. (line 379) * ginac-excompiler: Input/output. (line 379)
* ginsh: What it can do for you. * ginsh: What it can do for you.
(line 6) (line 6)
* ginsh <1>: Fundamental containers. * ginsh <1>: Fundamental containers.
(line 37) (line 37)
* ginsh <2>: What does not belong into GiNaC. * ginsh <2>: What does not belong into GiNaC.
(line 6) (line 6)
* GMP: Numbers. (line 6) * GMP: Numbers. (line 6)
* H(): Built-in functions. (line 48) * H(): Built-in functions. (line 49)
* harmonic polylogarithm: Multiple polylogarithms. * harmonic polylogarithm: Multiple polylogarithms.
(line 6) (line 6)
* has(): Expressions. (line 6) * has(): Expressions. (line 6)
* has() <1>: Pattern matching and advanced substitut ions. * has() <1>: Pattern matching and advanced substitut ions.
(line 150) (line 150)
* Hermite polynomial: How to use it from within C++. * Hermite polynomial: How to use it from within C++.
(line 39) (line 39)
* hierarchy of classes: Symbols. (line 6) * hierarchy of classes: Symbols. (line 6)
* hierarchy of classes <1>: Adding classes. (line 26) * hierarchy of classes <1>: Adding classes. (line 26)
* history of GiNaC: Introduction. (line 6) * history of GiNaC: Introduction. (line 6)
* hold(): Symbolic functions. (line 86) * hold(): Symbolic functions. (line 86)
* hold() <1>: Adding classes. (line 248) * hold() <1>: Adding classes. (line 248)
* hyperbolic function: Mathematical functions. * hyperbolic function: Mathematical functions.
(line 6) (line 6)
* I: Numbers. (line 46) * I: Numbers. (line 46)
* I/O: Input/output. (line 5) * I/O: Input/output. (line 5)
* idx (class): Indexed objects. (line 17) * idx (class): Indexed objects. (line 17)
* imag(): Numbers. (line 191) * imag(): Numbers. (line 191)
* imag_part(): Built-in functions. (line 20) * imag_part(): Built-in functions. (line 21)
* indexed (class): Indexed objects. (line 16) * indexed (class): Indexed objects. (line 16)
* index_dimensions: Input/output. (line 118) * index_dimensions: Input/output. (line 118)
* info(): Information about expressions. * info(): Information about expressions.
(line 9) (line 9)
* input of expressions: Input/output. (line 180) * input of expressions: Input/output. (line 180)
* installation: Installing GiNaC. (line 6) * installation: Installing GiNaC. (line 6)
* integral (class): Integrals. (line 6) * integral (class): Integrals. (line 6)
* integration_kernel(): Iterated integrals. (line 17)
* inverse() (matrix): Matrices. (line 226) * inverse() (matrix): Matrices. (line 226)
* inverse() (numeric): Numbers. (line 187) * inverse() (numeric): Numbers. (line 187)
* iquo(): Numbers. (line 234) * iquo(): Numbers. (line 234)
* irem(): Numbers. (line 231) * irem(): Numbers. (line 231)
* isqrt(): Numbers. (line 198) * isqrt(): Numbers. (line 198)
* is_a<...>(): Information about expressions. * is_a<...>(): Information about expressions.
(line 9) (line 9)
* is_equal(): Information about expressions. * is_equal(): Information about expressions.
(line 206) (line 206)
* is_equal_same_type(): Adding classes. (line 308) * is_equal_same_type(): Adding classes. (line 308)
* is_exactly_a<...>(): Information about expressions. * is_exactly_a<...>(): Information about expressions.
(line 9) (line 9)
* is_polynomial(): Polynomial arithmetic. * is_polynomial(): Polynomial arithmetic.
(line 9) (line 9)
* is_zero(): Information about expressions. * is_zero(): Information about expressions.
(line 206) (line 206)
* is_zero_matrix(): Matrices. (line 122) * is_zero_matrix(): Matrices. (line 122)
* iterated_integral(): Built-in functions. (line 54)
* iterated_integral() <1>: Built-in functions. (line 57)
* iterators: Information about expressions. * iterators: Information about expressions.
(line 124) (line 124)
* Kronecker_dtau_kernel(): Iterated integrals. (line 24)
* Kronecker_dz_kernel(): Iterated integrals. (line 26)
* latex: Input/output. (line 98) * latex: Input/output. (line 98)
* Laurent expansion: Series expansion. (line 6) * Laurent expansion: Series expansion. (line 6)
* LCM: Polynomial arithmetic. * LCM: Polynomial arithmetic.
(line 220) (line 220)
* lcm(): Polynomial arithmetic. * lcm(): Polynomial arithmetic.
(line 220) (line 220)
* ldegree(): Polynomial arithmetic. * ldegree(): Polynomial arithmetic.
(line 92) (line 92)
* let_op(): Structures. (line 247) * let_op(): Structures. (line 247)
* let_op() <1>: Adding classes. (line 338) * let_op() <1>: Adding classes. (line 338)
* lgamma(): Built-in functions. (line 54) * lgamma(): Built-in functions. (line 59)
* Li(): Built-in functions. (line 43) * Li(): Built-in functions. (line 44)
* Li2(): Built-in functions. (line 41) * Li2(): Built-in functions. (line 42)
* link_ex: Input/output. (line 343) * link_ex: Input/output. (line 343)
* lists: Lists. (line 6) * lists: Lists. (line 6)
* log(): Built-in functions. (line 38) * log(): Built-in functions. (line 39)
* lorentz_eps(): Indexed objects. (line 585) * lorentz_eps(): Indexed objects. (line 585)
* lorentz_g(): Indexed objects. (line 522) * lorentz_g(): Indexed objects. (line 522)
* lsolve(): Solving linear systems of equations. * lsolve(): Solving linear systems of equations.
(line 6) (line 6)
* lst (class): Lists. (line 6) * lst (class): Lists. (line 6)
* lst_to_clifford(): Non-commutative objects. * lst_to_clifford(): Non-commutative objects.
(line 327) (line 327)
* lst_to_matrix(): Matrices. (line 33) * lst_to_matrix(): Matrices. (line 33)
* Machin's formula: Series expansion. (line 38) * Machin's formula: Series expansion. (line 38)
* map(): Applying a function on subexpressions. * map(): Applying a function on subexpressions.
(line 6) (line 6)
* match(): Pattern matching and advanced substitut ions. * match(): Pattern matching and advanced substitut ions.
(line 50) (line 50)
* matrix (class): Matrices. (line 6) * matrix (class): Matrices. (line 6)
* metric_tensor(): Indexed objects. (line 489) * metric_tensor(): Indexed objects. (line 489)
* mod(): Numbers. (line 227) * mod(): Numbers. (line 227)
* modular_form_kernel(): Iterated integrals. (line 31)
* Monte Carlo integration: Input/output. (line 321) * Monte Carlo integration: Input/output. (line 321)
* mul: Fundamental containers. * mul: Fundamental containers.
(line 6) (line 6)
* mul <1>: Internal representation of products and sums. * mul <1>: Internal representation of products and sums.
(line 6) (line 6)
* multiple polylogarithm: Multiple polylogarithms. * multiple polylogarithm: Multiple polylogarithms.
(line 6) (line 6)
* multiple zeta value: Multiple polylogarithms. * multiple zeta value: Multiple polylogarithms.
(line 6) (line 6)
* multiple_polylog_kernel(): Iterated integrals. (line 19)
* ncmul (class): Non-commutative objects. * ncmul (class): Non-commutative objects.
(line 43) (line 43)
* Nielsen's generalized polylogarithm: Multiple polylogarithms. * Nielsen's generalized polylogarithm: Multiple polylogarithms.
(line 6) (line 6)
* nops(): Lists. (line 6) * nops(): Lists. (line 6)
* nops() <1>: Information about expressions. * nops() <1>: Information about expressions.
(line 112) (line 112)
* normal(): Rational expressions. * normal(): Rational expressions.
(line 9) (line 9)
* no_index_dimensions: Input/output. (line 118) * no_index_dimensions: Input/output. (line 118)
skipping to change at line 7701 skipping to change at line 7772
(line 42) (line 42)
* numerator: Rational expressions. * numerator: Rational expressions.
(line 42) (line 42)
* numeric (class): Numbers. (line 6) * numeric (class): Numbers. (line 6)
* numer_denom(): Rational expressions. * numer_denom(): Rational expressions.
(line 42) (line 42)
* op(): Lists. (line 6) * op(): Lists. (line 6)
* op() <1>: Information about expressions. * op() <1>: Information about expressions.
(line 112) (line 112)
* Order(): Series expansion. (line 6) * Order(): Series expansion. (line 6)
* Order() <1>: Built-in functions. (line 62) * Order() <1>: Built-in functions. (line 69)
* output of expressions: Input/output. (line 9) * output of expressions: Input/output. (line 9)
* pair-wise representation: Internal representation of products and sums. * pair-wise representation: Internal representation of products and sums.
(line 16) (line 16)
* Pattern matching: Pattern matching and advanced substitut ions. * Pattern matching: Pattern matching and advanced substitut ions.
(line 6) (line 6)
* Pi: Constants. (line 6) * Pi: Constants. (line 6)
* pole_error (class): Error handling. (line 6) * pole_error (class): Error handling. (line 6)
* polylogarithm: Multiple polylogarithms. * polylogarithm: Multiple polylogarithms.
(line 6) (line 6)
* polynomial: Fundamental containers. * polynomial: Fundamental containers.
skipping to change at line 7746 skipping to change at line 7817
* print_latex (class): Printing. (line 13) * print_latex (class): Printing. (line 13)
* print_tree (class): Printing. (line 13) * print_tree (class): Printing. (line 13)
* product rule: Symbolic differentiation. * product rule: Symbolic differentiation.
(line 6) (line 6)
* product rule <1>: Symbolic functions. (line 167) * product rule <1>: Symbolic functions. (line 167)
* pseries (class): Series expansion. (line 6) * pseries (class): Series expansion. (line 6)
* pseudo-remainder: Polynomial arithmetic. * pseudo-remainder: Polynomial arithmetic.
(line 167) (line 167)
* pseudo-vector: Non-commutative objects. * pseudo-vector: Non-commutative objects.
(line 337) (line 337)
* psi(): Built-in functions. (line 57) * psi(): Built-in functions. (line 62)
* quo(): Polynomial arithmetic. * quo(): Polynomial arithmetic.
(line 167) (line 167)
* quotient: Polynomial arithmetic. * quotient: Polynomial arithmetic.
(line 167) (line 167)
* radical: Internal representation of products and sums. * radical: Internal representation of products and sums.
(line 41) (line 41)
* rank(): Matrices. (line 200) * rank(): Matrices. (line 200)
* rational: Numbers. (line 6) * rational: Numbers. (line 6)
* real(): Numbers. (line 190) * real(): Numbers. (line 190)
* realsymbol(): Symbols. (line 158) * realsymbol(): Symbols. (line 158)
* real_part(): Built-in functions. (line 19) * real_part(): Built-in functions. (line 20)
* reduced_matrix(): Matrices. (line 62) * reduced_matrix(): Matrices. (line 62)
* reference counting: Expressions are reference counted. * reference counting: Expressions are reference counted.
(line 6) (line 6)
* REGISTER_FUNCTION: Symbolic functions. (line 10) * REGISTER_FUNCTION: Symbolic functions. (line 10)
* relational (class): Relations. (line 6) * relational (class): Relations. (line 6)
* relational (class) <1>: Information about expressions. * relational (class) <1>: Information about expressions.
(line 196) (line 196)
* rem(): Polynomial arithmetic. * rem(): Polynomial arithmetic.
(line 167) (line 167)
* remainder: Polynomial arithmetic. * remainder: Polynomial arithmetic.
skipping to change at line 7789 skipping to change at line 7860
(line 247) (line 247)
* return_type(): Non-commutative objects. * return_type(): Non-commutative objects.
(line 60) (line 60)
* return_type() <1>: Information about expressions. * return_type() <1>: Information about expressions.
(line 9) (line 9)
* return_type_tinfo(): Non-commutative objects. * return_type_tinfo(): Non-commutative objects.
(line 60) (line 60)
* return_type_tinfo() <1>: Information about expressions. * return_type_tinfo() <1>: Information about expressions.
(line 9) (line 9)
* rounding: Numbers. (line 104) * rounding: Numbers. (line 104)
* S(): Built-in functions. (line 47) * S(): Built-in functions. (line 48)
* series(): Series expansion. (line 6) * series(): Series expansion. (line 6)
* simplification: Rational expressions. * simplification: Rational expressions.
(line 9) (line 9)
* simplify_indexed(): Indexed objects. (line 401) * simplify_indexed(): Indexed objects. (line 401)
* sin(): Built-in functions. (line 24) * sin(): Built-in functions. (line 25)
* sinh(): Built-in functions. (line 31) * sinh(): Built-in functions. (line 32)
* smod(): Numbers. (line 229) * smod(): Numbers. (line 229)
* solve(): Matrices. (line 214) * solve(): Matrices. (line 214)
* spinidx (class): Indexed objects. (line 165) * spinidx (class): Indexed objects. (line 165)
* spinor_metric(): Indexed objects. (line 545) * spinor_metric(): Indexed objects. (line 545)
* sqrfree(): Polynomial arithmetic. * sqrfree(): Polynomial arithmetic.
(line 276) (line 276)
* sqrt(): Built-in functions. (line 23) * sqrt(): Built-in functions. (line 24)
* square-free decomposition: Polynomial arithmetic. * square-free decomposition: Polynomial arithmetic.
(line 276) (line 276)
* step(): Built-in functions. (line 17) * step(): Built-in functions. (line 18)
* STL: Advantages. (line 14) * STL: Advantages. (line 14)
* subs(): Symbols. (line 152) * subs(): Symbols. (line 152)
* subs() <1>: Mathematical functions. * subs() <1>: Mathematical functions.
(line 17) (line 17)
* subs() <2>: Indexed objects. (line 205) * subs() <2>: Indexed objects. (line 205)
* subs() <3>: Methods and functions. * subs() <3>: Methods and functions.
(line 18) (line 18)
* subs() <4>: Substituting expressions. * subs() <4>: Substituting expressions.
(line 6) (line 6)
* subs() <5>: Pattern matching and advanced substitut ions. * subs() <5>: Pattern matching and advanced substitut ions.
skipping to change at line 7827 skipping to change at line 7898
* sub_matrix(): Matrices. (line 62) * sub_matrix(): Matrices. (line 62)
* symbol (class): Symbols. (line 6) * symbol (class): Symbols. (line 6)
* symbolic_matrix(): Matrices. (line 41) * symbolic_matrix(): Matrices. (line 41)
* symmetrize(): Symmetrization. (line 6) * symmetrize(): Symmetrization. (line 6)
* symmetrize_cyclic(): Symmetrization. (line 6) * symmetrize_cyclic(): Symmetrization. (line 6)
* symmetry (class): Indexed objects. (line 254) * symmetry (class): Indexed objects. (line 254)
* sy_anti(): Indexed objects. (line 254) * sy_anti(): Indexed objects. (line 254)
* sy_cycl(): Indexed objects. (line 254) * sy_cycl(): Indexed objects. (line 254)
* sy_none(): Indexed objects. (line 254) * sy_none(): Indexed objects. (line 254)
* sy_symm(): Indexed objects. (line 254) * sy_symm(): Indexed objects. (line 254)
* tan(): Built-in functions. (line 26) * tan(): Built-in functions. (line 27)
* tanh(): Built-in functions. (line 33) * tanh(): Built-in functions. (line 34)
* Taylor expansion: Series expansion. (line 6) * Taylor expansion: Series expansion. (line 6)
* temporary replacement: Rational expressions. * temporary replacement: Rational expressions.
(line 9) (line 9)
* tensor (class): Indexed objects. (line 458) * tensor (class): Indexed objects. (line 458)
* tgamma(): Built-in functions. (line 53) * tgamma(): Built-in functions. (line 58)
* to_cl_N(): Numbers. (line 251) * to_cl_N(): Numbers. (line 251)
* to_double(): Numbers. (line 251) * to_double(): Numbers. (line 251)
* to_int(): Numbers. (line 251) * to_int(): Numbers. (line 251)
* to_long(): Numbers. (line 251) * to_long(): Numbers. (line 251)
* to_polynomial(): Rational expressions. * to_polynomial(): Rational expressions.
(line 59) (line 59)
* to_rational(): Rational expressions. * to_rational(): Rational expressions.
(line 59) (line 59)
* trace(): Matrices. (line 200) * trace(): Matrices. (line 200)
* transpose(): Matrices. (line 126) * transpose(): Matrices. (line 126)
skipping to change at line 7864 skipping to change at line 7935
(line 6) (line 6)
* Tree traversal: Input/output. (line 136) * Tree traversal: Input/output. (line 136)
* trigonometric function: Mathematical functions. * trigonometric function: Mathematical functions.
(line 6) (line 6)
* unit(): Polynomial arithmetic. * unit(): Polynomial arithmetic.
(line 193) (line 193)
* unitcontprim(): Polynomial arithmetic. * unitcontprim(): Polynomial arithmetic.
(line 193) (line 193)
* unit_matrix(): Matrices. (line 41) * unit_matrix(): Matrices. (line 41)
* unlink_ex: Input/output. (line 361) * unlink_ex: Input/output. (line 361)
* user_defined_kernel(): Iterated integrals. (line 33)
* variance: Indexed objects. (line 23) * variance: Indexed objects. (line 23)
* varidx (class): Indexed objects. (line 133) * varidx (class): Indexed objects. (line 133)
* viewgar: Input/output. (line 425) * viewgar: Input/output. (line 425)
* visit(): Visitors and tree traversal. * visit(): Visitors and tree traversal.
(line 6) (line 6)
* visitor (class): Visitors and tree traversal. * visitor (class): Visitors and tree traversal.
(line 6) (line 6)
* wildcard (class): Pattern matching and advanced substitut ions. * wildcard (class): Pattern matching and advanced substitut ions.
(line 6) (line 6)
* Zeta function: What it can do for you. * Zeta function: What it can do for you.
(line 129) (line 129)
* zeta(): Built-in functions. (line 50) * zeta(): Built-in functions. (line 51)
* zeta() <1>: Built-in functions. (line 51) * zeta() <1>: Built-in functions. (line 52)
 End of changes. 41 change blocks. 
43 lines changed or deleted 116 lines changed or added

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