[C++20] Clean up using-declarations.

[ginac.git] / doc / tutorial / ginac.texi

diff --git a/doc/tutorial/ginac.texi b/doc/tutorial/ginac.texi
index 386dfeafd1d6c92aa8ea84572b1e371eabcb0252..1ba42151e86a39898c9c2dbef89cc4268c1d5a3f 100644 (file)
--- a/doc/tutorial/ginac.texi
+++ b/doc/tutorial/ginac.texi
@@ -372,8 +372,8 @@ lambda^2-3*lambda+11
 @end example
 
 Multivariate polynomials and rational functions may be expanded,
-collected and normalized (i.e. converted to a ratio of two coprime 
-polynomials):
+collected, factorized, and normalized (i.e. converted to a ratio of
+two coprime polynomials):
 
 @example
 > a = x^4 + 2*x^2*y^2 + 4*x^3*y + 12*x*y^3 - 3*y^4;
@@ -382,6 +382,8 @@ polynomials):
 4*x*y-y^2+x^2
 > expand(a*b);
 8*x^5*y+17*x^4*y^2+43*x^2*y^4-24*x*y^5+16*x^3*y^3+3*y^6+x^6
+> factor(%);
+(4*x*y+x^2-y^2)^2*(x^2+3*y^2)
 > collect(a+b,x);
 4*x^3*y-y^2-3*y^4+(12*y^3+4*y)*x+x^4+x^2*(1+2*y^2)
 > collect(a+b,y);
@@ -390,6 +392,9 @@ polynomials):
 3*y^2+x^2
 @end example
 
+Here we have made use of the @command{ginsh}-command @code{%} to pop the
+previously evaluated element from @command{ginsh}'s internal stack.
+
 You can differentiate functions and expand them as Taylor or Laurent
 series in a very natural syntax (the second argument of @code{series} is
 a relation defining the evaluation point, the third specifies the
@@ -414,9 +419,6 @@ x^(-1)-0.5772156649015328606+(0.9890559953279725555)*x
 -Euler-1/12+Order((x-1/2*Pi)^3)
 @end example
 
-Here we have made use of the @command{ginsh}-command @code{%} to pop the
-previously evaluated element from @command{ginsh}'s internal stack.
-
 Often, functions don't have roots in closed form.  Nevertheless, it's
 quite easy to compute a solution numerically, to arbitrary precision:
 
@@ -5980,7 +5982,7 @@ calls of several @code{expand()} methods with desired flags.
 The multiple polylogarithm is the most generic member of a family of functions,
 to which others like the harmonic polylogarithm, Nielsen's generalized
 polylogarithm and the multiple zeta value belong.
-Everyone of these functions can also be written as a multiple polylogarithm with specific
+Each of these functions can also be written as a multiple polylogarithm with specific
 parameters. This whole family of functions is therefore often referred to simply as
 multiple polylogarithms, containing @code{Li}, @code{G}, @code{H}, @code{S} and @code{zeta}.
 The multiple polylogarithm itself comes in two variants: @code{Li} and @code{G}. While
@@ -6694,8 +6696,8 @@ expression a unique name:
 
 @example
 #include <fstream>
-using namespace std;
 #include <ginac/ginac.h>
+using namespace std;
 using namespace GiNaC;
 
 int main()
@@ -7611,9 +7613,8 @@ product in a C++ @code{struct}:
 
 @example
 #include <iostream>
-using namespace std;
-
 #include <ginac/ginac.h>
+using namespace std;
 using namespace GiNaC;
 
 struct sprod_s @{
@@ -8005,9 +8006,8 @@ as follows:
 #include <iostream>
 #include <string>   
 #include <stdexcept>
-using namespace std;
-
 #include <ginac/ginac.h>
+using namespace std;
 using namespace GiNaC;
 @end example