Problem with tkz-tab and its the PSTricks package equivalent
up vote
4
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My code:(use tkz-tab)
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb,newcent}
usepackage{tkz-tab}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-dfrac{1}{4}$ , +/$g(1)$, -/$m$, +/$+infty$}
end{tikzpicture}
end{document}
The result compiling:
Question:
- How to move
m-1/4lower thanm? It means
2. Does PSTricks has the command or package(I don't know) the same as tkz-tab? Assume without any the command or package, so how to create it only use the pspicture environment?
pstricks pspicture tkz-tab
edited Dec 6 at 6:56
Artificial Stupidity
4,78111039
asked Dec 4 at 8:43
chishimotoji
543212
add a comment |
up vote
4
down vote
favorite
My code:(use tkz-tab)
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb,newcent}
usepackage{tkz-tab}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-dfrac{1}{4}$ , +/$g(1)$, -/$m$, +/$+infty$}
end{tikzpicture}
end{document}
The result compiling:
Question:
- How to move
m-1/4lower thanm? It means
2. Does PSTricks has the command or package(I don't know) the same as tkz-tab? Assume without any the command or package, so how to create it only use the pspicture environment?
pstricks pspicture tkz-tab
edited Dec 6 at 6:56
Artificial Stupidity
4,78111039
asked Dec 4 at 8:43
chishimotoji
543212
You're right, it seems I misunderstood. I've added a new code (leaving the previous one, as it might be of interest to some). Is it better?
– Bernard
Dec 4 at 14:09
add a comment |
up vote
4
down vote
favorite
up vote
4
down vote
favorite
My code:(use tkz-tab)
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb,newcent}
usepackage{tkz-tab}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-dfrac{1}{4}$ , +/$g(1)$, -/$m$, +/$+infty$}
end{tikzpicture}
end{document}
The result compiling:
Question:
- How to move
m-1/4lower thanm? It means
2. Does PSTricks has the command or package(I don't know) the same as tkz-tab? Assume without any the command or package, so how to create it only use the pspicture environment?
pstricks pspicture tkz-tab
edited Dec 6 at 6:56
Artificial Stupidity
4,78111039
asked Dec 4 at 8:43
chishimotoji
543212
My code:(use tkz-tab)
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb,newcent}
usepackage{tkz-tab}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-dfrac{1}{4}$ , +/$g(1)$, -/$m$, +/$+infty$}
end{tikzpicture}
end{document}
The result compiling:
Question:
- How to move
m-1/4lower thanm? It means
2. Does PSTricks has the command or package(I don't know) the same as tkz-tab? Assume without any the command or package, so how to create it only use the pspicture environment?
pstricks pspicture tkz-tab
pstricks pspicture tkz-tab
edited Dec 6 at 6:56
Artificial Stupidity
4,78111039
asked Dec 4 at 8:43
chishimotoji
543212
edited Dec 6 at 6:56
Artificial Stupidity
4,78111039
asked Dec 4 at 8:43
chishimotoji
543212
edited Dec 6 at 6:56
Artificial Stupidity
4,78111039
edited Dec 6 at 6:56
Artificial Stupidity
4,78111039
edited Dec 6 at 6:56
Artificial Stupidity
4,78111039
4,78111039
asked Dec 4 at 8:43
chishimotoji
543212
asked Dec 4 at 8:43
chishimotoji
543212
asked Dec 4 at 8:43
chishimotoji
543212
543212
You're right, it seems I misunderstood. I've added a new code (leaving the previous one, as it might be of interest to some). Is it better?
– Bernard
Dec 4 at 14:09
add a comment |
You're right, it seems I misunderstood. I've added a new code (leaving the previous one, as it might be of interest to some). Is it better?
– Bernard
Dec 4 at 14:09
You're right, it seems I misunderstood. I've added a new code (leaving the previous one, as it might be of interest to some). Is it better?
– Bernard
Dec 4 at 14:09
You're right, it seems I misunderstood. I've added a new code (leaving the previous one, as it might be of interest to some). Is it better?
– Bernard
Dec 4 at 14:09
add a comment |
3 Answers
3
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oldest
votes
up vote
4
down vote
accepted
For your first question, is this what you want? Note I replaced $dfrac{1}{4}$ with $mfrac{1}{4}$ (medium sized fraction from nccmath) , as I think that, used as a numerical coefficient, it looks nicer. The level alignment of m-mfrac{1}{4} and m is obtained adding a vphantom{mfrac{1}{4}} to the latter, so the boxes containing both minima have the same height.
documentclass[12pt,a4paper]{article}
usepackage{amsmath, amssymb, newcent}
usepackage{tkz-tab}
usepackage{nccmath}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/$mvphantom{mfrac 14}$, +/$+infty$}
end{tikzpicture}
end{document}
Edit:
It seems I misundertood the requirement. I hope the following code is more like you want (same preamble):
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/raisebox{4ex}{$mvphantom{mfrac{1}4}$}, +/$+infty$}
end{tikzpicture}
answered Dec 4 at 11:55
Bernard
164k769192
I see it seem was not changed. " m-1/4 " = " m "
– chishimotoji
Dec 4 at 12:02
Both ms are now at the same level, using avphantom{mfrac{1}{4}for the rightmost m
– Bernard
Dec 4 at 12:40
You can edit your answer to show what you say. My english is not good.
– chishimotoji
Dec 4 at 12:58
I've added some explanations. Is that clearer?
– Bernard
Dec 4 at 13:28
smash[t]{vphantom{mfrac{1}{4}}}seems better. Ordfrac, of course, ifdfracis used in the other place.
– egreg
Dec 4 at 13:40
|
show 3 more comments
up vote
4
down vote
You can use the new tablvar package (documentation in French):
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{m} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
Removing the tikz option makes the package rely on PSTricks.
If you want to easily move the m up (which I can't see the necessity of), raise it.
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{raisebox{2ex}{$mmathstrut$}} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
answered Dec 4 at 14:00
egreg
704k8618753154
Thank you but you should see edited question! My aim is to set "m" at that position.
– chishimotoji
Dec 4 at 14:02
@chishimotoji Added. Probably there are more features in the package documentation.
– egreg
Dec 4 at 14:04
add a comment |
up vote
4
down vote
Out of topic!
For polynomials and rational functions with minimum one rational zero
we can do the factorization and finding the (only) rational roots with the help of
the xintexpr and polexpr packages. Then the table is build automatic.
So we can type for e.g. in the document
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
Here is the code for a full example.
documentclass[fleqn,svgnames,x11names,dvipsnames]{article}
usepackage[a4paper,margin=1.25cm]{geometry}
usepackage{amsmath,amssymb}
usepackage{xintexpr}
usepackage{polexpr}
usepackage{xfp,xparse,expl3,xstring}
usepackage{booktabs}
usepackage{siunitx}
sisetup{group-separator={,},output-decimal-marker={,},exponent-product=cdot,per-mode=fraction,product-units=single,
input-product=*,output-product=cdot,unit-mode=text,quotient-mode=fraction,locale=DE,range-phrase = { bis }}
usepackage{pstricks}
letfunkPolToExpr
letSTZeroLPolSturmIsolatedZeroLeft
NewExpandableDocumentCommand{XcalcR}{sO{8}m}{% calculates #3 and rounds to #2 decimals; non formating: for formated output use Xcalc not XcalcR
PolDecToString{xintREZ{xinttheiexpr[#2]reduce(#3)relax}}%
}
NewDocumentCommand{Xcalc}{t+stDsO{2}m}{% calculates #5, rounds to #4 decimals (trailing zeros are supressed) and typesets the number with num-macro formatting
IfBooleanT{#1}{xintifSgn{xinttheexpr #6relax}{}{+}{+}}% or as fraction, reduced or not reduced and with #3=D is a boolean in displaystyle
IfBooleanTF{#2}%
{IfBooleanT{#3}{displaystyle}%
IfBooleanTF{#4}%
{xintSignedFrac{xinttheexpr #6relax}}%
{xintSignedFrac{xinttheexpr reduce(#6)relax}}%
}%
{num{PolDecToString{xintREZ{xinttheiexpr[#5]reduce(#6)relax}}}}%
}
newcommand{NullStellen}[1]{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{x_{##1} = Xcalc{PolSturmIsolatedZeroLeft{#1}{##1}}%
xintifForLast{}{; }}%
}
NewDocumentCommand{hackNst}{O{nt}D(){x}m}{%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#3}=0}{}{%
xintiloop [1+1]
expandafterdefexpandaftertmpxintbracediloopindex
xintdefvar #2tmp = PolSturmIsolatedZeroLeft{#3}tmp;
xintdefvar #1tmp = PolSturmIsolatedZeroMultiplicity{#3}tmp;
ifnumxintiloopindex<numexprPolSturmNbOfIsolatedZeros{#3}relax
repeat
}%
}
NewDocumentCommand{GenFloatSturmZeros}{ssO{9}D(){x}m}{%
IfBooleanTF{#1}%
{%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
IfBooleanF{#2}{hackNst(#4){#5}}%
}%
{%
PolGenFloatVariant{#5}%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
}%
}
defsmstext{steigt}%
defsmftext{fällt}
defHPtext{Hochpkt.}
defTPtext{Tiefpkt.}
defSPtext{Terrassenpkt.}
defDiffRowtxt{f^{prime}}
deflueckColor#1{colorlet{lckColor}{#1}}
lueckColor{BrickRed}
defPoltxt{textcolor{lckColor}{Polstelle}}
defLueckentxt{textcolor{lckColor}{Def.-Lücke}}
defNotdeftxt{textcolor{lckColor}{diagup}}
NewDocumentCommand{VZTabelle}{sO{f}O{x}O{#21}!d()}{%
IfValueTF{#5}{%
PolGCD{#4}{#5}{commonfactorA}%
poldef erwzaehlerA(x) := (#4(x)#5(x))/(commonfactorA(x))^2 ;%
poldef gekuerzterNenner(x) := #5(x)/commonfactorA(x) ;%
PolGCD{erwzaehlerA}{commonfactorA}{commonfactorB}%
poldef commonfactorC(x) := commonfactorA(x)/commonfactorB(x) ;%
poldef erwzaehler(x) := erwzaehlerA(x)commonfactorC(x) ;%
}%
{%
poldef erwzaehler(x) := #4(x) ;%
poldef commonfactorC(x) := 1 ;%
}%
PolToSturm{erwzaehler}{erwzaehler}%
PolSturmIsolateZeros**{erwzaehler}%
PolEnsureIntervalLengths{erwzaehler}{-9}%
begin{tabular}{*{XcalcR{2+2*PolSturmNbOfIsolatedZeros{erwzaehler}}}{>{$}c<{$}}}
IfBooleanF{#1}{toprule}
IfValueTF{#5}{Firstrow{erwzaehler}{#3}{#5}}{Firstrow{erwzaehler}{#3}{1}}
IfValueTF{#5}{VZmakerows{erwzaehler}{#5}{#4}{#3}}{VZmakerows{erwzaehler}{1}{#4}{#3}}
IfValueTF{#5}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{#5}}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{1}}
IfValueTF{#5}{Graphrow{#2}{erwzaehler}{#5}}{Graphrow{#2}{erwzaehler}{1}}
IfBooleanF{#1}{ \ bottomrule}
end{tabular}%
}
defleadcoef#1{xintifboolexpr{PolLeadingCoeff{#1}=1}{}{Xcalc{PolLeadingCoeff{#1}}cdot}}
defnst#1#2{Xcalc*{STZeroL{#1}{#2}}}
defVZfktwertL#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax}{-}{0}{+}}
defVZfktwertR#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}+0.001relax)relax}{-}{0}{+}}
defVZfktwert#1#2{xintifSgn{xinttheexpr #1(STZeroL{#1}{#2})relax}{-}{0}{+}}
deftabfactor#1#2#3#4{%
xintifboolexpr{STZeroL{#1}{#2}=0}%
{#4}%
{(#4Xcalc+*{-STZeroL{#1}{#2}})
xintifboolexpr{PolSturmIsolatedZeroMultiplicity{#1}{#2}=1}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{^{0}}%
{}%
}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{%
^{0}
}%
{%
^{PolSturmIsolatedZeroMultiplicity{#1}{#2}}
}%
}%
}%
}
defMonotonieL#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}-0.001})relax}{text{smftext}}{0}{text{smstext}}}
defMonotonieR#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}+0.001})relax}{text{smftext}}{0}{text{smstext}}}
defExtrema#1#2{xintifboolexpr{xinttheexpr (#1(XcalcR{STZeroL{#1}{#2}-0.001}))*(#1(XcalcR{STZeroL{#1}{#2}+0.001}))relax >0}%
{text{SPtext}}%
{%
xintifboolexpr{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax>0}%
{text{HPtext}}%
{text{TPtext}}%
}%
}%
NewDocumentCommand{VZtabfac}{st{>}mmm}{%
IfBooleanTF{#1}%
{xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5})^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}}%
{%
IfBooleanTF{#2}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}+0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}-0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
}%
}
newcommand{VZmakerows}[4]{%
xintifboolexpr{PolLeadingCoeff{#1}<0}%
{%
xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}} & -
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do { & - & - } \ midrule
}%
{}%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{tabfactor{#1}{##1}{#3}{#4} & VZtabfac{#1}{##1}{1}%
xintFor* ##2 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{ &
IfDecimal{#2}%
{VZtabfac*{#1}{##1}{##2}}%
{%
xintifboolexpr{xinttheexpr #2(STZeroL{#1}{##2})relax = 0}%
{Notdeftxt}
{VZtabfac*{#1}{##1}{##2}}%
}%
& VZtabfac>{#1}{##1}{##2}
}
\ midrule
}
}
newcommand{Firstrow}[3]{%
#2 & -infty<#2<nst{#1}{1} & IfDecimal{#3}{#2=nst{#1}{1}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{1})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{1}}}%
{#2=nst{#1}{1}}%
}%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#1}>1}{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}-1}}
do
{%
& nst{#1}{xinttheexpr ##1relax}<#2<nst{#1}{xinttheexpr ##1+1relax} &
IfDecimal{#3}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{xinttheexpr ##1+1relax})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{xinttheexpr ##1+1relax}}}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
}%
}%
}%
{}%
& nst{#1}{PolSturmNbOfIsolatedZeros{#1}}<#2<+infty\ midrule
}
NewDocumentCommand{Diffrow}{mmmm}{%
#1(#3)
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& VZfktwertL{#2}{##1} &
IfDecimal{#4}%
{VZfktwert{#2}{##1}}%
{%
xintifboolexpr{xinttheexpr #4(STZeroL{#2}{##1})relax = 0}%
{Notdeftxt}
{VZfktwert{#2}{##1}}%
}%
}%
& VZfktwertR{#2}{PolSturmNbOfIsolatedZeros{#2}} \ midrule
}
newcommand{Graphrow}[3]{%
text{G}_{#1}
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& MonotonieL{#2}{##1} & IfDecimal{#3}{Extrema{#2}{##1}}{%
xintifboolexpr{xinttheexpr #3(STZeroL{#2}{##1})relax = 0}%
{%
xintifboolexpr{xinttheexpr gekuerzterNenner(STZeroL{#2}{##1})relax =0}%
{text{Poltxt}}
{text{Lueckentxt}}
}%
{Extrema{#2}{##1}}%
}%
}%
& MonotonieR{#2}{PolSturmNbOfIsolatedZeros{#2}}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Begin of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Counter for LOOP over the roots of f
newcountzeroCount
% Set counter to 1
zeroCount=1
% Proof if a root is "0" then only write x and not (x-0) ...
% factor the roots like (x-x_1) if x_1>0, (x+x_1) if x_1<0
newcommand{factorZeros}[1]{%
xintifZero{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}%
{varname}
{left(varnamexintiiifSgn{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}{+}{-}{-}
xintSignedFrac{xintIrr{xintAbs{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}}}right)
}
}
% Don't write multiplicity of a root in case it is =1
newcommand{ignoreMultiplicityOne}[1]{%
xintifOne{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}{}{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}
}
% Hide leading coefficient if leading coefficient is + or - 1.
newcommand{hideLeadingCoeff}[1]{%
xintifOne{xintAbs{PolLeadingCoeff{#1}}}%
{xintiiifSgn{PolLeadingCoeff{#1}}{-}{}{}}
{xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}}}
}
% Typeset polynomial as factors of its rational roots: f(x)=a_n(x-x_1)(x-x_2)...(x-x_n)(quotient function)
newcommand{typesetFactors}[2][x]{%
defvarname{#1}%
% #1 = function
PolToSturm{#2}{#2}%
PolSturmIsolateZeros**{#2}%
xintifboolexpr{PolDegree{#2}<2}{xintifboolexpr{PolDegree{#2}<1}{PolTypeset[varname]{#2}}{left(PolTypeset[varname]{#2}right)}}{%
xintifboolexpr{PolDegree{#2}=PolDegree{#2_norr}}{PolTypeset[varname]{#2}}{%
poldef #2_rat(x) := #2(x)/#2_norr(x);
poldef #2_norr_VZ(x) := xintSgn{PolLeadingCoeff{#2_norr}}#2_norr(x);
PolToSturm{#2_rat}{#2_rat}%
PolSturmIsolateZeros**{#2_rat}%
%
hideLeadingCoeff{#2}%
% Loop over all rational zeros of the function f = #2 ; #2_rat is the part with only rational zeros
% #2_rat is defined as #2(x)/#2_norr(x), and #2_norr is the part of #2 with no rational roots (square-free version is #2_irr )
loop
factorZeros{#2_rat}^{ignoreMultiplicityOne{#2_rat}}%
advance zeroCount 1
ifnum zeroCount<xinttheexprPolSturmNbOfIsolatedZeros{#2_rat}+1relax
repeat
xintifZero{PolDegree{#2_norr}}%
{}%
{left(PolTypeset[varname]{#2_norr_VZ}right)}%
}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
defsmstext{$nearrow$}
defsmftext{$searrow$}
defHPtext{H}
defTPtext{T}
defSPtext{Ter.}
defDiffRowtxt{f^{prime}}
defPoltxt{textcolor{lckColor}{Pol}}
defLueckentxt{textcolor{lckColor}{Loch}}
begin{document}
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
bigskip
poldef zf1(x):=(x^2-4x+3)*(x-2);
PolReduceCoeffs{zf1}
poldef nf1(x):=(x^2+1)(x-2)^2;
PolReduceCoeffs{nf1}
Given is the derivative:
[f^{prime}:x mapsto frac{PolTypeset{zf1}}{PolTypeset{nf1}} = frac{typesetFactors{zf1}}{typesetFactors{nf1}}]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][x][zf1](nf1)
end{document}
answered Dec 6 at 5:43
Thomas
694212
(+1) Thank you for your problem expansion. But truly I don't understand anything. hihi :-))
– chishimotoji
Dec 6 at 8:32
very impressive but you seem to be using some development version ofpolexprwhich finds rational roots as current version0.6does not have the double starredPolSturmIsolateZeros**.... I am very jealous!
– jfbu
Dec 6 at 9:51
@jfbu yes, you are right. I actually forgot that.
– Thomas
Dec 6 at 11:17
I will make my best to ping thepolexprauthor to make soon a release of dev version....
– jfbu
Dec 6 at 13:41
The code has been adapted to the new syntax of polexpr 0.7, which will appear shortly on ctan. Many thanks to the diligent author of polexpr.
– Thomas
2 days ago
|
show 1 more comment
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
For your first question, is this what you want? Note I replaced $dfrac{1}{4}$ with $mfrac{1}{4}$ (medium sized fraction from nccmath) , as I think that, used as a numerical coefficient, it looks nicer. The level alignment of m-mfrac{1}{4} and m is obtained adding a vphantom{mfrac{1}{4}} to the latter, so the boxes containing both minima have the same height.
documentclass[12pt,a4paper]{article}
usepackage{amsmath, amssymb, newcent}
usepackage{tkz-tab}
usepackage{nccmath}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/$mvphantom{mfrac 14}$, +/$+infty$}
end{tikzpicture}
end{document}
Edit:
It seems I misundertood the requirement. I hope the following code is more like you want (same preamble):
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/raisebox{4ex}{$mvphantom{mfrac{1}4}$}, +/$+infty$}
end{tikzpicture}
answered Dec 4 at 11:55
Bernard
164k769192
I see it seem was not changed. " m-1/4 " = " m "
– chishimotoji
Dec 4 at 12:02
Both ms are now at the same level, using avphantom{mfrac{1}{4}for the rightmost m
– Bernard
Dec 4 at 12:40
You can edit your answer to show what you say. My english is not good.
– chishimotoji
Dec 4 at 12:58
I've added some explanations. Is that clearer?
– Bernard
Dec 4 at 13:28
smash[t]{vphantom{mfrac{1}{4}}}seems better. Ordfrac, of course, ifdfracis used in the other place.
– egreg
Dec 4 at 13:40
|
show 3 more comments
up vote
4
down vote
accepted
For your first question, is this what you want? Note I replaced $dfrac{1}{4}$ with $mfrac{1}{4}$ (medium sized fraction from nccmath) , as I think that, used as a numerical coefficient, it looks nicer. The level alignment of m-mfrac{1}{4} and m is obtained adding a vphantom{mfrac{1}{4}} to the latter, so the boxes containing both minima have the same height.
documentclass[12pt,a4paper]{article}
usepackage{amsmath, amssymb, newcent}
usepackage{tkz-tab}
usepackage{nccmath}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/$mvphantom{mfrac 14}$, +/$+infty$}
end{tikzpicture}
end{document}
Edit:
It seems I misundertood the requirement. I hope the following code is more like you want (same preamble):
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/raisebox{4ex}{$mvphantom{mfrac{1}4}$}, +/$+infty$}
end{tikzpicture}
answered Dec 4 at 11:55
Bernard
164k769192
I see it seem was not changed. " m-1/4 " = " m "
– chishimotoji
Dec 4 at 12:02
Both ms are now at the same level, using avphantom{mfrac{1}{4}for the rightmost m
– Bernard
Dec 4 at 12:40
You can edit your answer to show what you say. My english is not good.
– chishimotoji
Dec 4 at 12:58
I've added some explanations. Is that clearer?
– Bernard
Dec 4 at 13:28
smash[t]{vphantom{mfrac{1}{4}}}seems better. Ordfrac, of course, ifdfracis used in the other place.
– egreg
Dec 4 at 13:40
|
show 3 more comments
up vote
4
down vote
accepted
up vote
4
down vote
accepted
For your first question, is this what you want? Note I replaced $dfrac{1}{4}$ with $mfrac{1}{4}$ (medium sized fraction from nccmath) , as I think that, used as a numerical coefficient, it looks nicer. The level alignment of m-mfrac{1}{4} and m is obtained adding a vphantom{mfrac{1}{4}} to the latter, so the boxes containing both minima have the same height.
documentclass[12pt,a4paper]{article}
usepackage{amsmath, amssymb, newcent}
usepackage{tkz-tab}
usepackage{nccmath}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/$mvphantom{mfrac 14}$, +/$+infty$}
end{tikzpicture}
end{document}
Edit:
It seems I misundertood the requirement. I hope the following code is more like you want (same preamble):
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/raisebox{4ex}{$mvphantom{mfrac{1}4}$}, +/$+infty$}
end{tikzpicture}
answered Dec 4 at 11:55
Bernard
164k769192
For your first question, is this what you want? Note I replaced $dfrac{1}{4}$ with $mfrac{1}{4}$ (medium sized fraction from nccmath) , as I think that, used as a numerical coefficient, it looks nicer. The level alignment of m-mfrac{1}{4} and m is obtained adding a vphantom{mfrac{1}{4}} to the latter, so the boxes containing both minima have the same height.
documentclass[12pt,a4paper]{article}
usepackage{amsmath, amssymb, newcent}
usepackage{tkz-tab}
usepackage{nccmath}
begin{document}
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/$mvphantom{mfrac 14}$, +/$+infty$}
end{tikzpicture}
end{document}
Edit:
It seems I misundertood the requirement. I hope the following code is more like you want (same preamble):
begin{tikzpicture}
tkzTabInit{$x$ /1, $g'(x)$ /1, $g(x)$ /3.5}
{$-infty$,$a$,$1$,$3$, $+infty$}
tkzTabLine{,-,0,+,0,-,0,+ }
tkzTabVar{+/$+infty$, -/$m-mfrac{1}{4}$ , +/$g(1)$, -/raisebox{4ex}{$mvphantom{mfrac{1}4}$}, +/$+infty$}
end{tikzpicture}
answered Dec 4 at 11:55
Bernard
164k769192
edited Dec 4 at 14:14
answered Dec 4 at 11:55
Bernard
164k769192
answered Dec 4 at 11:55
Bernard
164k769192
answered Dec 4 at 11:55
Bernard
164k769192
164k769192
I see it seem was not changed. " m-1/4 " = " m "
– chishimotoji
Dec 4 at 12:02
Both ms are now at the same level, using avphantom{mfrac{1}{4}for the rightmost m
– Bernard
Dec 4 at 12:40
You can edit your answer to show what you say. My english is not good.
– chishimotoji
Dec 4 at 12:58
I've added some explanations. Is that clearer?
– Bernard
Dec 4 at 13:28
smash[t]{vphantom{mfrac{1}{4}}}seems better. Ordfrac, of course, ifdfracis used in the other place.
– egreg
Dec 4 at 13:40
|
show 3 more comments
I see it seem was not changed. " m-1/4 " = " m "
– chishimotoji
Dec 4 at 12:02
Both ms are now at the same level, using avphantom{mfrac{1}{4}for the rightmost m
– Bernard
Dec 4 at 12:40
You can edit your answer to show what you say. My english is not good.
– chishimotoji
Dec 4 at 12:58
I've added some explanations. Is that clearer?
– Bernard
Dec 4 at 13:28
smash[t]{vphantom{mfrac{1}{4}}}seems better. Ordfrac, of course, ifdfracis used in the other place.
– egreg
Dec 4 at 13:40
I see it seem was not changed. " m-1/4 " = " m "
– chishimotoji
Dec 4 at 12:02
I see it seem was not changed. " m-1/4 " = " m "
– chishimotoji
Dec 4 at 12:02
Both ms are now at the same level, using a
vphantom{mfrac{1}{4} for the rightmost m– Bernard
Dec 4 at 12:40
Both ms are now at the same level, using a
vphantom{mfrac{1}{4} for the rightmost m– Bernard
Dec 4 at 12:40
You can edit your answer to show what you say. My english is not good.
– chishimotoji
Dec 4 at 12:58
You can edit your answer to show what you say. My english is not good.
– chishimotoji
Dec 4 at 12:58
I've added some explanations. Is that clearer?
– Bernard
Dec 4 at 13:28
I've added some explanations. Is that clearer?
– Bernard
Dec 4 at 13:28
smash[t]{vphantom{mfrac{1}{4}}} seems better. Or dfrac, of course, if dfrac is used in the other place.– egreg
Dec 4 at 13:40
smash[t]{vphantom{mfrac{1}{4}}} seems better. Or dfrac, of course, if dfrac is used in the other place.– egreg
Dec 4 at 13:40
|
show 3 more comments
up vote
4
down vote
You can use the new tablvar package (documentation in French):
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{m} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
Removing the tikz option makes the package rely on PSTricks.
If you want to easily move the m up (which I can't see the necessity of), raise it.
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{raisebox{2ex}{$mmathstrut$}} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
answered Dec 4 at 14:00
egreg
704k8618753154
Thank you but you should see edited question! My aim is to set "m" at that position.
– chishimotoji
Dec 4 at 14:02
@chishimotoji Added. Probably there are more features in the package documentation.
– egreg
Dec 4 at 14:04
add a comment |
up vote
4
down vote
You can use the new tablvar package (documentation in French):
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{m} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
Removing the tikz option makes the package rely on PSTricks.
If you want to easily move the m up (which I can't see the necessity of), raise it.
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{raisebox{2ex}{$mmathstrut$}} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
answered Dec 4 at 14:00
egreg
704k8618753154
Thank you but you should see edited question! My aim is to set "m" at that position.
– chishimotoji
Dec 4 at 14:02
@chishimotoji Added. Probably there are more features in the package documentation.
– egreg
Dec 4 at 14:04
add a comment |
up vote
4
down vote
up vote
4
down vote
You can use the new tablvar package (documentation in French):
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{m} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
Removing the tikz option makes the package rely on PSTricks.
If you want to easily move the m up (which I can't see the necessity of), raise it.
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{raisebox{2ex}{$mmathstrut$}} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
answered Dec 4 at 14:00
egreg
704k8618753154
You can use the new tablvar package (documentation in French):
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{m} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
Removing the tikz option makes the package rely on PSTricks.
If you want to easily move the m up (which I can't see the necessity of), raise it.
documentclass[12pt,a4paper]{article}
usepackage{amsmath,amssymb}
usepackage[tikz]{tablvar}
begin{document}
[
begin{tablvar}{4}
hline
x & -infty && a && 1 && 3 && +infty \
hline
g'(x) && - & 0 & + & 0 & - & 0 & + & \
hline
variations{
mil{g(x)} &
haut{+infty} &&
bas{m-dfrac{1}{4mathstrut}} &&
haut{g(1)} &&
bas{raisebox{2ex}{$mmathstrut$}} &&
haut{+infty}
}
hline
end{tablvar}
]
end{document}
answered Dec 4 at 14:00
egreg
704k8618753154
edited Dec 4 at 14:04
answered Dec 4 at 14:00
egreg
704k8618753154
answered Dec 4 at 14:00
egreg
704k8618753154
answered Dec 4 at 14:00
egreg
704k8618753154
704k8618753154
Thank you but you should see edited question! My aim is to set "m" at that position.
– chishimotoji
Dec 4 at 14:02
@chishimotoji Added. Probably there are more features in the package documentation.
– egreg
Dec 4 at 14:04
add a comment |
Thank you but you should see edited question! My aim is to set "m" at that position.
– chishimotoji
Dec 4 at 14:02
@chishimotoji Added. Probably there are more features in the package documentation.
– egreg
Dec 4 at 14:04
Thank you but you should see edited question! My aim is to set "m" at that position.
– chishimotoji
Dec 4 at 14:02
Thank you but you should see edited question! My aim is to set "m" at that position.
– chishimotoji
Dec 4 at 14:02
@chishimotoji Added. Probably there are more features in the package documentation.
– egreg
Dec 4 at 14:04
@chishimotoji Added. Probably there are more features in the package documentation.
– egreg
Dec 4 at 14:04
add a comment |
up vote
4
down vote
Out of topic!
For polynomials and rational functions with minimum one rational zero
we can do the factorization and finding the (only) rational roots with the help of
the xintexpr and polexpr packages. Then the table is build automatic.
So we can type for e.g. in the document
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
Here is the code for a full example.
documentclass[fleqn,svgnames,x11names,dvipsnames]{article}
usepackage[a4paper,margin=1.25cm]{geometry}
usepackage{amsmath,amssymb}
usepackage{xintexpr}
usepackage{polexpr}
usepackage{xfp,xparse,expl3,xstring}
usepackage{booktabs}
usepackage{siunitx}
sisetup{group-separator={,},output-decimal-marker={,},exponent-product=cdot,per-mode=fraction,product-units=single,
input-product=*,output-product=cdot,unit-mode=text,quotient-mode=fraction,locale=DE,range-phrase = { bis }}
usepackage{pstricks}
letfunkPolToExpr
letSTZeroLPolSturmIsolatedZeroLeft
NewExpandableDocumentCommand{XcalcR}{sO{8}m}{% calculates #3 and rounds to #2 decimals; non formating: for formated output use Xcalc not XcalcR
PolDecToString{xintREZ{xinttheiexpr[#2]reduce(#3)relax}}%
}
NewDocumentCommand{Xcalc}{t+stDsO{2}m}{% calculates #5, rounds to #4 decimals (trailing zeros are supressed) and typesets the number with num-macro formatting
IfBooleanT{#1}{xintifSgn{xinttheexpr #6relax}{}{+}{+}}% or as fraction, reduced or not reduced and with #3=D is a boolean in displaystyle
IfBooleanTF{#2}%
{IfBooleanT{#3}{displaystyle}%
IfBooleanTF{#4}%
{xintSignedFrac{xinttheexpr #6relax}}%
{xintSignedFrac{xinttheexpr reduce(#6)relax}}%
}%
{num{PolDecToString{xintREZ{xinttheiexpr[#5]reduce(#6)relax}}}}%
}
newcommand{NullStellen}[1]{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{x_{##1} = Xcalc{PolSturmIsolatedZeroLeft{#1}{##1}}%
xintifForLast{}{; }}%
}
NewDocumentCommand{hackNst}{O{nt}D(){x}m}{%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#3}=0}{}{%
xintiloop [1+1]
expandafterdefexpandaftertmpxintbracediloopindex
xintdefvar #2tmp = PolSturmIsolatedZeroLeft{#3}tmp;
xintdefvar #1tmp = PolSturmIsolatedZeroMultiplicity{#3}tmp;
ifnumxintiloopindex<numexprPolSturmNbOfIsolatedZeros{#3}relax
repeat
}%
}
NewDocumentCommand{GenFloatSturmZeros}{ssO{9}D(){x}m}{%
IfBooleanTF{#1}%
{%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
IfBooleanF{#2}{hackNst(#4){#5}}%
}%
{%
PolGenFloatVariant{#5}%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
}%
}
defsmstext{steigt}%
defsmftext{fällt}
defHPtext{Hochpkt.}
defTPtext{Tiefpkt.}
defSPtext{Terrassenpkt.}
defDiffRowtxt{f^{prime}}
deflueckColor#1{colorlet{lckColor}{#1}}
lueckColor{BrickRed}
defPoltxt{textcolor{lckColor}{Polstelle}}
defLueckentxt{textcolor{lckColor}{Def.-Lücke}}
defNotdeftxt{textcolor{lckColor}{diagup}}
NewDocumentCommand{VZTabelle}{sO{f}O{x}O{#21}!d()}{%
IfValueTF{#5}{%
PolGCD{#4}{#5}{commonfactorA}%
poldef erwzaehlerA(x) := (#4(x)#5(x))/(commonfactorA(x))^2 ;%
poldef gekuerzterNenner(x) := #5(x)/commonfactorA(x) ;%
PolGCD{erwzaehlerA}{commonfactorA}{commonfactorB}%
poldef commonfactorC(x) := commonfactorA(x)/commonfactorB(x) ;%
poldef erwzaehler(x) := erwzaehlerA(x)commonfactorC(x) ;%
}%
{%
poldef erwzaehler(x) := #4(x) ;%
poldef commonfactorC(x) := 1 ;%
}%
PolToSturm{erwzaehler}{erwzaehler}%
PolSturmIsolateZeros**{erwzaehler}%
PolEnsureIntervalLengths{erwzaehler}{-9}%
begin{tabular}{*{XcalcR{2+2*PolSturmNbOfIsolatedZeros{erwzaehler}}}{>{$}c<{$}}}
IfBooleanF{#1}{toprule}
IfValueTF{#5}{Firstrow{erwzaehler}{#3}{#5}}{Firstrow{erwzaehler}{#3}{1}}
IfValueTF{#5}{VZmakerows{erwzaehler}{#5}{#4}{#3}}{VZmakerows{erwzaehler}{1}{#4}{#3}}
IfValueTF{#5}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{#5}}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{1}}
IfValueTF{#5}{Graphrow{#2}{erwzaehler}{#5}}{Graphrow{#2}{erwzaehler}{1}}
IfBooleanF{#1}{ \ bottomrule}
end{tabular}%
}
defleadcoef#1{xintifboolexpr{PolLeadingCoeff{#1}=1}{}{Xcalc{PolLeadingCoeff{#1}}cdot}}
defnst#1#2{Xcalc*{STZeroL{#1}{#2}}}
defVZfktwertL#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax}{-}{0}{+}}
defVZfktwertR#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}+0.001relax)relax}{-}{0}{+}}
defVZfktwert#1#2{xintifSgn{xinttheexpr #1(STZeroL{#1}{#2})relax}{-}{0}{+}}
deftabfactor#1#2#3#4{%
xintifboolexpr{STZeroL{#1}{#2}=0}%
{#4}%
{(#4Xcalc+*{-STZeroL{#1}{#2}})
xintifboolexpr{PolSturmIsolatedZeroMultiplicity{#1}{#2}=1}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{^{0}}%
{}%
}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{%
^{0}
}%
{%
^{PolSturmIsolatedZeroMultiplicity{#1}{#2}}
}%
}%
}%
}
defMonotonieL#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}-0.001})relax}{text{smftext}}{0}{text{smstext}}}
defMonotonieR#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}+0.001})relax}{text{smftext}}{0}{text{smstext}}}
defExtrema#1#2{xintifboolexpr{xinttheexpr (#1(XcalcR{STZeroL{#1}{#2}-0.001}))*(#1(XcalcR{STZeroL{#1}{#2}+0.001}))relax >0}%
{text{SPtext}}%
{%
xintifboolexpr{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax>0}%
{text{HPtext}}%
{text{TPtext}}%
}%
}%
NewDocumentCommand{VZtabfac}{st{>}mmm}{%
IfBooleanTF{#1}%
{xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5})^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}}%
{%
IfBooleanTF{#2}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}+0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}-0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
}%
}
newcommand{VZmakerows}[4]{%
xintifboolexpr{PolLeadingCoeff{#1}<0}%
{%
xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}} & -
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do { & - & - } \ midrule
}%
{}%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{tabfactor{#1}{##1}{#3}{#4} & VZtabfac{#1}{##1}{1}%
xintFor* ##2 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{ &
IfDecimal{#2}%
{VZtabfac*{#1}{##1}{##2}}%
{%
xintifboolexpr{xinttheexpr #2(STZeroL{#1}{##2})relax = 0}%
{Notdeftxt}
{VZtabfac*{#1}{##1}{##2}}%
}%
& VZtabfac>{#1}{##1}{##2}
}
\ midrule
}
}
newcommand{Firstrow}[3]{%
#2 & -infty<#2<nst{#1}{1} & IfDecimal{#3}{#2=nst{#1}{1}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{1})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{1}}}%
{#2=nst{#1}{1}}%
}%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#1}>1}{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}-1}}
do
{%
& nst{#1}{xinttheexpr ##1relax}<#2<nst{#1}{xinttheexpr ##1+1relax} &
IfDecimal{#3}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{xinttheexpr ##1+1relax})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{xinttheexpr ##1+1relax}}}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
}%
}%
}%
{}%
& nst{#1}{PolSturmNbOfIsolatedZeros{#1}}<#2<+infty\ midrule
}
NewDocumentCommand{Diffrow}{mmmm}{%
#1(#3)
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& VZfktwertL{#2}{##1} &
IfDecimal{#4}%
{VZfktwert{#2}{##1}}%
{%
xintifboolexpr{xinttheexpr #4(STZeroL{#2}{##1})relax = 0}%
{Notdeftxt}
{VZfktwert{#2}{##1}}%
}%
}%
& VZfktwertR{#2}{PolSturmNbOfIsolatedZeros{#2}} \ midrule
}
newcommand{Graphrow}[3]{%
text{G}_{#1}
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& MonotonieL{#2}{##1} & IfDecimal{#3}{Extrema{#2}{##1}}{%
xintifboolexpr{xinttheexpr #3(STZeroL{#2}{##1})relax = 0}%
{%
xintifboolexpr{xinttheexpr gekuerzterNenner(STZeroL{#2}{##1})relax =0}%
{text{Poltxt}}
{text{Lueckentxt}}
}%
{Extrema{#2}{##1}}%
}%
}%
& MonotonieR{#2}{PolSturmNbOfIsolatedZeros{#2}}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Begin of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Counter for LOOP over the roots of f
newcountzeroCount
% Set counter to 1
zeroCount=1
% Proof if a root is "0" then only write x and not (x-0) ...
% factor the roots like (x-x_1) if x_1>0, (x+x_1) if x_1<0
newcommand{factorZeros}[1]{%
xintifZero{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}%
{varname}
{left(varnamexintiiifSgn{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}{+}{-}{-}
xintSignedFrac{xintIrr{xintAbs{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}}}right)
}
}
% Don't write multiplicity of a root in case it is =1
newcommand{ignoreMultiplicityOne}[1]{%
xintifOne{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}{}{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}
}
% Hide leading coefficient if leading coefficient is + or - 1.
newcommand{hideLeadingCoeff}[1]{%
xintifOne{xintAbs{PolLeadingCoeff{#1}}}%
{xintiiifSgn{PolLeadingCoeff{#1}}{-}{}{}}
{xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}}}
}
% Typeset polynomial as factors of its rational roots: f(x)=a_n(x-x_1)(x-x_2)...(x-x_n)(quotient function)
newcommand{typesetFactors}[2][x]{%
defvarname{#1}%
% #1 = function
PolToSturm{#2}{#2}%
PolSturmIsolateZeros**{#2}%
xintifboolexpr{PolDegree{#2}<2}{xintifboolexpr{PolDegree{#2}<1}{PolTypeset[varname]{#2}}{left(PolTypeset[varname]{#2}right)}}{%
xintifboolexpr{PolDegree{#2}=PolDegree{#2_norr}}{PolTypeset[varname]{#2}}{%
poldef #2_rat(x) := #2(x)/#2_norr(x);
poldef #2_norr_VZ(x) := xintSgn{PolLeadingCoeff{#2_norr}}#2_norr(x);
PolToSturm{#2_rat}{#2_rat}%
PolSturmIsolateZeros**{#2_rat}%
%
hideLeadingCoeff{#2}%
% Loop over all rational zeros of the function f = #2 ; #2_rat is the part with only rational zeros
% #2_rat is defined as #2(x)/#2_norr(x), and #2_norr is the part of #2 with no rational roots (square-free version is #2_irr )
loop
factorZeros{#2_rat}^{ignoreMultiplicityOne{#2_rat}}%
advance zeroCount 1
ifnum zeroCount<xinttheexprPolSturmNbOfIsolatedZeros{#2_rat}+1relax
repeat
xintifZero{PolDegree{#2_norr}}%
{}%
{left(PolTypeset[varname]{#2_norr_VZ}right)}%
}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
defsmstext{$nearrow$}
defsmftext{$searrow$}
defHPtext{H}
defTPtext{T}
defSPtext{Ter.}
defDiffRowtxt{f^{prime}}
defPoltxt{textcolor{lckColor}{Pol}}
defLueckentxt{textcolor{lckColor}{Loch}}
begin{document}
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
bigskip
poldef zf1(x):=(x^2-4x+3)*(x-2);
PolReduceCoeffs{zf1}
poldef nf1(x):=(x^2+1)(x-2)^2;
PolReduceCoeffs{nf1}
Given is the derivative:
[f^{prime}:x mapsto frac{PolTypeset{zf1}}{PolTypeset{nf1}} = frac{typesetFactors{zf1}}{typesetFactors{nf1}}]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][x][zf1](nf1)
end{document}
answered Dec 6 at 5:43
Thomas
694212
(+1) Thank you for your problem expansion. But truly I don't understand anything. hihi :-))
– chishimotoji
Dec 6 at 8:32
very impressive but you seem to be using some development version ofpolexprwhich finds rational roots as current version0.6does not have the double starredPolSturmIsolateZeros**.... I am very jealous!
– jfbu
Dec 6 at 9:51
@jfbu yes, you are right. I actually forgot that.
– Thomas
Dec 6 at 11:17
I will make my best to ping thepolexprauthor to make soon a release of dev version....
– jfbu
Dec 6 at 13:41
The code has been adapted to the new syntax of polexpr 0.7, which will appear shortly on ctan. Many thanks to the diligent author of polexpr.
– Thomas
2 days ago
|
show 1 more comment
up vote
4
down vote
Out of topic!
For polynomials and rational functions with minimum one rational zero
we can do the factorization and finding the (only) rational roots with the help of
the xintexpr and polexpr packages. Then the table is build automatic.
So we can type for e.g. in the document
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
Here is the code for a full example.
documentclass[fleqn,svgnames,x11names,dvipsnames]{article}
usepackage[a4paper,margin=1.25cm]{geometry}
usepackage{amsmath,amssymb}
usepackage{xintexpr}
usepackage{polexpr}
usepackage{xfp,xparse,expl3,xstring}
usepackage{booktabs}
usepackage{siunitx}
sisetup{group-separator={,},output-decimal-marker={,},exponent-product=cdot,per-mode=fraction,product-units=single,
input-product=*,output-product=cdot,unit-mode=text,quotient-mode=fraction,locale=DE,range-phrase = { bis }}
usepackage{pstricks}
letfunkPolToExpr
letSTZeroLPolSturmIsolatedZeroLeft
NewExpandableDocumentCommand{XcalcR}{sO{8}m}{% calculates #3 and rounds to #2 decimals; non formating: for formated output use Xcalc not XcalcR
PolDecToString{xintREZ{xinttheiexpr[#2]reduce(#3)relax}}%
}
NewDocumentCommand{Xcalc}{t+stDsO{2}m}{% calculates #5, rounds to #4 decimals (trailing zeros are supressed) and typesets the number with num-macro formatting
IfBooleanT{#1}{xintifSgn{xinttheexpr #6relax}{}{+}{+}}% or as fraction, reduced or not reduced and with #3=D is a boolean in displaystyle
IfBooleanTF{#2}%
{IfBooleanT{#3}{displaystyle}%
IfBooleanTF{#4}%
{xintSignedFrac{xinttheexpr #6relax}}%
{xintSignedFrac{xinttheexpr reduce(#6)relax}}%
}%
{num{PolDecToString{xintREZ{xinttheiexpr[#5]reduce(#6)relax}}}}%
}
newcommand{NullStellen}[1]{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{x_{##1} = Xcalc{PolSturmIsolatedZeroLeft{#1}{##1}}%
xintifForLast{}{; }}%
}
NewDocumentCommand{hackNst}{O{nt}D(){x}m}{%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#3}=0}{}{%
xintiloop [1+1]
expandafterdefexpandaftertmpxintbracediloopindex
xintdefvar #2tmp = PolSturmIsolatedZeroLeft{#3}tmp;
xintdefvar #1tmp = PolSturmIsolatedZeroMultiplicity{#3}tmp;
ifnumxintiloopindex<numexprPolSturmNbOfIsolatedZeros{#3}relax
repeat
}%
}
NewDocumentCommand{GenFloatSturmZeros}{ssO{9}D(){x}m}{%
IfBooleanTF{#1}%
{%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
IfBooleanF{#2}{hackNst(#4){#5}}%
}%
{%
PolGenFloatVariant{#5}%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
}%
}
defsmstext{steigt}%
defsmftext{fällt}
defHPtext{Hochpkt.}
defTPtext{Tiefpkt.}
defSPtext{Terrassenpkt.}
defDiffRowtxt{f^{prime}}
deflueckColor#1{colorlet{lckColor}{#1}}
lueckColor{BrickRed}
defPoltxt{textcolor{lckColor}{Polstelle}}
defLueckentxt{textcolor{lckColor}{Def.-Lücke}}
defNotdeftxt{textcolor{lckColor}{diagup}}
NewDocumentCommand{VZTabelle}{sO{f}O{x}O{#21}!d()}{%
IfValueTF{#5}{%
PolGCD{#4}{#5}{commonfactorA}%
poldef erwzaehlerA(x) := (#4(x)#5(x))/(commonfactorA(x))^2 ;%
poldef gekuerzterNenner(x) := #5(x)/commonfactorA(x) ;%
PolGCD{erwzaehlerA}{commonfactorA}{commonfactorB}%
poldef commonfactorC(x) := commonfactorA(x)/commonfactorB(x) ;%
poldef erwzaehler(x) := erwzaehlerA(x)commonfactorC(x) ;%
}%
{%
poldef erwzaehler(x) := #4(x) ;%
poldef commonfactorC(x) := 1 ;%
}%
PolToSturm{erwzaehler}{erwzaehler}%
PolSturmIsolateZeros**{erwzaehler}%
PolEnsureIntervalLengths{erwzaehler}{-9}%
begin{tabular}{*{XcalcR{2+2*PolSturmNbOfIsolatedZeros{erwzaehler}}}{>{$}c<{$}}}
IfBooleanF{#1}{toprule}
IfValueTF{#5}{Firstrow{erwzaehler}{#3}{#5}}{Firstrow{erwzaehler}{#3}{1}}
IfValueTF{#5}{VZmakerows{erwzaehler}{#5}{#4}{#3}}{VZmakerows{erwzaehler}{1}{#4}{#3}}
IfValueTF{#5}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{#5}}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{1}}
IfValueTF{#5}{Graphrow{#2}{erwzaehler}{#5}}{Graphrow{#2}{erwzaehler}{1}}
IfBooleanF{#1}{ \ bottomrule}
end{tabular}%
}
defleadcoef#1{xintifboolexpr{PolLeadingCoeff{#1}=1}{}{Xcalc{PolLeadingCoeff{#1}}cdot}}
defnst#1#2{Xcalc*{STZeroL{#1}{#2}}}
defVZfktwertL#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax}{-}{0}{+}}
defVZfktwertR#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}+0.001relax)relax}{-}{0}{+}}
defVZfktwert#1#2{xintifSgn{xinttheexpr #1(STZeroL{#1}{#2})relax}{-}{0}{+}}
deftabfactor#1#2#3#4{%
xintifboolexpr{STZeroL{#1}{#2}=0}%
{#4}%
{(#4Xcalc+*{-STZeroL{#1}{#2}})
xintifboolexpr{PolSturmIsolatedZeroMultiplicity{#1}{#2}=1}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{^{0}}%
{}%
}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{%
^{0}
}%
{%
^{PolSturmIsolatedZeroMultiplicity{#1}{#2}}
}%
}%
}%
}
defMonotonieL#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}-0.001})relax}{text{smftext}}{0}{text{smstext}}}
defMonotonieR#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}+0.001})relax}{text{smftext}}{0}{text{smstext}}}
defExtrema#1#2{xintifboolexpr{xinttheexpr (#1(XcalcR{STZeroL{#1}{#2}-0.001}))*(#1(XcalcR{STZeroL{#1}{#2}+0.001}))relax >0}%
{text{SPtext}}%
{%
xintifboolexpr{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax>0}%
{text{HPtext}}%
{text{TPtext}}%
}%
}%
NewDocumentCommand{VZtabfac}{st{>}mmm}{%
IfBooleanTF{#1}%
{xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5})^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}}%
{%
IfBooleanTF{#2}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}+0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}-0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
}%
}
newcommand{VZmakerows}[4]{%
xintifboolexpr{PolLeadingCoeff{#1}<0}%
{%
xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}} & -
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do { & - & - } \ midrule
}%
{}%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{tabfactor{#1}{##1}{#3}{#4} & VZtabfac{#1}{##1}{1}%
xintFor* ##2 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{ &
IfDecimal{#2}%
{VZtabfac*{#1}{##1}{##2}}%
{%
xintifboolexpr{xinttheexpr #2(STZeroL{#1}{##2})relax = 0}%
{Notdeftxt}
{VZtabfac*{#1}{##1}{##2}}%
}%
& VZtabfac>{#1}{##1}{##2}
}
\ midrule
}
}
newcommand{Firstrow}[3]{%
#2 & -infty<#2<nst{#1}{1} & IfDecimal{#3}{#2=nst{#1}{1}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{1})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{1}}}%
{#2=nst{#1}{1}}%
}%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#1}>1}{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}-1}}
do
{%
& nst{#1}{xinttheexpr ##1relax}<#2<nst{#1}{xinttheexpr ##1+1relax} &
IfDecimal{#3}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{xinttheexpr ##1+1relax})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{xinttheexpr ##1+1relax}}}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
}%
}%
}%
{}%
& nst{#1}{PolSturmNbOfIsolatedZeros{#1}}<#2<+infty\ midrule
}
NewDocumentCommand{Diffrow}{mmmm}{%
#1(#3)
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& VZfktwertL{#2}{##1} &
IfDecimal{#4}%
{VZfktwert{#2}{##1}}%
{%
xintifboolexpr{xinttheexpr #4(STZeroL{#2}{##1})relax = 0}%
{Notdeftxt}
{VZfktwert{#2}{##1}}%
}%
}%
& VZfktwertR{#2}{PolSturmNbOfIsolatedZeros{#2}} \ midrule
}
newcommand{Graphrow}[3]{%
text{G}_{#1}
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& MonotonieL{#2}{##1} & IfDecimal{#3}{Extrema{#2}{##1}}{%
xintifboolexpr{xinttheexpr #3(STZeroL{#2}{##1})relax = 0}%
{%
xintifboolexpr{xinttheexpr gekuerzterNenner(STZeroL{#2}{##1})relax =0}%
{text{Poltxt}}
{text{Lueckentxt}}
}%
{Extrema{#2}{##1}}%
}%
}%
& MonotonieR{#2}{PolSturmNbOfIsolatedZeros{#2}}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Begin of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Counter for LOOP over the roots of f
newcountzeroCount
% Set counter to 1
zeroCount=1
% Proof if a root is "0" then only write x and not (x-0) ...
% factor the roots like (x-x_1) if x_1>0, (x+x_1) if x_1<0
newcommand{factorZeros}[1]{%
xintifZero{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}%
{varname}
{left(varnamexintiiifSgn{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}{+}{-}{-}
xintSignedFrac{xintIrr{xintAbs{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}}}right)
}
}
% Don't write multiplicity of a root in case it is =1
newcommand{ignoreMultiplicityOne}[1]{%
xintifOne{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}{}{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}
}
% Hide leading coefficient if leading coefficient is + or - 1.
newcommand{hideLeadingCoeff}[1]{%
xintifOne{xintAbs{PolLeadingCoeff{#1}}}%
{xintiiifSgn{PolLeadingCoeff{#1}}{-}{}{}}
{xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}}}
}
% Typeset polynomial as factors of its rational roots: f(x)=a_n(x-x_1)(x-x_2)...(x-x_n)(quotient function)
newcommand{typesetFactors}[2][x]{%
defvarname{#1}%
% #1 = function
PolToSturm{#2}{#2}%
PolSturmIsolateZeros**{#2}%
xintifboolexpr{PolDegree{#2}<2}{xintifboolexpr{PolDegree{#2}<1}{PolTypeset[varname]{#2}}{left(PolTypeset[varname]{#2}right)}}{%
xintifboolexpr{PolDegree{#2}=PolDegree{#2_norr}}{PolTypeset[varname]{#2}}{%
poldef #2_rat(x) := #2(x)/#2_norr(x);
poldef #2_norr_VZ(x) := xintSgn{PolLeadingCoeff{#2_norr}}#2_norr(x);
PolToSturm{#2_rat}{#2_rat}%
PolSturmIsolateZeros**{#2_rat}%
%
hideLeadingCoeff{#2}%
% Loop over all rational zeros of the function f = #2 ; #2_rat is the part with only rational zeros
% #2_rat is defined as #2(x)/#2_norr(x), and #2_norr is the part of #2 with no rational roots (square-free version is #2_irr )
loop
factorZeros{#2_rat}^{ignoreMultiplicityOne{#2_rat}}%
advance zeroCount 1
ifnum zeroCount<xinttheexprPolSturmNbOfIsolatedZeros{#2_rat}+1relax
repeat
xintifZero{PolDegree{#2_norr}}%
{}%
{left(PolTypeset[varname]{#2_norr_VZ}right)}%
}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
defsmstext{$nearrow$}
defsmftext{$searrow$}
defHPtext{H}
defTPtext{T}
defSPtext{Ter.}
defDiffRowtxt{f^{prime}}
defPoltxt{textcolor{lckColor}{Pol}}
defLueckentxt{textcolor{lckColor}{Loch}}
begin{document}
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
bigskip
poldef zf1(x):=(x^2-4x+3)*(x-2);
PolReduceCoeffs{zf1}
poldef nf1(x):=(x^2+1)(x-2)^2;
PolReduceCoeffs{nf1}
Given is the derivative:
[f^{prime}:x mapsto frac{PolTypeset{zf1}}{PolTypeset{nf1}} = frac{typesetFactors{zf1}}{typesetFactors{nf1}}]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][x][zf1](nf1)
end{document}
answered Dec 6 at 5:43
Thomas
694212
(+1) Thank you for your problem expansion. But truly I don't understand anything. hihi :-))
– chishimotoji
Dec 6 at 8:32
very impressive but you seem to be using some development version ofpolexprwhich finds rational roots as current version0.6does not have the double starredPolSturmIsolateZeros**.... I am very jealous!
– jfbu
Dec 6 at 9:51
@jfbu yes, you are right. I actually forgot that.
– Thomas
Dec 6 at 11:17
I will make my best to ping thepolexprauthor to make soon a release of dev version....
– jfbu
Dec 6 at 13:41
The code has been adapted to the new syntax of polexpr 0.7, which will appear shortly on ctan. Many thanks to the diligent author of polexpr.
– Thomas
2 days ago
|
show 1 more comment
up vote
4
down vote
up vote
4
down vote
Out of topic!
For polynomials and rational functions with minimum one rational zero
we can do the factorization and finding the (only) rational roots with the help of
the xintexpr and polexpr packages. Then the table is build automatic.
So we can type for e.g. in the document
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
Here is the code for a full example.
documentclass[fleqn,svgnames,x11names,dvipsnames]{article}
usepackage[a4paper,margin=1.25cm]{geometry}
usepackage{amsmath,amssymb}
usepackage{xintexpr}
usepackage{polexpr}
usepackage{xfp,xparse,expl3,xstring}
usepackage{booktabs}
usepackage{siunitx}
sisetup{group-separator={,},output-decimal-marker={,},exponent-product=cdot,per-mode=fraction,product-units=single,
input-product=*,output-product=cdot,unit-mode=text,quotient-mode=fraction,locale=DE,range-phrase = { bis }}
usepackage{pstricks}
letfunkPolToExpr
letSTZeroLPolSturmIsolatedZeroLeft
NewExpandableDocumentCommand{XcalcR}{sO{8}m}{% calculates #3 and rounds to #2 decimals; non formating: for formated output use Xcalc not XcalcR
PolDecToString{xintREZ{xinttheiexpr[#2]reduce(#3)relax}}%
}
NewDocumentCommand{Xcalc}{t+stDsO{2}m}{% calculates #5, rounds to #4 decimals (trailing zeros are supressed) and typesets the number with num-macro formatting
IfBooleanT{#1}{xintifSgn{xinttheexpr #6relax}{}{+}{+}}% or as fraction, reduced or not reduced and with #3=D is a boolean in displaystyle
IfBooleanTF{#2}%
{IfBooleanT{#3}{displaystyle}%
IfBooleanTF{#4}%
{xintSignedFrac{xinttheexpr #6relax}}%
{xintSignedFrac{xinttheexpr reduce(#6)relax}}%
}%
{num{PolDecToString{xintREZ{xinttheiexpr[#5]reduce(#6)relax}}}}%
}
newcommand{NullStellen}[1]{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{x_{##1} = Xcalc{PolSturmIsolatedZeroLeft{#1}{##1}}%
xintifForLast{}{; }}%
}
NewDocumentCommand{hackNst}{O{nt}D(){x}m}{%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#3}=0}{}{%
xintiloop [1+1]
expandafterdefexpandaftertmpxintbracediloopindex
xintdefvar #2tmp = PolSturmIsolatedZeroLeft{#3}tmp;
xintdefvar #1tmp = PolSturmIsolatedZeroMultiplicity{#3}tmp;
ifnumxintiloopindex<numexprPolSturmNbOfIsolatedZeros{#3}relax
repeat
}%
}
NewDocumentCommand{GenFloatSturmZeros}{ssO{9}D(){x}m}{%
IfBooleanTF{#1}%
{%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
IfBooleanF{#2}{hackNst(#4){#5}}%
}%
{%
PolGenFloatVariant{#5}%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
}%
}
defsmstext{steigt}%
defsmftext{fällt}
defHPtext{Hochpkt.}
defTPtext{Tiefpkt.}
defSPtext{Terrassenpkt.}
defDiffRowtxt{f^{prime}}
deflueckColor#1{colorlet{lckColor}{#1}}
lueckColor{BrickRed}
defPoltxt{textcolor{lckColor}{Polstelle}}
defLueckentxt{textcolor{lckColor}{Def.-Lücke}}
defNotdeftxt{textcolor{lckColor}{diagup}}
NewDocumentCommand{VZTabelle}{sO{f}O{x}O{#21}!d()}{%
IfValueTF{#5}{%
PolGCD{#4}{#5}{commonfactorA}%
poldef erwzaehlerA(x) := (#4(x)#5(x))/(commonfactorA(x))^2 ;%
poldef gekuerzterNenner(x) := #5(x)/commonfactorA(x) ;%
PolGCD{erwzaehlerA}{commonfactorA}{commonfactorB}%
poldef commonfactorC(x) := commonfactorA(x)/commonfactorB(x) ;%
poldef erwzaehler(x) := erwzaehlerA(x)commonfactorC(x) ;%
}%
{%
poldef erwzaehler(x) := #4(x) ;%
poldef commonfactorC(x) := 1 ;%
}%
PolToSturm{erwzaehler}{erwzaehler}%
PolSturmIsolateZeros**{erwzaehler}%
PolEnsureIntervalLengths{erwzaehler}{-9}%
begin{tabular}{*{XcalcR{2+2*PolSturmNbOfIsolatedZeros{erwzaehler}}}{>{$}c<{$}}}
IfBooleanF{#1}{toprule}
IfValueTF{#5}{Firstrow{erwzaehler}{#3}{#5}}{Firstrow{erwzaehler}{#3}{1}}
IfValueTF{#5}{VZmakerows{erwzaehler}{#5}{#4}{#3}}{VZmakerows{erwzaehler}{1}{#4}{#3}}
IfValueTF{#5}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{#5}}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{1}}
IfValueTF{#5}{Graphrow{#2}{erwzaehler}{#5}}{Graphrow{#2}{erwzaehler}{1}}
IfBooleanF{#1}{ \ bottomrule}
end{tabular}%
}
defleadcoef#1{xintifboolexpr{PolLeadingCoeff{#1}=1}{}{Xcalc{PolLeadingCoeff{#1}}cdot}}
defnst#1#2{Xcalc*{STZeroL{#1}{#2}}}
defVZfktwertL#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax}{-}{0}{+}}
defVZfktwertR#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}+0.001relax)relax}{-}{0}{+}}
defVZfktwert#1#2{xintifSgn{xinttheexpr #1(STZeroL{#1}{#2})relax}{-}{0}{+}}
deftabfactor#1#2#3#4{%
xintifboolexpr{STZeroL{#1}{#2}=0}%
{#4}%
{(#4Xcalc+*{-STZeroL{#1}{#2}})
xintifboolexpr{PolSturmIsolatedZeroMultiplicity{#1}{#2}=1}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{^{0}}%
{}%
}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{%
^{0}
}%
{%
^{PolSturmIsolatedZeroMultiplicity{#1}{#2}}
}%
}%
}%
}
defMonotonieL#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}-0.001})relax}{text{smftext}}{0}{text{smstext}}}
defMonotonieR#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}+0.001})relax}{text{smftext}}{0}{text{smstext}}}
defExtrema#1#2{xintifboolexpr{xinttheexpr (#1(XcalcR{STZeroL{#1}{#2}-0.001}))*(#1(XcalcR{STZeroL{#1}{#2}+0.001}))relax >0}%
{text{SPtext}}%
{%
xintifboolexpr{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax>0}%
{text{HPtext}}%
{text{TPtext}}%
}%
}%
NewDocumentCommand{VZtabfac}{st{>}mmm}{%
IfBooleanTF{#1}%
{xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5})^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}}%
{%
IfBooleanTF{#2}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}+0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}-0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
}%
}
newcommand{VZmakerows}[4]{%
xintifboolexpr{PolLeadingCoeff{#1}<0}%
{%
xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}} & -
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do { & - & - } \ midrule
}%
{}%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{tabfactor{#1}{##1}{#3}{#4} & VZtabfac{#1}{##1}{1}%
xintFor* ##2 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{ &
IfDecimal{#2}%
{VZtabfac*{#1}{##1}{##2}}%
{%
xintifboolexpr{xinttheexpr #2(STZeroL{#1}{##2})relax = 0}%
{Notdeftxt}
{VZtabfac*{#1}{##1}{##2}}%
}%
& VZtabfac>{#1}{##1}{##2}
}
\ midrule
}
}
newcommand{Firstrow}[3]{%
#2 & -infty<#2<nst{#1}{1} & IfDecimal{#3}{#2=nst{#1}{1}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{1})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{1}}}%
{#2=nst{#1}{1}}%
}%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#1}>1}{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}-1}}
do
{%
& nst{#1}{xinttheexpr ##1relax}<#2<nst{#1}{xinttheexpr ##1+1relax} &
IfDecimal{#3}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{xinttheexpr ##1+1relax})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{xinttheexpr ##1+1relax}}}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
}%
}%
}%
{}%
& nst{#1}{PolSturmNbOfIsolatedZeros{#1}}<#2<+infty\ midrule
}
NewDocumentCommand{Diffrow}{mmmm}{%
#1(#3)
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& VZfktwertL{#2}{##1} &
IfDecimal{#4}%
{VZfktwert{#2}{##1}}%
{%
xintifboolexpr{xinttheexpr #4(STZeroL{#2}{##1})relax = 0}%
{Notdeftxt}
{VZfktwert{#2}{##1}}%
}%
}%
& VZfktwertR{#2}{PolSturmNbOfIsolatedZeros{#2}} \ midrule
}
newcommand{Graphrow}[3]{%
text{G}_{#1}
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& MonotonieL{#2}{##1} & IfDecimal{#3}{Extrema{#2}{##1}}{%
xintifboolexpr{xinttheexpr #3(STZeroL{#2}{##1})relax = 0}%
{%
xintifboolexpr{xinttheexpr gekuerzterNenner(STZeroL{#2}{##1})relax =0}%
{text{Poltxt}}
{text{Lueckentxt}}
}%
{Extrema{#2}{##1}}%
}%
}%
& MonotonieR{#2}{PolSturmNbOfIsolatedZeros{#2}}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Begin of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Counter for LOOP over the roots of f
newcountzeroCount
% Set counter to 1
zeroCount=1
% Proof if a root is "0" then only write x and not (x-0) ...
% factor the roots like (x-x_1) if x_1>0, (x+x_1) if x_1<0
newcommand{factorZeros}[1]{%
xintifZero{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}%
{varname}
{left(varnamexintiiifSgn{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}{+}{-}{-}
xintSignedFrac{xintIrr{xintAbs{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}}}right)
}
}
% Don't write multiplicity of a root in case it is =1
newcommand{ignoreMultiplicityOne}[1]{%
xintifOne{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}{}{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}
}
% Hide leading coefficient if leading coefficient is + or - 1.
newcommand{hideLeadingCoeff}[1]{%
xintifOne{xintAbs{PolLeadingCoeff{#1}}}%
{xintiiifSgn{PolLeadingCoeff{#1}}{-}{}{}}
{xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}}}
}
% Typeset polynomial as factors of its rational roots: f(x)=a_n(x-x_1)(x-x_2)...(x-x_n)(quotient function)
newcommand{typesetFactors}[2][x]{%
defvarname{#1}%
% #1 = function
PolToSturm{#2}{#2}%
PolSturmIsolateZeros**{#2}%
xintifboolexpr{PolDegree{#2}<2}{xintifboolexpr{PolDegree{#2}<1}{PolTypeset[varname]{#2}}{left(PolTypeset[varname]{#2}right)}}{%
xintifboolexpr{PolDegree{#2}=PolDegree{#2_norr}}{PolTypeset[varname]{#2}}{%
poldef #2_rat(x) := #2(x)/#2_norr(x);
poldef #2_norr_VZ(x) := xintSgn{PolLeadingCoeff{#2_norr}}#2_norr(x);
PolToSturm{#2_rat}{#2_rat}%
PolSturmIsolateZeros**{#2_rat}%
%
hideLeadingCoeff{#2}%
% Loop over all rational zeros of the function f = #2 ; #2_rat is the part with only rational zeros
% #2_rat is defined as #2(x)/#2_norr(x), and #2_norr is the part of #2 with no rational roots (square-free version is #2_irr )
loop
factorZeros{#2_rat}^{ignoreMultiplicityOne{#2_rat}}%
advance zeroCount 1
ifnum zeroCount<xinttheexprPolSturmNbOfIsolatedZeros{#2_rat}+1relax
repeat
xintifZero{PolDegree{#2_norr}}%
{}%
{left(PolTypeset[varname]{#2_norr_VZ}right)}%
}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
defsmstext{$nearrow$}
defsmftext{$searrow$}
defHPtext{H}
defTPtext{T}
defSPtext{Ter.}
defDiffRowtxt{f^{prime}}
defPoltxt{textcolor{lckColor}{Pol}}
defLueckentxt{textcolor{lckColor}{Loch}}
begin{document}
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
bigskip
poldef zf1(x):=(x^2-4x+3)*(x-2);
PolReduceCoeffs{zf1}
poldef nf1(x):=(x^2+1)(x-2)^2;
PolReduceCoeffs{nf1}
Given is the derivative:
[f^{prime}:x mapsto frac{PolTypeset{zf1}}{PolTypeset{nf1}} = frac{typesetFactors{zf1}}{typesetFactors{nf1}}]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][x][zf1](nf1)
end{document}
answered Dec 6 at 5:43
Thomas
694212
Out of topic!
For polynomials and rational functions with minimum one rational zero
we can do the factorization and finding the (only) rational roots with the help of
the xintexpr and polexpr packages. Then the table is build automatic.
So we can type for e.g. in the document
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
Here is the code for a full example.
documentclass[fleqn,svgnames,x11names,dvipsnames]{article}
usepackage[a4paper,margin=1.25cm]{geometry}
usepackage{amsmath,amssymb}
usepackage{xintexpr}
usepackage{polexpr}
usepackage{xfp,xparse,expl3,xstring}
usepackage{booktabs}
usepackage{siunitx}
sisetup{group-separator={,},output-decimal-marker={,},exponent-product=cdot,per-mode=fraction,product-units=single,
input-product=*,output-product=cdot,unit-mode=text,quotient-mode=fraction,locale=DE,range-phrase = { bis }}
usepackage{pstricks}
letfunkPolToExpr
letSTZeroLPolSturmIsolatedZeroLeft
NewExpandableDocumentCommand{XcalcR}{sO{8}m}{% calculates #3 and rounds to #2 decimals; non formating: for formated output use Xcalc not XcalcR
PolDecToString{xintREZ{xinttheiexpr[#2]reduce(#3)relax}}%
}
NewDocumentCommand{Xcalc}{t+stDsO{2}m}{% calculates #5, rounds to #4 decimals (trailing zeros are supressed) and typesets the number with num-macro formatting
IfBooleanT{#1}{xintifSgn{xinttheexpr #6relax}{}{+}{+}}% or as fraction, reduced or not reduced and with #3=D is a boolean in displaystyle
IfBooleanTF{#2}%
{IfBooleanT{#3}{displaystyle}%
IfBooleanTF{#4}%
{xintSignedFrac{xinttheexpr #6relax}}%
{xintSignedFrac{xinttheexpr reduce(#6)relax}}%
}%
{num{PolDecToString{xintREZ{xinttheiexpr[#5]reduce(#6)relax}}}}%
}
newcommand{NullStellen}[1]{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{x_{##1} = Xcalc{PolSturmIsolatedZeroLeft{#1}{##1}}%
xintifForLast{}{; }}%
}
NewDocumentCommand{hackNst}{O{nt}D(){x}m}{%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#3}=0}{}{%
xintiloop [1+1]
expandafterdefexpandaftertmpxintbracediloopindex
xintdefvar #2tmp = PolSturmIsolatedZeroLeft{#3}tmp;
xintdefvar #1tmp = PolSturmIsolatedZeroMultiplicity{#3}tmp;
ifnumxintiloopindex<numexprPolSturmNbOfIsolatedZeros{#3}relax
repeat
}%
}
NewDocumentCommand{GenFloatSturmZeros}{ssO{9}D(){x}m}{%
IfBooleanTF{#1}%
{%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
IfBooleanF{#2}{hackNst(#4){#5}}%
}%
{%
PolGenFloatVariant{#5}%
PolToSturm{#5}{#5}%
PolSturmIsolateZeros**{#5}%
PolEnsureIntervalLengths{#5}{-#3}%
}%
}
defsmstext{steigt}%
defsmftext{fällt}
defHPtext{Hochpkt.}
defTPtext{Tiefpkt.}
defSPtext{Terrassenpkt.}
defDiffRowtxt{f^{prime}}
deflueckColor#1{colorlet{lckColor}{#1}}
lueckColor{BrickRed}
defPoltxt{textcolor{lckColor}{Polstelle}}
defLueckentxt{textcolor{lckColor}{Def.-Lücke}}
defNotdeftxt{textcolor{lckColor}{diagup}}
NewDocumentCommand{VZTabelle}{sO{f}O{x}O{#21}!d()}{%
IfValueTF{#5}{%
PolGCD{#4}{#5}{commonfactorA}%
poldef erwzaehlerA(x) := (#4(x)#5(x))/(commonfactorA(x))^2 ;%
poldef gekuerzterNenner(x) := #5(x)/commonfactorA(x) ;%
PolGCD{erwzaehlerA}{commonfactorA}{commonfactorB}%
poldef commonfactorC(x) := commonfactorA(x)/commonfactorB(x) ;%
poldef erwzaehler(x) := erwzaehlerA(x)commonfactorC(x) ;%
}%
{%
poldef erwzaehler(x) := #4(x) ;%
poldef commonfactorC(x) := 1 ;%
}%
PolToSturm{erwzaehler}{erwzaehler}%
PolSturmIsolateZeros**{erwzaehler}%
PolEnsureIntervalLengths{erwzaehler}{-9}%
begin{tabular}{*{XcalcR{2+2*PolSturmNbOfIsolatedZeros{erwzaehler}}}{>{$}c<{$}}}
IfBooleanF{#1}{toprule}
IfValueTF{#5}{Firstrow{erwzaehler}{#3}{#5}}{Firstrow{erwzaehler}{#3}{1}}
IfValueTF{#5}{VZmakerows{erwzaehler}{#5}{#4}{#3}}{VZmakerows{erwzaehler}{1}{#4}{#3}}
IfValueTF{#5}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{#5}}{Diffrow{DiffRowtxt}{erwzaehler}{#3}{1}}
IfValueTF{#5}{Graphrow{#2}{erwzaehler}{#5}}{Graphrow{#2}{erwzaehler}{1}}
IfBooleanF{#1}{ \ bottomrule}
end{tabular}%
}
defleadcoef#1{xintifboolexpr{PolLeadingCoeff{#1}=1}{}{Xcalc{PolLeadingCoeff{#1}}cdot}}
defnst#1#2{Xcalc*{STZeroL{#1}{#2}}}
defVZfktwertL#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax}{-}{0}{+}}
defVZfktwertR#1#2{xintifSgn{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}+0.001relax)relax}{-}{0}{+}}
defVZfktwert#1#2{xintifSgn{xinttheexpr #1(STZeroL{#1}{#2})relax}{-}{0}{+}}
deftabfactor#1#2#3#4{%
xintifboolexpr{STZeroL{#1}{#2}=0}%
{#4}%
{(#4Xcalc+*{-STZeroL{#1}{#2}})
xintifboolexpr{PolSturmIsolatedZeroMultiplicity{#1}{#2}=1}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{^{0}}%
{}%
}%
{%
xintifboolexpr{commonfactorC(STZeroL{#1}{#2})=0}%
{%
^{0}
}%
{%
^{PolSturmIsolatedZeroMultiplicity{#1}{#2}}
}%
}%
}%
}
defMonotonieL#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}-0.001})relax}{text{smftext}}{0}{text{smstext}}}
defMonotonieR#1#2{xintifSgn{xinttheexpr #1(XcalcR{STZeroL{#1}{#2}+0.001})relax}{text{smftext}}{0}{text{smstext}}}
defExtrema#1#2{xintifboolexpr{xinttheexpr (#1(XcalcR{STZeroL{#1}{#2}-0.001}))*(#1(XcalcR{STZeroL{#1}{#2}+0.001}))relax >0}%
{text{SPtext}}%
{%
xintifboolexpr{xinttheexpr #1(xinttheexpr STZeroL{#1}{#2}-0.001relax)relax>0}%
{text{HPtext}}%
{text{TPtext}}%
}%
}%
NewDocumentCommand{VZtabfac}{st{>}mmm}{%
IfBooleanTF{#1}%
{xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5})^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}}%
{%
IfBooleanTF{#2}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}+0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
{%
xintifSgn{xinttheexpr (-STZeroL{#3}{#4}+STZeroL{#3}{#5}-0.001)^xintifboolexpr{commonfactorC(STZeroL{#3}{#4})=0}{0}%
{PolSturmIsolatedZeroMultiplicity{#3}{#4}}relax}{-}{0}{+}%
}%
}%
}
newcommand{VZmakerows}[4]{%
xintifboolexpr{PolLeadingCoeff{#1}<0}%
{%
xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}} & -
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do { & - & - } \ midrule
}%
{}%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{tabfactor{#1}{##1}{#3}{#4} & VZtabfac{#1}{##1}{1}%
xintFor* ##2 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}}}
do
{ &
IfDecimal{#2}%
{VZtabfac*{#1}{##1}{##2}}%
{%
xintifboolexpr{xinttheexpr #2(STZeroL{#1}{##2})relax = 0}%
{Notdeftxt}
{VZtabfac*{#1}{##1}{##2}}%
}%
& VZtabfac>{#1}{##1}{##2}
}
\ midrule
}
}
newcommand{Firstrow}[3]{%
#2 & -infty<#2<nst{#1}{1} & IfDecimal{#3}{#2=nst{#1}{1}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{1})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{1}}}%
{#2=nst{#1}{1}}%
}%
xintifboolexpr{PolSturmNbOfIsolatedZeros{#1}>1}{%
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#1}-1}}
do
{%
& nst{#1}{xinttheexpr ##1relax}<#2<nst{#1}{xinttheexpr ##1+1relax} &
IfDecimal{#3}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
{xintifboolexpr{xinttheexpr #3(STZeroL{#1}{xinttheexpr ##1+1relax})relax = 0 }%
{textcolor{lckColor}{#2=nst{#1}{xinttheexpr ##1+1relax}}}%
{#2=nst{#1}{xinttheexpr ##1+1relax}}%
}%
}%
}%
{}%
& nst{#1}{PolSturmNbOfIsolatedZeros{#1}}<#2<+infty\ midrule
}
NewDocumentCommand{Diffrow}{mmmm}{%
#1(#3)
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& VZfktwertL{#2}{##1} &
IfDecimal{#4}%
{VZfktwert{#2}{##1}}%
{%
xintifboolexpr{xinttheexpr #4(STZeroL{#2}{##1})relax = 0}%
{Notdeftxt}
{VZfktwert{#2}{##1}}%
}%
}%
& VZfktwertR{#2}{PolSturmNbOfIsolatedZeros{#2}} \ midrule
}
newcommand{Graphrow}[3]{%
text{G}_{#1}
xintFor* ##1 in {xintSeq{1}{PolSturmNbOfIsolatedZeros{#2}}}
do
{& MonotonieL{#2}{##1} & IfDecimal{#3}{Extrema{#2}{##1}}{%
xintifboolexpr{xinttheexpr #3(STZeroL{#2}{##1})relax = 0}%
{%
xintifboolexpr{xinttheexpr gekuerzterNenner(STZeroL{#2}{##1})relax =0}%
{text{Poltxt}}
{text{Lueckentxt}}
}%
{Extrema{#2}{##1}}%
}%
}%
& MonotonieR{#2}{PolSturmNbOfIsolatedZeros{#2}}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Begin of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Counter for LOOP over the roots of f
newcountzeroCount
% Set counter to 1
zeroCount=1
% Proof if a root is "0" then only write x and not (x-0) ...
% factor the roots like (x-x_1) if x_1>0, (x+x_1) if x_1<0
newcommand{factorZeros}[1]{%
xintifZero{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}%
{varname}
{left(varnamexintiiifSgn{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}{+}{-}{-}
xintSignedFrac{xintIrr{xintAbs{PolSturmIsolatedZeroLeft{#1}{thezeroCount}}}}right)
}
}
% Don't write multiplicity of a root in case it is =1
newcommand{ignoreMultiplicityOne}[1]{%
xintifOne{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}{}{PolSturmIsolatedZeroMultiplicity{#1}{thezeroCount}}
}
% Hide leading coefficient if leading coefficient is + or - 1.
newcommand{hideLeadingCoeff}[1]{%
xintifOne{xintAbs{PolLeadingCoeff{#1}}}%
{xintiiifSgn{PolLeadingCoeff{#1}}{-}{}{}}
{xintSignedFrac{xintIrr{PolLeadingCoeff{#1}}}}
}
% Typeset polynomial as factors of its rational roots: f(x)=a_n(x-x_1)(x-x_2)...(x-x_n)(quotient function)
newcommand{typesetFactors}[2][x]{%
defvarname{#1}%
% #1 = function
PolToSturm{#2}{#2}%
PolSturmIsolateZeros**{#2}%
xintifboolexpr{PolDegree{#2}<2}{xintifboolexpr{PolDegree{#2}<1}{PolTypeset[varname]{#2}}{left(PolTypeset[varname]{#2}right)}}{%
xintifboolexpr{PolDegree{#2}=PolDegree{#2_norr}}{PolTypeset[varname]{#2}}{%
poldef #2_rat(x) := #2(x)/#2_norr(x);
poldef #2_norr_VZ(x) := xintSgn{PolLeadingCoeff{#2_norr}}#2_norr(x);
PolToSturm{#2_rat}{#2_rat}%
PolSturmIsolateZeros**{#2_rat}%
%
hideLeadingCoeff{#2}%
% Loop over all rational zeros of the function f = #2 ; #2_rat is the part with only rational zeros
% #2_rat is defined as #2(x)/#2_norr(x), and #2_norr is the part of #2 with no rational roots (square-free version is #2_irr )
loop
factorZeros{#2_rat}^{ignoreMultiplicityOne{#2_rat}}%
advance zeroCount 1
ifnum zeroCount<xinttheexprPolSturmNbOfIsolatedZeros{#2_rat}+1relax
repeat
xintifZero{PolDegree{#2_norr}}%
{}%
{left(PolTypeset[varname]{#2_norr_VZ}right)}%
}}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% End of code for factorization %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
defsmstext{$nearrow$}
defsmftext{$searrow$}
defHPtext{H}
defTPtext{T}
defSPtext{Ter.}
defDiffRowtxt{f^{prime}}
defPoltxt{textcolor{lckColor}{Pol}}
defLueckentxt{textcolor{lckColor}{Loch}}
begin{document}
poldef zf1(x):=(-1/4)(x^2-4x+3)(x-2)^2(x^2+1)(x^2+2);
PolReduceCoeffs{zf1}
Given is the derivative:
[f^{prime}:t mapsto PolTypeset[t]{zf1} = typesetFactors[t]{zf1} ]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][t][zf1]
bigskip
poldef zf1(x):=(x^2-4x+3)*(x-2);
PolReduceCoeffs{zf1}
poldef nf1(x):=(x^2+1)(x-2)^2;
PolReduceCoeffs{nf1}
Given is the derivative:
[f^{prime}:x mapsto frac{PolTypeset{zf1}}{PolTypeset{nf1}} = frac{typesetFactors{zf1}}{typesetFactors{nf1}}]
medskip
renewcommand{arraystretch}{1.2}%
VZTabelle[f][x][zf1](nf1)
end{document}
answered Dec 6 at 5:43
Thomas
694212
edited 2 days ago
answered Dec 6 at 5:43
Thomas
694212
answered Dec 6 at 5:43
Thomas
694212
answered Dec 6 at 5:43
Thomas
694212
694212
(+1) Thank you for your problem expansion. But truly I don't understand anything. hihi :-))
– chishimotoji
Dec 6 at 8:32
very impressive but you seem to be using some development version ofpolexprwhich finds rational roots as current version0.6does not have the double starredPolSturmIsolateZeros**.... I am very jealous!
– jfbu
Dec 6 at 9:51
@jfbu yes, you are right. I actually forgot that.
– Thomas
Dec 6 at 11:17
I will make my best to ping thepolexprauthor to make soon a release of dev version....
– jfbu
Dec 6 at 13:41
The code has been adapted to the new syntax of polexpr 0.7, which will appear shortly on ctan. Many thanks to the diligent author of polexpr.
– Thomas
2 days ago
|
show 1 more comment
(+1) Thank you for your problem expansion. But truly I don't understand anything. hihi :-))
– chishimotoji
Dec 6 at 8:32
very impressive but you seem to be using some development version ofpolexprwhich finds rational roots as current version0.6does not have the double starredPolSturmIsolateZeros**.... I am very jealous!
– jfbu
Dec 6 at 9:51
@jfbu yes, you are right. I actually forgot that.
– Thomas
Dec 6 at 11:17
I will make my best to ping thepolexprauthor to make soon a release of dev version....
– jfbu
Dec 6 at 13:41
The code has been adapted to the new syntax of polexpr 0.7, which will appear shortly on ctan. Many thanks to the diligent author of polexpr.
– Thomas
2 days ago
(+1) Thank you for your problem expansion. But truly I don't understand anything. hihi :-))
– chishimotoji
Dec 6 at 8:32
(+1) Thank you for your problem expansion. But truly I don't understand anything. hihi :-))
– chishimotoji
Dec 6 at 8:32
very impressive but you seem to be using some development version of
polexpr which finds rational roots as current version 0.6 does not have the double starred PolSturmIsolateZeros**.... I am very jealous!– jfbu
Dec 6 at 9:51
very impressive but you seem to be using some development version of
polexpr which finds rational roots as current version 0.6 does not have the double starred PolSturmIsolateZeros**.... I am very jealous!– jfbu
Dec 6 at 9:51
@jfbu yes, you are right. I actually forgot that.
– Thomas
Dec 6 at 11:17
@jfbu yes, you are right. I actually forgot that.
– Thomas
Dec 6 at 11:17
I will make my best to ping the
polexpr author to make soon a release of dev version....– jfbu
Dec 6 at 13:41
I will make my best to ping the
polexpr author to make soon a release of dev version....– jfbu
Dec 6 at 13:41
The code has been adapted to the new syntax of polexpr 0.7, which will appear shortly on ctan. Many thanks to the diligent author of polexpr.
– Thomas
2 days ago
The code has been adapted to the new syntax of polexpr 0.7, which will appear shortly on ctan. Many thanks to the diligent author of polexpr.
– Thomas
2 days ago
|
show 1 more comment
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You're right, it seems I misunderstood. I've added a new code (leaving the previous one, as it might be of interest to some). Is it better?
– Bernard
Dec 4 at 14:09