|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
-- This file:
-- http://angg.twu.net/LATEX/2021-1-C2-critical-points.lua.html
-- http://angg.twu.net/LATEX/2021-1-C2-critical-points.lua
-- (find-angg "LATEX/2021-1-C2-critical-points.lua")
-- Author: Eduardo Ochs <eduardoochs@gmail.com>
--
-- (defun l () (interactive) (find-LATEX "2021-1-C2-critical-points.lua"))
--
-- Used by: (find-LATEX "2021-1-C2-integral-figuras.tex")
-- Superseded by: (find-LATEX "2021pict2e.lua")
-- «.Points» (to "Points")
-- «.Points-test» (to "Points-test")
--
-- «.f_do_slide_8» (to "f_do_slide_8")
-- «.f_parabola_preferida» (to "f_parabola_preferida")
-- «.Partition» (to "Partition")
-- «.Partition» (to "Partition")
--
-- «.Partition-tests» (to "Partition-tests")
-- «.CriticalPoints-tests» (to "CriticalPoints-tests")
-- «.Approxer» (to "Approxer")
-- «.Approxer-tests» (to "Approxer-tests")
-- «.Piecewisify» (to "Piecewisify")
-- «.Piecewisify-tests» (to "Piecewisify-tests")
require "edrxpict" -- (find-LATEX "edrxpict.lua")
-- ____ _ _
-- | _ \ ___ (_)_ __ | |_ ___
-- | |_) / _ \| | '_ \| __/ __|
-- | __/ (_) | | | | | |_\__ \
-- |_| \___/|_|_| |_|\__|___/
--
-- «Points» (to ".Points")
-- A class for sequences of points.
-- Copied from: (find-LATEX "edrxpict.lua" "Points")
--
Points = Class {
type = "Points",
fromfx = function (a, b, n, f)
local ps = Points {}
for i=0,n do
local x = a + (b-a)*(i/n)
table.insert(ps, v(x,f(x)))
end
return ps
end,
fromFt = function (a, b, n, F)
local ps = Points {}
for i=0,n do
local t = a + (b-a)*(i/n)
table.insert(ps, v(F(t)))
end
return ps
end,
__tostring = function (li) return li:tostring(" ") end,
__index = {
tostring = function (ps, sep) return mapconcat(tostring, ps, sep or "") end,
add = function (ps, ...)
for _,p in ipairs({...}) do table.insert(ps, p) end
return ps
end,
copy = function (ps) return Points(shallowcopy(ps)) end,
polyunder = function (ps)
local x0, x1 = ps[1][1], ps[#ps][1]
return ps:copy():add(v(x1,0), v(x0,0))
end,
pict = function (ps, pre) return (pre or "")..ps:tostring() end,
--
line = function (ps) return ps:pict"\\polyline" end,
poly = function (ps) return ps:pict"\\polygon" end,
fill = function (ps) return ps:pict"\\polygon*" end,
},
}
-- «Points-test» (to ".Points-test")
--[[
* (eepitch-lua51)
* (eepitch-kill)
* (eepitch-lua51)
dofile "edrxpict.lua"
ps = Points.fromfx(2, 3, 5, function (x) return 10*x end)
= ps
= ps:polyunder()
= ps:polyunder():pict"\\polyline"
= ps:polyunder():pict"\\polygon"
= ps:polyunder():pict"\\polygon*"
= Points{v(1,2), v(3,4)}:polyunder():pict"\\polygon*"
= ps:line()
= ps:poly()
= ps:fill()
-- (find-pict2epage 9 "2.4 Extensions")
--]]
-- «f_do_slide_8» (to ".f_do_slide_8")
-- (c2m211somas2p 8 "imagens-de-intervalos")
-- (c2m211somas2a "imagens-de-intervalos")
f_do_slide_8 = function (x)
if x <= 3 then return x+3 end
if 3 < x and x < 8 then return 9-x end
if 8 <= x then return x-7 end
end
-- «f_parabola_preferida» (to ".f_parabola_preferida")
-- (c2m211somas1p 6 "exercicio-1")
-- (c2m211somas1a "exercicio-1")
f_parabola_preferida = function (x) return 4 - (x-2)^2 end
--[[
* (eepitch-lua51)
* (eepitch-kill)
* (eepitch-lua51)
dofile "2021-1-C2-critical-points.lua"
f = f_do_slide_8
= f(0)
= f(2)
= f(3)
= f(4)
= f(7)
= f(8)
= f(9)
f = f_parabola_preferida
= f(0), f(1), f(2), f(3), f(4)
--]]
-- «Partition» (to ".Partition")
--
Partition = Class {
type = "Partition",
new = function (a, b) return Partition {points={a,b}} end,
__index = {
a = function (ptn) return ptn.points[1] end,
b = function (ptn) return ptn.points[#ptn.points] end,
N = function (ptn) return #ptn.points - 1 end,
ai = function (ptn, i) return ptn.points[i] end,
bi = function (ptn, i) return ptn.points[i+1] end,
bminusa = function (ptn) return ptn:b() - ptn:a() end,
splitn = function (ptn, N)
local points = {}
local Delta = ptn:bminusa()
for k=0,N do table.insert(points, ptn:a() + Delta*(k/N)) end
return Partition {points=points}
end,
},
}
-- «Partition-tests» (to ".Partition-tests")
--[[
* (eepitch-lua51)
* (eepitch-kill)
* (eepitch-lua51)
dofile "2021-1-C2-critical-points.lua"
ptn = Partition.new(2, 10)
PPV(ptn)
PPV(ptn:splitn(4))
ptn = Partition.new(2, 10):splitn(4)
for i=1,ptn:N() do
print(i, ptn:ai(i), ptn:bi(i))
end
--]]
CriticalPoints = Class {
type = "CriticalPoints",
new = function (f, allcriticalpoints)
return CriticalPoints {f=f, allcriticalpoints=allcriticalpoints}
end,
__index = {
criticalpointsin = function (cp, a, b)
local A = {}
for _,x in ipairs(cp.allcriticalpoints) do
if a < x and x < b then table.insert(A, x) end
end
return A
end,
infin = function (cp, a, b)
local inf = cp.f(a)
local add = function (x) inf = min(inf, cp.f(x)) end
add(b)
for _,x in ipairs(cp:criticalpointsin(a, b)) do add(x) end
return inf
end,
supin = function (cp, a, b)
local sup = cp.f(a)
local add = function (x) sup = max(sup, cp.f(x)) end
add(b)
for _,x in ipairs(cp:criticalpointsin(a, b)) do add(x) end
return sup
end,
method = function (cp, method, a, b)
return cp[method](cp, a, b)
end,
},
}
-- «CriticalPoints-tests» (to ".CriticalPoints-tests")
--[[
* (eepitch-lua51)
* (eepitch-kill)
* (eepitch-lua51)
dofile "2021-1-C2-critical-points.lua"
cpf = CriticalPoints.new(f_do_slide_8, {3,8})
= cpf:supin(7, 9)
= cpf:infin(7, 9)
= cpf:method("supin", 7, 9)
--]]
-- (find-es "tex" "pict2e")
-- TODO: add support for using different colours for filling and
-- contour
--
Rect0 = Class {
type = "Rect0",
new = function (a, b, y) return Rect0 {a=a, b=b, y=y} end,
__index = {
corners = function (r0)
local a, b, y = r0.a, r0.b, r0.y
return pformat("(%s,%s)(%s,%s)(%s,%s)(%s,%s)", a,y, b,y, b,0, a,0)
end,
pict = function (r0, what)
str = ""
corners = r0:corners()
if what:match"c" then str = str .. "\\polygon*" .. corners .. "%\n" end
if what:match"a" then str = str .. "\\polygon" .. corners .. "%\n" end
return str
end,
},
}
--[[
* (eepitch-lua51)
* (eepitch-kill)
* (eepitch-lua51)
dofile "2021-1-C2-critical-points.lua"
r0 = Rect0.new(1, 2, 3)
= r0:pict("a")
= r0:pict("c")
= r0:pict("ac")
--]]
-- «Approxer» (to ".Approxer")
--
Approxer = Class {
type = "Approxer",
__index = {
pict = function (ap, N, method, what)
bigstr = ""
local cp = CriticalPoints.new(ap.f, ap.allcps)
local pt = Partition.new(ap.a, ap.b):splitn(N or ap.N)
for i=1,pt:N() do
local ai,bi = pt:ai(i), pt:bi(i)
local y = cp:method(method or ap.method, ai, bi)
local pict = Rect0.new(ai, bi, y):pict(what or ap.what)
bigstr = bigstr .. pict
end
return bigstr
end,
},
}
-- «Approxer-tests» (to ".Approxer-tests")
--[[
* (eepitch-lua51)
* (eepitch-kill)
* (eepitch-lua51)
dofile "2021-1-C2-critical-points.lua"
appr = Approxer {
f = f_do_slide_8,
allcps = {3,8},
a = 2,
b = 10,
N = 4,
method = "supin",
what = "ac",
}
= appr:pict()
= appr:pict(2, "infin", "c")
--]]
-- criticalpointsin = function (cp, a, b)
-- local A = {}
-- for _,x in ipairs(cp.allcriticalpoints) do
-- if a < x and x < b then table.insert(A, x) end
-- end
-- return A
-- end,
-- infin = function (cp, a, b)
-- local inf = cp.f(a)
-- local add = function (x) inf = min(inf, cp.f(x)) end
-- add(b)
-- for _,x in ipairs(cp:criticalpointsin(a, b)) do add(x) end
-- return inf
-- end,
-- supin = function (cp, a, b)
-- local sup = cp.f(a)
-- local add = function (x) sup = max(sup, cp.f(x)) end
-- add(b)
-- for _,x in ipairs(cp:criticalpointsin(a, b)) do add(x) end
-- return sup
-- end,
-- method = function (cp, method, a, b)
-- return cp[method](cp, a, b)
-- end,
-- «Piecewisify» (to ".Piecewisify")
--
Piecewisify = Class {
type = "Piecewisify",
new = function (f, ...)
return Piecewisify({f=f}):setpoints(...)
end,
__index = {
addpoint = function (pwi, p)
if type(p) ~= "number" then PP("Not a number:", p); error() end
pwi.points:add(p)
return pwi
end,
addlistofpoints = function (pwi, A)
for _,p in ipairs(A) do pwi:addpoint(p) end
return pwi
end,
addpoints = function (pwi, ...)
for _,o in ipairs({...}) do
if type(o) == "number" then pwi:addpoint(o)
elseif type(o) == "table" then pwi:addlistofpoints(o)
else PP("not a point or a list of points:", o); error()
end
end
return pwi
end,
setpoints = function (pwi, ...)
pwi.points = SetL.new()
return pwi:addpoints(...)
end,
-- See: (find-angg "LUA/lua50init.lua" "SetL")
--
pointsin = function (pwi, a, b)
local A = {}
for _,x in ipairs(pwi.points:ks()) do
if a < x and x < b then table.insert(A, x) end
end
return A
end,
--
eps = 1/32768,
delta = 1/128,
hasjump = function (pwi, x0, x1)
local y0,y1 = pwi.f(x0), pwi.f(x1)
return math.abs(y1-y0) > pwi.delta
end,
hasjumpl = function (pwi, x) return pwi:hasjump(x - pwi.eps, x) end,
hasjumpr = function (pwi, x) return pwi:hasjump(x, x + pwi.eps) end,
--
xy = function (pwi, x, y) return pformat("(%s,%s)", x, y or pwi.f(x)) end,
xyl = function (pwi, xc) return pwi:xy(xc - pwi.eps) end,
xyr = function (pwi, xc) return pwi:xy(xc + pwi.eps) end,
jumps_and_xys = function (pwi, xc)
local xyl, xyc, xyr = pwi:xyl(xc), pwi:xy(xc), pwi:xyr(xc)
local jumpl = pwi:hasjumpl(xc) and "l" or ""
local jumpr = pwi:hasjumpr(xc) and "r" or ""
local jumps = jumpl..jumpr
return jumps, xyl, xyc, xyr
end,
piecewise_m = function (pwi, xc) -- "m" is for "middle"
local jumps, xyl, xyc, xyr = pwi:jumps_and_xys(xc)
if jumps == "" then str = xyc
elseif jumps == "l" then str = format("%so %sc", xyl, xyc)
elseif jumps == "r" then str = format("%sc %so", xyc, xyr)
elseif jumps == "lr" then str = format("%so %sc %so", xyl, xyc, xyr)
end
return str
end,
piecewise = function (pwi, a, b, method, sep)
local method, sep = method or "piecewise_m", sep or "--"
local f = function (x) return pwi[method](pwi, x) end
local str = pwi:xyr(a)
-- for _,x in ipairs(pwi.points:ks()) do
-- if a < x and x < b then
-- str = str..sep..f(x)
-- end
-- end
for _,x in ipairs(pwi:pointsin(a, b)) do
str = str..sep..f(x)
end
str = str..sep..pwi:xyl(b)
return str
end,
pw = function (pwi, a, b)
return pictpiecewise(pwi:piecewise(a, b))
end,
-- See: (find-LATEX "edrxpict.lua" "Piecewise")
-- (find-LATEX "edrxpict.lua" "Piecewise" "pictpiecewise =")
--
piecewise_pol1 = function (pwi, xc) -- for polygons
local jumps, xyl, xyc, xyr = pwi:jumps_and_xys(xc)
if jumps == ""
then return xyc
else return format("%s%s", xyl, xyr)
end
end,
piecewise_pol = function (pwi, a, b)
local a0, b0 = pwi:xy(a, 0), pwi:xy(b, 0)
return a0..pwi:piecewise(a, b, "piecewise_pol1", "")..b0
end,
pol = function (pwi, a, b, star)
return "\\polygon"..(star or "")..pwi:piecewise_pol(a, b)
end,
--
inforsup = function (pwi, maxormin, a, b)
local y = pwi.f(a)
local consider = function (x) y = maxormin(y, pwi.f(x)) end
consider(a + pwi.eps)
for _,x in ipairs(pwi:pointsin(a, b)) do
consider(x - pwi.eps)
consider(x)
consider(x + pwi.eps)
end
consider(b - pwi.eps)
consider(b)
return y
end,
inf = function (pwi, a, b) return pwi:inforsup(min, a, b) end,
sup = function (pwi, a, b) return pwi:inforsup(max, a, b) end,
max = function (pwi, a, b) return max(pwi.f(a), pwi.f(b)) end,
min = function (pwi, a, b) return min(pwi.f(a), pwi.f(b)) end,
zero = function (pwi) return 0 end,
method = function (pwi, mname, a, b) return pwi[mname](pwi, a, b) end,
rct = function (pwi, mname1, mname2, a, b)
local y1 = pwi:method(mname1, a, b)
local y2 = pwi:method(mname2 or "zero", a, b)
return pformat("(%s,%s)(%s,%s)(%s,%s)(%s,%s)", a,y1, a,y2, b,y2, b,y1)
end,
rects = function (pwi, ptn, mname1, mname2, star)
local str = ""
for i=1,ptn:N() do
local ai,bi = ptn:ai(i), ptn:bi(i)
local rct = pwi:rct(mname1, mname2, ai, bi)
str = str .. format("\\polygon%s%s\n", star or "*", rct)
end
return str
end,
},
}
-- «Piecewisify-tests» (to ".Piecewisify-tests")
--[[
* (eepitch-lua51)
* (eepitch-kill)
* (eepitch-lua51)
dofile "2021-1-C2-critical-points.lua"
f_parabola_2 = function (x)
if x <= 4 then return f_parabola_preferida(x) end
if x < 5 then return 2 end
if x == 5 then return 3 end
if x < 6 then return 1 end
return 0
end
pwi = Piecewisify {f = f_parabola_2}
pwi:setpoints(2, 3, seq(0, 1, 0.25))
pwi = Piecewisify.new(f_parabola_2, seq(0, 4, 0.25), 5, 6)
pwi = Piecewisify.new(f_parabola_2, seq(0, 4, 1 ), 5, 6)
= pwi.points:ksc(" ")
= pwi:piecewise_m(2)
= pwi:piecewise_m(4)
= pwi:piecewise_m(5)
= pwi:piecewise_m(6)
= pwi:piecewise(0, 7)
= pwi:pw(0, 7)
= pwi:piecewise_pol(0, 7)
= pwi:pol(0, 7)
= pwi:pol(0, 7, "*")
= pwi:sup(0.5, 2.5)
= pwi:inf(0.5, 2.5)
= pwi:rct("sup", "inf", 0.5, 2.5)
= pwi:rects(Partition.new(0, 2):splitn(4), "sup")
= pwi:rects(Partition.new(0, 2):splitn(4), "sup", "inf")
--]]
--[[
* (eepitch-lua51)
* (eepitch-kill)
* (eepitch-lua51)
dofile "2021-1-C2-critical-points.lua"
dirichlet_incl_Q = SetL.new()
dirichlet_incl = function (x)
if dirichlet_incl_Q:has(x) then return x end
return x+1
end
for _,x in ipairs(seq(0, 1, 1/16)) do
dirichlet_incl_Q:add(x)
end
pwid = Piecewisify.new(dirichlet_incl, seq(0, 1, 1/16))
= dirichlet_incl(0.25)
= dirichlet_incl(0.2)
--]]