mmcdara

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3 years, 183 days

MaplePrimes Activity


These are answers submitted by mmcdara

Luckily I had already done it, I just wasted a little time finding the code.
From memory I believe I recoded in Maple the code from R (not completely sure).

Here it is. 
I remember having done some tests and it seemed ok by the time , but feel free to inform me if it's not.


 

restart:

LSD_Sampling := proc(alpha, N)
  local c, V, X, n, u, q:

  uses Statistics:

  c     := log(1-alpha):
  V     := Sample(Uniform(0, 1), N):
  X     := Vector[column](N):
  for n from 1 to N do
     if V[n] > alpha then
        X[n] := 1:
     else
        u := Sample(Uniform(0, 1), 1)[1]:
        q := 1 - exp(c*u):
        if V[n] < q^2 then
           X[n] := ceil(1+log(V[n])/log(q)):
        elif V[n] < q then
           X[n] := 1:
        else
           X[n] := 2:
        end if:
     end if:
  end do:

  return X:
end proc:

Statistics:-Histogram(LSD_Sampling(0.9, 10000))

 

 


 

Download LogarithmicSeriesSampling.mw


BTW: I'm curious, what application of the Logarithmic Series distribution are you interested in?
 

Even Re(code) doesn't work. The reason is: Maple doesn't know if the variables are real or complex !

Assuming all the quanties are real a simple way is

subs(J=I, collect(subs(I=J, code), J));

Other possibility: use assume, or assuming (see help pages)

I already ask this question a few years ago, you can trace the answers here

Question: I'd like to have some tips about Version Control (source control) process

The solution I use:

  1. I assume you have 2 mw files, let's say File1.mw and File2.mw whoch correspond to different versions of the same code.
     
  2. Open File1.mw and Export it to maple input format (I choose the same name as the mw file)
     
  3. Open File2.mw and Export it to maple input format; two files File1.mpl and File2.mpl are created
     
  4. If not already installed, download NotePad++ (free on Windows ; Notepad++ has contextual coloring for Maple syntax)
     
  5. Open NotePad++ and load File1.mpl and File2.mpl
     
  6. In the menu of NotePad++ choose Run > Compare (this from memory because I do not have NotePad++ on this laptop)
    The differences between the two versions are then easily visble

Once done, you have the choice to keep developping with the Java interface of Maple (as usual), or to develop "within" NotePad++.
In this latter case you will have to load an mpl file in a Maple session to verify that its content gives the expected result.
It's up to you, I personally prefer developping "within" a worksheet, saving in a new mw file, and exporting it in a new mpl file.

PS : there exist probably tools apart NotePad++ that can "understand" Maple's syntax and propose contextual coloring.

I would like to complete acer's answer. Personally I like to use notations like theta__i instead of theta[i], because i is not only limited to integer values.
But using theta__i instead of theta[i] requires some precautions.
You will find below a few examples.

Be careful, when used in a procedure, variables of the form theta__||n (last example) are not implicitely considered as local but as global: this may cause unwanted behaviours.

 

# differences :

a := theta[2]:
b := theta__2:

whattype(a), whattype(b);

printf("%a, %a", theta[2], theta__2);
print(a,b);

indexed, symbol

 

theta[2], theta__2

 

theta[2], theta__2

(1)

# use of theta__  ; first case: wrong way

restart:

vars := seq(theta__i, i=1..4);

mean := Array(1..4, i -> diff(w(vars), theta__i));

 

vars := `#msub(mi("&theta;",fontstyle = "normal"),mi("i"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("i"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("i"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("i"))`

 

Array(%id = 18446744078232662014)

(2)

# use of theta__  ; second case: good way

restart:

vars := seq(theta__||i, i=1..4);

mean := Array(1..4, i -> diff(w(vars), theta__||i));

 

vars := `#msub(mi("&theta;",fontstyle = "normal"),mi("1"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("2"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("3"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("4"))`

 

Array(%id = 18446744078232662014)

(3)

# Why using theta__ instead of theta[..] can be useful?
# Because the index n in theta__n can be of non numeric type:

restart:
MyIndices := [a, b, c, d];
vars := seq(theta__||n, n in MyIndices);

mean := Array(1..4, i -> diff(w(vars), vars[i]));

MyIndices := [a, b, c, d]

 

vars := `#msub(mi("&theta;",fontstyle = "normal"),mi("a"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("b"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("c"))`, `#msub(mi("&theta;",fontstyle = "normal"),mi("d"))`

 

Array(%id = 18446744078232662014)

(4)

restart:

f := proc()
local n:
for n from 1 to 4 do
  theta__||n := n
end do:
end proc:

f():
seq(theta__||i, i=1..4);

1, 2, 3, 4

(5)

 


 

Download indexed_theta.mw

 

Two ways (probably among others)

restart:
vars := seq(theta[i], i=1..4);
mean := Array(1..4, i -> diff(w(vars), theta[i]));

# or
mean := Array(1..4, i -> diff(w(vars), vars[i]));

Download mean.mw

In case you would be interested, give a lookk to this site :

https://oeis.org/?language=english

Next just copy-paste 2,3,4,6,8,9,10,12,14,15,16,18 in the search field and enjoy

 

I personally prefer to define f, f2 and f3 defore instanciate x and y (even if the result is the same).
The main point is in the definition of f3 which should had been f3 := (x, y) -> f(x, y)+f2(x, y)


 

restart:

f  := (x,y) -> x+y;
f2 := (x,y) -> x^2+y^3;
f3 := (x,y) -> f(x,y)+f2(x,y);

proc (x, y) options operator, arrow; x+y end proc

 

proc (x, y) options operator, arrow; x^2+y^3 end proc

 

proc (x, y) options operator, arrow; f(x, y)+f2(x, y) end proc

(1)

f(x,y) ; f2(x,y) ; f3(x,y)

x+y

 

y^3+x^2

 

y^3+x^2+x+y

(2)

x := 5;
y := 10;
f(x, y) ; f2(x, y) ; f3(x, y)

5

 

10

 

15

 

1025

 

1040

(3)

x := 'x':
y := 'y':
x;
y;
f(x, y) ; f2(x, y) ; f3(x, y)

x

 

y

 

x+y

 

y^3+x^2

 

y^3+x^2+x+y

(4)

restart:

x := 5;
y := 10;

5

 

10

(5)

f  := (x,y) -> x+y;
f2 := (x,y) -> x^2+y^3;
f3 := (x,y) -> f(x,y)+f2(x,y);

proc (x, y) options operator, arrow; x+y end proc

 

proc (x, y) options operator, arrow; x^2+y^3 end proc

 

proc (x, y) options operator, arrow; f(x, y)+f2(x, y) end proc

(6)

f(x, y) ; f2(x, y) ; f3(x, y)

15

 

1025

 

1040

(7)

 


 

Download f.mw

Maybe this?


 

restart

u := sin(x+y):

MyPlot := plot3d(u, x = 0 .. 2*Pi, y = 0 .. 2*Pi):

plottools:-getdata(MyPlot);

["grid", [0. .. 6.28318530717958, 0. .. 6.28318530717958, -1. .. 1.], Vector(4, {(1) = ` 1..49 x 1..49 `*Array, (2) = `Data Type: `*float[8], (3) = `Storage: `*rectangular, (4) = `Order: `*C_order})]

(1)

M := plottools:-getdata(MyPlot)[-1];

M := Vector(4, {(1) = ` 1..49 x 1..49 `*Array, (2) = `Data Type: `*float[8], (3) = `Storage: `*rectangular, (4) = `Order: `*C_order})

(2)

# search Export in the help pages for other formats

MyFile := FileTools:-JoinPath(["Desktop/Maple/M.csv"], base=homedir):
Export(MyFile, M):

56423

(3)

 


 

Download Export.mw

I believe Christopher has put the finger on the problem when saying "...  is it the y portion? "

The structure of the pde is rather simple with the form F . grad(Phi) = 0 (I use Phi instead of w).
For these equations the key is in finding the characteristic curves. The problem seems to be: Maple cannot find a closed form expression for the second component of these curves.

(Nota: Maple find the characteristic curves in closed form if alpha=0 ... and successes too in solving the pde)


 

restart:

pde :=  diff(Phi(x,y,z),x)
        +
        (y^2- a*exp(alpha*x)*(x*y-1))*diff(Phi(x,y,z),y)
        +
        (c*exp(beta*x)*z^2+b*exp(-beta*x))*diff(Phi(x,y,z),z)
        =
        0;

diff(Phi(x, y, z), x)+(y^2-a*exp(alpha*x)*(x*y-1))*(diff(Phi(x, y, z), y))+(c*exp(beta*x)*z^2+b*exp(-beta*x))*(diff(Phi(x, y, z), z)) = 0

(1)

# Write the pde as F . grad(Phi) = 0 where F is the vector (A, B, C)
# Next parameterize F as a function of s : F(s) = (A(s), B(s), C(s))

A := s -> 1;
B := s -> (y(s)^2- a*exp(alpha*x(s))*(x(s)*y(s)-1));
C := s -> (c*exp(beta*x(s))*z(s)^2+b*exp(-beta*x(s)));

proc (s) options operator, arrow; 1 end proc

 

proc (s) options operator, arrow; y(s)^2-a*exp(alpha*x(s))*(x(s)*y(s)-1) end proc

 

proc (s) options operator, arrow; c*exp(beta*x(s))*z(s)^2+b*exp(-beta*x(s)) end proc

(2)

# Define the characteristic curve associated to the pde as the curve defined by (U(s), V(s), W(s))
# where :
#   diff(U(s), s) = A(s)
#   diff(V(s), s) = B(s)
#   diff(W(s), s) = C(s)
#
# Then Phi(s) is constant along each characteristic curve
#
# Characteristic curve, component 1

eq1 := diff(U(s), s) = A(s);
dsolve(eq1, U(s)):
U := unapply(rhs(%), s);

diff(U(s), s) = 1

 

proc (s) options operator, arrow; s+_C1 end proc

(3)

# Characteristic curve, component 3


eq3 := subs({x(s)=U(s), z(s)=W(s)}, diff(W(s), s) = C(s));

dsolve(eq3, W(s))

diff(W(s), s) = c*exp(beta*(s+_C1))*W(s)^2+b*exp(-beta*(s+_C1))

 

W(s) = -(1/2)*(exp(beta*_C1)*exp(beta*s)*beta+tan((1/2)*((exp(beta*_C1))^2*(exp(beta*s))^2*(4*b*c-beta^2))^(1/2)*(_C2-s)/(exp(beta*_C1)*exp(beta*s)))*((exp(beta*_C1))^2*(exp(beta*s))^2*(4*b*c-beta^2))^(1/2))/((exp(beta*_C1))^2*(exp(beta*s))^2*c)

(4)

# Characteristic curve, component 2


eq2 := subs({x(s)=U(s), y(s)=V(s)}, diff(V(s), s) = expand(B(s)));

infolevel[dsolve] := 4:
dsolve(eq2, V(s))

diff(V(s), s) = V(s)^2-a*exp(alpha*(s+_C1))*(s+_C1)*V(s)+a*exp(alpha*(s+_C1))

 

Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
trying Riccati
trying inverse_Riccati
trying 1st order ODE linearizable_by_differentiation
--- Trying Lie symmetry methods, 1st order ---
 -> Computing symmetries using: way = 4
 -> Computing symmetries using: way = 2
 -> Computing symmetries using: way = 6
trying symmetry patterns for 1st order ODEs
-> trying a symmetry pattern of the form [F(x)*G(y), 0]
-> trying a symmetry pattern of the form [0, F(x)*G(y)]
-> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)]

 -> Computing symmetries using: way = HINT
   -> Calling odsolve with the ODE diff(y(x) x) = 2*y(x)/x y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear

      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x) = y(x)/x y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)+y(x)*alpha y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful

   -> Calling odsolve with the ODE diff(y(x) x)+exp(alpha*(x+_C1))*a*K[1]*(x+_C1) y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature

      <- quadrature successful
   -> Calling odsolve with the ODE diff(y(x) x)+(y(x)*_C1*alpha+y(x)*alpha*x+y(x)+2*K[1])/(x+_C1) y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear

      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)+K[1] y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)+y(x)*alpha-K[1] y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)+y(x)*(_C1*alpha+alpha*x+1)/(x+_C1) y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x) = 0 y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x) = -y(x)*alpha y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)-2*y(x)/x y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)-(x*alpha*K[1]+y(x))/x y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)-(x*_C1*alpha*K[1]+y(x)*_C1+x*K[1]-K[1]*alpha)/(_C1*x-1) y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful

   -> Calling odsolve with the ODE diff(y(x) x)-y(x)/x y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)-y(x)*_C1/(_C1*x-1) y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
   -> Calling odsolve with the ODE diff(y(x) x)+(exp(alpha*_C1)*a*K[1]-2*y(x))/x y(x)
      *** Sublevel 2 ***
      Methods for first order ODEs:
      --- Trying classification methods ---
      trying a quadrature
      trying 1st order linear
      <- 1st order linear successful
-> trying a symmetry pattern of the form [F(x),G(x)]
-> trying a symmetry pattern of the form [F(y),G(y)]
-> trying a symmetry pattern of the form [F(x)+G(y), 0]

-> trying a symmetry pattern of the form [0, F(x)+G(y)]
-> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)]

-> trying a symmetry pattern of conformal type

 

# Note there is no explicit solution found: I suppose this is the reason why Maple cannot solve the pde

eq2 := subs({x(s)=U(s), y(s)=V(s), alpha=0}, diff(V(s), s) = expand(B(s)));

infolevel[dsolve] := 4:
dsolve(eq2, V(s))

diff(V(s), s) = V(s)^2-a*exp(0)*(s+_C1)*V(s)+a*exp(0)

 

Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
Chini's absolute invariant is: a*(s+_C1)^2
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries

Found potential symmetries: [0 1] [exp(-V)/a exp(-V)*(s+_C1)]
Proceeding with integration step.

 

V(s) = -(-_C1*exp(-(1/2)*a*s^2)/exp(s*_C1*a)-s*exp(-(1/2)*a*s^2)/exp(s*_C1*a)+1/(-(1/2)*exp(-(1/2)*_C1^2*a)*erf((1/2)*(-2*a)^(1/2)*(s+_C1))*(-2*Pi*a)^(1/2)+_C2))*exp(s*_C1*a)*a/exp(-(1/2)*a*s^2)

(5)

pdsolve(subs(alpha=0, pde), Phi(x,y,z))

Methods for first order ODEs:

--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
Chini's absolute invariant is: a*x^2
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
Found potential symmetries: [0 1] [exp(-y) exp(-y)*a*x]
Proceeding with integration step.

Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
Chini's absolute invariant is: beta^2/(c*b)
<- Chini successful

Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
<- 1st order linear successful

 

Phi(x, y, z) = _F1(-(-erf((1/2)*(-2*a)^(1/2)*x)*a*x+y*erf((1/2)*(-2*a)^(1/2)*x)+(-2*a/Pi)^(1/2)*exp((1/2)*a*x^2))/((-2*a/Pi)^(1/2)*(-a*x+y)), beta*(2*beta*arctan(beta*(2*c*exp(beta*x)*z+beta)/(4*b*beta^2*c-beta^4)^(1/2))-(beta^2*(4*b*c-beta^2))^(1/2)*x)/(beta^2*(4*b*c-beta^2))^(1/2))

(6)

 


 

Download CharacteristicCurves.mw


 

Maybe this ?

Watchout: you did not say what to do if A inter B is the empty set
 


 

restart:

p := proc(A::set, B::set, C::set)
   local AB, ABC:
   AB := A intersect B:
   if AB <> {} then
      ABC := AB intersect C:
      if ABC <> {} then
         return ABC
      else
         return `union`(seq(AB *~ C[n], n=1..nops(c)))
      end if;
   else
      error  "It's not specified what to do when A inter B = {} "
   end if:
end proc:

a := {x6,x4,x2,x7,x8,x9,x10}:  
b := {x2,x3,x5,x8}:
c := {x4,x9,x11,x12,x13}:

p(a, b, c)

{x2*x11, x2*x12, x2*x13, x2*x4, x2*x9, x8*x11, x8*x12, x8*x13, x8*x4, x8*x9}

(1)

a := {x6,x4}:  
b := {x5,x8}:
c := {x12,x13}:

p(a, b, c)

Error, (in p) It's not specified what to do when A inter B = {}

 

a := {x6,x4}:  
b := {x4, x5,x8}:
c := {x4, x12,x13}:

p(a, b, c)

{x4}

(2)

 


 

Download sets.mw

Sorry, I do not know about this game and I can give an answer because this later depends on the strategy used to form the new teams.
I consider here 2 strategies (one could probably imagine a few others) . One gives the result proposed by CarlLove after Kitonum's correction, the second a rather different result.

You will find the correct values and a numerical simulation for each strategy


 

restart:

with(Statistics):
with(combinat):

# How is the game managed (I do not know about it)?
#
# Strategy 1:
# One decides to form the red team first by randomly picking 5 out of 15 members
# Next one decides to form the green team by randomly picking 5 out of the 10 remaining members
# The 5 remaining ones form the last (orange) team
#
#
# Strategy 2
# One randomly form 3 new teams of 5 members and randomly decide to color thes teams as red, green, orange
 

 

STRATEGY 1

 

ProbabilityToPutFiveBluesInTheRedTeam := mul((6-n)/(15-n), n=0..4);
ProbabilityToPutTheLastBlueInTheOrangeTeam := 1/2;

ProbabilityStrategy1 := ProbabilityToPutFiveBluesInTheRedTeam * ProbabilityToPutTheLastBlueInTheOrangeTeam ;

2/1001

 

1/2

 

1/1001

(1)

# simulation strategy 1

OriginalBlueTeam   := [B$6]:
OriginalYellowTeam := [Y$9]:
candidates         := [OriginalBlueTeam[], OriginalYellowTeam[]]:


N := 10^6:


NewTeams := Matrix(N, 3):
for n from 1 to N do
   x := randperm(candidates):
   NewTeams[n, 1] := numelems(select(has, x[ 1.. 5], B)):
   NewTeams[n, 2] := numelems(select(has, x[ 6..10], B)):
   NewTeams[n, 3] := numelems(select(has, x[11..15], B)):
end do:


DesiredNumberOfBluesInNewTeams := [5, 0, 1]; # teams ordered red, green, orange

FrequencyStrategy1 := add( map(n -> if [entries(NewTeams[n,..], nolist)] = [5, 0, 1] then 1 end if, [$1..N]) ) / N

[5, 0, 1]

 

201/200000

(2)

evalf([ProbabilityStrategy1, FrequencyStrategy1]);

[0.9990009990e-3, 0.1005000000e-2]

(3)

 

STRATEGY 2

 

ProbabilityToPutFiveBluesInTheSameTeam   := mul((6-n)/(15-n), n=0..4) * 3;
ProbabilityToPutTheLastBlueInAnotherTeam := 1;

ProbabilityStrategy2 := ProbabilityToPutFiveBluesInTheSameTeam * ProbabilityToPutTheLastBlueInAnotherTeam ;

6/1001

 

1

 

6/1001

(4)

# simulation strategy 2

OriginalBlueTeam   := [B$6]:
OriginalYellowTeam := [Y$9]:
candidates         := [OriginalBlueTeam[], OriginalYellowTeam[]]:


N := 10^6:


NewTeams := Matrix(N, 3):
for n from 1 to N do
   x := randperm(candidates):
   NewTeams[n, 1] := numelems(select(has, x[ 1.. 5], B)):
   NewTeams[n, 2] := numelems(select(has, x[ 6..10], B)):
   NewTeams[n, 3] := numelems(select(has, x[11..15], B)):
end do:


DesiredPartitions := { permute([5, 0, 1])[] };

FrequencyStrategy2 := add( map(n -> if member([entries(NewTeams[n,..], nolist)], DesiredPartitions) then 1 end if, [$1..N]) ) / N;

{[0, 1, 5], [0, 5, 1], [1, 0, 5], [1, 5, 0], [5, 0, 1], [5, 1, 0]}

 

3003/500000

(5)

evalf([ProbabilityStrategy2, FrequencyStrategy2]);

[0.5994005994e-2, 0.6006000000e-2]

(6)

 


 

Download Teams.mw

plots:-implicitplot(
     X^3-24.478115*X^2/Y-(0.2038793409e-2+19.08455282*Y/(Y-97.539))*X-.2550630228/Y,
     X=0..50,
     Y=0..1
);

You can add some options (see implicitplot help page), for onstance  gridrefine=2 to get a smoother plot.
 

The Grid Package and the Grid Computing Toolbox are two differents things :

  • the former is included in your Maple license
  • the latter has to be purchased separatetly

In the Maple 2015.2 help pages it's written 
Before installing the Grid Computing Toolbox, you must install and activate Maple 18. For details and installation instructions, see the Install.html file on the product disc.  
which seems to suggest that the Grid Computing Toolbox was introduced with Maple 18 (???)


What I understand is that you solve numerically the Heat equation with a time explicit finite difference scheme:

  • At each time t[j+1] the solution is explicitely constructed from the knowledge of the solution y(x,t[j]) at previous time t[j] and from the boundary conditions at x=0 and x=2
  •  
  • You use a forward first order approximation of the evolution diff(y(x,t), t) term:
  • diff(y(x,t), t) at t=t[j+1] ~ (y(x, t[j+1])-y(x, t[j]) / (t[j+1]-t[j])
  •  
  • You use a centered finite difference approximation of diff(y(x, t), x, x) at point x[k]:
  • diff(y(x,t), x, x) at x=x[k] ~ (-y(x[k-1], t) +2*y(x[k], t) - y(x[k+1], t)) /dx^2  where dx = x[k+1]-x[k] = x[k]-x[k-1]

Then your are right  saying that the there is somewhere a tridiagonal matrix (maybe you just did not explain this point clearly enough and that this is which led to all these exchanges).

In fact this tridiagonal matrix represents the discretization of  diff(y(x, t), x, x)  over the whole spatial domain (it would be pentadiagonal if the spatial domain was 2D)

 

I prepared this worksheet and tried to wrote it the more pedagogical way I could.

I other people here want to contribute to your problem, I hope this worksheet will help them to better understand your question


 

restart;

with(plots):

interface(rtablesize=10):

f:=unapply(-x^2+1,x):

mu[1]:=unapply(1/(t^2+1),t):

mu[2]:=unapply(1/(t-5),t):

g:=unapply(t^3-7*x,[t,x]):

l   :=  2: T := 3:

n   := 10: m := n:

h   := l/n;

tau := T/m;

CFL := tau/h^2;
if CFL >= 1/2 then WARNING("The CFL condition is not verified, the scheme is unstable") end if;

1/5

 

3/10

 

15/2

 

Warning, The CFL condition is not verified, the scheme is unstable

 

# Finite differences approcimations
#
# y  stands for the solution at some time j
# yF stands for the solution at next time j+1 (F = Future)

FD_T  := k -> (yF[k]-y[k]) / tau:
FD_X  := k -> (y[k+1]-y[k]) / h:
CDD_X := k -> simplify((FD_X(k,j)-FD_X(k-1,j)) / h):

# examples

FD_T(k);
FD_X(k);
CDD_X(k);

(10/3)*yF[k]-(10/3)*y[k]

 

5*y[k+1]-5*y[k]

 

25*y[k+1]-50*y[k]+25*y[k-1]

(1)

# Evolution equation from time j to time j+1, at centered location k

eq := (k,j) -> FD_T(k) = CDD_X(k) + g(t[j], x[k]):
Updator := (k, j) -> solve(eq(k,j), yF[k]):
# examples

eq(k, j);
Updator(k, j);

(10/3)*yF[k]-(10/3)*y[k] = t[j]^3-7*x[k]-50*y[k]+25*y[k-1]+25*y[k+1]

 

(3/10)*t[j]^3-(21/10)*x[k]-14*y[k]+(15/2)*y[k-1]+(15/2)*y[k+1]

(2)

# Stationary matrix (boundary conditions not already accounted for)

# interface(rtablesize=n+1):

StationaryMatrix := Matrix(n+1, n+1, (k, kk) -> coeff(Updator(k, j), y[2*k-kk])):

# Source Vector at any time j > 0

SourceVector := Vector[column](n+1, k -> g(t[j], x[k-1])):
 

# Variables at time j  (boundary conditions not already accounted for)

Vars := Vector[column](n+1, k -> y[k-1]):

# Boundary conditions at k=0

StationaryMatrix[1, 1]      := 1:
StationaryMatrix[1, 2..n+1] := 0:
SourceVector[1]             := mu[1](t[j]):

# Boundary conditions at k=n

StationaryMatrix[n+1, 1..n] := 0:
StationaryMatrix[n+1, n+1]  := 1:
SourceVector[n+1]           := mu[2](t[j]):
  

# uncomment to vizualize

# StationaryMatrix , Vars,  SourceVector

# spatial mesh and time marching

x := h *~ [$0..n];
t := tau *~ [$0..m];

[0, 1/5, 2/5, 3/5, 4/5, 1, 6/5, 7/5, 8/5, 9/5, 2]

 

[0, 3/10, 3/5, 9/10, 6/5, 3/2, 9/5, 21/10, 12/5, 27/10, 3]

(3)

# Initial value of y (j=0)

Y0 := [seq(f(x[k]), k=1..n+1)]
 

[1, 24/25, 21/25, 16/25, 9/25, 0, -11/25, -24/25, -39/25, -56/25, -3]

(4)

# Value of y at future time j+1 given the solution at time j

Y := Matrix(n+1, evalf(Y0)):

for j from 2 to m do
Y[.., j-1];
  Y := < Y | StationaryMatrix . Y[.., j-1] + SourceVector >
end do:

# uncomment to vizualize

# evalf[4](Y);

plots:-matrixplot(Y, heights=histogram, labels=['x', 't', 'Y'])

 

Let's do the same thing while respecting the CFL condition
Divide the time step by 20:

restart;

with(plots):

interface(rtablesize=10):

f:=unapply(-x^2+1,x):

mu[1]:=unapply(1/(t^2+1),t):

mu[2]:=unapply(1/(t-5),t):

g:=unapply(t^3-7*x, [t,x]):

l   :=  2: T := 3:

n   := 10: m := n*16:

h   := l/n:

tau := T/m:

CFL := tau/h^2;
if CFL >= 1/2 then WARNING("The CFL condition is not verified, the scheme is unstable") end if;

15/32

(5)

FD_T  := k -> (yF[k]-y[k]) / tau:
FD_X  := k -> (y[k+1]-y[k]) / h:
CDD_X := k -> simplify((FD_X(k,j)-FD_X(k-1,j)) / h):

eq := (k,j) -> FD_T(k) = CDD_X(k) + g(t[j], x[k]):
Updator := (k, j) -> solve(eq(k,j), yF[k]):

StationaryMatrix := Matrix(n+1, n+1, (k, kk) -> coeff(Updator(k, j), y[2*k-kk])):
SourceVector     := Vector[column](n+1, k -> g(t[j], x[k-1])):
Vars             := Vector[column](n+1, k -> y[k-1]):

StationaryMatrix[1, 1]      := 1:
StationaryMatrix[1, 2..n+1] := 0:
SourceVector[1]             := mu[1](t[j]):

StationaryMatrix[n+1, 1..n] := 0:
StationaryMatrix[n+1, n+1]  := 1:
SourceVector[n+1]           := mu[2](t[j]):

x := h *~ [$0..n]:
t := tau *~ [$0..m]:

Y0 := [seq(f(x[k]), k=1..n+1)]:
Y := Matrix(n+1, evalf(Y0)):

for j from 2 to m do
Y[.., j-1];
  Y := < Y | StationaryMatrix . Y[.., j-1] + SourceVector >
end do:


plots:-matrixplot(Y, labels=['x', 't', 'Y'])

 

``

(6)


 

Download HeatEquation.mw

To complete tomleslie's answer...

I do not know if you are aware of the possibility to parameterize ode systems (ic problems only, not bv problems)?
This is a very interesting feature when the system to solve depends on a large number of parameters, or when you want to study the sensitivity of the solution to those parameters.

In case you would be interested, here is a modification of tomleslie's previous code ("new" commands are blue written).

 

  restart:

  interface(rtablesize=10):

#
# Define gamma as local (don't like doing this!)
#
  local gamma:local pi:

#
# Replaced 'indexed' parameters with 'inert subscript'
# parameters - otherwise one gets a problem defining
# both the unindexed 'phi' and the indexed phi[c]
#

if false then
  M__h := 100: beta__o := 0.034: beta__j := .025: mu__1 := 0.0004:
  epsilon := .7902: alpha := 0.11: psi := 0.000136: phi := 0.05:
  omega := .7: eta := .134: delta := .245: f := 0.21:
  M__v := 1000: beta__k := 0.09:   mu__v := .0005: M__c := .636:
  beta__g := 0.15: mu__c := 0.0019: pi :=0.01231: theta := 0.12: mu__e := 0.005:
end if:

#
# D() is Maple's differential operator replated D(T)
# with DD(T) in the following to avoid confusion
#
  ODE1 := diff(B(T), T) = M__h-beta__o*B(T)-beta__j*B(T)-mu__1*B(T)+epsilon*G(T)+alpha*F(T):
  ODE2 := diff(C(T), T) = beta__o*B(T)*J(T)-beta__j*C(T)-(psi+mu__1+phi)*C(T):
  ODE3 := diff(DD(T), T) = beta__j*B(T)*L(T)- beta__o*E(T)-(omega+mu__1+eta)*DD(T):
  ODE4 := diff(E(T), T) = beta__o*E(T)-beta__j*C(T)-(delta+mu__1+eta+phi)*E(T):
  ODE5 := diff(F(T), T) = psi*C(T)-(alpha+mu__1)*F(T)+f*delta*E(T):
  ODE6 := diff(G(T), T) = omega*DD(T)-(epsilon+mu__1)*G(T)+(1-f)*delta*E(T):
  ODE7 := diff(H(T), T) = M__v-beta__k*H(T)-mu__v*H(T):
  ODE8 := diff(J(T), T) = beta__k*H(T)-mu__v*J(T):
  ODE9 := diff(K(T), T) = M__c-beta__g*K(T)-mu__c*K(T):
  ODE10:= diff(L(T), T) = beta__g*K(T)-mu__c*L(T):
  ODE11:= diff(M(T), T) = pi*(DD(T)+ theta*E(T))-mu__e*M(T):
 

 
if false then
  B0 := 100: C0 := 90: D0 := 45: E0 := 38:
  F0 := 10: G0 := 45: H0 := 50: J0 := 70: K0 :=20: L0:= 65: M0 :=22:
end if:

# system + ic


sys := { ODE1, ODE2, ODE3, ODE4, ODE5, ODE6, ODE7, ODE8, ODE9, ODE10, ODE11,
                   B(0) = B0, C(0) = C0, DD(0) = D0, E(0) = E0,
                   F(0) = F0, G(0) = G0, H(0) = H0, J(0) = J0, K(0) = K0, L(0) = L0, M(0) = M0
                 }:

params := convert(indets(sys, name) minus {T}, list);

[B0, C0, D0, E0, F0, G0, H0, J0, K0, L0, M0, M__c, M__h, M__v, alpha, beta__g, beta__j, beta__k, beta__o, delta, epsilon, eta, f, mu__1, mu__c, mu__e, mu__v, omega, phi, pi, psi, theta]

(1)

#
# Solve system
#
  ans := dsolve( { ODE1, ODE2, ODE3, ODE4, ODE5, ODE6, ODE7, ODE8, ODE9, ODE10, ODE11,
                   B(0) = B0, C(0) = C0, DD(0) = D0, E(0) = E0,
                   F(0) = F0, G(0) = G0, H(0) = H0, J(0) = J0, K(0) = K0, L(0) = L0, M(0) = M0
                 },
                 parameters = params,
                 numeric
               );

proc (x_rkf45) local _res, _dat, _vars, _solnproc, _xout, _ndsol, _pars, _n, _i; option `Copyright (c) 2000 by Waterloo Maple Inc. All rights reserved.`; if 1 < nargs then error "invalid input: too many arguments" end if; _EnvDSNumericSaveDigits := Digits; Digits := 15; if _EnvInFsolve = true then _xout := evalf[_EnvDSNumericSaveDigits](x_rkf45) else _xout := evalf(x_rkf45) end if; _dat := Array(1..4, {(1) = proc (_xin) local _xout, _dtbl, _dat, _vmap, _x0, _y0, _val, _dig, _n, _ne, _nd, _nv, _pars, _ini, _par, _i, _j, _k, _src; option `Copyright (c) 2002 by Waterloo Maple Inc. All rights reserved.`; table( [( "complex" ) = false ] ) _xout := _xin; _pars := [B0 = B0, C0 = C0, D0 = D0, E0 = E0, F0 = F0, G0 = G0, H0 = H0, J0 = J0, K0 = K0, L0 = L0, M0 = M0, M__c = M__c, M__h = M__h, M__v = M__v, alpha = alpha, beta__g = beta__g, beta__j = beta__j, beta__k = beta__k, beta__o = beta__o, delta = delta, epsilon = epsilon, eta = eta, f = f, mu__1 = mu__1, mu__c = mu__c, mu__e = mu__e, mu__v = mu__v, omega = omega, phi = phi, pi = pi, psi = psi, theta = theta]; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 24, [( 1 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 2 ) = (datatype = float[8], order = C_order, storage = rectangular), ( 3 ) = ([0, 0, 0, Array(1..0, 1..2, {}, datatype = float[8], order = C_order)]), ( 4 ) = (Array(1..54, {(1) = 11, (2) = 11, (3) = 0, (4) = 0, (5) = 32, (6) = 0, (7) = 0, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 0, (19) = 30000, (20) = 0, (21) = 0, (22) = 1, (23) = 4, (24) = 0, (25) = 1, (26) = 15, (27) = 1, (28) = 0, (29) = 1, (30) = 3, (31) = 3, (32) = 0, (33) = 1, (34) = 0, (35) = 0, (36) = 0, (37) = 0, (38) = 0, (39) = 0, (40) = 0, (41) = 0, (42) = 0, (43) = 1, (44) = 0, (45) = 0, (46) = 0, (47) = 0, (48) = 0, (49) = 0, (50) = 50, (51) = 1, (52) = 0, (53) = 0, (54) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = .0, (7) = .0, (8) = 0.10e-5, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = 1.0, (14) = .0, (15) = .49999999999999, (16) = .0, (17) = 1.0, (18) = 1.0, (19) = .0, (20) = .0, (21) = 1.0, (22) = 1.0, (23) = .0, (24) = .0, (25) = 0.10e-14, (26) = .0, (27) = .0, (28) = .0}, datatype = float[8], order = C_order)), ( 6 ) = (Array(1..43, {(1) = B0, (2) = C0, (3) = D0, (4) = E0, (5) = F0, (6) = G0, (7) = H0, (8) = J0, (9) = K0, (10) = L0, (11) = M0, (12) = Float(undefined), (13) = Float(undefined), (14) = Float(undefined), (15) = Float(undefined), (16) = Float(undefined), (17) = Float(undefined), (18) = Float(undefined), (19) = Float(undefined), (20) = Float(undefined), (21) = Float(undefined), (22) = Float(undefined), (23) = Float(undefined), (24) = Float(undefined), (25) = Float(undefined), (26) = Float(undefined), (27) = Float(undefined), (28) = Float(undefined), (29) = Float(undefined), (30) = Float(undefined), (31) = Float(undefined), (32) = Float(undefined), (33) = Float(undefined), (34) = Float(undefined), (35) = Float(undefined), (36) = Float(undefined), (37) = Float(undefined), (38) = Float(undefined), (39) = Float(undefined), (40) = Float(undefined), (41) = Float(undefined), (42) = Float(undefined), (43) = Float(undefined)})), ( 7 ) = ([Array(1..4, 1..7, {(1, 1) = .0, (1, 2) = .203125, (1, 3) = .3046875, (1, 4) = .75, (1, 5) = .8125, (1, 6) = .40625, (1, 7) = .8125, (2, 1) = 0.6378173828125e-1, (2, 2) = .0, (2, 3) = .279296875, (2, 4) = .27237892150878906, (2, 5) = -0.9686851501464844e-1, (2, 6) = 0.1956939697265625e-1, (2, 7) = .5381584167480469, (3, 1) = 0.31890869140625e-1, (3, 2) = .0, (3, 3) = -.34375, (3, 4) = -.335235595703125, (3, 5) = .2296142578125, (3, 6) = .41748046875, (3, 7) = 11.480712890625, (4, 1) = 0.9710520505905151e-1, (4, 2) = .0, (4, 3) = .40350341796875, (4, 4) = 0.20297467708587646e-1, (4, 5) = -0.6054282188415527e-2, (4, 6) = -0.4770040512084961e-1, (4, 7) = .77858567237854}, datatype = float[8], order = C_order), Array(1..6, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = 1.0, (2, 1) = .25, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = 1.0, (3, 1) = .1875, (3, 2) = .5625, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = 2.0, (4, 1) = .23583984375, (4, 2) = -.87890625, (4, 3) = .890625, (4, 4) = .0, (4, 5) = .0, (4, 6) = .2681884765625, (5, 1) = .1272735595703125, (5, 2) = -.5009765625, (5, 3) = .44921875, (5, 4) = -0.128936767578125e-1, (5, 5) = .0, (5, 6) = 0.626220703125e-1, (6, 1) = -0.927734375e-1, (6, 2) = .626220703125, (6, 3) = -.4326171875, (6, 4) = .1418304443359375, (6, 5) = -0.861053466796875e-1, (6, 6) = .3131103515625}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .386, (3) = .21, (4) = .63, (5) = 1.0, (6) = 1.0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .25, (2) = -.1043, (3) = .1035, (4) = -0.362e-1, (5) = .0, (6) = .0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 1.544, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = .9466785280815533, (3, 2) = .25570116989825814, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = 3.3148251870684886, (4, 2) = 2.896124015972123, (4, 3) = .9986419139977808, (4, 4) = .0, (4, 5) = .0, (5, 1) = 1.2212245092262748, (5, 2) = 6.019134481287752, (5, 3) = 12.537083329320874, (5, 4) = -.687886036105895, (5, 5) = .0, (6, 1) = 1.2212245092262748, (6, 2) = 6.019134481287752, (6, 3) = 12.537083329320874, (6, 4) = -.687886036105895, (6, 5) = 1.0}, datatype = float[8], order = C_order), Array(1..6, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = -5.6688, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (3, 1) = -2.4300933568337584, (3, 2) = -.20635991570891224, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (4, 1) = -.10735290581452621, (4, 2) = -9.594562251021896, (4, 3) = -20.470286148096154, (4, 4) = .0, (4, 5) = .0, (5, 1) = 7.496443313968615, (5, 2) = -10.246804314641219, (5, 3) = -33.99990352819906, (5, 4) = 11.708908932061595, (5, 5) = .0, (6, 1) = 8.083246795922411, (6, 2) = -7.981132988062785, (6, 3) = -31.52159432874373, (6, 4) = 16.319305431231363, (6, 5) = -6.0588182388340535}, datatype = float[8], order = C_order), Array(1..3, 1..5, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (2, 1) = 10.126235083446911, (2, 2) = -7.487995877607633, (2, 3) = -34.800918615557414, (2, 4) = -7.9927717075687275, (2, 5) = 1.0251377232956207, (3, 1) = -.6762803392806898, (3, 2) = 6.087714651678606, (3, 3) = 16.43084320892463, (3, 4) = 24.767225114183653, (3, 5) = -6.5943891257167815}, datatype = float[8], order = C_order)]), ( 9 ) = ([Array(1..11, {(1) = .1, (2) = .1, (3) = .1, (4) = .1, (5) = .1, (6) = .1, (7) = .1, (8) = .1, (9) = .1, (10) = .1, (11) = .1}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), Array(1..11, 1..11, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (1, 8) = .0, (1, 9) = .0, (1, 10) = .0, (1, 11) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (2, 8) = .0, (2, 9) = .0, (2, 10) = .0, (2, 11) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (3, 8) = .0, (3, 9) = .0, (3, 10) = .0, (3, 11) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (4, 8) = .0, (4, 9) = .0, (4, 10) = .0, (4, 11) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (5, 8) = .0, (5, 9) = .0, (5, 10) = .0, (5, 11) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (6, 8) = .0, (6, 9) = .0, (6, 10) = .0, (6, 11) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (7, 7) = .0, (7, 8) = .0, (7, 9) = .0, (7, 10) = .0, (7, 11) = .0, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (8, 7) = .0, (8, 8) = .0, (8, 9) = .0, (8, 10) = .0, (8, 11) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (9, 7) = .0, (9, 8) = .0, (9, 9) = .0, (9, 10) = .0, (9, 11) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (10, 7) = .0, (10, 8) = .0, (10, 9) = .0, (10, 10) = .0, (10, 11) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0, (11, 7) = .0, (11, 8) = .0, (11, 9) = .0, (11, 10) = .0, (11, 11) = .0}, datatype = float[8], order = C_order), Array(1..11, 1..11, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (1, 8) = .0, (1, 9) = .0, (1, 10) = .0, (1, 11) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (2, 8) = .0, (2, 9) = .0, (2, 10) = .0, (2, 11) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (3, 8) = .0, (3, 9) = .0, (3, 10) = .0, (3, 11) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (4, 8) = .0, (4, 9) = .0, (4, 10) = .0, (4, 11) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (5, 8) = .0, (5, 9) = .0, (5, 10) = .0, (5, 11) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (6, 8) = .0, (6, 9) = .0, (6, 10) = .0, (6, 11) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (7, 7) = .0, (7, 8) = .0, (7, 9) = .0, (7, 10) = .0, (7, 11) = .0, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (8, 7) = .0, (8, 8) = .0, (8, 9) = .0, (8, 10) = .0, (8, 11) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (9, 7) = .0, (9, 8) = .0, (9, 9) = .0, (9, 10) = .0, (9, 11) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (10, 7) = .0, (10, 8) = .0, (10, 9) = .0, (10, 10) = .0, (10, 11) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0, (11, 7) = .0, (11, 8) = .0, (11, 9) = .0, (11, 10) = .0, (11, 11) = .0}, datatype = float[8], order = C_order), Array(1..11, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (7, 1) = .0, (7, 2) = .0, (7, 3) = .0, (7, 4) = .0, (7, 5) = .0, (7, 6) = .0, (8, 1) = .0, (8, 2) = .0, (8, 3) = .0, (8, 4) = .0, (8, 5) = .0, (8, 6) = .0, (9, 1) = .0, (9, 2) = .0, (9, 3) = .0, (9, 4) = .0, (9, 5) = .0, (9, 6) = .0, (10, 1) = .0, (10, 2) = .0, (10, 3) = .0, (10, 4) = .0, (10, 5) = .0, (10, 6) = .0, (11, 1) = .0, (11, 2) = .0, (11, 3) = .0, (11, 4) = .0, (11, 5) = .0, (11, 6) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = 0, (2) = 0, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 0, (8) = 0, (9) = 0, (10) = 0, (11) = 0}, datatype = integer[8]), Array(1..43, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0}, datatype = float[8], order = C_order), Array(1..43, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0}, datatype = float[8], order = C_order), Array(1..43, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0}, datatype = float[8], order = C_order), Array(1..43, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..43, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0}, datatype = float[8], order = C_order), Array(1..43, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0, (12) = .0, (13) = .0, (14) = .0, (15) = .0, (16) = .0, (17) = .0, (18) = .0, (19) = .0, (20) = .0, (21) = .0, (22) = .0, (23) = .0, (24) = .0, (25) = .0, (26) = .0, (27) = .0, (28) = .0, (29) = .0, (30) = .0, (31) = .0, (32) = .0, (33) = .0, (34) = .0, (35) = .0, (36) = .0, (37) = .0, (38) = .0, (39) = .0, (40) = .0, (41) = .0, (42) = .0, (43) = .0}, datatype = float[8], order = C_order), Array(1..11, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0, (7) = .0, (8) = .0, (9) = .0, (10) = .0, (11) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..11, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0, (1, 7) = .0, (1, 8) = .0, (1, 9) = .0, (1, 10) = .0, (1, 11) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (2, 4) = .0, (2, 5) = .0, (2, 6) = .0, (2, 7) = .0, (2, 8) = .0, (2, 9) = .0, (2, 10) = .0, (2, 11) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (3, 4) = .0, (3, 5) = .0, (3, 6) = .0, (3, 7) = .0, (3, 8) = .0, (3, 9) = .0, (3, 10) = .0, (3, 11) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (4, 4) = .0, (4, 5) = .0, (4, 6) = .0, (4, 7) = .0, (4, 8) = .0, (4, 9) = .0, (4, 10) = .0, (4, 11) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (5, 4) = .0, (5, 5) = .0, (5, 6) = .0, (5, 7) = .0, (5, 8) = .0, (5, 9) = .0, (5, 10) = .0, (5, 11) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0, (6, 4) = .0, (6, 5) = .0, (6, 6) = .0, (6, 7) = .0, (6, 8) = .0, (6, 9) = .0, (6, 10) = .0, (6, 11) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = B(T), Y[2] = C(T), Y[3] = DD(T), Y[4] = E(T), Y[5] = F(T), Y[6] = G(T), Y[7] = H(T), Y[8] = J(T), Y[9] = K(T), Y[10] = L(T), Y[11] = M(T)]`; YP[1] := -Y[1]*Y[28]-Y[1]*Y[30]-Y[1]*Y[35]+Y[5]*Y[26]+Y[6]*Y[32]+Y[24]; YP[2] := Y[30]*Y[1]*Y[8]-Y[28]*Y[2]-(Y[42]+Y[35]+Y[40])*Y[2]; YP[3] := Y[28]*Y[1]*Y[10]-Y[30]*Y[4]-(Y[39]+Y[35]+Y[33])*Y[3]; YP[4] := Y[30]*Y[4]-Y[28]*Y[2]-(Y[31]+Y[35]+Y[33]+Y[40])*Y[4]; YP[5] := Y[42]*Y[2]-(Y[26]+Y[35])*Y[5]+Y[34]*Y[31]*Y[4]; YP[6] := Y[39]*Y[3]-(Y[32]+Y[35])*Y[6]+(1-Y[34])*Y[31]*Y[4]; YP[7] := -Y[7]*Y[29]-Y[7]*Y[38]+Y[25]; YP[8] := Y[7]*Y[29]-Y[8]*Y[38]; YP[9] := -Y[9]*Y[27]-Y[9]*Y[36]+Y[23]; YP[10] := Y[9]*Y[27]-Y[10]*Y[36]; YP[11] := Y[41]*(Y[4]*Y[43]+Y[3])-Y[37]*Y[11]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = B(T), Y[2] = C(T), Y[3] = DD(T), Y[4] = E(T), Y[5] = F(T), Y[6] = G(T), Y[7] = H(T), Y[8] = J(T), Y[9] = K(T), Y[10] = L(T), Y[11] = M(T)]`; YP[1] := -Y[1]*Y[28]-Y[1]*Y[30]-Y[1]*Y[35]+Y[5]*Y[26]+Y[6]*Y[32]+Y[24]; YP[2] := Y[30]*Y[1]*Y[8]-Y[28]*Y[2]-(Y[42]+Y[35]+Y[40])*Y[2]; YP[3] := Y[28]*Y[1]*Y[10]-Y[30]*Y[4]-(Y[39]+Y[35]+Y[33])*Y[3]; YP[4] := Y[30]*Y[4]-Y[28]*Y[2]-(Y[31]+Y[35]+Y[33]+Y[40])*Y[4]; YP[5] := Y[42]*Y[2]-(Y[26]+Y[35])*Y[5]+Y[34]*Y[31]*Y[4]; YP[6] := Y[39]*Y[3]-(Y[32]+Y[35])*Y[6]+(1-Y[34])*Y[31]*Y[4]; YP[7] := -Y[7]*Y[29]-Y[7]*Y[38]+Y[25]; YP[8] := Y[7]*Y[29]-Y[8]*Y[38]; YP[9] := -Y[9]*Y[27]-Y[9]*Y[36]+Y[23]; YP[10] := Y[9]*Y[27]-Y[10]*Y[36]; YP[11] := Y[41]*(Y[4]*Y[43]+Y[3])-Y[37]*Y[11]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 24 ) = (0)  ] ))  ] ); _y0 := Array(0..43, {(1) = 0., (2) = B0, (3) = C0, (4) = D0, (5) = E0, (6) = F0, (7) = G0, (8) = H0, (9) = J0, (10) = K0, (11) = L0, (12) = M0, (13) = undefined, (14) = undefined, (15) = undefined, (16) = undefined, (17) = undefined, (18) = undefined, (19) = undefined, (20) = undefined, (21) = undefined, (22) = undefined, (23) = undefined, (24) = undefined, (25) = undefined, (26) = undefined, (27) = undefined, (28) = undefined, (29) = undefined, (30) = undefined, (31) = undefined, (32) = undefined, (33) = undefined, (34) = undefined, (35) = undefined, (36) = undefined, (37) = undefined, (38) = undefined, (39) = undefined, (40) = undefined, (41) = undefined, (42) = undefined, (43) = undefined}); _vmap := array( 1 .. 11, [( 1 ) = (1), ( 2 ) = (2), ( 3 ) = (3), ( 4 ) = (4), ( 5 ) = (5), ( 6 ) = (6), ( 7 ) = (7), ( 9 ) = (9), ( 8 ) = (8), ( 11 ) = (11), ( 10 ) = (10)  ] ); _x0 := _dtbl[1][5][5]; _n := _dtbl[1][4][1]; _ne := _dtbl[1][4][3]; _nd := _dtbl[1][4][4]; _nv := _dtbl[1][4][16]; if not type(_xout, 'numeric') then if member(_xout, ["start", "left", "right"]) then if _Env_smart_dsolve_numeric = true or _dtbl[1][4][10] = 1 then if _xout = "left" then if type(_dtbl[2], 'table') then return _dtbl[2][5][1] end if elif _xout = "right" then if type(_dtbl[3], 'table') then return _dtbl[3][5][1] end if end if end if; return _dtbl[1][5][5] elif _xout = "method" then return _dtbl[1][15] elif _xout = "storage" then return evalb(_dtbl[1][4][10] = 1) elif _xout = "leftdata" then if not type(_dtbl[2], 'array') then return NULL else return eval(_dtbl[2]) end if elif _xout = "rightdata" then if not type(_dtbl[3], 'array') then return NULL else return eval(_dtbl[3]) end if elif _xout = "enginedata" then return eval(_dtbl[1]) elif _xout = "enginereset" then _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); return NULL elif _xout = "initial" then return procname(_y0[0]) elif _xout = "laxtol" then return _dtbl[`if`(member(_dtbl[4], {2, 3}), _dtbl[4], 1)][5][18] elif _xout = "numfun" then return `if`(member(_dtbl[4], {2, 3}), _dtbl[_dtbl[4]][4][18], 0) elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return procname(_y0[0]), [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] elif _xout = "last" then if _dtbl[4] <> 2 and _dtbl[4] <> 3 or _x0-_dtbl[_dtbl[4]][5][1] = 0. then error "no information is available on last computed point" else _xout := _dtbl[_dtbl[4]][5][1] end if elif _xout = "function" then if _dtbl[1][4][33]-2. = 0 then return eval(_dtbl[1][10], 1) else return eval(_dtbl[1][10][1], 1) end if elif _xout = "map" then return copy(_vmap) elif type(_xin, `=`) and type(rhs(_xin), 'list') and member(lhs(_xin), {"initial", "parameters", "initial_and_parameters"}) then _ini, _par := [], []; if lhs(_xin) = "initial" then _ini := rhs(_xin) elif lhs(_xin) = "parameters" then _par := rhs(_xin) elif select(type, rhs(_xin), `=`) <> [] then _par, _ini := selectremove(type, rhs(_xin), `=`) elif nops(rhs(_xin)) < nops(_pars)+1 then error "insufficient data for specification of initial and parameters" else _par := rhs(_xin)[-nops(_pars) .. -1]; _ini := rhs(_xin)[1 .. -nops(_pars)-1] end if; _xout := lhs(_xout); if _par <> [] then `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) end if; `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars); if _Env_smart_dsolve_numeric = true and type(_y0[0], 'numeric') and _dtbl[1][4][10] <> 1 then procname("right") := _y0[0]; procname("left") := _y0[0] end if; if _xout = "initial" then return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)] elif _xout = "parameters" then return [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] else return [_y0[0], seq(_y0[_vmap[_i]], _i = 1 .. _n-_ne)], [seq(_y0[_n+_i], _i = 1 .. nops(_pars))] end if elif _xin = "eventstop" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then return 0 end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 <= _dtbl[5-_i][4][9] then _i := 5-_i; _dtbl[4] := _i; _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) elif 100 <= _dtbl[_i][4][9] then _j := round(_dtbl[_i][4][17]); return round(_dtbl[_i][3][1][_j, 1]) else return 0 end if elif _xin = "eventstatus" then if _nv = 0 then error "this solution has no events" end if; _i := [selectremove(proc (a) options operator, arrow; _dtbl[1][3][1][a, 7] = 1 end proc, {seq(_j, _j = 1 .. round(_dtbl[1][3][1][_nv+1, 1]))})]; return ':-enabled' = _i[1], ':-disabled' = _i[2] elif _xin = "eventclear" then if _nv = 0 then error "this solution has no events" end if; _i := _dtbl[4]; if _i <> 2 and _i <> 3 then error "no events to clear" end if; if _dtbl[_i][4][10] = 1 and assigned(_dtbl[5-_i]) and _dtbl[_i][4][9] < 100 and 100 < _dtbl[5-_i][4][9] then _dtbl[4] := 5-_i; _i := 5-_i end if; if _dtbl[_i][4][9] < 100 then error "no events to clear" elif _nv < _dtbl[_i][4][9]-100 then error "event error condition cannot be cleared" else _j := _dtbl[_i][4][9]-100; if irem(round(_dtbl[_i][3][1][_j, 4]), 2) = 1 then error "retriggerable events cannot be cleared" end if; _j := round(_dtbl[_i][3][1][_j, 1]); for _k to _nv do if _dtbl[_i][3][1][_k, 1] = _j then if _dtbl[_i][3][1][_k, 2] = 3 then error "range events cannot be cleared" end if; _dtbl[_i][3][1][_k, 8] := _dtbl[_i][3][1][_nv+1, 8] end if end do; _dtbl[_i][4][17] := 0; _dtbl[_i][4][9] := 0; if _dtbl[1][4][10] = 1 then if _i = 2 then try procname(procname("left")) catch:  end try else try procname(procname("right")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and member(lhs(_xin), {"eventdisable", "eventenable"}) then if _nv = 0 then error "this solution has no events" end if; if type(rhs(_xin), {('list')('posint'), ('set')('posint')}) then _i := {op(rhs(_xin))} elif type(rhs(_xin), 'posint') then _i := {rhs(_xin)} else error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; if select(proc (a) options operator, arrow; _nv < a end proc, _i) <> {} then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _k := {}; for _j to _nv do if member(round(_dtbl[1][3][1][_j, 1]), _i) then _k := `union`(_k, {_j}) end if end do; _i := _k; if lhs(_xin) = "eventdisable" then _dtbl[4] := 0; _j := [evalb(assigned(_dtbl[2]) and member(_dtbl[2][4][17], _i)), evalb(assigned(_dtbl[3]) and member(_dtbl[3][4][17], _i))]; for _k in _i do _dtbl[1][3][1][_k, 7] := 0; if assigned(_dtbl[2]) then _dtbl[2][3][1][_k, 7] := 0 end if; if assigned(_dtbl[3]) then _dtbl[3][3][1][_k, 7] := 0 end if end do; if _j[1] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[2][3][4][_k, 1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to defined init `, _dtbl[2][3][4][_k, 1]); _dtbl[2][3][1][_k, 8] := _dtbl[2][3][4][_k, 1] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to rate hysteresis init `, _dtbl[2][5][24]); _dtbl[2][3][1][_k, 8] := _dtbl[2][5][24] elif _dtbl[2][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[2][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to initial init `, _x0); _dtbl[2][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #2, event code `, _k, ` to fireinitial init `, _x0-1); _dtbl[2][3][1][_k, 8] := _x0-1 end if end do; _dtbl[2][4][17] := 0; _dtbl[2][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("left")) end if end if; if _j[2] then for _k to _nv+1 do if _k <= _nv and not type(_dtbl[3][3][4][_k, 2], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to defined init `, _dtbl[3][3][4][_k, 2]); _dtbl[3][3][1][_k, 8] := _dtbl[3][3][4][_k, 2] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to rate hysteresis init `, _dtbl[3][5][24]); _dtbl[3][3][1][_k, 8] := _dtbl[3][5][24] elif _dtbl[3][3][1][_k, 2] = 0 and irem(iquo(round(_dtbl[3][3][1][_k, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to initial init `, _x0); _dtbl[3][3][1][_k, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #3, event code `, _k, ` to fireinitial init `, _x0+1); _dtbl[3][3][1][_k, 8] := _x0+1 end if end do; _dtbl[3][4][17] := 0; _dtbl[3][4][9] := 0; if _dtbl[1][4][10] = 1 then procname(procname("right")) end if end if else for _k in _i do _dtbl[1][3][1][_k, 7] := 1 end do; _dtbl[2] := evaln(_dtbl[2]); _dtbl[3] := evaln(_dtbl[3]); _dtbl[4] := 0; if _dtbl[1][4][10] = 1 then if _x0 <= procname("right") then try procname(procname("right")) catch:  end try end if; if procname("left") <= _x0 then try procname(procname("left")) catch:  end try end if end if end if; return  elif type(_xin, `=`) and lhs(_xin) = "eventfired" then if not type(rhs(_xin), 'list') then error "'eventfired' must be specified as a list" end if; if _nv = 0 then error "this solution has no events" end if; if _dtbl[4] <> 2 and _dtbl[4] <> 3 then error "'direction' must be set prior to calling/setting 'eventfired'" end if; _i := _dtbl[4]; _val := NULL; if not assigned(_EnvEventRetriggerWarned) then _EnvEventRetriggerWarned := false end if; for _k in rhs(_xin) do if type(_k, 'integer') then _src := _k elif type(_k, 'integer' = 'anything') and type(evalf(rhs(_k)), 'numeric') then _k := lhs(_k) = evalf[max(Digits, 18)](rhs(_k)); _src := lhs(_k) else error "'eventfired' entry is not valid: %1", _k end if; if _src < 1 or round(_dtbl[1][3][1][_nv+1, 1]) < _src then error "event identifiers must be integers in the range 1..%1", round(_dtbl[1][3][1][_nv+1, 1]) end if; _src := {seq(`if`(_dtbl[1][3][1][_j, 1]-_src = 0., _j, NULL), _j = 1 .. _nv)}; if nops(_src) <> 1 then error "'eventfired' can only be set/queried for root-finding events and time/interval events" end if; _src := _src[1]; if _dtbl[1][3][1][_src, 2] <> 0. and _dtbl[1][3][1][_src, 2]-2. <> 0. then error "'eventfired' can only be set/queried for root-finding events and time/interval events" elif irem(round(_dtbl[1][3][1][_src, 4]), 2) = 1 then if _EnvEventRetriggerWarned = false then WARNING(`'eventfired' has no effect on events that retrigger`) end if; _EnvEventRetriggerWarned := true end if; if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then _val := _val, undefined elif type(_dtbl[_i][3][4][_src, _i-1], 'undefined') or _i = 2 and _dtbl[2][3][1][_src, 8] < _dtbl[2][3][4][_src, 1] or _i = 3 and _dtbl[3][3][4][_src, 2] < _dtbl[3][3][1][_src, 8] then _val := _val, _dtbl[_i][3][1][_src, 8] else _val := _val, _dtbl[_i][3][4][_src, _i-1] end if; if type(_k, `=`) then if _dtbl[_i][3][1][_src, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_src, 4]), 32), 2) = 1 then error "cannot set event code for a rate hysteresis event" end if; userinfo(3, {'events', 'eventreset'}, `manual set event code `, _src, ` to value `, rhs(_k)); _dtbl[_i][3][1][_src, 8] := rhs(_k); _dtbl[_i][3][4][_src, _i-1] := rhs(_k) end if end do; return [_val] elif type(_xin, `=`) and lhs(_xin) = "direction" then if not member(rhs(_xin), {-1, 1, ':-left', ':-right'}) then error "'direction' must be specified as either '1' or 'right' (positive) or '-1' or 'left' (negative)" end if; _src := `if`(_dtbl[4] = 2, -1, `if`(_dtbl[4] = 3, 1, undefined)); _i := `if`(member(rhs(_xin), {1, ':-right'}), 3, 2); _dtbl[4] := _i; _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #4, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if; return _src elif _xin = "eventcount" then if _dtbl[1][3][1] = 0 or _dtbl[4] <> 2 and _dtbl[4] <> 3 then return 0 else return round(_dtbl[_dtbl[4]][3][1][_nv+1, 12]) end if else return "procname" end if end if; if _xout = _x0 then return [_x0, seq(evalf(_dtbl[1][6][_vmap[_i]]), _i = 1 .. _n-_ne)] end if; _i := `if`(_x0 <= _xout, 3, 2); if _xin = "last" and 0 < _dtbl[_i][4][9] and _dtbl[_i][4][9] < 100 then _dat := eval(_dtbl[_i], 2); _j := _dat[4][20]; return [_dat[11][_j, 0], seq(_dat[11][_j, _vmap[_i]], _i = 1 .. _n-_ne-_nd), seq(_dat[8][1][_vmap[_i]], _i = _n-_ne-_nd+1 .. _n-_ne)] end if; if not type(_dtbl[_i], 'array') then _dtbl[_i] := `dsolve/numeric/SC/IVPdcopy`(_dtbl[1], `if`(assigned(_dtbl[_i]), _dtbl[_i], NULL)); if 0 < _nv then for _j to _nv+1 do if _j <= _nv and not type(_dtbl[_i][3][4][_j, _i-1], 'undefined') then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to defined init `, _dtbl[_i][3][4][_j, _i-1]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][3][4][_j, _i-1] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 32), 2) = 1 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to rate hysteresis init `, _dtbl[_i][5][24]); _dtbl[_i][3][1][_j, 8] := _dtbl[_i][5][24] elif _dtbl[_i][3][1][_j, 2] = 0 and irem(iquo(round(_dtbl[_i][3][1][_j, 4]), 2), 2) = 0 then userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to initial init `, _x0); _dtbl[_i][3][1][_j, 8] := _x0 else userinfo(3, {'events', 'eventreset'}, `reinit #5, event code `, _j, ` to fireinitial init `, _x0-2*_i+5.0); _dtbl[_i][3][1][_j, 8] := _x0-2*_i+5.0 end if end do end if end if; if _xin <> "last" then if 0 < 0 then if `dsolve/numeric/checkglobals`(op(_dtbl[1][14]), _pars, _n, _y0) then `dsolve/numeric/SC/reinitialize`(_dtbl, _y0, _n, procname, _pars, _i) end if end if; if _dtbl[1][4][7] = 0 then error "parameters must be initialized before solution can be computed" end if end if; _dat := eval(_dtbl[_i], 2); _dtbl[4] := _i; try _src := `dsolve/numeric/SC/IVPrun`(_dat, _xout) catch: userinfo(2, `dsolve/debug`, print(`Exception in solnproc:`, [lastexception][2 .. -1])); error  end try; if _src = 0 and 100 < _dat[4][9] then _val := _dat[3][1][_nv+1, 8] else _val := _dat[11][_dat[4][20], 0] end if; if _src <> 0 or _dat[4][9] <= 0 then _dtbl[1][5][1] := _xout else _dtbl[1][5][1] := _val end if; if _i = 3 and _val < _xout then Rounding := -infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further right of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further right of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further right of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further right of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further right of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further right of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further right of %1", evalf[8](_val) end if elif _i = 2 and _xout < _val then Rounding := infinity; if _dat[4][9] = 1 then error "cannot evaluate the solution further left of %1, probably a singularity", evalf[8](_val) elif _dat[4][9] = 2 then error "cannot evaluate the solution further left of %1, maxfun limit exceeded (see ?dsolve,maxfun for details)", evalf[8](_val) elif _dat[4][9] = 3 then if _dat[4][25] = 3 then error "cannot evaluate the solution past the initial point, problem may be initially singular or improperly set up" else error "cannot evaluate the solution past the initial point, problem may be complex, initially singular or improperly set up" end if elif _dat[4][9] = 4 then error "cannot evaluate the solution further left of %1, accuracy goal cannot be achieved with specified 'minstep'", evalf[8](_val) elif _dat[4][9] = 5 then error "cannot evaluate the solution further left of %1, too many step failures, tolerances may be too loose for problem", evalf[8](_val) elif _dat[4][9] = 6 then error "cannot evaluate the solution further left of %1, cannot downgrade delay storage for problems with delay derivative order > 1, try increasing delaypts", evalf[8](_val) elif _dat[4][9] = 10 then error "cannot evaluate the solution further right of %1, interrupt requested", evalf[8](_val) elif 100 < _dat[4][9] then if _dat[4][9]-100 = _nv+1 then error "constraint projection failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+2 then error "index-1 and derivative evaluation failure on event at t=%1", evalf[8](_val) elif _dat[4][9]-100 = _nv+3 then error "maximum number of event iterations reached (%1) at t=%2", round(_dat[3][1][_nv+1, 3]), evalf[8](_val) else if _Env_dsolve_nowarnstop <> true then `dsolve/numeric/warning`(StringTools:-FormatMessage("cannot evaluate the solution further left of %1, event #%2 triggered a halt", evalf[8](_val), round(_dat[3][1][_dat[4][9]-100, 1]))) end if; Rounding := 'nearest'; _xout := _val end if else error "cannot evaluate the solution further left of %1", evalf[8](_val) end if end if; if _EnvInFsolve = true then _dig := _dat[4][26]; _dat[4][26] := _EnvDSNumericSaveDigits; _Env_dsolve_SC_native := true; if _dat[4][25] = 1 then _i := 1; _dat[4][25] := 2 else _i := _dat[4][25] end if; _val := `dsolve/numeric/SC/IVPval`(_dat, _xout, _src); _dat[4][25] := _i; _dat[4][26] := _dig; [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] else Digits := _dat[4][26]; _val := `dsolve/numeric/SC/IVPval`(eval(_dat, 2), _xout, _src); [_xout, seq(_val[_vmap[_i]], _i = 1 .. _n-_ne)] end if end proc, (2) = Array(0..0, {}), (3) = [T, B(T), C(T), DD(T), E(T), F(T), G(T), H(T), J(T), K(T), L(T), M(T)], (4) = [B0 = B0, C0 = C0, D0 = D0, E0 = E0, F0 = F0, G0 = G0, H0 = H0, J0 = J0, K0 = K0, L0 = L0, M0 = M0, M__c = M__c, M__h = M__h, M__v = M__v, alpha = alpha, beta__g = beta__g, beta__j = beta__j, beta__k = beta__k, beta__o = beta__o, delta = delta, epsilon = epsilon, eta = eta, f = f, mu__1 = mu__1, mu__c = mu__c, mu__e = mu__e, mu__v = mu__v, omega = omega, phi = phi, pi = pi, psi = psi, theta = theta]}); _vars := _dat[3]; _pars := map(rhs, _dat[4]); _n := nops(_vars)-1; _solnproc := _dat[1]; if not type(_xout, 'numeric') then if member(x_rkf45, ["start", 'start', "method", 'method', "left", 'left', "right", 'right', "leftdata", "rightdata", "enginedata", "eventstop", 'eventstop', "eventclear", 'eventclear', "eventstatus", 'eventstatus', "eventcount", 'eventcount', "laxtol", 'laxtol', "numfun", 'numfun', NULL]) then _res := _solnproc(convert(x_rkf45, 'string')); if 1 < nops([_res]) then return _res elif type(_res, 'array') then return eval(_res, 1) elif _res <> "procname" then return _res end if elif member(x_rkf45, ["last", 'last', "initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(x_rkf45, 'string'); _res := _solnproc(_xout); if _xout = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] elif _xout = "initial_and_parameters" then return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] end if elif type(_xout, `=`) and member(lhs(_xout), ["initial", 'initial', "parameters", 'parameters', "initial_and_parameters", 'initial_and_parameters', NULL]) then _xout := convert(lhs(x_rkf45), 'string') = rhs(x_rkf45); if type(rhs(_xout), 'list') then _res := _solnproc(_xout) else error "initial and/or parameter values must be specified in a list" end if; if lhs(_xout) = "initial" then return [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] elif lhs(_xout) = "parameters" then return [seq(_pars[_i] = _res[_i], _i = 1 .. nops(_pars))] else return [seq(_vars[_i+1] = [_res][1][_i+1], _i = 0 .. _n), seq(_pars[_i] = [_res][2][_i], _i = 1 .. nops(_pars))] end if elif type(_xout, `=`) and member(lhs(_xout), ["eventdisable", 'eventdisable', "eventenable", 'eventenable', "eventfired", 'eventfired', "direction", 'direction', NULL]) then return _solnproc(convert(lhs(x_rkf45), 'string') = rhs(x_rkf45)) elif _xout = "solnprocedure" then return eval(_solnproc) elif _xout = "sysvars" then return _vars end if; if procname <> unknown then return ('procname')(x_rkf45) else _ndsol; _ndsol := pointto(_dat[2][0]); return ('_ndsol')(x_rkf45) end if end if; try _res := _solnproc(_xout); [seq(_vars[_i+1] = _res[_i+1], _i = 0 .. _n)] catch: error  end try end proc

(2)

# define parameter values

DefaultValues := [
   M__h    = 100,
   beta__o = 0.034,
   beta__j = .025,
   mu__1   = 0.0004,
   epsilon = .7902,
   alpha   = 0.11,
   psi     = 0.000136,
   phi     = 0.05,
   omega   = .7,
   eta     = .134,
   delta   = .245,
   f       = 0.21,
   M__v    = 1000,
   beta__k = 0.09,
   mu__v   = .0005,
   M__c    = .636,
   beta__g = 0.15,
   mu__c   = 0.0019,
   pi      = 0.01231,
   theta   = 0.12,
   mu__e   = 0.005,
  
   B0 = 100,
   C0 = 90,
   D0 = 45,
   E0 = 38,  
   F0 = 10,
   G0 = 45,
   H0 = 50,
   J0 = 70,
   K0 = 20,
   L0 = 65,
   M0 = 22
]:

# Form the list of numerical values ordered as params

ValuatedParams := subs(DefaultValues, params)
   

[100, 90, 45, 38, 10, 45, 50, 70, 20, 65, 22, .636, 100, 1000, .11, .15, 0.25e-1, 0.9e-1, 0.34e-1, .245, .7902, .134, .21, 0.4e-3, 0.19e-2, 0.5e-2, 0.5e-3, .7, 0.5e-1, 0.1231e-1, 0.136e-3, .12]

(3)

# Instanciate the solution for this set of values

ans(parameters=ValuatedParams)

[B0 = 100., C0 = 90., D0 = 45., E0 = 38., F0 = 10., G0 = 45., H0 = 50., J0 = 70., K0 = 20., L0 = 65., M0 = 22., M__c = .636, M__h = 100., M__v = 1000., alpha = .11, beta__g = .15, beta__j = 0.25e-1, beta__k = 0.9e-1, beta__o = 0.34e-1, delta = .245, epsilon = .7902, eta = .134, f = .21, mu__1 = 0.4e-3, mu__c = 0.19e-2, mu__e = 0.5e-2, mu__v = 0.5e-3, omega = .7, phi = 0.5e-1, pi = 0.1231e-1, psi = 0.136e-3, theta = .12]

(4)

#
# Plot solutions for a few of the dependent variablss
# just to show everything is working (more-or-less!)
#

  plots:-odeplot( ans, [T, B(T)] , T=0..5);

if false then
  plots:-odeplot( ans, [T, C(T)] , T=0..5);
  plots:-odeplot( ans, [T, DD(T)], T=0..5);
  plots:-odeplot( ans, [T, E(T)] , T=0..5);
  plots:-odeplot( ans, [T, F(T)] , T=0..5);
  plots:-odeplot( ans, [T, G(T)] , T=0..5);
  plots:-odeplot( ans, [T, H(T)] , T=0..5);
  plots:-odeplot( ans, [T, J(T)] , T=0..5);
  plots:-odeplot( ans, [T, K(T)] , T=0..5);
  plots:-odeplot( ans, [T, L(T)] , T=0..5);
  plots:-odeplot( ans, [T, M(T)] , T=0..5);
end if:

 

# How two use the parametric solution?
#
# Here an example with only one variable parameter, but this could be generalized to many
# (the main problem is then more a problem of "readability" than a technical one).
#

f := proc(VariableName::symbol, ParamName::symbol, ParamValues::list, params, ParamDefault, sol)
  local V, k, p, MyChoice, ValuatedParams:
  V := unapply(VariableName(T), T):
  k := 0:
  for p in ParamValues do
    k := k+1:
    MyChoice       := map(u -> if lhs(u)=ParamName then lhs(u)=p else u end if, ParamDefault):
    ValuatedParams := subs(MyChoice, params):
    sol(parameters=ValuatedParams):
    plot||k := plots:-odeplot(
                               sol,
                               [T, V(T)] ,
                               T=0..5,
                               legend=p,
                               color=ColorTools:-Color([rand()/10^12, rand()/10^12, rand()/10^12])
                             );
  end do:
  plots:-display(seq(plot||k, k=1..numelems(ParamValues)));
end proc:

f(C, beta__o, [0.030, 0.034, 0.038], params, DefaultValues, ans)

 

 


 

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