mmcdara

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3 years, 228 days

MaplePrimes Activity


These are answers submitted by mmcdara


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)

 

 


 

Download odeStuff_Parametric.mw

Here are a few simple ways to answer your question

 

restart

randomize():

with(LinearAlgebra):

A := RandomMatrix(3)

A := Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = -1, (2, 1) = 0, (2, 2) = -2, (2, 3) = 2, (3, 1) = 1, (3, 2) = 1, (3, 3) = -2})

(1)

# Theses lines to see what happens
 
1 /~ Diagonal(A):
DiagonalMatrix(%):

A . DiagonalMatrix(1 /~ Diagonal(A) );

Matrix(3, 3, {(1, 1) = 1, (1, 2) = 76/81, (1, 3) = 3/47, (2, 1) = -70/57, (2, 2) = 1, (2, 3) = 21/94, (3, 1) = -68/57, (3, 2) = 11/27, (3, 3) = 1})

(2)

Probably better to do this

restart:

NormalizeMatrix := proc(A::Matrix)
  local NR, NL:
  NR, NL := LinearAlgebra:-Dimension(A):

  if NR <> NL then
     error "the input matrix must be a square matrix":
     return NULL:
  end if:

  if mul(A[i,i], i=1..NR) = 0 then
     error "the diagonal of the input matrix contains zeroes":
     return NULL:
  end if:

  return Matrix(NR, NR, (i,j) -> A[i,j] / A[i,i]);
end proc:

randomize():

N := 3:
A := LinearAlgebra:-RandomMatrix(N, generator=-2..2);

A := Matrix(3, 3, {(1, 1) = 0, (1, 2) = 1, (1, 3) = -1, (2, 1) = 0, (2, 2) = -2, (2, 3) = 2, (3, 1) = 1, (3, 2) = 1, (3, 3) = -2})

(3)

NormalizeMatrix(A)

Error, (in NormalizeMatrix) the diagonal of the input matrix contains zeroes

 

 


 

Download SomeSolutions.mw

A solution "by hand"


 

restart;

sys_Pde := [ diff(V(x, t), x, x) = 0, diff(T(x, t), x, x) = 0 ]:

BC := [ eval(diff(V(x, t), x), x = 1) = 0, eval(V(x, t), x = 0) = 1+cos(w*t), eval(T(x, t), x = 0) = 0, eval(T(x, t), x = 1) = 1 ]:
 

sol__1 := pdsolve(sys_Pde);

{T(x, t) = _F1(t)*x+_F2(t), V(x, t) = _F3(t)*x+_F4(t)}

(1)

bc__0 := eval(sol__1, x=0);
bc__1 := eval(sol__1, x=1);

{T(0, t) = _F2(t), V(0, t) = _F4(t)}

 

{T(1, t) = _F1(t)+_F2(t), V(1, t) = _F3(t)+_F4(t)}

(2)

subs(bc__0 union bc__1, BC):
F__cond := simplify(subs(sol__1, %));

[_F3(t) = 0, _F4(t) = 1+cos(w*t), _F2(t) = 0, _F1(t)+_F2(t) = 1]

(3)

sol__2 := subs(F__cond, sol__1):
print~(sol__2):  # _F1(t) is an arbitrary function.

T(x, t) = _F1(t)*x

 

V(x, t) = 1+cos(w*t)

(4)

# check the partial diff. equations

eval(sys_Pde, sol__2)

[0 = 0, 0 = 0]

(5)

# check the boundary conditions

eval(BC, sol__2)

[0 = 0, V(0, t) = 1+cos(w*t), T(0, t) = 0, T(1, t) = 1]

(6)

 


 

Download PDE_solved.mw


 

Fourier coefficients are defined for periodic Lebesgue  integrable  functions.
Your function f does not belong to this class. 

A common practice in signal processing for a signal f(t) recorded on a finite interval D, let's say D=[0, T >0] is to periodize f outside of D by replicating it by translations f(t+kT) where t is in Z. 
Assuming then f is Lebesgue integrable then computing the Fourier coefficients makes sense (which makes new problems because of the dubbed "aliasing" phenomenon).
Another trick is to consider that a non periodic function is a function of perior=+infinity ... but even in this case f(t)=t^2 doen(t belong to L^1(R)

Your main problem is not to integrate f(t)*cos(omega*t) or f(t)*sin(omega*t), but to reshape f in order that it becomes a periodic Lebesgue square integrable  function.

Note that using Fourier transformation is of no help

Constructing a plot like the one you give is not a problem.

But I really doubt this kind of 3D plots could be of great interest in PCA.
In PCA, "Biplot" graphics not only represent individuals as points in the plan spanned by two principal directions, but these same individuals as numbers or names, plus the projections (as vectors) of the original factors with their names.
This is fundamental for the analysis of the PCA results where we are not concerned by the shape of a cloud of points, but by the closeness of each of them, considered as a particular individual, to the vector factors.
If you all of this together, it will be a real mess.

The only limit situation where a 2D plot seems more or less relevant is when the number of factors is equal to three
(in this case you can associate a color to each "factor vector", which avoids writting a 3D text to identify it, and interpret the PCA results)  and when the number of individuals is small enough to make the observation of one of them a simple task.
 

@LichengZhang   @Carl Love

Carl Love already gave an elegant answer to the question "How many cycles of length n does a given Graph contain?" (which obiously includes your first one " I'd like to know  whether it contains a C4 (cycle of 4) as its subgraph"

I authorise myself to reproduce Carl's code here 

restart:

with(GraphTheory):
with(RandomGraphs):

randomize():

NV := 8:
NE := 12:
G  := RandomGraph(NV, NE):

# The code below has been written by Carl Love
# https://www.mapleprimes.com/questions/226523-How-To-Determine-The-Number-Of-Circles-Of--Graph-#answer257002

kCycles:= (A::Matrix, n::posint, k::And(posint, Not({1,2})))->
   add(
       (-1)^(k-i)*binomial(n-i,n-k) *
         add(LinearAlgebra:-Trace(A[S,S]^k), S= combinat:-choose(n,i)),
       i= 2..k
   )/2/k:

# How to use it (cf above link)

A:= GraphTheory:-AdjacencyMatrix(G):
seq(printf("G contains %2d cycles of length %2d\n", kCycles(A, NV, k), k), k= 3..NV);

G contains  2 cycles of length  3
G contains  5 cycles of length  4
G contains  6 cycles of length  5

G contains  5 cycles of length  6
G contains  4 cycles of length  7

G contains  0 cycles of length  8

 

 

 

Download Does_G_contains_a_given_cycle.mw

 

Does a given Graph contain K(n) where K(n) is the complete graph with n vertices?
GraphTheory:-CliqueNumber(G) returns the highest value of N fpr wich K(N) is a complete subgraph of G. 
Then, G contains at least one complete subgraph K(n) for n=3..N.
Conversely, G contains no complete subgraph K(n) such that n > N

Here is an example (your own file is treated later)

Be carefull: as recently mentioned in another thread, Maple doesn't support loops (FromVertex = ToVertex)


PS: if your graph is not oriented, you can replace 
edges := { convert(M, listlist)[] }
by
edges := { convert~(convert(M, listlist), set)[] }

restart:

kernelopts(homedir):
file := cat(%, "/Desktop/graph.xlsx"):

M := Import(file);

M := Matrix(9, 2, {(1, 1) = 1.0, (1, 2) = 2.0, (2, 1) = 1.0, (2, 2) = 3.0, (3, 1) = 2.0, (3, 2) = 4.0, (4, 1) = 2.0, (4, 2) = 5.0, (5, 1) = 2.0, (5, 2) = 1.0, (6, 1) = 3.0, (6, 2) = 2.0, (7, 1) = 4.0, (7, 2) = 5.0, (8, 1) = 5.0, (8, 2) = 4.0, (9, 1) = 5.0, (9, 2) = 1.0}, datatype = float[8])

(1)

M := map(round, M);

M := Matrix(9, 2, {(1, 1) = 1, (1, 2) = 2, (2, 1) = 1, (2, 2) = 3, (3, 1) = 2, (3, 2) = 4, (4, 1) = 2, (4, 2) = 5, (5, 1) = 2, (5, 2) = 1, (6, 1) = 3, (6, 2) = 2, (7, 1) = 4, (7, 2) = 5, (8, 1) = 5, (8, 2) = 4, (9, 1) = 5, (9, 2) = 1})

(2)

edges := { convert(M, listlist)[] }

{[1, 2], [1, 3], [2, 1], [2, 4], [2, 5], [3, 2], [4, 5], [5, 1], [5, 4]}

(3)

G :=GraphTheory:-Graph(edges)

GRAPHLN(directed, unweighted, [1, 2, 3, 4, 5], Array(%id = 18446744078100176518), `GRAPHLN/table/1`, 0)

(4)

GraphTheory:-DrawGraph(G)

 

 


 

Download Excel2Graph.mw



Now the worksheet with your own file (2896 Vertices graph)
You will have a lot of work to do to obtain a readable display !


Excel2Graph_bis.mw

Here are plots for 1 and 2 categorical variables.
They were done with Maple 2015. I do not have Maple 2018 on this laptop and I do not remember if Maple 2018 has, or not,  specific  features to plot categorical variables

For an unknown reason I can't load the content of the worksheet

CategoricalVariables.mw


PS to customize the 3D plots you might try other commands than
plottools:-line( [i, j, 0], [i, j, Sizes[i, j]], thickness=10, color=GenderColor[i]),
for instance, having set GenderColor := [blue, red]:
plottools:-cylinder([i, j, 0], 0.2, Sizes[i, j], color=GenderColor[i], style=surface),

 

Why not solving  symbolically with formal parameters instead of numeric ones, and next substituting the formal parameters by the suitable numerical values?


 

ode1 := (2/3)*diff(theta(t), t$2)+A__1*theta(t)-A__2*x(t) = 0;

(2/3)*(diff(diff(theta(t), t), t))+A__1*theta(t)-A__2*x(t) = 0

(1)

ode2 := 2*diff(x(t),t$2)+B__1*x(t)-B__2*theta(t) =B__3*sin(B__4*t);

2*(diff(diff(x(t), t), t))+B__1*x(t)-B__2*theta(t) = B__3*sin(B__4*t)

(2)

ics := theta(0) = 0, (D(theta))(0) = D__1, x(0) = 0, (D(x))(0) = 0

theta(0) = 0, (D(theta))(0) = D__1, x(0) = 0, (D(x))(0) = 0

(3)

solution := dsolve([ode1, ode2, ics]):

simplify~(eval(solution, [A__1=1250, A__2=500, B__1=1000, B__2=500, B__3=50, B__4=20, D__1=1/100]))

{theta(t) = (12001/97500)*5^(1/2)*3^(1/2)*sin(5*5^(1/2)*3^(1/2)*t)+(399/260000)*5^(1/2)*sin(20*5^(1/2)*t)-(15/32)*sin(20*t), x(t) = (12001/48750)*5^(1/2)*3^(1/2)*sin(5*5^(1/2)*3^(1/2)*t)-(133/520000)*5^(1/2)*sin(20*5^(1/2)*t)-(59/64)*sin(20*t)}

(4)

simplify~(eval(solution, [A__1=1250, A__2=500, B__1=1000, B__2=500, B__3=50, B__4=90, D__1=1/100]))

{theta(t) = (18103/10042500)*5^(1/2)*3^(1/2)*sin(5*5^(1/2)*3^(1/2)*t)-(798/495625)*5^(1/2)*sin(20*5^(1/2)*t)+(5/12566)*sin(90*t), x(t) = (18103/5021250)*5^(1/2)*3^(1/2)*sin(5*5^(1/2)*3^(1/2)*t)+(133/495625)*5^(1/2)*sin(20*5^(1/2)*t)-(83/25132)*sin(90*t)}

(5)

 


 

Download TryThis.mw

Here an answer obviously less concise than vv's answer, but I found funny to adress the problem from a pure geometrical point of view.

I had already done this exercise with Geogebra... and I wondered if Maple could do the same ?


Still this f... error
Maple Worksheet - Error
Failed to load the worksheet /maplenet/convert/geometry.mw .

 

Download geometry.mw

 

The problem is such general, with such many potential situations, that no universal answer can be given (it's my opinion).
So, if you have a particular example in mind, you should load the mw file by using the big green arrow in order a more suited answer might be delivered.

Nevertheless here are two examples of what it is possible to do


ATTENTION : I am not sure that 'allvalues' always returns the solutions in the same order order... and this is essential here to ensure that the curves are not tangled.
This is a point I will verify and, possibly modify the code accordingly

 

restart:

interface(rtablesize=10):

# lets's start with a simpple case
# Suppose that x and y are two functions of z of KNOWN explicit forms
# Example

x := z -> cos(z):
y := z -> z*sin(z):

# then the "solution curve" C(z) = { (x(z), y(z)), z=a..b } can be drawn with

plot([x(z), y(z), z=0..20*Pi]);

 

# Now a harder case:
# x and y are two functions of z of UNKNOWN explicit forms
# Example

vars := [x, y, z, t];
randpoly(vars, degree = 2);

[x, y, z, t]

 

-73*t^2-56*x^2-62*y^2+97*z^2+87*x

(1)

sys := [ seq(randpoly(vars, degree = 2)=0, n=1..3) ]:
print~(sys):

71*x*z-17*y^2-75*y*z-44*t+80*x-82 = 0

 

74*t*y+37*t*z-92*y^2+6*y*z+72*z^2+75*x = 0

 

-47*t*z-29*x^2+95*x*y+11*y*z-49*z^2-8 = 0

(2)

# Exact solutions

ExactSolutions := solve(evalf(sys),vars):

# The solution is made of 1 functional solution and 10 "numeric" 4-uples

NumberOfSolutions := numelems(ExactSolutions);

for n from 1 to NumberOfSolutions do
 # printf("List of functions the solution %2d contains %a\n", n, select~(has, rhs~(ExactSolutions[n]), 'function') )
  printf("List of functions the solution %2d contains %a\n", n, numelems~(indets~(rhs~(ExactSolutions[n]), function)) )
end do:

11

 

List of functions the solution  1 contains [1, 1, 0, 1]
List of functions the solution  2 contains [0, 0, 0, 0]
List of functions the solution  3 contains [0, 0, 0, 0]
List of functions the solution  4 contains [0, 0, 0, 0]
List of functions the solution  5 contains [0, 0, 0, 0]
List of functions the solution  6 contains [0, 0, 0, 0]
List of functions the solution  7 contains [0, 0, 0, 0]
List of functions the solution  8 contains [0, 0, 0, 0]
List of functions the solution  9 contains [0, 0, 0, 0]
List of functions the solution 10 contains [0, 0, 0, 0]
List of functions the solution 11 contains [0, 0, 0, 0]

 

# Only solution 1 is functional with x, y and t expressed as functions of z:

ExactSolutions[1][3]

z = z

(3)

# Here are numerical values of the couple (x, y)  if z=0 (for instance)

evalf~(subs~(z=0, rhs~(ExactSolutions[1][1..2])))

[[.4652284527, .3230260837]]

(4)

# doing so for fifferent values of z enables ploting the "solution curve" C(z) = { (x(z), y(z)), z=a..b }
# for some real range a..b
# Example

f := u -> evalf~(subs~(z=u, rhs~(ExactSolutions[1][1..2]))):

Cxy := [seq(f(u), u in seq(0..10, 0.1))]:

plot(Cxy, labels=['x(z)', 'y(z)']);



Cxyz := [seq([f(u)[], u], u in seq(-10..10, 0.1))]:

PLOT3D(CURVES(Cxyz), AXESLABELS('x(z)', 'y(z)', 'z'))

 

 

# The problem is that x(z) is not unique and that among all the solutions some are complex

[ evalf(allvalues(subs(z=0, rhs(ExactSolutions[1][1])))) ];
numelems(%);

# and neither is y(z)
[ evalf(allvalues(subs(z=0, rhs(ExactSolutions[1][2])))) ];
numelems(%);

[.4652284527, 40.93176235, -0.2722500350e-1-0.2586203384e-1*I, -.6917623044+.5507283075*I, -.6917623044-.5507283075*I, -0.2722500350e-1+0.2586203384e-1*I]

 

6

 

[.3230260837, 12.49701637, -1.634238234+1.536633637*I, -.2856782808+.1087988986*I, -.2856782808-.1087988986*I, -1.634238234-1.536633637*I]

 

6

(5)

# Test if Cxyz(s) is real before doing the plot.

realCxyz := proc(a, b, step, s, verbose)
   local g, c, u, h:

   g := (u, s) -> evalf~(allvalues(subs~(z=u, rhs~(ExactSolutions[1][1..2])))[s]):

   c := NULL:
   for u from a to b by step do
     h := g(u, s):  
     if verbose then print(u, h, `and`(op(is~(Im~(h) =~ 0.))) ); end if;
     if `and`(op(is~(Im~(h) =~ 0.))) then
        c := c, [h[], u]
     end if:
   end do:
   [c]
end proc:

# realCxyz(-1, 1, 1, 3, true)

PLOT3D( seq( CURVES(realCxyz(-2, 2, 0.005, s, false), COLOR(RGB, s/6, 0, 1-s/6), THICKNESS(s)), s=1..6), AXESLABELS('x(z)', 'y(z)', 'z'))

 

 


 

Download Example.mw

In a lot of languages the function to draw a graph names "plot" or "draw".
In Maple it's "plot", so I advice you to search the name "plot" in the help pages to get all the possibilitues of the function.

I don't understand what "relative" and "absolute" maxima mean to you.
So, look to the attached file, maybe it will help you.

 

Download Plot.mw

Here are Approximate and Exact poltynomial interpolations.

As I mentioned to tomleslie, tout data do not correspond to the plot your initial question contains


 

restart:

# OP's code after a few cosmetic arrangements

t := proc (x) options operator, arrow; b*x^10+c*x^9+d*x^8+e*x^7+f*x^6+g*x^5+h*x^4+i*x^3+j*x^2+k*x+l end proc;

solve([
        t(-2)  = 2,
        t(-1)  = 0,
        t(0)   = -2,
        t(1)   = 0,
        t(2.5) = -3,
        t(2.8) = 0,
        t(3.5) = 0,
        eval(diff(t(x), x), x = -2)  = 0,
        eval(diff(t(x), x), x = 0)   = 0,  
        eval(diff(t(x), x), x = 1)   = 0,
        eval(diff(t(x), x), x = 2.5) = 0
      ],
      [b, c, d, e, f, g, h, i, j, k, l]
);

plot(-.2688965311*x^10+.1687428810*x^9-.3166922031*x^8-1.490599337*x^7+1.885667707*x^6+3.845330330*x^5-6.939129440*x^4-2.523473874*x^3+7.112020606*x^2-2, x = -2..3.5):  # change of the range

proc (x) options operator, arrow; b*x^10+c*x^9+d*x^8+e*x^7+f*x^6+g*x^5+h*x^4+i*x^3+j*x^2+k*x+l end proc

 

[[b = -0.2688965311e-1, c = .1687428810, d = -0.3166922031e-1, e = -1.490599337, f = 1.885667707, g = 3.845330330, h = -6.939129440, i = -2.523473874, j = 7.112020606, k = 0., l = -2.]]

(1)

# My code

t  := proc (x) options operator, arrow; b*x^10+c*x^9+d*x^8+e*x^7+f*x^6+g*x^5+h*x^4+i*x^3+j*x^2+k*x+l end proc;
dt := unapply(diff(t(x), x), x);

proc (x) options operator, arrow; b*x^10+c*x^9+d*x^8+e*x^7+f*x^6+g*x^5+h*x^4+i*x^3+j*x^2+k*x+l end proc

 

proc (x) options operator, arrow; 10*b*x^9+9*c*x^8+8*d*x^7+7*e*x^6+6*f*x^5+5*g*x^4+4*h*x^3+3*i*x^2+2*j*x+k end proc

(2)

# data

# 1/ points

X  := [-2, -1,  0, 1, 2.5, 2.8, 3.5];
Y  := [ 2,  0, -2, 0,  -3,   0,   0];

# 2/ derivatives

dX := [-2, 0, 1, 2.5];
dY := [ 0, 0, 0,   0];

[-2, -1, 0, 1, 2.5, 2.8, 3.5]

 

[2, 0, -2, 0, -3, 0, 0]

 

[-2, 0, 1, 2.5]

 

[0, 0, 0, 0]

(3)

# sys  : system to solve
# vars : unknowns

sys  := [ (t~(X) =~ Y)[], (dt~(dX) =~ dY)[] ];
vars := convert(indets(t(1)), list);

[1024*b-512*c+256*d-128*e+64*f-32*g+16*h-8*i+4*j-2*k+l = 2, b-c+d-e+f-g+h-i+j-k+l = 0, l = -2, b+c+d+e+f+g+h+i+j+k+l = 0, 9536.743164*b+3814.697266*c+1525.878906*d+610.3515625*e+244.140625*f+97.65625*g+39.0625*h+15.625*i+6.25*j+2.5*k+l = -3, 29619.67667*b+10578.45595*c+3778.019983*d+1349.292851*e+481.890304*f+172.10368*g+61.4656*h+21.952*i+7.84*j+2.8*k+l = 0, 275854.7354*b+78815.63867*c+22518.75391*d+6433.929688*e+1838.265625*f+525.21875*g+150.0625*h+42.875*i+12.25*j+3.5*k+l = 0, -5120*b+2304*c-1024*d+448*e-192*f+80*g-32*h+12*i-4*j+k = 0, k = 0, 10*b+9*c+8*d+7*e+6*f+5*g+4*h+3*i+2*j+k = 0, 38146.97266*b+13732.91015*c+4882.812500*d+1708.984375*e+585.93750*f+195.3125*g+62.500*h+18.75*i+5.0*j+k = 0]

 

[b, c, d, e, f, g, h, i, j, k, l]

(4)

sol := solve(sys, vars)[] ;  # identical to the OP's

[b = -0.2688965311e-1, c = .1687428810, d = -0.3166922031e-1, e = -1.490599337, f = 1.885667707, g = 3.845330330, h = -6.939129440, i = -2.523473874, j = 7.112020606, k = 0., l = -2.]

(5)

# checking

T  := unapply(eval( t(x), sol), x):
dT := unapply(eval(dt(x), sol), x):

 T~( X) -~  Y;  # should be close to 0
dT~(dX) -~ dY;  # should be close to 0

[-0.9e-7, 0., 0., 0., -0.9e-7, 0.77e-6, 0.312e-5]

 

[0.19e-6, 0., 0., -0.2e-7]

(6)

plot(eval(t(x), sol), x=min(X[], dX[])..max(X[], dX[]))

 

# 1/ points

X  := [-2, -1,  0, 1, 5/2, 14/5, 7/2];
Y  := [ 2,  0, -2, 0,  -3,   0,   0];

# 2/ derivatives

dX := [-2, 0, 1, 5/2];
dY := [ 0, 0, 0,   0];

# 3/ exact solution

sys := [ (t~(X) =~ Y)[], (dt~(dX) =~ dY)[] ]:

sol := solve(sys, vars)[];

# 4/ checking

T  := unapply(eval( t(x), sol), x):
dT := unapply(eval(dt(x), sol), x):

 T~( X) -~  Y;
dT~(dX) -~ dY;

[-2, -1, 0, 1, 5/2, 14/5, 7/2]

 

[2, 0, -2, 0, -3, 0, 0]

 

[-2, 0, 1, 5/2]

 

[0, 0, 0, 0]

 

[b = -8786501831/326761419600, c = 551386670273/3267614196000, d = -206965577023/6535228392000, e = -19482815598053/13070456784000, f = 203690408213/108020304000, g = 50260227787849/13070456784000, h = -8245235930497/1188223344000, i = -4122869858861/1633807098000, j = 663981147407/93360405600, k = 0, l = -2]

 

[0, 0, 0, 0, 0, 0, 0]

 

[0, 0, 0, 0]

(7)

 


 

Download PolynomialFit.mw

If I understant correctly your problem, you want to assess the value of y2[n] = F(x2[n]) for each n=1..maxx.

If it is so here is a solution.
I have no doubt that someone else will prvide you a more concise/elegant solution


Sorry, I still get this ?!*&?!!  error when I try to load the content of the worksheet 
Maple Worksheet - Error
Failed to load the worksheet /maplenet/convert/Spline.mw .

 

Download Spline.mw

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