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tomleslie

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15 years, 175 days
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These are answers submitted by tomleslie

I'm guessing you mean exp() not 'e'...

tomleslie 13876
April 08 2018
0 0

So you need

y:=sin((x^2)/((x^10)+4))*exp(3*x);
evalf(subs(x=7,diff(y,x$8)));

Extra square brackets...

tomleslie 13876
April 08 2018
1 12

in the return statement of the procedure TETRA() - change this to

Tetra := proc(verts)
       local a,b,c,d;
       a := verts[1];
       b := verts[2];
       c := verts[3];
       d := verts[4];
       RETURN( [a,b,c], [a,b,d], [a,c,d], [b,c,d]);# removed a pair of [ ]
end:

Note also that the use of RETURN (ie capitalised has been deprecated for some time). Probably better to change all 'RETURN' statements to lower case, just to be safe

This has been answered before...

tomleslie 13876
April 07 2018
1 4

at least once

 See https://www.mapleprimes.com/questions/224310-Having-Trouble-Sorting-My-Solutions

I have no idea why the OP does not accept answers previously given: however more or less the same answer I gave at that time would be

> 

  restart:
#
# Set up expression an plot it
#
  S3 := -(1/2*I)*(-(2*I)*exp(I*Pi*k*tau/T)*Pi*k-exp(I*Pi*k*tau/T)*T+I*exp(I*Pi*k*tau/T)*Pi*k*tau+(4*I)*Pi*k-(2*I)*exp(-I*Pi*k*tau/T)*Pi*k+exp(-I*Pi*k*tau/T)*T+I*exp(-I*Pi*k*tau/T)*Pi*k*tau)*sin(2*Pi*k*x/T)/(Pi^2*k^2):
  S6 := (m, x)-> a[0]+Sum(S3, k = 1 .. m):
  m := 10: a[0] := 0: T := 4: tau := 2:
  Q1 := unapply( diff(S6(m, x), x), x):
  plot( Q1, 0..10);
#
# generate the first numRoots (set to 50, but this
# is arbitrary) of Q1
#
  with(RootFinding):
  first:=0:
  numRoots:=50:
  rts:=Vector(numRoots):
  for j from 1 by 1 to numRoots do
      rts[j]:= NextZero( Q1, first):
      first:=rts[j]:
  od:
  rts;

 

RTABLE(18446744074186145094, anything, Vector[column], rectangular, Fortran_order, [], 1, 1 .. 50)

 (1)

Download getRoots2.mw

I have no idea why the OP considers these answers unsatisfactory, or keeps reposting the same question without referenceing these answers!

My $0.02...

tomleslie 13876
April 06 2018
0 0

Based on the request in your original post, see the attached

> 

restart;
#
# Define expression
#
expr:=x*c*(diff(f(eta), eta))*epsilon/(-epsilon*t+1)^2+(1/2)*x*c^2*(diff(f(eta), eta, eta))*y*epsilon/((-epsilon*t+1)^3*sqrt(c/(nu*(-epsilon*t+1)))*nu)+x*c^2*(diff(f(eta), eta))^2/(-epsilon*t+1)^2-sqrt(c*nu/(-epsilon*t+1))*f(eta)*x*c*(diff(f(eta), eta, eta))*sqrt(c/(nu*(-epsilon*t+1)))/(-epsilon*t+1) = a*x*epsilon/(-epsilon*t+1)^2+a^2*x/(-epsilon*t+1)^2+x*c^2*(diff(f(eta), eta, eta, eta))/(-epsilon*t+1)^2+sqrt(2)*GAMMA*x^2*c^3*(diff(f(eta), eta, eta))*sqrt(c/(nu*(-epsilon*t+1)))*(diff(f(eta), eta, eta, eta))/(-epsilon*t+1)^3+A*g*beta[T]*theta(eta)*T[w]-A*g*beta[T]*theta(eta)*T[infinity]+A*g*beta[C]*phi(eta)*C[w]-A*g*beta[C]*phi(eta)*C[infinity]-sigma*B^2*x*c*(diff(f(eta), eta))/(rho*(-epsilon*t+1))+sigma*B^2*a*x/(rho*(-epsilon*t+1)):
#
# Isolate the requested term x*c^2*diff(f(eta), eta$3)
#
  subs( K=x*c^2*diff(f(eta), eta$3),
        isolate
        ( algsubs
          ( x*c^2*diff(f(eta), eta$3)=K,
            expr
          ),
          K
        )/(-epsilon*t+1)^2
      ):
  expr1:=lhs(%)=simplify(expand( rhs(%)));

x*c^2*(diff(diff(diff(f(eta), eta), eta), eta))/(-epsilon*t+1)^2 = ((epsilon*t-1)*nu*(-rho*(diff(f(eta), eta))^2*c^2*x-x*(epsilon*rho-sigma*B^2*(epsilon*t-1))*c*(diff(f(eta), eta))+A*rho*beta[T]*g*(epsilon*t-1)^2*(T[w]-T[infinity])*theta(eta)+A*rho*beta[C]*g*(epsilon*t-1)^2*(C[w]-C[infinity])*phi(eta)+((epsilon+a)*rho-sigma*B^2*(epsilon*t-1))*a*x)*(-c/(nu*(epsilon*t-1)))^(1/2)+(diff(diff(f(eta), eta), eta))*x*rho*(f(eta)*(epsilon*t-1)*(-nu*c/(epsilon*t-1))^(1/2)+(1/2)*y*epsilon)*c^2)/((-c/(nu*(epsilon*t-1)))^(1/2)*nu*(GAMMA*(diff(diff(f(eta), eta), eta))*(-c/(nu*(epsilon*t-1)))^(1/2)*2^(1/2)*c*x-epsilon*t+1)*rho*(epsilon*t-1)^2)

 (1)
> 

 

Download simpProb.mw

Many ways...

tomleslie 13876
April 04 2018
0 1

one of which would be

  restart;
  L := 0.5e-1: M := 6: h:=15: k := 180:
  b := 0.1e-1: deltaX := L/(M-1): theta:= arctan(b/(2*L)):
  eqs := T[0]=200, T[6]=25:
  for m to M-1 do
        eqs := eqs,
             (1-(m-1/2)*deltaX/L)*(T[m-1]-T[m])+(1-(m+1/2)*deltaX/L)*(T[m+1]-T[m])+h*deltaX^2/(k*L*sin(theta)) = 0
  end do:
  T:=Array(0..6, rhs~(convert( solve([eqs]), list)));

 

"Quick and dirty" version...

tomleslie 13876
April 04 2018
1 1
> 

  restart;
  with(plots):
#
# Set up problem: note that the selected initial
# conditions are arbitrary. The quantity 'mu'
# controls the non-linearity and the damping
#
  vdp:=diff(x(t),t$2)-mu*(1-x(t)^2)*diff(x(t),t)+x(t)=0:
  ics:=x(0)=0, D(x)(0)=-0.1:
  sol:=dsolve( [vdp, ics],
               numeric,
               parameters=[mu],
               maxfun=0
             ):
#
# Set up a list of values for the parameter 'mu'
# and plot the associated limit cycles
#
  muVals:= [0.2, 1.0, 2.0, 3.0, 4.0, 5]:
  colors:= [red, green, blue, orange, grey, purple, black]:
  for j from 1 by 1 to numelems(muVals) do
      sol( parameters=[ muVals[j] ] );
      p[j]:= odeplot
             ( sol,
               [x(t), diff(x(t),t)],
               t=100..200,
               numpoints=10000,
               color=colors[j],
               scaling=constrained
             );
  od:
  display( convert(p, list)
         );

 
> 

 
> 

 

Download vdp.mw

It depends...

tomleslie 13876
April 03 2018
0 1

You claim that

{C1 = -1/2*Pi, G = A, x = x}, {C1 = 1/2*Pi, G = -A, x = x}, {C1 = C1, G = G, x = arctan(cos(C1)*G/(G*sin(C1)+A))}

is the "correct, expected result". Actually it isn't "correct" because it omits an infinite number of possible solutions, which would be returned by the code

 restart;
  eq := G*cos(x+C1) = A*sin(x):
  sol2:=solve(eq, {G, C1, x}, allsolutions=true);

at the expense of introducing three (arbitrary) integers.

In your second case, removing 'x' from the list of variables to be solved for, Maple returns {C1 = C1, G = A*sin(x)/cos(x+C1)} , when you *want* {C1 = -1/2*Pi, G = A}, {C1 = 1/2*Pi, G = -A}. Again these are two *special* cases of the solution returned by Maple. You can try substituting your two "special" cases into the solution returned by Maple - and yes, they are two solutions out of an ininite number of others.

A similar argument can be applied to your third example

As a general rule, when an equation has an infinite number of possible solutions, Maple is not very good at guessing which two or three you happen to be thinking of.

Rossler System...

tomleslie 13876
April 03 2018
1 0

The Rossler system is actuallly used as one of the examples on the help page for IterativeMaps/Bifurcation.

Just type ?Bifurcation at the Maple prompt

Typos...

tomleslie 13876
April 03 2018
0 4

Notice that in your first output list of pairs, the lowest integer is 3, but in the second list it is 2 - as a result of a difference in the definition of 'GenerateOddPrimeNumberPairs'

Fixed 'signs' in the second definition of 'Identity'

Now both methods give the same answer - see attached

> 

NULL

 

 

delta(x, y) = piecewise(x = y, 1, x <> y, 0)

 

 

{x}*is*the*fractional*part*of*x

 

NULL

 

 

`&Mscr;`(p, q) = {`mod`(p, q)+(1/4)*p*(q-1)}

"`&Dscr;`(p,q)="
1-delta({(1/4)*p}, 0)-delta({(1/4)*q}, 0)+delta({(1/4)*q}, 0)*delta({(q-1)*(1/4)}, 0)+1/2*(delta({(q-2)*(1/4)}, 0)+delta({(1/4)*q}, 0)-delta({(q-2)*(1/4)}, 0)*delta({(q-3)*(1/4)}, 0))

 

For*all*odd*primes*p, q

 

WHEN ALL TERMS ARE PLACED ON THE LHS AND 0 ON THE OTHER, THE OUTPUT PLACES ALL p,q IN THE SET T, AS I EXPECTED.

> 

restart; S := {}; with(combinat); with(numtheory); T := {}; F := {}; C := {}; AlphaTotal := {}; BetaTotal := {}
> 

delta := proc (x, y) options operator, arrow; piecewise(x = y, 1, x <> y, 0) end proc
> 

Identity := proc (p, q) options operator, arrow; frac(`mod`(p, q)+(1/4)*p*(q-1))+delta(frac((1/4)*p), 0)+delta(frac((1/4)*q-1/4), 0)-delta(frac((1/4)*q), 0)*delta(frac((1/4)*q-1/4), 0)+(1/2)*delta(frac((1/4)*q-1/2), 0)+(1/2)*delta(frac((1/4)*q-3/4), 0)-(1/2)*delta(frac((1/4)*q-1/2), 0)*delta(frac((1/4)*q-3/4), 0)-1 = 0 end proc

proc (p, q) options operator, arrow; frac(`mod`(p, q)+(1/4)*p*(q-1))+delta(frac((1/4)*p), 0)+delta(frac((1/4)*q-1/4), 0)-delta(frac((1/4)*q), 0)*delta(frac((1/4)*q-1/4), 0)+(1/2)*delta(frac((1/4)*q-1/2), 0)+(1/2)*delta(frac((1/4)*q-3/4), 0)-(1/2)*delta(frac((1/4)*q-1/2), 0)*delta(frac((1/4)*q-3/4), 0)-1 = 0 end proc

 (1)
> 

GenerateOddPrimeNumberPairs := proc (N) options operator, arrow; choose([seq(ithprime(k), k = 2 .. N)], 2) end proc
> 

AssignToTrueOrFalseSet := proc (x, y) global T, F; if is(Identity(x, y)) = true then T := `union`({[x, y]}, T) else F := `union`({[x, y]}, F) end if end proc
> 

QueryIdentity := proc (N) local P, k; P := GenerateOddPrimeNumberPairs(N); for k to nops(P) do AssignToTrueOrFalseSet(P[k][1], P[k][2]) end do end proc
> 

QueryIdentity(20); T; F

{[3, 5], [3, 7], [3, 11], [3, 13], [3, 17], [3, 19], [3, 23], [3, 29], [3, 31], [3, 37], [3, 41], [3, 43], [3, 47], [3, 53], [3, 59], [3, 61], [3, 67], [3, 71], [5, 7], [5, 11], [5, 13], [5, 17], [5, 19], [5, 23], [5, 29], [5, 31], [5, 37], [5, 41], [5, 43], [5, 47], [5, 53], [5, 59], [5, 61], [5, 67], [5, 71], [7, 11], [7, 13], [7, 17], [7, 19], [7, 23], [7, 29], [7, 31], [7, 37], [7, 41], [7, 43], [7, 47], [7, 53], [7, 59], [7, 61], [7, 67], [7, 71], [11, 13], [11, 17], [11, 19], [11, 23], [11, 29], [11, 31], [11, 37], [11, 41], [11, 43], [11, 47], [11, 53], [11, 59], [11, 61], [11, 67], [11, 71], [13, 17], [13, 19], [13, 23], [13, 29], [13, 31], [13, 37], [13, 41], [13, 43], [13, 47], [13, 53], [13, 59], [13, 61], [13, 67], [13, 71], [17, 19], [17, 23], [17, 29], [17, 31], [17, 37], [17, 41], [17, 43], [17, 47], [17, 53], [17, 59], [17, 61], [17, 67], [17, 71], [19, 23], [19, 29], [19, 31], [19, 37], [19, 41], [19, 43], [19, 47], [19, 53], [19, 59], [19, 61], [19, 67], [19, 71], [23, 29], [23, 31], [23, 37], [23, 41], [23, 43], [23, 47], [23, 53], [23, 59], [23, 61], [23, 67], [23, 71], [29, 31], [29, 37], [29, 41], [29, 43], [29, 47], [29, 53], [29, 59], [29, 61], [29, 67], [29, 71], [31, 37], [31, 41], [31, 43], [31, 47], [31, 53], [31, 59], [31, 61], [31, 67], [31, 71], [37, 41], [37, 43], [37, 47], [37, 53], [37, 59], [37, 61], [37, 67], [37, 71], [41, 43], [41, 47], [41, 53], [41, 59], [41, 61], [41, 67], [41, 71], [43, 47], [43, 53], [43, 59], [43, 61], [43, 67], [43, 71], [47, 53], [47, 59], [47, 61], [47, 67], [47, 71], [53, 59], [53, 61], [53, 67], [53, 71], [59, 61], [59, 67], [59, 71], [61, 67], [61, 71], [67, 71]}

  

{}

 (2)

BUT WHEN THE EQUALITY IS DEFINED AS FOLLOWS, SOME RESULTS ARE PLACED IN THE F SET. (INDICATING THE EQUALITY FOR ALL [p,q] is FALSE)

> 

restart; S := {}; with(combinat); with(numtheory); T := {}; F := {}; C := {}; AlphaTotal := {}; BetaTotal := {}
> 

delta := proc (x, y) options operator, arrow; piecewise(x = y, 1, x <> y, 0) end proc
> 

Identity := proc (p, q) options operator, arrow; frac(`mod`(p, q)+(1/4)*p*(q-1)) = -delta(frac((1/4)*p), 0)-delta(frac((1/4)*q-1/4), 0)+delta(frac((1/4)*q), 0)*delta(frac((1/4)*q-1/4), 0)-(1/2)*delta(frac((1/4)*q-1/2), 0)-(1/2)*delta(frac((1/4)*q-3/4), 0)+(1/2)*delta(frac((1/4)*q-1/2), 0)*delta(frac((1/4)*q-3/4), 0)+1 end proc

proc (p, q) options operator, arrow; frac(`mod`(p, q)+(1/4)*p*(q-1)) = -delta(frac((1/4)*p), 0)-delta(frac((1/4)*q-1/4), 0)+delta(frac((1/4)*q), 0)*delta(frac((1/4)*q-1/4), 0)-(1/2)*delta(frac((1/4)*q-1/2), 0)-(1/2)*delta(frac((1/4)*q-3/4), 0)+(1/2)*delta(frac((1/4)*q-1/2), 0)*delta(frac((1/4)*q-3/4), 0)+1 end proc

 (3)
> 

GenerateNumberPairs := proc (N) options operator, arrow; choose([seq(ithprime(k), k = 2 .. N)], 2) end proc
> 

AssignToTrueOrFalseSet := proc (x, y) global T, F; if is(Identity(x, y)) = true then T := `union`({[x, y]}, T) else F := `union`({[x, y]}, F) end if end proc
> 

QueryIdentity := proc (N) local P, k; P := GenerateNumberPairs(N); for k to nops(P) do AssignToTrueOrFalseSet(P[k][1], P[k][2]) end do end proc
> 

QueryIdentity(20); T; F

{[3, 5], [3, 7], [3, 11], [3, 13], [3, 17], [3, 19], [3, 23], [3, 29], [3, 31], [3, 37], [3, 41], [3, 43], [3, 47], [3, 53], [3, 59], [3, 61], [3, 67], [3, 71], [5, 7], [5, 11], [5, 13], [5, 17], [5, 19], [5, 23], [5, 29], [5, 31], [5, 37], [5, 41], [5, 43], [5, 47], [5, 53], [5, 59], [5, 61], [5, 67], [5, 71], [7, 11], [7, 13], [7, 17], [7, 19], [7, 23], [7, 29], [7, 31], [7, 37], [7, 41], [7, 43], [7, 47], [7, 53], [7, 59], [7, 61], [7, 67], [7, 71], [11, 13], [11, 17], [11, 19], [11, 23], [11, 29], [11, 31], [11, 37], [11, 41], [11, 43], [11, 47], [11, 53], [11, 59], [11, 61], [11, 67], [11, 71], [13, 17], [13, 19], [13, 23], [13, 29], [13, 31], [13, 37], [13, 41], [13, 43], [13, 47], [13, 53], [13, 59], [13, 61], [13, 67], [13, 71], [17, 19], [17, 23], [17, 29], [17, 31], [17, 37], [17, 41], [17, 43], [17, 47], [17, 53], [17, 59], [17, 61], [17, 67], [17, 71], [19, 23], [19, 29], [19, 31], [19, 37], [19, 41], [19, 43], [19, 47], [19, 53], [19, 59], [19, 61], [19, 67], [19, 71], [23, 29], [23, 31], [23, 37], [23, 41], [23, 43], [23, 47], [23, 53], [23, 59], [23, 61], [23, 67], [23, 71], [29, 31], [29, 37], [29, 41], [29, 43], [29, 47], [29, 53], [29, 59], [29, 61], [29, 67], [29, 71], [31, 37], [31, 41], [31, 43], [31, 47], [31, 53], [31, 59], [31, 61], [31, 67], [31, 71], [37, 41], [37, 43], [37, 47], [37, 53], [37, 59], [37, 61], [37, 67], [37, 71], [41, 43], [41, 47], [41, 53], [41, 59], [41, 61], [41, 67], [41, 71], [43, 47], [43, 53], [43, 59], [43, 61], [43, 67], [43, 71], [47, 53], [47, 59], [47, 61], [47, 67], [47, 71], [53, 59], [53, 61], [53, 67], [53, 71], [59, 61], [59, 67], [59, 71], [61, 67], [61, 71], [67, 71]}

  

{}

 (4)
> 

NULL

Download KDR.mw

Numerous syntax issues...

tomleslie 13876
April 03 2018
0 1

and I have to guess what might be intended in places. Some of these 'guesses may be wrong!

See my notes in the attached

> 

restart;

with(LinearAlgebra):
with(linalg):
with(DifferentialGeometry):
with(VariationalCalculus):
with(Tensor):
with(tensor):
with(Tools):
DGsetup([t, x, y, z], M):
g1 := evalDG(N(t)^2*`&t`(dt, dt)-a(t)^2*(`&t`(dx, dx)+`&t`(dy, dy)+`&t`(dz, dz))):
g1inv := InverseMetric(g1):
C1 := Christoffel(g1):
RM1 := CurvatureTensor(C1):
RM1g := CurvatureTensor(g1):
ContractIndices(RM1, [[1, 3]]):
ContractIndices(RM1g, [[1, 3]]):
RT := RicciTensor(g1, RM1):
ContractIndices(RT, g1inv, [[1, 1], [2, 2]]):
S1 := RicciScalar(g1, RM1):
ContractIndices(RT, g1inv, [[1, 1], [2, 2]]):
RM1contra := RaiseLowerIndices(g1inv, RM1, [2, 3, 4]):
RM1cov := RaiseLowerIndices(g1, RM1, [1]):
Kret1 := ContractIndices(RM1contra, RM1cov, [[1, 1], [2, 2], [3, 3], [4, 4]]):
eval(simplify(subs(N(t) = 1, Kret1))):
Lag := a(t)^3*N(t)*Kret1:
phi = phi(t):
Lam := -(1/2)*a(t)^3*N(t)*((diff(phi, t))^2/N(t)^2+m^2*phi):
Ltot := Lag+Lam:
EqELN := EulerLagrange(Ltot, t, N(t)):
EqN := eval(simplify(subs(N(t) = 1, EqELN))):
Eqa := subs(N(t) = 1, Ltot):
EqELa := EulerLagrange(EqaD, t, a(t)):
tini := 0:
with(DEtools):
with(plots):
tfin = 10:
(D(a))(tini) = 0:
((D@@2)(a))(tini) = 0:
((D@@3)(a))(tini) = 0:
phi := a(t)^(-3): #changed 'a' to 'a(t)' NB overrides definition of phi above
Dphi := 0:
(D(N))(tini) = 0:
((D@@2)(N))(tini) = 0:
((D@@3)(N))(tini) = 0:
> 

tfin:= 2:
#
# EqN uses the variable 'm' which is unassigned
# So seet this equalt to 1 for now
#
  m:=1;
#
# EqN uses the dependent variable a(t) AND the
# unassigned variable 'a'. Should the latter
# be assigned to a numeric? or is a(t) intended?
# Only OP knows - but I set a:=a(t) in the definition
# of phi above. If it is meant to be a simple
# variable - then don't use 'a' as the name, and
# give it a value
#
# ode is of order 3, so needs 3 boundary conditions.
# OP has supplied 4, so I 'removed the last one
#
# With all of these edits, dsolve() now 'works', but
# hits a singularity at t= .81969892 so kill the
# range setting in the dsolve() command, and just
# control the range with the odeplot() command
#
# Not hard to see why Maple considers the system
# to be unstable at t=~.8 or so
#
  sys1 := {EqN[], a(tini) = 0.1e-2, D(a)(tini) = 0.1e-2, (D@@2)(a)(tini) = 0.1e-2}:# removed this boundary condition, (D@@3)(a)(tini) = 1}:
  p1 := dsolve(sys1, numeric, abserr = 1.*10^(-8), relerr = 1.*10^(-8));
  figN := odeplot(p1, [t, a(t)], t=0..0.8);
 

1

  

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 := []; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 26, [( 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..63, {(1) = 3, (2) = 3, (3) = 0, (4) = 0, (5) = 0, (6) = 0, (7) = 1, (8) = 0, (9) = 0, (10) = 0, (11) = 0, (12) = 0, (13) = 0, (14) = 0, (15) = 0, (16) = 0, (17) = 0, (18) = 1, (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, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = .0, (2) = 0.10e-7, (3) = .0, (4) = 0.500001e-14, (5) = .0, (6) = 0.9655289077439908e-6, (7) = .0, (8) = 0.10e-7, (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..3, {(1) = 0.10e-2, (2) = 0.10e-2, (3) = 0.10e-2}, datatype = float[8], order = C_order)), ( 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..3, {(1) = 1.0, (2) = 1.0, (3) = 1.0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, 1..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0}, datatype = float[8], order = C_order), Array(1..3, 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}, datatype = float[8], order = C_order), Array(1..3, {(1) = 0, (2) = 0, (3) = 0}, datatype = integer[8]), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..6, {(1) = .0, (2) = .0, (3) = .0, (4) = .0, (5) = .0, (6) = .0}, datatype = float[8], order = C_order)]), ( 8 ) = ([Array(1..3, {(1) = 0.10e-2, (2) = 0.10e-2, (3) = 0.10e-2}, datatype = float[8], order = C_order), Array(1..3, {(1) = .0, (2) = .0, (3) = .0}, datatype = float[8], order = C_order), Array(1..3, {(1) = 0.10e-2, (2) = 0.10e-2, (3) = 20833.33433333333}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..3, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (2, 0) = .0, (2, 1) = .0, (2, 2) = .0, (2, 3) = .0, (3, 0) = .0, (3, 1) = .0, (3, 2) = .0, (3, 3) = .0, (4, 0) = .0, (4, 1) = .0, (4, 2) = .0, (4, 3) = .0, (5, 0) = .0, (5, 1) = .0, (5, 2) = .0, (5, 3) = .0, (6, 0) = .0, (6, 1) = .0, (6, 2) = .0, (6, 3) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = a(t), Y[2] = diff(a(t),t), Y[3] = diff(diff(a(t),t),t)]`; YP[3] := -(1/48)*(-24*Y[1]^2*Y[3]^2+48*Y[1]*Y[2]^2*Y[3]-72*Y[2]^4-Y[1])/(Y[1]^2*Y[2]); YP[1] := Y[2]; YP[2] := Y[3]; 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] = a(t), Y[2] = diff(a(t),t), Y[3] = diff(diff(a(t),t),t)]`; YP[3] := -(1/48)*(-24*Y[1]^2*Y[3]^2+48*Y[1]*Y[2]^2*Y[3]-72*Y[2]^4-Y[1])/(Y[1]^2*Y[2]); YP[1] := Y[2]; YP[2] := Y[3]; 0 end proc, -1, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0) ] )) ] ); _y0 := Array(0..3, {(1) = 0., (2) = 0.1e-2, (3) = 0.1e-2}); _vmap := array( 1 .. 3, [( 1 ) = (1), ( 2 ) = (2), ( 3 ) = (3) ] ); _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 _dat[17] <> _dtbl[1][17] then _dtbl[1][17] := _dat[17]; _dtbl[1][10] := _dat[10] end if; 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, a(t), diff(a(t), t), diff(diff(a(t), t), t)], (4) = []}); _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 := 1; _ndsol := _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

  

 
M > 

#
# EqELa is nothing like an ODE - so no idea what to
# do with this, but rather obvious that dsolve()
# won't have a clue either!
#
EqELa;
sys2 := {EqELa, a(tini) = 0.1e-6, (D(a))(tini) = 0.1e-4, ((D@@2)(a))(tini) = 0.1e-5, ((D@@3)(a))(tini) = 1, ((D@@4)(a))(tini) = 0.1e-5};
p2 := dsolve(sys2, type = numeric, abserr = 1.*10^(-8), relerr = 1.*10^(-8), range = tini .. tfin);
figa := odeplot(p2, [t, a(t)]);

{0, 0 = K[1], EqaD = K[2]}

  

{a(0) = 0.1e-6, (D(a))(0) = 0.1e-4, ((D@@2)(a))(0) = 0.1e-5, ((D@@3)(a))(0) = 1, ((D@@4)(a))(0) = 0.1e-5, {0, 0 = K[1], EqaD = K[2]}}

  

Error, (in dsolve/numeric/process_input) system must be entered as a set/list of expressions/equations

  

Error, (in plots/odeplot) input is not a valid dsolve/numeric solution

  
M > 

 

Download odeProb.mw

The first 50 real roots in ascending ord...

tomleslie 13876
April 02 2018
0 1

can be obtained with the following. (BTW I have a feeling  have done this before - but I can't find it!?)

> 

restart;
S3 := -(1/2*I)*(-(2*I)*exp(I*Pi*k*tau/T)*Pi*k-exp(I*Pi*k*tau/T)*T+I*exp(I*Pi*k*tau/T)*Pi*k*tau+(4*I)*Pi*k-(2*I)*exp(-I*Pi*k*tau/T)*Pi*k+exp(-I*Pi*k*tau/T)*T+I*exp(-I*Pi*k*tau/T)*Pi*k*tau)*sin(2*Pi*k*x/T)/(Pi^2*k^2):
> 

S5 := (m, x)->a[0]+sum(S3, k = 1 .. m):
> 

T := N*tau:
m := 10: a[0] := 0: N := 2: tau := 2:
Q1 := unapply(diff(S5(m, x), x),x):
plot(Q1, 0..10);
with(RootFinding):
first:=0:
numRoots:=50:
sol:=Vector(numRoots):
for i from 1 by 1 to numRoots do
    sol[i]:=NextZero(Q1, first);
    first:=sol[i]:
od:
sol;

 

RTABLE(18446744074187322974, anything, Vector[column], rectangular, Fortran_order, [], 1, 1 .. 50)

 (1)
> 

 

       

``

Download getRoots.mw

backticks...

tomleslie 13876
April 02 2018
1 0

Enclose the offending symbol in backticks, to avoid MAple trying teh "evaluate" it, as in

> 

with(StringTools)
> 

Join([convert(A, 'string'), "=", convert(B, 'string'), "(mod", convert(Set2[1], 'string'), ")"])

"A = B (mod Set2[1] )"

 (1)
> 

convert(`&equiv;`, 'string')

"&equiv;"

 (2)
> 

 

Download badsymb.mw

Hmmm...

tomleslie 13876
April 02 2018
1 4

I can't generate a difference between Maple 18 and Maple 2017. For both versions, ImportVector() seems to use "hardware floats" by default. For 64-bit OS and 64-bit Maple versions, this means that I get about 15 digits from ImportVector.

Relatively easy to fix, however - just set the datatype on the import command to 'sfloat', as in

ImportVector( filename, datatype=sfloat);

If you are really getting a difference between Maple 18 and Maple 2017, then maybe you had something set up in your maple.ini file for the former, which "forced" the use of software floats at all times???

Another alternative...

tomleslie 13876
April 01 2018
0 0

   restart;
   randomize():
   with(RandomTools):
   P:= Generate(choose({-1,1}), makeproc=true):
#
# 10 random numbers in a list
#
   L:=[seq( P(), k=1..10)];
#
# or another 10 random numbers as a vector
#
   V:=Vector[row]( [seq( P(), k=1..10)] );
#
# or 100 random numbers in a 10x10 matrix
#
   Matrix(10, 10, (i,j)->P());

 

 

My $0.02...

tomleslie 13876
March 31 2018
1 1

Kitonum's trick is great - until you change the plot size, in whihc case the number of spaces required in the tile varies - cos for a specified font size, the widht of the text string will be a fixed number of pixels

Rouben's also fine, except you have to "know" the coordinates of the top-right corner of the data plot in order for the textplot 'align' option to work correctly. Probably better to get the top-right corrdinates of the data plot using getdata()

So my solution

> 

restart;
with(plots):
with(plottools):
params := beta=1, alpha[1]=.5:
f := beta+alpha[1]+sin(beta*x):
> 

p1:= plot( eval(f, [params]),
           x = 0..Pi,
           labels = [x, "f'(x)"],
           labeldirections = [horizontal, vertical]
         ):
p2:= textplot( [ op( [2,1,2], getdata(p1)),
                 op( [2,2,2], getdata(p1)),
                 typeset( "%1=1, %2=0.5\n\n",
                          beta, alpha[1]
                        )
               ],
               align = [left],
               font = [ "Arial", 10, Bold ]
             ):
display( [ p1, p2 ],
         axes = boxed,
         size = [300, 270]
       );

 
> 

 

Download placeTitle.mw

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