Carl Love

Carl Love

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11 years, 280 days
Himself
Wayland, Massachusetts, United States
My name was formerly Carl Devore.

MaplePrimes Activity


These are answers submitted by Carl Love

In your Case 2, solve chose (somewhat arbitrarily) two different branches of LambertW, but any branch would work. If you change all instances of solve(...to solve(..., allsolutions), I think you'll understand what's happening. If not, I'll explain further.

First, pick a value of Kr to use. I chose 0.1. Then do

Kr:= 0.1:
F:= [f, Theta, Phi](eta):
Frame:= Inf-> plots:-odeplot(
    dsolve({OdeSys, eval([Cond], 10= Inf)[]}, numeric),
    `[]`~(eta, F), legend= F
):
plots:-animate(Frame, [Inf], Inf= 0.1..10, frames= 50, size= 500*[2,1]); 


 

All you need is 

(A,B):= convert~(Solution, list, nested)[]: print('A'=A, 'B'= B);

6Here's a procedure for it:

#Bins takes a histogram and its data sample and returns a list of "bin-bounds" = count.
Bins:= (
    H::And(specfunc(PLOT), patfunc(specfunc(POLYGONS), anything)),
    S::{rtable,list}(numeric)
)-> 
local `&<`:= curry, `&>`:= rcurry;
    (Statistics:-TallyInto &< S &> _rest @ (op&<[1,1]..op&<[2,1])~)(
        [indets(op(1,H), And(listlist(numeric), 4 &under nops))[]]
    )
:

Applying it to the histogram in your worksheet:

Bins(Q,A);   

[-2.71396014591046 .. -2.66663223217829 = 2, -2.66663223217829 .. -2.61930431844611 = 0, -2.61930431844611 .. -2.57197640471394 = 0, 
-2.57197640471394 .. -2.52464849098177 = 2, -2.52464849098177 .. 2.4773205772496 = 0, -2.4773205772496 .. -2.42999266351743 = 1, -2.42999266351743 .. -2.38266474978526 = 1, -2.38266474978526 .. 
-2.33533683605308 = 2, -2.33533683605308 .. -2.28800892232091 = 1, 
-2.28800892232091 .. -2.24068100858874 = 3, -2.24068100858874 .. 
-2.19335309485657 = 1, -2.19335309485657 .. -2.1460251811244 = 0, 
-2.1460251811244 .. -2.09869726739223 = 4, -2.09869726739223 .. 
-2.05136935366005 = 0, -2.05136935366005 .. -2.00404143992788 = 1, 
-2.00404143992788 .. -1.95671352619571 = 3, -1.95671352619571 .. 
-1.90938561246354 = 5, -1.90938561246354 .. -1.86205769873137 = 2, 
-1.86205769873137 .. -1.8147297849992 = 4, -1.8147297849992 .. 
-1.76740187126702 = 4, -1.76740187126702 .. -1.72007395753485 = 2, 
-1.72007395753485 .. -1.67274604380268 = 3, -1.67274604380268 .. 
-1.62541813007051 = 4, -1.62541813007051 .. -1.57809021633834 = 2, 
-1.57809021633834 .. -1.53076230260617 = 3, -1.53076230260617 .. 
-1.48343438887399 = 5, -1.48343438887399 .. -1.43610647514182 = 7, 
-1.43610647514182 .. -1.38877856140965 = 7, -1.38877856140965 .. 
-1.34145064767748 = 7, -1.34145064767748 .. -1.29412273394531 = 6, 
-1.29412273394531 .. -1.24679482021313 = 4, -1.24679482021313 .. 
-1.19946690648096 = 12, -1.19946690648096 .. -1.15213899274879 = 6, 
-1.15213899274879 .. -1.10481107901662 = 9, -1.10481107901662 .. 
-1.05748316528445 = 6, -1.05748316528445 .. -1.01015525155228 = 13, 
-1.01015525155228 .. -.962827337820104 = 18, -.962827337820104 .. 
-.915499424087933 = 8, -.915499424087933 .. -.868171510355761 = 19, 
-.868171510355761 .. -.820843596623589 = 15, -.820843596623589 .. 
-.773515682891417 = 9, -.773515682891417 .. -.726187769159246 = 15, 
-.726187769159246 .. -.678859855427074 = 17, -.678859855427074 .. 
-.631531941694902 = 11, -.631531941694902 .. -.584204027962731 = 19, 
-.584204027962731 .. -.536876114230559 = 11, -.536876114230559 .. 
-.489548200498387 = 21, -.489548200498387 .. -.442220286766215 = 13, 
-.442220286766215 .. -.394892373034044 = 20, -.394892373034044 .. 
-.347564459301872 = 15, -.347564459301872 .. -.3002365455697 = 24, 
-.3002365455697 .. -.252908631837528 = 12, -.252908631837528 .. 
-.205580718105357 = 21, -.205580718105357 .. -.158252804373185 = 16, 
-.158252804373185 .. -.110924890641013 = 22, -.110924890641013 .. 
-.0635969769088414 = 23, -.0635969769088414 .. -.0162690631766695 = 25, 
-.0162690631766695 .. .0310588505555018 = 14, .0310588505555018 .. 
.0783867642876737 = 18, .0783867642876737 .. .125714678019845 = 22, 
.125714678019845 .. .173042591752017 = 18, .173042591752017 .. .220370505484189
= 19, .220370505484189 .. .267698419216361 = 22, .267698419216361 .. 
.315026332948532 = 14, .315026332948532 .. .362354246680704 = 22, 
.362354246680704 .. .409682160412876 = 10, .409682160412876 .. .457010074145048
= 20, .457010074145048 .. .504337987877219 = 10, .504337987877219 .. 
.551665901609391 = 7, .551665901609391 .. .598993815341563 = 17, 
.598993815341563 .. .646321729073735 = 21, .646321729073735 .. .693649642805906
= 11, .693649642805906 .. .740977556538078 = 15, .740977556538078 .. 
.78830547027025 = 15, .78830547027025 .. .835633384002421 = 12, 
.835633384002421 .. .882961297734593 = 13, .882961297734593 .. .930289211466765
= 13, .930289211466765 .. .977617125198937 = 14, .977617125198937 .. 
1.02494503893111 = 10, 1.02494503893111 .. 1.07227295266328 = 11, 
1.07227295266328 .. 1.11960086639545 = 13, 1.11960086639545 .. 1.16692878012762
= 7, 1.16692878012762 .. 1.2142566938598 = 8, 1.2142566938598 .. 
1.26158460759197 = 7, 1.26158460759197 .. 1.30891252132414 = 13, 
1.30891252132414 .. 1.35624043505631 = 11, 1.35624043505631 .. 1.40356834878848
= 11, 1.40356834878848 .. 1.45089626252065 = 9, 1.45089626252065 .. 
1.49822417625283 = 6, 1.49822417625283 .. 1.545552089985 = 9, 1.545552089985 ..
1.59288000371717 = 6, 1.59288000371717 .. 1.64020791744934 = 9, 
1.64020791744934 .. 1.68753583118151 = 8, 1.68753583118151 .. 1.73486374491368
= 4, 1.73486374491368 .. 1.78219165864586 = 4, 1.78219165864586 .. 
1.82951957237803 = 3, 1.82951957237803 .. 1.8768474861102 = 3, 1.8768474861102
.. 1.92417539984237 = 3, 1.92417539984237 .. 1.97150331357454 = 6, 
1.97150331357454 .. 2.01883122730671 = 6, 2.01883122730671 .. 2.06615914103889
= 4, 2.06615914103889 .. 2.11348705477106 = 1, 2.11348705477106 .. 
2.16081496850323 = 1, 2.16081496850323 .. 2.2081428822354 = 3, 2.2081428822354
.. 2.25547079596757 = 2, 2.25547079596757 .. 2.30279870969975 = 0, 
2.30279870969975 .. 2.35012662343192 = 2, 2.35012662343192 .. 2.39745453716409
= 2, 2.39745453716409 .. 2.44478245089626 = 0, 2.44478245089626 .. 
2.49211036462843 = 0, 2.49211036462843 .. 2.5394382783606 = 1, 2.5394382783606
.. 2.58676619209278 = 1, 2.58676619209278 .. 2.63409410582495 = 1, 
2.63409410582495 .. 2.68142201955712 = 0, 2.68142201955712 .. 2.72874993328929
= 0, 2.72874993328929 .. 2.77607784702146 = 0, 2.77607784702146 .. 
2.82340576075363 = 0, 2.82340576075363 .. 2.87073367448581 = 1, 
2.87073367448581 .. 2.91806158821798 = 0, 2.91806158821798 .. 2.96538950195015
= 1]

You asked:

  • I am unable to add a constraint with if statement.

The if syntax that you were using isn't allowed in 2D Input. Anyway the same thing can be better expressed with max because it'll remain symbolic until the solver supplies values for the variables, whereas an if evaluates immediately.

eq1:= Q2 = max(0, d*h*(x - delta*(alpha+beta)));

  • I have to minimize the function TRC, i am unable to do. 

You need a comma after the I2 = 0..8.

  • Check whether i have represented I1 and I2 with respect to alpha and beta right or wrong.

I think that I'd need substantial background information on the problem to answer that.

Anyway, making the two minor syntax changes that I've given, the solver will at least return an answer, which is a step in the right direction, even though the ensuing discussion indicates that there are still some logic errors to work out.

You've been making the same mistakes over and over, for years now. In your 2D Input, you need to remove any spaces after displaytextplot, and draw. Also, the arguments to display need to be in parentheses, not square brackets, just like all other procedural commands. These are exactly the same problems as the last Question of yours that I Answered.

Even a crystal ball cannot solve a nonsensical problem. You cannot simultaneously solve n equations for more than n variables (n=1 in this case). That is a fundamental principle of mathematics. It is beyond being an undecidable problem (such as the Continuum Hypothesis or the propositions whose existence is guaranteed by Godel's Theorem). It makes no sense at all.

Following the directions given on the Wikipedia page "Weibull distribution", section "Parameter estimation", subsection "Ordinary least square using Weibull plot", I only made a single small change to your computation: I changed (i - 0.5)/n to (i - 0.3)/(n + 0.4). Anything else that may appear to be a change is merely a mathematically equivalent simplification. (In particular, there's no need to use both sort and Rank; using either one is sufficient.) Then I used Statistics:-LinearFit to do the regression.

restart
:

St:= Statistics:

S:= -<-255.172, -235.249, -196.935, -132.567, -77.3946, -32.1839, -0.766284>;

Vector(7, {(1) = 255.172, (2) = 235.249, (3) = 196.935, (4) = 132.567, (5) = 77.3946, (6) = 32.1839, (7) = .766284})

n:= St:-Count(S);

7

lnpof:= map(i-> ln(-ln(1-(i-0.3)/(n+0.4))), St:-Rank(S)^+);

Vector(7, {(1) = .8578795099, (2) = .3858416541, (3) = 0.8194559696e-2, (4) = -.3665129205, (5) = -.7898398342, (6) = -1.343181903, (7) = -2.308880127})

lnsigma:= ln~(S);

Vector(7, {(1) = 5.541937827582851, (2) = 5.46064452765943, (3) = 5.282873725054183, (4) = 4.887088178000508, (5) = 4.34891701071996, (6) = 3.4714663275086806, (7) = -.26620242077484896})

plot((data:= <lnsigma | lnpof>), style= point);

PV:= St:-LinearFit([1, ln_x], data, ln_x, output= parametervector, summarize);

Summary:
----------------
Model: -2.4778919+.47999708*ln_x
----------------
Coefficients:
              Estimate  Std. Error  t-value  P(>|t|)
Parameter 1   -2.4779    0.4212     -5.8834   0.0020
Parameter 2    0.4800    0.0931      5.1576   0.0036
----------------
R-squared: 0.8418, Adjusted R-squared: 0.8101

Vector(2, {(1) = -2.4778918861403914, (2) = .47999707789937385})

k:= PV[2]; lambda:= exp(-PV[1]/k);

HFloat(0.47999707789937385)

HFloat(174.56657533276328)

On the Maple help page ?Weibull, k is called c, and lambda is called b.

X:= St:-RandomVariable(Weibull(lambda,k)):

plot(St:-PDF(X,t), t= 0..9);

 

Download WeibullParameters.mw

This Answer is fundamentally the same as the one by @nm ; I just want to give some background information.

The command evalb, when applied to an equation, checks whether the two sides of the equation are syntactically identical; but you want to check whether the two sides are mathematically equivalent.

Two expressions are syntactically identical if they are stored in exactly the same way in the computer's memory. (In Maple, in practice, this means that only one copy is actually stored at all.) Via a process that Maple calls automatic simplification, some expressions which are typed in differently will become syntactically identical:

evalb(sin(a+b) = sin(b+a));
             
true

I think that you (and most other readers here) already understand basically what it means for two expressions to be mathematically equivalent. Let me try to be more precise (although this is still not a perfect definition): Two expressions are mathematically equivalent if they are numerically equal for all possible numeric substitutions of their variables for which they're both defined and which satisfy the "current assumptions". The "current assumptions", if unspecified, are all complex numbers. In Maple, assuming or assume can be used to specify assumptions. 

Of course, if two expressions are syntactically identical they are also mathematically equivalent. But the converse isn't true. It can be proven (see Richardson's Theorem) that it's impossible to create a perfect algorithm to determine mathematical equivalence. So, whether you use isexpandsimplify, or numerous other commands, there will always be undecidable cases. However, one may hope that in the cases where the equivalence is already established via a well-known and elementary theorem (such as the case you present), Maple will be able to verify it. The best command to try first is is:

is(sin(x + y) = sin(x)*cos(y) + cos(x)*sin(y));
              true

The reason that is is the best first choice is that it will give up and return FAIL if after some reasonable amount of time it cannot decide true or false. Other commands may run for an infeasible amount of time.

Regarding checking integrations: Suppose that f is the expression that you want to integrate (let's say with respect to x), and is your proposed antiderivative that you want to check. Then do

is(f = diff(F,x));

You may need to include assumptions on x. Appropriate assumptions are likely obvious. For example,

is(f = diff(F,x)) assuming x > -1, x < 1;

There's usually no need to exclude isolated singularities such as specific values of x that make a denominator 0.

Keep in mind that the default assumption in Maple is "all complex numbers", not "all real numbers". The calculus of functions of complex variables is sufficiently similar to that of real variables that you usually do not need to make the distinction.

 

Level curves and orthogonal trajectories via numeric solution of an ODE

Author: Carl Love <carl.j.love@gmail.com>, 2024-May-24

 

restart
:

interface(prompt= "")
:

We are working with the level curves in the plane for this function:

F:= (x,y)-> x*y^2 - x^2 - y^2;

proc (x, y) options operator, arrow; x*y^2-x^2-y^2 end proc

In other words, the curves specified implicitly by the equation

F(x,y) = k;

x*y^2-x^2-y^2 = k

for any real constant k. Trajectories orthogonal to the level curves are important because they are the projections into the coordinate plane of the paths of steepest ascent/descent on the corresponding surface. Slopes of orthogonal trajectories are the negative reciprocals of the implicit derivative dy/dx, which we find by partial differentiation: dy/dx = - (dF/dx)/(dF/dy), so the orthogonal direction is (dF/dy)/(dF/dx). An unknown (aka dependent) function needs to be expressed "as a function of" its independent variable (x in this case). In other words (i.e., without symbolic-computation jargon), we need to change y to y(x).

Tj_ode:= diff(y(x),x) = (D[2]/D[1])(F)(x, y(x));

diff(y(x), x) = (2*y(x)*x-2*y(x))/(y(x)^2-2*x)

Note that when using D operator for partial derivatives, you don't specify the differentiation variable directly; instead, you specify its position in the function's argument sequence. So, D[2] is the derivative with respect to (w.r.t.) y, and D[1] is w.r.t. x.

 

It's reasonable to guess that Maple can solve this ODE symbolically, but it can't do it directly:

dsolve(Tj_ode);

The NULL response to the above command shows that Maple cannot solve it symbolically, unless, perhaps some other options are given. I choose not to pursue that possibility, because a numeric solve is totally adequate for this.

 

I name the starting point (x0, y0) and ask dsolve to construct a numeric solver for this generic starting point:

Tj_sol:= dsolve({Tj_ode, y(x0)=y0}, numeric, parameters= [x0, y0]);

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 := [x0 = x0, y0 = y0]; _dtbl := array( 1 .. 4, [( 1 ) = (array( 1 .. 28, [( 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..65, {(1) = 1, (2) = 1, (3) = 0, (4) = 0, (5) = 2, (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, (55) = 0, (56) = 0, (57) = 0, (58) = 0, (59) = 10000, (60) = 0, (61) = 1000, (62) = 0, (63) = 0, (64) = -1, (65) = 0}, datatype = integer[8])), ( 5 ) = (Array(1..28, {(1) = undefined, (2) = 0.10e-5, (3) = .0, (4) = 0.500001e-14, (5) = undefined, (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..3, {(1) = y0, (2) = Float(undefined), (3) = 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..1, {(1) = .1}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..1, {(1, 1) = .0}, datatype = float[8], order = C_order), Array(1..1, 1..6, {(1, 1) = .0, (1, 2) = .0, (1, 3) = .0, (1, 4) = .0, (1, 5) = .0, (1, 6) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 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..1, {(1) = .0}, datatype = float[8], order = C_order), Array(1..2, {(1) = .0, (2) = .0}, datatype = float[8], order = C_order), Array(1..1, {(1) = 0}, datatype = integer[8])]), ( 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..1, {(1) = .0}, datatype = float[8], order = C_order), 0, 0]), ( 11 ) = (Array(1..6, 0..1, {(1, 1) = .0, (2, 0) = .0, (2, 1) = .0, (3, 0) = .0, (3, 1) = .0, (4, 0) = .0, (4, 1) = .0, (5, 0) = .0, (5, 1) = .0, (6, 0) = .0, (6, 1) = .0}, datatype = float[8], order = C_order)), ( 10 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x)]`; YP[1] := (2*X*Y[1]-2*Y[1])/(Y[1]^2-2*X); 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), ( 13 ) = (), ( 12 ) = (), ( 15 ) = ("rkf45"), ( 14 ) = ([0, 0]), ( 18 ) = ([]), ( 19 ) = (0), ( 16 ) = ([0, 0, 0, 0, 0, []]), ( 17 ) = ([proc (N, X, Y, YP) option `[Y[1] = y(x)]`; YP[1] := (2*X*Y[1]-2*Y[1])/(Y[1]^2-2*X); 0 end proc, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), ( 22 ) = (0), ( 23 ) = (0), ( 20 ) = ([]), ( 21 ) = (0), ( 27 ) = (""), ( 26 ) = (Array(1..0, {})), ( 25 ) = (Array(1..0, {})), ( 24 ) = (0), ( 28 ) = (0)  ] ))  ] ); _y0 := Array(0..3, {(1) = x0, (2) = y0, (3) = undefined}); _vmap := array( 1 .. 1, [( 1 ) = (1)  ] ); _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); _i := false; if _par <> [] then _i := `dsolve/numeric/process_parameters`(_n, _pars, _par, _y0) end if; if _ini <> [] then _i := `dsolve/numeric/process_initial`(_n-_ne, _ini, _y0, _pars, _vmap) or _i end if; if _i then `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 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 elif type(_xin, `=`) and lhs(_xin) = "setdatacallback" then if not type(rhs(_xin), 'nonegint') then error "data callback must be a nonnegative integer (address)" end if; _dtbl[1][28] := rhs(_xin) 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]; if type(_EnvDSNumericSaveDigits, 'posint') then _dat[4][26] := _EnvDSNumericSaveDigits else _dat[4][26] := Digits end if; _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) = [x, y(x)], (4) = [x0 = x0, y0 = y0]}); _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

Note that the order in which I specified the parameters will be important later. Maple's cryptic response to the above command simply means everything is fine and the numeric colver has been constructed.

 

Get a 2D contour plot, choosing somewhat arbitrary ranges for x and y:

plots:-contourplot(
    (Cspec:= (F(x,y):= lhs(C)), (x_rng:= x= -2..2), (y_rng:= y= -2..2))
);

The color bar shows the value of k for the level curve of the corresponding color.  The color bar clutters the final plot (IMO), so I remove it, and I add some additional k values to fill in the center. Suitable values of k can be guessed by mosuing over the above plot (clickiing on any level curve shows its k value).

Cplot:= plots:-contourplot(
    (Cspec:= Cspec, 'contours'= [seq(-0.4..-2, -0.4), seq(-3..-15, -2)]),
    'colorbar'= false, thickness= 0.7
);

Even without the color bar, clicking on level curves still shows their k values.

 

A 3D-plot will help with understanding what a "steepest path" means:

surf_plot:= plot3d(Cspec, 'style'= 'surfacecontour', 'transparency'= 0.3);

Make up any two starting points (there's no limit on how many points you use), and record the count of them for convenience:

Pts:= [[-1, 1/2], [1, -1]]:
npts:= nops(Pts);

2

Plot the orthogonal trajectories in the coordinate plane:

plots:-display(  #command to merge plots
    Cplot,
    (
        #Iterate over Pts, setting P= [x0, y0] and setting j to the numeric
        #position of P in Pts:
        for j,P in Pts do
            Tj_sol('parameters'= P);  #Reset 'parameters' for each point.
            plots:-odeplot(  #plotter for numeric dsolve solutions
                Tj_sol, [x,y(x)], x_rng,
                'thickness'= 3, 'linestyle'= 'dash',
                #Use pure chromatic colors ("HUEs") evenly spaced on the
                #visible spectrum:
                'color'= 'COLOR'('HUE', 0.86*(j-1)/npts),
                'legend'= 'typeset'(
                    ``(x[0],y[0]) = P[],
                    #Punctuate _between_ legends, but not at end:
                    `if`(j<npts, ";   ", "")
                )
            )
        od
    ),
    'view'= rhs~([x_rng, y_rng]),
    #Scale so that right-angle intersections display "true":
    'scaling'= 'constrained',
    'title'= 'typeset'(
        "Level curves of ", F(x,y), "and some orthogonal trajectories"
    ),
    'titlefont'= ['TIMES', 'BOLD', 16],
    'caption'= "trajectories for starting points",
    'captionfont'= ['TIMES', 14],
    'size'= [800$2]  #800x800 pixels
);

By doing essentially the same thing, we can plot the corresponding "steepest-descent paths" on the surface

plots:-display(  
    surf_plot,
    (
        for j,P in Pts do
            Tj_sol('parameters'= P);  
            plots:-odeplot(
                Tj_sol, [x,y(x),F(x,y(x))], x_rng,
               'thickness'= 2, 'color'= 'COLOR'('HUE', 0.86*(j-1)/npts)
            )
        od
    ),
    'view'= [rhs~([x_rng, y_rng])[], -16..0]
);

 

Download PlotSteepestPath.mw

printlevel is an "environment variable". That means that if its value is changed within a procedure, then it is automatically reset to its previous value when the procedure exits. Thus, you don't need to store the previous value or reset printlevel yourself.

What you're calling the "depth" can be measured by kernelopts(level).

I ignore your weird use of RETURN. Without it, the second integral gives true*x, whereas I'd expect false*x. There are reasonable limitations on the "depth" within an expression to which assuming can penetrate. Compare:

(coulditbe(y=1) assuming y>=0, y<=x) assuming additionally, x>0, x<1;
                             
true

coulditbe(y=1) assuming y>=0, y<=x, x>0, x<1;
                           
 false

The first case seems more akin to your integral.

Your graph_list can be partitioned into a table of equivalence classes under isomorphism by

ListTools:-Classify(g-> [seq](op(4, GraphTheory:-CanonicalGraph(g))), graph_list)

Define the function with unapply:

f:= unapply(a*x, x)

A synonym for unapply is MakeFunction.

I assume that you're using 2D Input which you've converted to 1D for posting here. If so, then you need to remove any space characters you have after displaytextplot, and draw. Also, the arguments to display need to be in parentheses. If I change your display command to

display(
    textplot(
       [
           [coordinates(S)[], "S"], [coordinates(P)[], "P"], [coordinates(H)[], "H"], 
           [coordinates(K)[], "K"], [coordinates(M)[], "M"]
       ], font = [times, bold, 16], align = [above, right]
    ), 
    draw([
        delta(color = blue), deltap(color = blue), D(color = red), c1(color = black),
        S(color = black, symbol = solidcircle, symbolsize = 16),
        P(color = black, symbol = solidcircle, symbolsize = 16)
    ]), 
    scaling = constrained, axes = none, view = [-15 .. 15, -15 .. 15]
);

 then I get this plot (for which you'll likely want to change some of the sizes):

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