MaplePrimes - answers and comments on Question, Problem finding all positive roots of equations using fsolve

with DirectSearch

Markiyan Hirnyk — Sat, 21 Jan 2012 12:12:09 Z

Applying the DirectSearch package, I obtain ( sys is rewritten as a list up to the SolveEquations command syntax):
>with(DirectSearch):
>SolveEquations(eval(sys,w=2),{a1 = 0 .. 10, a2 = 0 .. 10, a3 = 0 .. 10, a4 = 0 .. 10, a5 = 0 .. 10, a6 = 0 .. 10, t = 0.1 .. 10},evaluationlimit=150000);
[
[2.82524552194119 10^(-14)   , Vector[column](%id = 165322972), [

a1 = HFloat(0.725100489728518038), a2 = HFloat(0.0950383729543587963),

a3 = HFloat(0.300076730499565014), a4 = HFloat(2.07480804148495012),

a5 = HFloat(0.0000457400995330662026), a6 = HFloat(0.604839203200166042),

                                        ]
t = HFloat(1.06795652011101994)], 6760]
See SolveEquations.mw
Try it with other values of w and the AllSolutions option on your own.

with fsolve

acer — Sat, 21 Jan 2012 13:13:08 Z

Since fsolve will only try a limited number of initial starting points, finding one whch converges can sometimes depend upon supplying ranges for the variables which are not "too" wide.

In the example, it is easy enough to find ranges which work for all seven entries of S (ie. values of w), by looking at the solutions for the few values of w for which convegence succeeds with the Poster's original ranges.

restart:

wvals:=[1.2,1.6,2,2.5,3,3.5,4]:
S:=[seq({w=wvals[i]},i=1..7)];

          [{w = 1.2}, {w = 1.6}, {w = 2}, {w = 2.5}, {w = 3}, {w = 3.5}, 

           {w = 4}]

sys := {exp(-.1204819277*(2.4039*t+15.44745000*t^2-11.03552334*t^3+2.595300000*t^4+.508258/t-44.6834-(2.40397*ln(t)+30.8949*t-16.55325000*t^2+3.460399999*t^3+.2541195000/t^2-49.812126)*t)/t)*a2*a4-a1*a3, exp(-.1204819277*(-2.071844454/t+.3999293136/t^2+2.897999999*t^3-0.2404762368e-1/t^3-6.278629824*t^2+1.49670934*t-.7274250000*t^4+4.532401680*ln(1000*t)+4.532401680*ln(298)+134.6934679-(-4.532276160/t-1.035931727/t^2-.9699000000*t^3+.2666044800/t^3+4.347000000*t^2-12.55719488*t-0.1794936000e-1/t^4-27.04489066*ln(1000*t)+27.04489066*ln(298)+28.54167*ln(t)+190.6774129)*t)/t)*a4*((1/2)*a1+(1/2)*a2+(1/2)*a3+(1/2)*a4+(1/2)*a5)-a1*a2, (1/4)*exp(-.1204819277*(95.3768*t-71.65195000*t^2+24.69369999*t^3-3.579500000*t^4+1.105339/t+179.76736-(95.37701*ln(t)-143.3039*t+37.04055000*t^2-4.772666666*t^3+.5525695000/t^2+363.377422)*t)/t)*(a1+a2+a3+a4+a5)^2*a5*a4-a1^3*a2, 2*a1+2*a4+4*a5-1.6-2*w, a2+a3+a5+a6-1, a2+2*a3+a4-.77-w, -202.86-180.476*w-a1*(33.0661*t-5.681700000*t^2+3.810933333*t^3-.6932000000*t^4+.158558/t-9.9807)-a2*(25.5675*t+3.048050000*t^2+1.351533333*t^3-.6678250000*t^4-.1310/t-118.0118)-a3*(24.9973*t+27.59345000*t^2-11.23045667*t^3+1.987075000*t^4+.1366/t-403.5951)-a4*(30.092*t+3.416250000*t^2+2.264466667*t^3-.6336000000*t^4-0.821e-1/t-250.8806)-a5*(-.703*t+54.23865000*t^2-14.17383333*t^3+1.465675000*t^4-.678565/t-76.84066)-a6*(27.04489066*t+.2287298242*t^2-4.532401680*ln(1000*t)+2.181502454/t-.3999293136/t^2+0.2404762368e-1/t^3-11.80536790-4.532401680*ln(298))}:

st:=time():
sol:='sol':
for i from 1 to nops(S) do
  sol[i]:=fsolve(eval(sys,S[i]),{a1=0..2,a2=0..2,a3=0..2,a4=0..10,a5=0..1,a6=0..2,t=0..2});
  if not type(eval(sol[i],1),specfunc(anything,fsolve)) then
     # i, max.abs. error, solution
     print([i, max(map(abs,evalf(eval(eval(sys,S[i]),sol[i])))), sol[i]]);
  end if;
end do:
time()-st;

 [         -11                                          
 [1, 5.1 10   , {a1 = 0.4745463021, a2 = 0.05174213138, 

   a3 = 0.1964321579, a4 = 1.525393553, a5 = 0.00003007250511, 

                                      ]
   a6 = 0.7517956383, t = 1.049462130}]
 [         -7                                          
 [2, 1.2 10  , {a1 = 0.5998025943, a2 = 0.07298483347, 

   a3 = 0.2484467941, a4 = 1.800121578, a5 = 0.00003791366446, 

                                      ]
   a6 = 0.6785304588, t = 1.060375417}]
 [         -7                                          
 [3, 1.5 10  , {a1 = 0.7251005506, a2 = 0.09503840192, 

   a3 = 0.3000768144, a4 = 2.074807969, a5 = 0.00004574008213, 

                                      ]
   a6 = 0.6048390436, t = 1.067956587}]
 [       -8                                         
 [4, 7 10  , {a1 = 0.8817218880, a2 = 0.1233029268, 

   a3 = 0.3642649879, a4 = 2.418167097, a5 = 0.00005550730265, 

                                      ]
   a6 = 0.5123765780, t = 1.074696743}]
 [       -8                                        
 [5, 4 10  , {a1 = 1.038324469, a2 = 0.1520588490, 

   a3 = 0.4281980719, a4 = 2.761545007, a5 = 0.00006526187271, 

                                      ]
   a6 = 0.4196778172, t = 1.079572257}]
 [          -7                                        
 [6, 1.29 10  , {a1 = 1.194906156, a2 = 0.1811351045, 

   a3 = 0.4919605334, a4 = 3.104943829, a5 = 0.00007500746271, 

                                      ]
   a6 = 0.3268293547, t = 1.083262425}]
 [         -8                                        
 [7, 1.3 10  , {a1 = 1.351469280, a2 = 0.2104315203, 

   a3 = 0.5556036263, a4 = 3.448361227, a5 = 0.00008474652456, 

                                      ]
   a6 = 0.2338801069, t = 1.086152488}]
                             4.150

One can also run this at higher working precision,

Digits:=20:

st:=time():
sol:='sol':
for i from 1 to nops(S) do
  sol[i]:=fsolve(eval(sys,S[i]),{a1=0..2,a2=0..2,a3=0..2,a4=0..10,a5=0..1,a6=0..2,t=0..2});
  if not type(eval(sol[i],1),specfunc(anything,fsolve)) then
     # i, max.abs. error, solution
     print([i, max(map(abs,evalf(eval(eval(sys,S[i]),sol[i])))), sol[i]]);
  end if;
end do:
time()-st;

[         -17                                
[1, 1.3 10   , {a1 = 0.47454630227099735486, 

  a2 = 0.051742131398933455701, a3 = 0.19643215794129892897, 

  a4 = 1.5253935527184686864, a5 = 0.000030072505266979386860, 

                                                         ]
  a6 = 0.75179563815450063595, t = 1.0494621299751955770}]
[         -17                                
[2, 2.6 10   , {a1 = 0.59980259451150934909, 

  a2 = 0.072984833487173778850, a3 = 0.24844679417683001007, 

  a4 = 1.8001215781591662010, a5 = 0.000037913664662224945514, 

                                                         ]
  a6 = 0.67853045867133398614, t = 1.0603754166160084808}]
[         -17                                
[3, 1.1 10   , {a1 = 0.72510055082673461650, 

  a2 = 0.095038401944926777977, a3 = 0.30007681452327544115, 

  a4 = 2.0748079690085223397, a5 = 0.000045740082371521882767, 

                                                         ]
  a6 = 0.60483904344942625899, t = 1.0679565867975334120}]
[         -17                                
[4, 1.3 10   , {a1 = 0.88172188825734010461, 

  a2 = 0.12330292680996198531, a3 = 0.36426498802662595140, 

  a4 = 2.4181670971367861119, a5 = 0.000055507302936891753254, 

                                                         ]
  a6 = 0.51237657786047517153, t = 1.0746967427846299290}]
[         -17                               
[5, 1.6 10   , {a1 = 1.0383244694955892758, 

  a2 = 0.15205884905244901302, a3 = 0.42819807209461689471, 

  a4 = 2.7615450067583171976, a5 = 0.000065261873046763296416, 

                                                         ]
  a6 = 0.41967781697988732897, t = 1.0795722568764767296}]
[          -17                               
[6, 1.86 10   , {a1 = 1.1949061567313915910, 

  a2 = 0.18113510449275594600, a3 = 0.49196053358241408129, 

  a4 = 3.1049438283424158914, a5 = 0.000075007463096258800517, 

                                                         ]
  a6 = 0.32682935446173371391, t = 1.0832624249502111602}]
[          -17                               
[7, 2.64 10   , {a1 = 1.3514692802731791716, 

  a2 = 0.21043152031278464376, a3 = 0.55560362650519318177, 

  a4 = 3.4483612266768289927, a5 = 0.000084746524995917846612, 

                                                         ]
  a6 = 0.23388010665702625662, t = 1.0861524880905071629}]
                             4.430

With a guess as to the nature of the problem (ranges too wide for a limited number of initial starting points) it is easier to find out that only the original range for a5 need be changed in order to find solutions for all seven w values.

Digits:=10:

st:=time():
sol:='sol':
for i from 1 to nops(S) do
  sol[i]:=fsolve(eval(sys,S[i]),{a1=0..10,a2=0..10,a3=0..10,a4=0..10,a5=0..0.1,a6=0..10,t=0..10});
  if not type(eval(sol[i],1),specfunc(anything,fsolve)) then
     # i, max.abs. error, solution
     print([i, max(map(abs,evalf(eval(eval(sys,S[i]),sol[i])))), sol[i]]);
  end if;
end do:
time()-st;

 [         -11                                          
 [1, 5.1 10   , {a1 = 0.4745463021, a2 = 0.05174213138, 

   a3 = 0.1964321579, a4 = 1.525393553, a5 = 0.00003007250511, 

                                      ]
   a6 = 0.7517956383, t = 1.049462130}]
 [         -7                                          
 [2, 1.2 10  , {a1 = 0.5998025943, a2 = 0.07298483347, 

   a3 = 0.2484467941, a4 = 1.800121578, a5 = 0.00003791366446, 

                                      ]
   a6 = 0.6785304588, t = 1.060375417}]
 [         -7                                          
 [3, 1.5 10  , {a1 = 0.7251005506, a2 = 0.09503840192, 

   a3 = 0.3000768144, a4 = 2.074807969, a5 = 0.00004574008213, 

                                      ]
   a6 = 0.6048390436, t = 1.067956587}]
 [       -8                                         
 [4, 7 10  , {a1 = 0.8817218880, a2 = 0.1233029268, 

   a3 = 0.3642649879, a4 = 2.418167097, a5 = 0.00005550730265, 

                                      ]
   a6 = 0.5123765780, t = 1.074696743}]
 [       -8                                        
 [5, 4 10  , {a1 = 1.038324469, a2 = 0.1520588490, 

   a3 = 0.4281980719, a4 = 2.761545007, a5 = 0.00006526187271, 

                                      ]
   a6 = 0.4196778172, t = 1.079572257}]
 [          -7                                        
 [6, 1.29 10  , {a1 = 1.194906156, a2 = 0.1811351045, 

   a3 = 0.4919605334, a4 = 3.104943829, a5 = 0.00007500746271, 

                                      ]
   a6 = 0.3268293547, t = 1.083262425}]
 [         -8                                        
 [7, 1.3 10  , {a1 = 1.351469280, a2 = 0.2104315203, 

   a3 = 0.5556036263, a4 = 3.448361227, a5 = 0.00008474652456, 

                                      ]
   a6 = 0.2338801069, t = 1.086152488}]
                             9.843

To be sure, fsolve could benefit from an additional option to specify the total number of loop iterations and also the total number of starting points, as well options for tolerances and working precision.

acer

some observations and suggestions

Axel Vogt — Sun, 22 Jan 2012 01:23:34 Z

The last 4 of your 7 equations are linear in a1 ... a6 (and w is a way to give
a nonhomogenity for that - it will be chosen by you anyway), parametrized by t,
and t only appears in the coefficients for that linear system (rational + log
in t (so t seems to be assumed as positive).

The first 2 are 'quadratic', the 3rd is of degree 4, the coefficients are 
exp( rational + log in t ), variables are a1 ... a5 and a6 is not present.

What I would try (after convert(sys, rational) to avoid floats) is to solve the
first 4 linear equations - giving a 3 dimensional vector space, parametrized by t,
which means you have *3* degrees of freedom (besides t)

And then feed it to equation 1 - 3 (may be even to complete squares for eq 1 + 2,
i.e. a quadratic normal form and go to eq 3 in case of deeper interest).

Note however that - what way ever you go - it seems to me, that you actually have
4 variables which you are going to feed to eq 1 - 3.

Thus generically I would expect a curve as solution. It may be, that in the Real
case (which I guess you are working with) it nevertheless may result in isolated
points, but that is not clear at all.

Hope that I got it right.

Solutions not matching when "w"is fixed and another parameter varied

elango8 — Sat, 12 May 2012 10:47:39 Z

I tried the same algorithm that was discussed here to vary one of the parameters other than "w" as i previously posted.It works fine but when i compare the common points between the two the solutions does not match i dont know why,all the inputs are the same.

I hace attached bot the files , and have marked the values in red that i am changing .It would be helpful to know if i have anything written wrong in the code.

Steam_1500_below_cri.mw - original

Varying_s.mw - modified

The modified file has a comon point at w=5 and s=-180.476 similar to original file...but the solution is totally different

List procedure

elango8 — Sat, 21 Jan 2012 12:56:54 Z

I am having a hard time understanding the list procedure , anyways i am gonna read some more on it and give it a try as you suggested.I tried the following link for understanding it better but its more confusing.

http://www.mapleprimes.com/questions/88377-Direct-Search-Optimization-Package

Tip

Markiyan Hirnyk — Sat, 21 Jan 2012 13:08:10 Z

Look at http://www.mapleprimes.com/posts/101374-DirectSearch-Optimization-Package-Version-2

This can be found by the "DirectSearch" search in MaplePrimes at the top of this page.