Find good starting candidates for Newton-Raphson knowing one of the solutions of a parametrized system of...












0












$begingroup$


I have a parameterized system of equations describing the crossed ladders problem.





  • $(x, y)$ are the $2$ horizontal distances respectively on the left/right of the junction of the ladders


  • $(a, b, c)$ are parameters for, respectively; the length of the first/second ladder and the distance between the ground and the junction.


Here is my system:



$$ (x + y)^2 (x^2 + c^2) - a^2 x^2 = 0$$
$$ (x + y)^2 (y^2 + c^2) - b^2 y^2 = 0$$



I've obtained the solution $(x, y) approx$ $(0.8572, 1.7461)$ for $(a, b, c) = (4, 3, 1)$.



Is there any way to find a function $f(a, b, c)$ that would give me a good starting candidate for my system?



Thank you.










share|cite|improve this question











$endgroup$












  • $begingroup$
    @Moo presumably one for which Newton-Raphson then converges
    $endgroup$
    – Federico
    Dec 4 '18 at 17:43










  • $begingroup$
    Yes I want a candidate for which N-R converges. I'm trying to figure out wether or not it's possible to get an analytical form for f(a, b, c)
    $endgroup$
    – FredV
    Dec 4 '18 at 19:16










  • $begingroup$
    Must you use the given set of equations or may you use the equivalent quartic equation given in your reference? Have you tried a continuation method? Instead of solving just one problem you can consider a class of problems where the defining parameters are $t(a,b,c)$ and $t=0.1,0.2,0.3, dotsc,1$. I suspect that the solution corresponding to t=0.1 might be a good initial guess for the case of t=0.2 and so on. Of course, this could be wrong!
    $endgroup$
    – Carl Christian
    Dec 6 '18 at 8:40










  • $begingroup$
    Thanks for the problem ! I had a lot of fun with it. Cheers.
    $endgroup$
    – Claude Leibovici
    Dec 9 '18 at 14:24










  • $begingroup$
    It would be a good exercise for you to implement the solution of the quartic equation in $T$.
    $endgroup$
    – Claude Leibovici
    Dec 10 '18 at 9:20
















0












$begingroup$


I have a parameterized system of equations describing the crossed ladders problem.





  • $(x, y)$ are the $2$ horizontal distances respectively on the left/right of the junction of the ladders


  • $(a, b, c)$ are parameters for, respectively; the length of the first/second ladder and the distance between the ground and the junction.


Here is my system:



$$ (x + y)^2 (x^2 + c^2) - a^2 x^2 = 0$$
$$ (x + y)^2 (y^2 + c^2) - b^2 y^2 = 0$$



I've obtained the solution $(x, y) approx$ $(0.8572, 1.7461)$ for $(a, b, c) = (4, 3, 1)$.



Is there any way to find a function $f(a, b, c)$ that would give me a good starting candidate for my system?



Thank you.










share|cite|improve this question











$endgroup$












  • $begingroup$
    @Moo presumably one for which Newton-Raphson then converges
    $endgroup$
    – Federico
    Dec 4 '18 at 17:43










  • $begingroup$
    Yes I want a candidate for which N-R converges. I'm trying to figure out wether or not it's possible to get an analytical form for f(a, b, c)
    $endgroup$
    – FredV
    Dec 4 '18 at 19:16










  • $begingroup$
    Must you use the given set of equations or may you use the equivalent quartic equation given in your reference? Have you tried a continuation method? Instead of solving just one problem you can consider a class of problems where the defining parameters are $t(a,b,c)$ and $t=0.1,0.2,0.3, dotsc,1$. I suspect that the solution corresponding to t=0.1 might be a good initial guess for the case of t=0.2 and so on. Of course, this could be wrong!
    $endgroup$
    – Carl Christian
    Dec 6 '18 at 8:40










  • $begingroup$
    Thanks for the problem ! I had a lot of fun with it. Cheers.
    $endgroup$
    – Claude Leibovici
    Dec 9 '18 at 14:24










  • $begingroup$
    It would be a good exercise for you to implement the solution of the quartic equation in $T$.
    $endgroup$
    – Claude Leibovici
    Dec 10 '18 at 9:20














0












0








0





$begingroup$


I have a parameterized system of equations describing the crossed ladders problem.





  • $(x, y)$ are the $2$ horizontal distances respectively on the left/right of the junction of the ladders


  • $(a, b, c)$ are parameters for, respectively; the length of the first/second ladder and the distance between the ground and the junction.


Here is my system:



$$ (x + y)^2 (x^2 + c^2) - a^2 x^2 = 0$$
$$ (x + y)^2 (y^2 + c^2) - b^2 y^2 = 0$$



I've obtained the solution $(x, y) approx$ $(0.8572, 1.7461)$ for $(a, b, c) = (4, 3, 1)$.



Is there any way to find a function $f(a, b, c)$ that would give me a good starting candidate for my system?



Thank you.










share|cite|improve this question











$endgroup$




I have a parameterized system of equations describing the crossed ladders problem.





  • $(x, y)$ are the $2$ horizontal distances respectively on the left/right of the junction of the ladders


  • $(a, b, c)$ are parameters for, respectively; the length of the first/second ladder and the distance between the ground and the junction.


Here is my system:



$$ (x + y)^2 (x^2 + c^2) - a^2 x^2 = 0$$
$$ (x + y)^2 (y^2 + c^2) - b^2 y^2 = 0$$



I've obtained the solution $(x, y) approx$ $(0.8572, 1.7461)$ for $(a, b, c) = (4, 3, 1)$.



Is there any way to find a function $f(a, b, c)$ that would give me a good starting candidate for my system?



Thank you.







real-analysis numerical-methods fixed-point-theorems newton-raphson






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 5 '18 at 12:25







FredV

















asked Dec 4 '18 at 17:32









FredVFredV

876




876












  • $begingroup$
    @Moo presumably one for which Newton-Raphson then converges
    $endgroup$
    – Federico
    Dec 4 '18 at 17:43










  • $begingroup$
    Yes I want a candidate for which N-R converges. I'm trying to figure out wether or not it's possible to get an analytical form for f(a, b, c)
    $endgroup$
    – FredV
    Dec 4 '18 at 19:16










  • $begingroup$
    Must you use the given set of equations or may you use the equivalent quartic equation given in your reference? Have you tried a continuation method? Instead of solving just one problem you can consider a class of problems where the defining parameters are $t(a,b,c)$ and $t=0.1,0.2,0.3, dotsc,1$. I suspect that the solution corresponding to t=0.1 might be a good initial guess for the case of t=0.2 and so on. Of course, this could be wrong!
    $endgroup$
    – Carl Christian
    Dec 6 '18 at 8:40










  • $begingroup$
    Thanks for the problem ! I had a lot of fun with it. Cheers.
    $endgroup$
    – Claude Leibovici
    Dec 9 '18 at 14:24










  • $begingroup$
    It would be a good exercise for you to implement the solution of the quartic equation in $T$.
    $endgroup$
    – Claude Leibovici
    Dec 10 '18 at 9:20


















  • $begingroup$
    @Moo presumably one for which Newton-Raphson then converges
    $endgroup$
    – Federico
    Dec 4 '18 at 17:43










  • $begingroup$
    Yes I want a candidate for which N-R converges. I'm trying to figure out wether or not it's possible to get an analytical form for f(a, b, c)
    $endgroup$
    – FredV
    Dec 4 '18 at 19:16










  • $begingroup$
    Must you use the given set of equations or may you use the equivalent quartic equation given in your reference? Have you tried a continuation method? Instead of solving just one problem you can consider a class of problems where the defining parameters are $t(a,b,c)$ and $t=0.1,0.2,0.3, dotsc,1$. I suspect that the solution corresponding to t=0.1 might be a good initial guess for the case of t=0.2 and so on. Of course, this could be wrong!
    $endgroup$
    – Carl Christian
    Dec 6 '18 at 8:40










  • $begingroup$
    Thanks for the problem ! I had a lot of fun with it. Cheers.
    $endgroup$
    – Claude Leibovici
    Dec 9 '18 at 14:24










  • $begingroup$
    It would be a good exercise for you to implement the solution of the quartic equation in $T$.
    $endgroup$
    – Claude Leibovici
    Dec 10 '18 at 9:20
















$begingroup$
@Moo presumably one for which Newton-Raphson then converges
$endgroup$
– Federico
Dec 4 '18 at 17:43




$begingroup$
@Moo presumably one for which Newton-Raphson then converges
$endgroup$
– Federico
Dec 4 '18 at 17:43












$begingroup$
Yes I want a candidate for which N-R converges. I'm trying to figure out wether or not it's possible to get an analytical form for f(a, b, c)
$endgroup$
– FredV
Dec 4 '18 at 19:16




$begingroup$
Yes I want a candidate for which N-R converges. I'm trying to figure out wether or not it's possible to get an analytical form for f(a, b, c)
$endgroup$
– FredV
Dec 4 '18 at 19:16












$begingroup$
Must you use the given set of equations or may you use the equivalent quartic equation given in your reference? Have you tried a continuation method? Instead of solving just one problem you can consider a class of problems where the defining parameters are $t(a,b,c)$ and $t=0.1,0.2,0.3, dotsc,1$. I suspect that the solution corresponding to t=0.1 might be a good initial guess for the case of t=0.2 and so on. Of course, this could be wrong!
$endgroup$
– Carl Christian
Dec 6 '18 at 8:40




$begingroup$
Must you use the given set of equations or may you use the equivalent quartic equation given in your reference? Have you tried a continuation method? Instead of solving just one problem you can consider a class of problems where the defining parameters are $t(a,b,c)$ and $t=0.1,0.2,0.3, dotsc,1$. I suspect that the solution corresponding to t=0.1 might be a good initial guess for the case of t=0.2 and so on. Of course, this could be wrong!
$endgroup$
– Carl Christian
Dec 6 '18 at 8:40












$begingroup$
Thanks for the problem ! I had a lot of fun with it. Cheers.
$endgroup$
– Claude Leibovici
Dec 9 '18 at 14:24




$begingroup$
Thanks for the problem ! I had a lot of fun with it. Cheers.
$endgroup$
– Claude Leibovici
Dec 9 '18 at 14:24












$begingroup$
It would be a good exercise for you to implement the solution of the quartic equation in $T$.
$endgroup$
– Claude Leibovici
Dec 10 '18 at 9:20




$begingroup$
It would be a good exercise for you to implement the solution of the quartic equation in $T$.
$endgroup$
– Claude Leibovici
Dec 10 '18 at 9:20










1 Answer
1






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3












$begingroup$

You could make at least life simpler writing $$(x + y)^2 (x^2 + c^2) = a^2 x^2tag 1$$
$$(x + y)^2 (y^2 + c^2) = b^2 y^2tag 2$$ Make the ratio
$$frac{x^2+c^2}{y^2+c^2}=frac{a^2 x^2 }{b^2 y^2 }tag 3$$ This gives two solutions
$$y_pm=pmfrac{a, c ,x}{sqrt{(b^2-a^2), x^2+b^2, c^2}}tag 4$$
Suppose that we use $y_+$ that we plug in $(1)$ to get as equation
$$left(a^2 c^4+b^2 c^4-a^2 b^2 c^2right)+x^2 left(a^4-a^2 b^2+2 b^2
c^2right)+x^4 left(b^2-a^2right)+2 a c left(c^2+x^2right) sqrt{(b^2-a^2)x^2 +b^2c^2}=0$$
and you can notice that we can solved for $x^2$ instead of $x$.



If $a >b$, this is limiting the search domain since we need $(b^2-a^2)x^2 +b^2c^2 geq 0$ that is to say (admitting that we look for $x>0$) $$0 leq x leq frac{b c}{sqrt{a^2-b^2}}$$ Next, what I should do is a Taylor expansion around $x=0$ which, assuming that $a,b,c$ are all positive to get
$$left(c^4 (a+b)^2-a^2 b^2 c^2right)+frac{ (a+b) left(a^2 b (a-b)-c^2 (a-2 b)
(a+b)right)}{b}x^2+Oleft(x^4right)$$
to get an estimate
$$x_0=sqrt{frac{a^2 b^3 c^2-b c^4 (a+b)^2}{(a+b) left(a^2 b (a-b)-c^2 (a-2 b) (a+b)right)}}$$



Applied to your case $(a, b, c) = (4, 3, 1)$, the upper bound would be $frac{3}{sqrt{7}}approx 1.134$ and $x_0=sqrt{frac{285}{434}}approx 0.810$.



Now, Newton method will generate the following iterates
$$ left(
begin{array}{cc}
n & x_n \
0 & 0.81035920 \
1 & 0.85758341 \
2 & 0.85721270 \
3 & 0.85721269
end{array}
right)$$
and, back to $(4)$, $y=1.74607506$.



Edit



We can even do better if, instead of using Taylor, we build the $[2,2]$ Padé approximant. Solving for $0$ th numerator, we get
$$x_0=sqrt{frac A {(a+b)B}}$$ where
$$A=4 b^2 c^2 left(a^4 b^3 (a-b)-a^2 b c^2 (a+b) left(a^2+a b-3 b^2right)+c^4 (a-2 b) (a+b)^3right)$$
$$B=4 a^4 b^3 (a-b)^2+c^4 (a+b)^2 left(a^3+2 a^2 b-11 a b^2+12 b^3right)+3 a^2 b^2
c^2 left(-3 a^3+6 a^2 b+a b^2-4 b^3right)$$
Applied to the example, this would give $x_0=sqrt{frac{26505}{37037}}approx 0.846$.



Newton iterates would then be
$$left(
begin{array}{cc}
n & x_n \
0 & 0.84595255 \
1 & 0.85722574 \
2 & 0.85721269
end{array}
right)$$



For the worked example, the function we look for the zero of is
$$f(x)=-7 x^4+2 left(4 sqrt{9-7 x^2}+65right) x^2+8 sqrt{9-7 x^2}-119$$ and its $[2,2]$ Padé approximant is
$$g(x)=frac{37037 x^2-26505}{35 x^2+279}$$ Just for curiosity, plot the two functions on the same graph. You would be amazed (I hope !).



We could still do better building the $[2,4]$ Padé approximant; the expression of $x_0$ would be really nasty but for the example we would have
$$h(x)=frac{74568309 x^2-54285660}{20650 x^4+85245 x^2+571428}$$ giving $x_0=sqrt{frac{18095220}{24856103} } approx 0.853$.



Update



We could make the problem simpler defining first $X=x^2$ and then $X=frac{b^2 c^2-T^2}{a^2-b^2}$. This makes the equation to be
$$-a^4 c^4-2 a^3 c^3T+ a^2left(a^2- b^2right)T^2+2 a c T^3+T^4=0tag5$$ which can be solved analytically for $T$. This quartic has two real roots since
$$Delta=-16 a^{12} c^4 (a-b)^2 (a+b)^2 left(left(a^2-b^2right)^2+27 c^4right) < 0qquad qquadforall a,b,c$$ (see here).



I suppose that the roots have opposite signs and that the one to keep is the positive root (very ugly expressions with awful radicals).



For the worked example, we should then have to solve
$$T^4+8 T^3+112 T^2-128 T-256=0$$ for which the real roots are $T=-1.06797$ and $T= 1.96375$



So, there is an explicit solution to the problem.






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    $begingroup$

    You could make at least life simpler writing $$(x + y)^2 (x^2 + c^2) = a^2 x^2tag 1$$
    $$(x + y)^2 (y^2 + c^2) = b^2 y^2tag 2$$ Make the ratio
    $$frac{x^2+c^2}{y^2+c^2}=frac{a^2 x^2 }{b^2 y^2 }tag 3$$ This gives two solutions
    $$y_pm=pmfrac{a, c ,x}{sqrt{(b^2-a^2), x^2+b^2, c^2}}tag 4$$
    Suppose that we use $y_+$ that we plug in $(1)$ to get as equation
    $$left(a^2 c^4+b^2 c^4-a^2 b^2 c^2right)+x^2 left(a^4-a^2 b^2+2 b^2
    c^2right)+x^4 left(b^2-a^2right)+2 a c left(c^2+x^2right) sqrt{(b^2-a^2)x^2 +b^2c^2}=0$$
    and you can notice that we can solved for $x^2$ instead of $x$.



    If $a >b$, this is limiting the search domain since we need $(b^2-a^2)x^2 +b^2c^2 geq 0$ that is to say (admitting that we look for $x>0$) $$0 leq x leq frac{b c}{sqrt{a^2-b^2}}$$ Next, what I should do is a Taylor expansion around $x=0$ which, assuming that $a,b,c$ are all positive to get
    $$left(c^4 (a+b)^2-a^2 b^2 c^2right)+frac{ (a+b) left(a^2 b (a-b)-c^2 (a-2 b)
    (a+b)right)}{b}x^2+Oleft(x^4right)$$
    to get an estimate
    $$x_0=sqrt{frac{a^2 b^3 c^2-b c^4 (a+b)^2}{(a+b) left(a^2 b (a-b)-c^2 (a-2 b) (a+b)right)}}$$



    Applied to your case $(a, b, c) = (4, 3, 1)$, the upper bound would be $frac{3}{sqrt{7}}approx 1.134$ and $x_0=sqrt{frac{285}{434}}approx 0.810$.



    Now, Newton method will generate the following iterates
    $$ left(
    begin{array}{cc}
    n & x_n \
    0 & 0.81035920 \
    1 & 0.85758341 \
    2 & 0.85721270 \
    3 & 0.85721269
    end{array}
    right)$$
    and, back to $(4)$, $y=1.74607506$.



    Edit



    We can even do better if, instead of using Taylor, we build the $[2,2]$ Padé approximant. Solving for $0$ th numerator, we get
    $$x_0=sqrt{frac A {(a+b)B}}$$ where
    $$A=4 b^2 c^2 left(a^4 b^3 (a-b)-a^2 b c^2 (a+b) left(a^2+a b-3 b^2right)+c^4 (a-2 b) (a+b)^3right)$$
    $$B=4 a^4 b^3 (a-b)^2+c^4 (a+b)^2 left(a^3+2 a^2 b-11 a b^2+12 b^3right)+3 a^2 b^2
    c^2 left(-3 a^3+6 a^2 b+a b^2-4 b^3right)$$
    Applied to the example, this would give $x_0=sqrt{frac{26505}{37037}}approx 0.846$.



    Newton iterates would then be
    $$left(
    begin{array}{cc}
    n & x_n \
    0 & 0.84595255 \
    1 & 0.85722574 \
    2 & 0.85721269
    end{array}
    right)$$



    For the worked example, the function we look for the zero of is
    $$f(x)=-7 x^4+2 left(4 sqrt{9-7 x^2}+65right) x^2+8 sqrt{9-7 x^2}-119$$ and its $[2,2]$ Padé approximant is
    $$g(x)=frac{37037 x^2-26505}{35 x^2+279}$$ Just for curiosity, plot the two functions on the same graph. You would be amazed (I hope !).



    We could still do better building the $[2,4]$ Padé approximant; the expression of $x_0$ would be really nasty but for the example we would have
    $$h(x)=frac{74568309 x^2-54285660}{20650 x^4+85245 x^2+571428}$$ giving $x_0=sqrt{frac{18095220}{24856103} } approx 0.853$.



    Update



    We could make the problem simpler defining first $X=x^2$ and then $X=frac{b^2 c^2-T^2}{a^2-b^2}$. This makes the equation to be
    $$-a^4 c^4-2 a^3 c^3T+ a^2left(a^2- b^2right)T^2+2 a c T^3+T^4=0tag5$$ which can be solved analytically for $T$. This quartic has two real roots since
    $$Delta=-16 a^{12} c^4 (a-b)^2 (a+b)^2 left(left(a^2-b^2right)^2+27 c^4right) < 0qquad qquadforall a,b,c$$ (see here).



    I suppose that the roots have opposite signs and that the one to keep is the positive root (very ugly expressions with awful radicals).



    For the worked example, we should then have to solve
    $$T^4+8 T^3+112 T^2-128 T-256=0$$ for which the real roots are $T=-1.06797$ and $T= 1.96375$



    So, there is an explicit solution to the problem.






    share|cite|improve this answer











    $endgroup$


















      3












      $begingroup$

      You could make at least life simpler writing $$(x + y)^2 (x^2 + c^2) = a^2 x^2tag 1$$
      $$(x + y)^2 (y^2 + c^2) = b^2 y^2tag 2$$ Make the ratio
      $$frac{x^2+c^2}{y^2+c^2}=frac{a^2 x^2 }{b^2 y^2 }tag 3$$ This gives two solutions
      $$y_pm=pmfrac{a, c ,x}{sqrt{(b^2-a^2), x^2+b^2, c^2}}tag 4$$
      Suppose that we use $y_+$ that we plug in $(1)$ to get as equation
      $$left(a^2 c^4+b^2 c^4-a^2 b^2 c^2right)+x^2 left(a^4-a^2 b^2+2 b^2
      c^2right)+x^4 left(b^2-a^2right)+2 a c left(c^2+x^2right) sqrt{(b^2-a^2)x^2 +b^2c^2}=0$$
      and you can notice that we can solved for $x^2$ instead of $x$.



      If $a >b$, this is limiting the search domain since we need $(b^2-a^2)x^2 +b^2c^2 geq 0$ that is to say (admitting that we look for $x>0$) $$0 leq x leq frac{b c}{sqrt{a^2-b^2}}$$ Next, what I should do is a Taylor expansion around $x=0$ which, assuming that $a,b,c$ are all positive to get
      $$left(c^4 (a+b)^2-a^2 b^2 c^2right)+frac{ (a+b) left(a^2 b (a-b)-c^2 (a-2 b)
      (a+b)right)}{b}x^2+Oleft(x^4right)$$
      to get an estimate
      $$x_0=sqrt{frac{a^2 b^3 c^2-b c^4 (a+b)^2}{(a+b) left(a^2 b (a-b)-c^2 (a-2 b) (a+b)right)}}$$



      Applied to your case $(a, b, c) = (4, 3, 1)$, the upper bound would be $frac{3}{sqrt{7}}approx 1.134$ and $x_0=sqrt{frac{285}{434}}approx 0.810$.



      Now, Newton method will generate the following iterates
      $$ left(
      begin{array}{cc}
      n & x_n \
      0 & 0.81035920 \
      1 & 0.85758341 \
      2 & 0.85721270 \
      3 & 0.85721269
      end{array}
      right)$$
      and, back to $(4)$, $y=1.74607506$.



      Edit



      We can even do better if, instead of using Taylor, we build the $[2,2]$ Padé approximant. Solving for $0$ th numerator, we get
      $$x_0=sqrt{frac A {(a+b)B}}$$ where
      $$A=4 b^2 c^2 left(a^4 b^3 (a-b)-a^2 b c^2 (a+b) left(a^2+a b-3 b^2right)+c^4 (a-2 b) (a+b)^3right)$$
      $$B=4 a^4 b^3 (a-b)^2+c^4 (a+b)^2 left(a^3+2 a^2 b-11 a b^2+12 b^3right)+3 a^2 b^2
      c^2 left(-3 a^3+6 a^2 b+a b^2-4 b^3right)$$
      Applied to the example, this would give $x_0=sqrt{frac{26505}{37037}}approx 0.846$.



      Newton iterates would then be
      $$left(
      begin{array}{cc}
      n & x_n \
      0 & 0.84595255 \
      1 & 0.85722574 \
      2 & 0.85721269
      end{array}
      right)$$



      For the worked example, the function we look for the zero of is
      $$f(x)=-7 x^4+2 left(4 sqrt{9-7 x^2}+65right) x^2+8 sqrt{9-7 x^2}-119$$ and its $[2,2]$ Padé approximant is
      $$g(x)=frac{37037 x^2-26505}{35 x^2+279}$$ Just for curiosity, plot the two functions on the same graph. You would be amazed (I hope !).



      We could still do better building the $[2,4]$ Padé approximant; the expression of $x_0$ would be really nasty but for the example we would have
      $$h(x)=frac{74568309 x^2-54285660}{20650 x^4+85245 x^2+571428}$$ giving $x_0=sqrt{frac{18095220}{24856103} } approx 0.853$.



      Update



      We could make the problem simpler defining first $X=x^2$ and then $X=frac{b^2 c^2-T^2}{a^2-b^2}$. This makes the equation to be
      $$-a^4 c^4-2 a^3 c^3T+ a^2left(a^2- b^2right)T^2+2 a c T^3+T^4=0tag5$$ which can be solved analytically for $T$. This quartic has two real roots since
      $$Delta=-16 a^{12} c^4 (a-b)^2 (a+b)^2 left(left(a^2-b^2right)^2+27 c^4right) < 0qquad qquadforall a,b,c$$ (see here).



      I suppose that the roots have opposite signs and that the one to keep is the positive root (very ugly expressions with awful radicals).



      For the worked example, we should then have to solve
      $$T^4+8 T^3+112 T^2-128 T-256=0$$ for which the real roots are $T=-1.06797$ and $T= 1.96375$



      So, there is an explicit solution to the problem.






      share|cite|improve this answer











      $endgroup$
















        3












        3








        3





        $begingroup$

        You could make at least life simpler writing $$(x + y)^2 (x^2 + c^2) = a^2 x^2tag 1$$
        $$(x + y)^2 (y^2 + c^2) = b^2 y^2tag 2$$ Make the ratio
        $$frac{x^2+c^2}{y^2+c^2}=frac{a^2 x^2 }{b^2 y^2 }tag 3$$ This gives two solutions
        $$y_pm=pmfrac{a, c ,x}{sqrt{(b^2-a^2), x^2+b^2, c^2}}tag 4$$
        Suppose that we use $y_+$ that we plug in $(1)$ to get as equation
        $$left(a^2 c^4+b^2 c^4-a^2 b^2 c^2right)+x^2 left(a^4-a^2 b^2+2 b^2
        c^2right)+x^4 left(b^2-a^2right)+2 a c left(c^2+x^2right) sqrt{(b^2-a^2)x^2 +b^2c^2}=0$$
        and you can notice that we can solved for $x^2$ instead of $x$.



        If $a >b$, this is limiting the search domain since we need $(b^2-a^2)x^2 +b^2c^2 geq 0$ that is to say (admitting that we look for $x>0$) $$0 leq x leq frac{b c}{sqrt{a^2-b^2}}$$ Next, what I should do is a Taylor expansion around $x=0$ which, assuming that $a,b,c$ are all positive to get
        $$left(c^4 (a+b)^2-a^2 b^2 c^2right)+frac{ (a+b) left(a^2 b (a-b)-c^2 (a-2 b)
        (a+b)right)}{b}x^2+Oleft(x^4right)$$
        to get an estimate
        $$x_0=sqrt{frac{a^2 b^3 c^2-b c^4 (a+b)^2}{(a+b) left(a^2 b (a-b)-c^2 (a-2 b) (a+b)right)}}$$



        Applied to your case $(a, b, c) = (4, 3, 1)$, the upper bound would be $frac{3}{sqrt{7}}approx 1.134$ and $x_0=sqrt{frac{285}{434}}approx 0.810$.



        Now, Newton method will generate the following iterates
        $$ left(
        begin{array}{cc}
        n & x_n \
        0 & 0.81035920 \
        1 & 0.85758341 \
        2 & 0.85721270 \
        3 & 0.85721269
        end{array}
        right)$$
        and, back to $(4)$, $y=1.74607506$.



        Edit



        We can even do better if, instead of using Taylor, we build the $[2,2]$ Padé approximant. Solving for $0$ th numerator, we get
        $$x_0=sqrt{frac A {(a+b)B}}$$ where
        $$A=4 b^2 c^2 left(a^4 b^3 (a-b)-a^2 b c^2 (a+b) left(a^2+a b-3 b^2right)+c^4 (a-2 b) (a+b)^3right)$$
        $$B=4 a^4 b^3 (a-b)^2+c^4 (a+b)^2 left(a^3+2 a^2 b-11 a b^2+12 b^3right)+3 a^2 b^2
        c^2 left(-3 a^3+6 a^2 b+a b^2-4 b^3right)$$
        Applied to the example, this would give $x_0=sqrt{frac{26505}{37037}}approx 0.846$.



        Newton iterates would then be
        $$left(
        begin{array}{cc}
        n & x_n \
        0 & 0.84595255 \
        1 & 0.85722574 \
        2 & 0.85721269
        end{array}
        right)$$



        For the worked example, the function we look for the zero of is
        $$f(x)=-7 x^4+2 left(4 sqrt{9-7 x^2}+65right) x^2+8 sqrt{9-7 x^2}-119$$ and its $[2,2]$ Padé approximant is
        $$g(x)=frac{37037 x^2-26505}{35 x^2+279}$$ Just for curiosity, plot the two functions on the same graph. You would be amazed (I hope !).



        We could still do better building the $[2,4]$ Padé approximant; the expression of $x_0$ would be really nasty but for the example we would have
        $$h(x)=frac{74568309 x^2-54285660}{20650 x^4+85245 x^2+571428}$$ giving $x_0=sqrt{frac{18095220}{24856103} } approx 0.853$.



        Update



        We could make the problem simpler defining first $X=x^2$ and then $X=frac{b^2 c^2-T^2}{a^2-b^2}$. This makes the equation to be
        $$-a^4 c^4-2 a^3 c^3T+ a^2left(a^2- b^2right)T^2+2 a c T^3+T^4=0tag5$$ which can be solved analytically for $T$. This quartic has two real roots since
        $$Delta=-16 a^{12} c^4 (a-b)^2 (a+b)^2 left(left(a^2-b^2right)^2+27 c^4right) < 0qquad qquadforall a,b,c$$ (see here).



        I suppose that the roots have opposite signs and that the one to keep is the positive root (very ugly expressions with awful radicals).



        For the worked example, we should then have to solve
        $$T^4+8 T^3+112 T^2-128 T-256=0$$ for which the real roots are $T=-1.06797$ and $T= 1.96375$



        So, there is an explicit solution to the problem.






        share|cite|improve this answer











        $endgroup$



        You could make at least life simpler writing $$(x + y)^2 (x^2 + c^2) = a^2 x^2tag 1$$
        $$(x + y)^2 (y^2 + c^2) = b^2 y^2tag 2$$ Make the ratio
        $$frac{x^2+c^2}{y^2+c^2}=frac{a^2 x^2 }{b^2 y^2 }tag 3$$ This gives two solutions
        $$y_pm=pmfrac{a, c ,x}{sqrt{(b^2-a^2), x^2+b^2, c^2}}tag 4$$
        Suppose that we use $y_+$ that we plug in $(1)$ to get as equation
        $$left(a^2 c^4+b^2 c^4-a^2 b^2 c^2right)+x^2 left(a^4-a^2 b^2+2 b^2
        c^2right)+x^4 left(b^2-a^2right)+2 a c left(c^2+x^2right) sqrt{(b^2-a^2)x^2 +b^2c^2}=0$$
        and you can notice that we can solved for $x^2$ instead of $x$.



        If $a >b$, this is limiting the search domain since we need $(b^2-a^2)x^2 +b^2c^2 geq 0$ that is to say (admitting that we look for $x>0$) $$0 leq x leq frac{b c}{sqrt{a^2-b^2}}$$ Next, what I should do is a Taylor expansion around $x=0$ which, assuming that $a,b,c$ are all positive to get
        $$left(c^4 (a+b)^2-a^2 b^2 c^2right)+frac{ (a+b) left(a^2 b (a-b)-c^2 (a-2 b)
        (a+b)right)}{b}x^2+Oleft(x^4right)$$
        to get an estimate
        $$x_0=sqrt{frac{a^2 b^3 c^2-b c^4 (a+b)^2}{(a+b) left(a^2 b (a-b)-c^2 (a-2 b) (a+b)right)}}$$



        Applied to your case $(a, b, c) = (4, 3, 1)$, the upper bound would be $frac{3}{sqrt{7}}approx 1.134$ and $x_0=sqrt{frac{285}{434}}approx 0.810$.



        Now, Newton method will generate the following iterates
        $$ left(
        begin{array}{cc}
        n & x_n \
        0 & 0.81035920 \
        1 & 0.85758341 \
        2 & 0.85721270 \
        3 & 0.85721269
        end{array}
        right)$$
        and, back to $(4)$, $y=1.74607506$.



        Edit



        We can even do better if, instead of using Taylor, we build the $[2,2]$ Padé approximant. Solving for $0$ th numerator, we get
        $$x_0=sqrt{frac A {(a+b)B}}$$ where
        $$A=4 b^2 c^2 left(a^4 b^3 (a-b)-a^2 b c^2 (a+b) left(a^2+a b-3 b^2right)+c^4 (a-2 b) (a+b)^3right)$$
        $$B=4 a^4 b^3 (a-b)^2+c^4 (a+b)^2 left(a^3+2 a^2 b-11 a b^2+12 b^3right)+3 a^2 b^2
        c^2 left(-3 a^3+6 a^2 b+a b^2-4 b^3right)$$
        Applied to the example, this would give $x_0=sqrt{frac{26505}{37037}}approx 0.846$.



        Newton iterates would then be
        $$left(
        begin{array}{cc}
        n & x_n \
        0 & 0.84595255 \
        1 & 0.85722574 \
        2 & 0.85721269
        end{array}
        right)$$



        For the worked example, the function we look for the zero of is
        $$f(x)=-7 x^4+2 left(4 sqrt{9-7 x^2}+65right) x^2+8 sqrt{9-7 x^2}-119$$ and its $[2,2]$ Padé approximant is
        $$g(x)=frac{37037 x^2-26505}{35 x^2+279}$$ Just for curiosity, plot the two functions on the same graph. You would be amazed (I hope !).



        We could still do better building the $[2,4]$ Padé approximant; the expression of $x_0$ would be really nasty but for the example we would have
        $$h(x)=frac{74568309 x^2-54285660}{20650 x^4+85245 x^2+571428}$$ giving $x_0=sqrt{frac{18095220}{24856103} } approx 0.853$.



        Update



        We could make the problem simpler defining first $X=x^2$ and then $X=frac{b^2 c^2-T^2}{a^2-b^2}$. This makes the equation to be
        $$-a^4 c^4-2 a^3 c^3T+ a^2left(a^2- b^2right)T^2+2 a c T^3+T^4=0tag5$$ which can be solved analytically for $T$. This quartic has two real roots since
        $$Delta=-16 a^{12} c^4 (a-b)^2 (a+b)^2 left(left(a^2-b^2right)^2+27 c^4right) < 0qquad qquadforall a,b,c$$ (see here).



        I suppose that the roots have opposite signs and that the one to keep is the positive root (very ugly expressions with awful radicals).



        For the worked example, we should then have to solve
        $$T^4+8 T^3+112 T^2-128 T-256=0$$ for which the real roots are $T=-1.06797$ and $T= 1.96375$



        So, there is an explicit solution to the problem.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 10 '18 at 8:02

























        answered Dec 9 '18 at 8:27









        Claude LeiboviciClaude Leibovici

        120k1157132




        120k1157132






























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