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Differentiating Functions and Root objects [ was Re: ArcCos[]]

• To: mathgroup at smc.vnet.net
• Subject: [mg24833] Differentiating Functions and Root objects [ was Re: ArcCos[]]
• From: "Allan Hayes" <hay at haystack.demon.co.uk>
• Date: Tue, 15 Aug 2000 03:04:05 -0400 (EDT)
• References: <8mqvd7$18j@smc.vnet.net> <8mtc0m$7qb@smc.vnet.net> <8n7q9r$1j7@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

John's posting stirred me into taking another look a this problem.
Here are four observations

1. We can differentiating a pure function with respect to a parameter

D[Function[x, Sin[y] x^2], y]

        Function[x, x^2*Cos[y]]


D[Sin[y] #^2 &, y]

        Cos[y]*#1^2 &

2. The original problem, differentiating Root objects

a = Root[-t + 2*#1 + 2*t^2*#1 + #1^3 &, 1]

Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1]

(b = D[a, t]) // InputForm

        (1 - 4*t*Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1])/
          (2 + 2*t^2 + 3*Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1]^
             2)

Make this output explicit

(rb = ToRadicals[b]) // InputForm

        (1 - 4*t*(-(2^(1/3)*(6 + 6*t^2))/
             (3*(27*t + Sqrt[729*t^2 + 4*(6 + 6*t^2)^3])^
               (1/3)) + (27*t + Sqrt[729*t^2 + 4*(6 + 6*t^2)^3])^
              (1/3)/(3*2^(1/3))))/(2 + 2*t^2 +
          3*(-(2^(1/3)*(6 + 6*t^2))/
              (3*(27*t + Sqrt[729*t^2 + 4*(6 + 6*t^2)^3])^
                (1/3)) +
             (27*t + Sqrt[729*t^2 + 4*(6 + 6*t^2)^3])^(1/3)/
              (3*2^(1/3)))^2)

Maybe  Solve and NSolve will work on rb == 0,   -- I gave up after a a brief
wait -- but we can quickly get

FindRoot[rb == 0, {t, 0}]

{t -> -818.986}

Is b correct for the rate of change of a respect to t?
We can check

rb - D[ToRadicals[a], t] // Simplify

0

We might expect with sol  the corresponding polynomial would have a multiple
root.

Check:

(poly = -t + 2*#1 + 2*t^2*#1 + #1^3 &[x] /. sol[[1]]) // InputForm

        818.985893312185 + 1.3414777868887153*^6*x + x^3


Solve[% == 0, x] // InputForm

        {{x -> -0.0006105102159100438},
         {x -> 0.0003052551079550219 - 1158.2218211072502*I},
         {x -> 0.0003052551079550219 + 1158.2218211072502*I}}


3. Can we still get the solution without using ToRadicals?

sol2 = FindRoot[b == 0, {t, 0}]

    FindRoot::"jsing": "Encountered a singular Jacobian at the point \!\(t\)
= \!\
    \(0.`\). Try perturbing the initial point(s)."

    FindRoot::"jsing": "Encountered a singular Jacobian at the point \!\(t\)
= \!\
    \(0.`\). Try perturbing the initial point(s)."

    FindRoot[b == 0, {t, 0}]

However

sol2 = FindRoot[b == 0, {t, 0.1}]

    {t -> -1.0150511440294088}

A different solution.

And we get

Solve[(-t + 2*#1 + 2*t^2*#1 + #1^3 &[x] /. sol2[[1]]) == 0, x]

{{x -> -0.2462928577523092},
  {x -> 0.1231464288761546 - 2.0263644239932934*I},
  {x -> 0.1231464288761546 + 2.0263644239932934*I}}

4. Is Differentiation of Root objects always valid? Is this method a general
one for finding which parameter values give multiple roots?

Allan
---------------------
Allan Hayes
Mathematica Training and Consulting
Leicester UK
www.haystack.demon.co.uk
hay at haystack.demon.co.uk
Voice: +44 (0)116 271 4198
Fax: +44 (0)870 164 0565

"John D. Hendrickson" <jdh at hend.net> wrote in message
news:8n7q9r$1j7 at smc.vnet.net...
> I've got Mathematica 4.0.2.0.  Since I'm running Win95 I don't mind
> crashing - thats pretty much a constant, so I tried your dare:)  But mine
> didn't crash - it gave me output.  Also - I have a substitute that might
be
> what you want.
>
> In[1]:=
> a = Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1];
>
> In[2]:=
> b = D[a, t];
>
> In[4]:=
> InputForm[ Solve[b == 0, t] ]
>
> Solve::"tdep": "The equations appear to involve the variables to be solved
\
> for in an essentially non-algebraic way."
>
> Out[4]//InputForm=
> Solve[(1 - 4*t*Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1])/
>    (2 + 2*t^2 + 3*Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1]^
>       2) == 0, t]
>
> ===================================================================
> Would the following be acceptible in your circumstance?
> I assumed by making the unexplicit root object explicit:
> ====================================================================
>
> In[3]:=
> Exit[]
>
> In[1]:=
>  a = Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1];
>
> In[2]:=
> ax = First[a][x]
>
> Out[2]=
> \!\(\(-t\) + 2\ x + 2\ t\^2\ x + x\^3\)
>
> In[3]:=
>  b = D[ax, t];
>
> In[4]:=
> InputForm[ Solve[b == 0, t] ]
>
> Out[4]//InputForm=
> {{t -> 1/(4*x)}}
>
> ===================================================================
>
> Allan Hayes wrote in message <8mtc0m$7qb at smc.vnet.net>...
> >
> >
> >
> >"Gianluca Gorni" <gorni at dimi.uniud.it> wrote in message
> >news:8mqvd7$18j at smc.vnet.net...
> >
> >
> >-----------  Snip -------------
> >> An unrelated problem: the following instructions consistently crash
> >> my Mac Mathematica 4 kernel:
> >>
> >> a = Root[-t + 2*#1 + 2*t^2*#1 + #1^3 & , 1];
> >> b = D[a, t];
> >> Solve[b == 0, t]
> >>
> >--
> >Gianluca ,
> >It crashes Mathematica 4.02  on Windows also.
> >
> >Allan
> >---------------------
> >Allan Hayes
> >Mathematica Training and Consulting
> >Leicester UK
> >www.haystack.demon.co.uk
> >hay at haystack.demon.co.uk
> >Voice: +44 (0)116 271 4198
> >Fax: +44 (0)870 164 0565
> >
> >
> >
> >
>
>
>

• Follow-Ups:
  • Re: Differentiating Functions and Root objects [ was Re: ArcCos[]]
    • From: Daniel Lichtblau <danl@wolfram.com>