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Signs of first and second derivatives of...

MaPal93 175 Maple 2024
differentiation numeric rootof
May 06 2024
1 2

I have a complicated bivariate function f(Gamma,rho) that is a RootOf of a quartic. I know that it is strictly positive (one of the four roots at least) for Gamma=0..10 and rho in (-1,+1), with bounds excluded.

I need to find the signs of its first and second derivatives (wrt to Gamma and wrt to rho: 4 derivatives in total).

I encounter numerical issues when I plot3d the derivatives using D[]() vs. fdiff() (numerical function evaluations of the RootOf). I was hoping for the two commands to produce the same output, but they don't it seems. What's going on?

Script:

> 

restart;
_quartic := RootOf(-8*(rho + 1)^4*_Z^4 + 12*(rho + 1)^3*Gamma*(rho - 1)*_Z^3 - 5*(rho + 1)^2*(-4/5 + Gamma^2*rho^2 + 2*(-2/5 - Gamma^2)*rho + Gamma^2)*_Z^2 - 4*(rho + 1)*Gamma*(rho^2 - 1)*_Z + Gamma^2*(rho + 1)*(rho - 1)^2);
convert(_quartic,radical):
f(Gamma,rho) := simplify(%):

RootOf((8*rho^3+24*rho^2+24*rho+8)*_Z^4+(-12*Gamma*rho^3-12*Gamma*rho^2+12*Gamma*rho+12*Gamma)*_Z^3+(5*Gamma^2*rho^3-5*Gamma^2*rho^2-5*Gamma^2*rho+5*Gamma^2-4*rho^2-8*rho-4)*_Z^2+(4*Gamma*rho^2-4*Gamma)*_Z-Gamma^2*rho^2+2*rho*Gamma^2-Gamma^2)

 (1)


Synthetic representation of derivatives

> 

der1_Gamma := diff(_quartic, Gamma):
der1_rho := diff(_quartic, rho):

Diff('f(Gamma,rho)', Gamma) = collect~(normal(eval(der1_Gamma, _quartic = 'f(Gamma,rho)')), 'f(Gamma,rho)');
Diff('f(Gamma,rho)', rho) = collect~(normal(eval(der1_rho, _quartic = 'f(Gamma,rho)')), 'f(Gamma,rho)');

der2_Gamma := diff(der1_Gamma, Gamma):
der2_rho := diff(der1_rho, rho):

Diff('f(Gamma,rho)', Gamma$2) = collect~(normal(eval(der2_Gamma, _quartic = 'f(Gamma,rho)')), 'f(Gamma,rho)');
Diff('f(Gamma,rho)', rho$2) = collect~(normal(eval(der2_rho, _quartic = 'f(Gamma,rho)')), 'f(Gamma,rho)');

Diff(f(Gamma, rho), Gamma) = -((-6*rho^3-6*rho^2+6*rho+6)*f(Gamma, rho)^3+(5*Gamma*rho^3-5*Gamma*rho^2-5*Gamma*rho+5*Gamma)*f(Gamma, rho)^2+(2*rho^2-2)*f(Gamma, rho)-Gamma*rho^2+2*Gamma*rho-Gamma)/((16*rho^3+48*rho^2+48*rho+16)*f(Gamma, rho)^3+(-18*Gamma*rho^3-18*Gamma*rho^2+18*Gamma*rho+18*Gamma)*f(Gamma, rho)^2+(5*Gamma^2*rho^3-5*Gamma^2*rho^2-5*Gamma^2*rho+5*Gamma^2-4*rho^2-8*rho-4)*f(Gamma, rho)+2*Gamma*rho^2-2*Gamma)

  

Diff(f(Gamma, rho), rho) = -(1/2)*((24*rho^2+48*rho+24)*f(Gamma, rho)^4+(-36*Gamma*rho^2-24*Gamma*rho+12*Gamma)*f(Gamma, rho)^3+(15*Gamma^2*rho^2-10*Gamma^2*rho-5*Gamma^2-8*rho-8)*f(Gamma, rho)^2+8*Gamma*rho*f(Gamma, rho)-2*rho*Gamma^2+2*Gamma^2)/((16*rho^3+48*rho^2+48*rho+16)*f(Gamma, rho)^3+(-18*Gamma*rho^3-18*Gamma*rho^2+18*Gamma*rho+18*Gamma)*f(Gamma, rho)^2+(5*Gamma^2*rho^3-5*Gamma^2*rho^2-5*Gamma^2*rho+5*Gamma^2-4*rho^2-8*rho-4)*f(Gamma, rho)+2*Gamma*rho^2-2*Gamma)

  

Diff(f(Gamma, rho), Gamma, Gamma) = ((448*rho^8+1792*rho^7+1792*rho^6-1792*rho^5-4480*rho^4-1792*rho^3+1792*rho^2+1792*rho+448)*f(Gamma, rho)^8+(-1632*Gamma*rho^8-3264*Gamma*rho^7+3264*Gamma*rho^6+9792*Gamma*rho^5-9792*Gamma*rho^3-3264*Gamma*rho^2+3264*Gamma*rho+1632*Gamma)*f(Gamma, rho)^7+(2120*Gamma^2*rho^8-8480*Gamma^2*rho^6-208*rho^7+12720*Gamma^2*rho^4-624*rho^6-208*rho^5-8480*Gamma^2*rho^2+1040*rho^4+1040*rho^3+2120*Gamma^2-208*rho^2-624*rho-208)*f(Gamma, rho)^6+(-1200*Gamma^3*rho^8+2400*Gamma^3*rho^7+2400*Gamma^3*rho^6-7200*Gamma^3*rho^5+640*Gamma*rho^7+640*Gamma*rho^6+7200*Gamma^3*rho^3-1920*Gamma*rho^5-2400*Gamma^3*rho^2-1920*Gamma*rho^4-2400*Gamma^3*rho+1920*Gamma*rho^3+1200*Gamma^3+1920*Gamma*rho^2-640*Gamma*rho-640*Gamma)*f(Gamma, rho)^5+(250*Gamma^4*rho^8-1000*Gamma^4*rho^7+1000*Gamma^4*rho^6+1000*Gamma^4*rho^5-632*Gamma^2*rho^7-2500*Gamma^4*rho^4+632*Gamma^2*rho^6+1000*Gamma^4*rho^3+1896*Gamma^2*rho^5+1000*Gamma^4*rho^2-1896*Gamma^2*rho^4+16*rho^6-1000*Gamma^4*rho-1896*Gamma^2*rho^3+32*rho^5+250*Gamma^4+1896*Gamma^2*rho^2-16*rho^4+632*Gamma^2*rho-64*rho^3-632*Gamma^2-16*rho^2+32*rho+16)*f(Gamma, rho)^4+(240*Gamma^3*rho^7-720*Gamma^3*rho^6+240*Gamma^3*rho^5+1200*Gamma^3*rho^4-32*Gamma*rho^6-1200*Gamma^3*rho^3-240*Gamma^3*rho^2+96*Gamma*rho^4+720*Gamma^3*rho-240*Gamma^3-96*Gamma*rho^2+32*Gamma)*f(Gamma, rho)^3+(-25*Gamma^4*rho^7+125*Gamma^4*rho^6-225*Gamma^4*rho^5+125*Gamma^4*rho^4+125*Gamma^4*rho^3-225*Gamma^4*rho^2+125*Gamma^4*rho-25*Gamma^4)*f(Gamma, rho)^2+(16*Gamma^3*rho^6-64*Gamma^3*rho^5+80*Gamma^3*rho^4-80*Gamma^3*rho^2+64*Gamma^3*rho-16*Gamma^3)*f(Gamma, rho)-5*Gamma^4*rho^6+30*Gamma^4*rho^5-75*Gamma^4*rho^4+100*Gamma^4*rho^3-75*Gamma^4*rho^2+30*Gamma^4*rho-5*Gamma^4)/(((16*rho^2+32*rho+16)*f(Gamma, rho)^3+(-18*Gamma*rho^2+18*Gamma)*f(Gamma, rho)^2+(5*Gamma^2*rho^2-10*Gamma^2*rho+5*Gamma^2-4*rho-4)*f(Gamma, rho)+2*Gamma*rho-2*Gamma)*((16*rho^3+48*rho^2+48*rho+16)*f(Gamma, rho)^3+(-18*Gamma*rho^3-18*Gamma*rho^2+18*Gamma*rho+18*Gamma)*f(Gamma, rho)^2+(5*Gamma^2*rho^3-5*Gamma^2*rho^2-5*Gamma^2*rho+5*Gamma^2-4*rho^2-8*rho-4)*f(Gamma, rho)+2*Gamma*rho^2-2*Gamma)^2)

  

Diff(f(Gamma, rho), rho, rho) = (1/4)*((21504*rho^6+129024*rho^5+322560*rho^4+430080*rho^3+322560*rho^2+129024*rho+21504)*f(Gamma, rho)^10+(-80640*Gamma*rho^6-347136*Gamma*rho^5-526080*Gamma*rho^4-245760*Gamma*rho^3+157440*Gamma*rho^2+199680*Gamma*rho+56064*Gamma)*f(Gamma, rho)^9+(127680*Gamma^2*rho^6+336512*Gamma^2*rho^5+122944*Gamma^2*rho^4-319744*Gamma^2*rho^3-18176*rho^5-246976*Gamma^2*rho^2-90880*rho^4+40576*Gamma^2*rho-181760*rho^3+53696*Gamma^2-181760*rho^2-90880*rho-18176)*f(Gamma, rho)^8+(-110016*Gamma^3*rho^6-112128*Gamma^3*rho^5+191808*Gamma^3*rho^4+172032*Gamma^3*rho^3+57344*Gamma*rho^5-105792*Gamma^3*rho^2+191488*Gamma*rho^4-59904*Gamma^3*rho+192512*Gamma*rho^3+24000*Gamma^3+2048*Gamma*rho^2-94208*Gamma*rho-37888*Gamma)*f(Gamma, rho)^7+(54960*Gamma^4*rho^6-28480*Gamma^4*rho^5-102480*Gamma^4*rho^4+56960*Gamma^4*rho^3-74176*Gamma^2*rho^5+40080*Gamma^4*rho^2-126144*Gamma^2*rho^4-28480*Gamma^4*rho+41088*Gamma^2*rho^3+7440*Gamma^4+138368*Gamma^2*rho^2+5120*rho^4+19776*Gamma^2*rho+20480*rho^3-25536*Gamma^2+30720*rho^2+20480*rho+5120)*f(Gamma, rho)^6+(-15300*Gamma^5*rho^6+30000*Gamma^5*rho^5-2100*Gamma^5*rho^4-24000*Gamma^5*rho^3+50048*Gamma^3*rho^5+14100*Gamma^5*rho^2+5632*Gamma^3*rho^4-6000*Gamma^5*rho-89856*Gamma^3*rho^3+3300*Gamma^5-1024*Gamma^3*rho^2-13056*Gamma*rho^4+39808*Gamma^3*rho-31232*Gamma*rho^3-4608*Gamma^3-15360*Gamma*rho^2+10752*Gamma*rho+7936*Gamma)*f(Gamma, rho)^5+(1875*Gamma^6*rho^6-6250*Gamma^6*rho^5+7125*Gamma^6*rho^4-3500*Gamma^6*rho^3-18316*Gamma^4*rho^5+2125*Gamma^6*rho^2+25540*Gamma^4*rho^4-2250*Gamma^6*rho+12072*Gamma^4*rho^3+875*Gamma^6-26520*Gamma^4*rho^2+12992*Gamma^2*rho^4+6244*Gamma^4*rho+10240*Gamma^2*rho^3+980*Gamma^4-15488*Gamma^2*rho^2-9728*Gamma^2*rho-512*rho^3+3008*Gamma^2-1536*rho^2-1536*rho-512)*f(Gamma, rho)^4+(3320*Gamma^5*rho^5-9240*Gamma^5*rho^4+7600*Gamma^5*rho^3-560*Gamma^5*rho^2-6144*Gamma^3*rho^4-1320*Gamma^5*rho+4992*Gamma^3*rho^3+200*Gamma^5+6784*Gamma^3*rho^2-4992*Gamma^3*rho+1024*Gamma*rho^3-640*Gamma^3+1536*Gamma*rho^2-512*Gamma)*f(Gamma, rho)^3+(-200*Gamma^6*rho^5+800*Gamma^6*rho^4-1200*Gamma^6*rho^3+800*Gamma^6*rho^2+1248*Gamma^4*rho^4-200*Gamma^6*rho-3136*Gamma^4*rho^3+1920*Gamma^4*rho^2+576*Gamma^4*rho-768*Gamma^2*rho^3-608*Gamma^4+768*Gamma^2*rho)*f(Gamma, rho)^2+(-16*Gamma^5*rho^4+192*Gamma^5*rho^3-480*Gamma^5*rho^2+448*Gamma^5*rho+256*Gamma^3*rho^3-144*Gamma^5-384*Gamma^3*rho^2+128*Gamma^3)*f(Gamma, rho)-20*Gamma^6*rho^4+80*Gamma^6*rho^3-120*Gamma^6*rho^2+80*Gamma^6*rho-32*Gamma^4*rho^3-20*Gamma^6+96*Gamma^4*rho^2-96*Gamma^4*rho+32*Gamma^4)/(((16*rho^2+32*rho+16)*f(Gamma, rho)^3+(-18*Gamma*rho^2+18*Gamma)*f(Gamma, rho)^2+(5*Gamma^2*rho^2-10*Gamma^2*rho+5*Gamma^2-4*rho-4)*f(Gamma, rho)+2*Gamma*rho-2*Gamma)*((16*rho^3+48*rho^2+48*rho+16)*f(Gamma, rho)^3+(-18*Gamma*rho^3-18*Gamma*rho^2+18*Gamma*rho+18*Gamma)*f(Gamma, rho)^2+(5*Gamma^2*rho^3-5*Gamma^2*rho^2-5*Gamma^2*rho+5*Gamma^2-4*rho^2-8*rho-4)*f(Gamma, rho)+2*Gamma*rho^2-2*Gamma)^2)

 (2)


Signs of derivatives: fdiff (numerical function evaluations of the RootOf) vs. D[]()

> 

restart;
with(plots):
> 

_quartic := RootOf(-8*(rho + 1)^4*_Z^4 + 12*(rho + 1)^3*Gamma*(rho - 1)*_Z^3 - 5*(rho + 1)^2*(-4/5 + Gamma^2*rho^2 + 2*(-2/5 - Gamma^2)*rho + Gamma^2)*_Z^2 - 4*(rho + 1)*Gamma*(rho^2 - 1)*_Z + Gamma^2*(rho + 1)*(rho - 1)^2):
> 

plot3d(_quartic, Gamma=0..10, rho=-1..+1, labels=[Gamma,rho,Lambda(Gamma,rho)],axesfont=["helvetica","roman",20],labelfont=["helvetica","roman",30]);
 

 

Define it as a f and test it for Gamma=1 and rho=0.5

> 

f := (Gamma,rho) -> RootOf(-8*(rho + 1)^4*_Z^4 + 12*(rho + 1)^3*Gamma*(rho - 1)*_Z^3 - 5*(rho + 1)^2*(-4/5 + Gamma^2*rho^2 + 2*(-2/5 - Gamma^2)*rho + Gamma^2)*_Z^2 - 4*(rho + 1)*Gamma*(rho^2 - 1)*_Z + Gamma^2*(rho + 1)*(rho - 1)^2):
evalf(f(1.0,0.5));

HFloat(0.5110796212870378)

 (3)

Value at zero:

> 

f(0,0):
allvalues(%):
fl := select(is, [allvalues(f(0,0))], positive)[];evalf(%);

(1/2)*2^(1/2)

  

.7071067810

 (4)

Value at infinity (commented out because too slow)

> 

#limit(f(x,y), {x = infinity, y = 0}):
#fh := select(is, [allvalues(%)], positive)[];evalf(%);

Derivative at zero:

> 

allvalues([D[1](f)(0,0)]):
Dfl := %[1][];

-1/4

 (5)

Derivative at a point, evaluated, vs numerical derivative at a point:

> 

D[1](f)(1,0.5):
evalf(%);
fdiff(f(x,y), x, {x = 1.0, y = 0.5});
fdiff(f, [1], [1.0,0.5]);

D[2](f)(1,0.5):
evalf(%);
fdiff(f(x,y), y, {x = 1.0, y = 0.5});
fdiff(f, [2], [1.0,0.5]);

HFloat(-0.05086932918910799)

  

-0.5086932919e-1

  

-0.5086932919e-1

  

HFloat(-0.05166477232109392)

  

-0.5166477232e-1

  

-0.5166477232e-1

 (6)

Can make a function out of fdiff

> 

fDfG := (Gamma,rho) -> fdiff(f, [1], [Gamma,rho]);
fDfr := (Gamma,rho) -> fdiff(f, [2], [Gamma,rho]);

proc (Gamma, rho) options operator, arrow; fdiff(f, [1], [Gamma, rho]) end proc

  

proc (Gamma, rho) options operator, arrow; fdiff(f, [2], [Gamma, rho]) end proc

 (7)

Check for numerical values close to thresholds:

> 

Digits := 15:
evalf('D[1]'(f)(0.1e-8,0.5));fdiff(f, [1], [0.1e-8,0.5]);
evalf('D[1]'(f)(0.1e-7,0.5));fdiff(f, [1], [0.1e-7,0.5]);
evalf('D[1]'(f)(0.1e-5,0.5));fdiff(f, [1], [0.1e-5,0.5]);
evalf('D[1]'(f)(0.00001,0.5));fdiff(f, [1], [0.00001,0.5]);
evalf('D[1]'(f)(0.001,0.5));fdiff(f, [1], [0.001,0.5]);


evalf('D[2]'(f)(1,-0.99));fdiff(f, [2], [1,-0.99]);
evalf('D[2]'(f)(1,-0.97));fdiff(f, [2], [1,-0.97]);
evalf('D[2]'(f)(1,-0.1));fdiff(f, [2], [1,-0.1]);
evalf('D[2]'(f)(1,0.98));fdiff(f, [2], [1,0.98]);
evalf('D[2]'(f)(1,-0.99));fdiff(f, [2], [1,-0.99]);

57735026.8022959

  

57735026.8022959

  

-0.833333329724894e-1

  

-0.833333329724894e-1

  

-0.833332972489415e-1

  

-0.833332972489415e-1

  

-0.833329724894151e-1

  

-0.833329724894151e-1

  

-0.832972489466445e-1

  

-0.832972489466445e-1

  

-223.615892086941

  

-223.615892086941

  

-43.0236130145893

  

-43.0236130145893

  

-.212392503268663

  

-.212392503268663

  

-0.127828146340716e-2

  

-0.127828146340716e-2

  

-223.615892086941

  

-223.615892086941

 (8)

Compare with D (vertical range here to prevent effect of large values from fdiff near zero):

> 

d1G := plot3d([D[1](f), fDfG], 0..10, -0.95..+0.95, view=-0.3..0, color = [red, blue]);
d1r := plot3d([D[2](f), fDfr], 0..10, -0.95..+0.95, color = [red, blue]);

 

 

 

Second derivatives:

> 

evalf('D[1,1]'(f)(1.0,0.5));
fdiff(f, [1, 1], [1.0,0.5]);

evalf('D[2,2]'(f)(1.0,0.5));
fdiff(f, [2, 2], [1.0,0.5]);

fD2fG := (Gamma,rho) -> fdiff(f, [1, 1], [Gamma]);
fD2fr := (Gamma,rho) -> fdiff(f, [2, 2], [Gamma]);

0.266607527050519e-1

  

0.266607527050519e-1

  

.151600577769391

  

.151600577769391

  

proc (Gamma, rho) options operator, arrow; fdiff(f, [1, 1], [Gamma]) end proc

  

proc (Gamma, rho) options operator, arrow; fdiff(f, [2, 2], [Gamma]) end proc

 (9)
> 

d2G:= plot3d([D[1,1](f), fD2fG], 0..10, -0.9..+0.9, color = [red, blue]);
d2r:= plot3d([D[2,2](f), fD2fr], 0..10, -0.9..+0.9, color = [red, blue]);
 

Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct

  

 

Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct

  

 
> 

d1d2G := plot3d([fDfG, fD2fG], 0.1e-6 .. 10, -0.98 .. +0.98, axesfont=["helvetica","roman",20],labelfont=["helvetica","roman",30], size=[1000,1000]);
d1d2r := plot3d([fDfr, fD2fr], 0.1e-6 .. 10, -0.98 .. +0.98, axesfont=["helvetica","roman",20],labelfont=["helvetica","roman",30], size=[1000,1000]);

Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct

  

 

Warning, unable to evaluate the function to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct

  

 
 

NULL

Download signs_derivatves_bivariate.mw

Robustness of plot(eval(something,[])) v...

MaPal93 175 Maple 2023
May 04 2024
2 7

Hi,

I want to analyze a quartic equation: rho-analysis.mw. I am interested in positive roots, which I need expressed in explicit/closed form.

I include four questions in the script.

Thanks!

EDIT (since my question was originally tagged as incomplete and even duplicate):
Above all I am trying to understand:

  1. Why implicitplot3d returns a unique strictly positive root for all rho and Gamma but plot() of the quartic evaluated for a specific pair of value returns three positive roots?
  2. rho is a correlation coefficient, thus bounded in (-1,+1): how to adjust the precision of the slider in Explore() so that I can play around with multiple rho values within said range?

How to "show" this inequality?...

MaPal93 175 Maple 2024
solve inequality duplicate-question
April 09 2024
1 14

I have two expressions, wo_theta and with_theta, which depend on multiple variables.

I would need your help to:

  1. Verify, as formally as possible, that wo_theta > with_theta always, i.e., for any value of theta different from zero (and regardless of the values taken up by the other variables)
  2. Show the above in a way that is easy and immediate to interpret (perhaps using some type of plot?)

In other words, I want to verify that as soon as I introduce any theta in my expression such expression becomes smaller:

> 

restart;
> 

local gamma;

gamma

 (1)
> 

assume(0 < gamma, 0 < nu__02, 0 < nu__01, 0 <= sigma__v, delta__1::real, delta__2::real, delta__3::real, theta::real);
interface(showassumed=0);

1

 (2)
> 

wo_theta := X__3*(-X__3*lambda__3 - delta__3*lambda__3 + DEV) + X__2*(-X__2*lambda__2 - delta__2*lambda__2 - nu__02) + X__1*(-X__1*lambda__1 - delta__1*lambda__1 - nu__01) + X__2*(nu__02 + DEV/2) + X__1*(nu__01 + DEV/2) - gamma*X__2^2*sigma__v^2/4 - gamma*X__1^2*sigma__v^2/4 + gamma*X__2*X__1*sigma__v^2/2;

X__3*(-X__3*lambda__3-delta__3*lambda__3+DEV)+X__2*(-X__2*lambda__2-delta__2*lambda__2-nu__02)+X__1*(-X__1*lambda__1-delta__1*lambda__1-nu__01)+X__2*(nu__02+(1/2)*DEV)+X__1*(nu__01+(1/2)*DEV)-(1/4)*gamma*X__2^2*sigma__v^2-(1/4)*gamma*X__1^2*sigma__v^2+(1/2)*gamma*X__2*X__1*sigma__v^2

 (3)
> 

with_theta := X__3*(-X__3*lambda__3 - theta*lambda__3 - delta__3*lambda__3 + DEV) + X__2*(-X__2*lambda__2 + theta*lambda__2 - delta__2*lambda__2 - nu__02) + X__1*(-X__1*lambda__1 + theta*lambda__1 - delta__1*lambda__1 - nu__01) + X__2*(nu__02 + DEV/2) + X__1*(nu__01 + DEV/2) - gamma*X__2^2*sigma__v^2/4 - gamma*X__1^2*sigma__v^2/4 + gamma*X__2*X__1*sigma__v^2/2 + theta*(lambda__1*(X__1 + delta__1 - theta) + lambda__2*(X__2 + delta__2 - theta) - lambda__3*(X__3 + delta__3 + theta));

X__3*(-X__3*lambda__3-theta*lambda__3-delta__3*lambda__3+DEV)+X__2*(-X__2*lambda__2+theta*lambda__2-delta__2*lambda__2-nu__02)+X__1*(-X__1*lambda__1+theta*lambda__1-delta__1*lambda__1-nu__01)+X__2*(nu__02+(1/2)*DEV)+X__1*(nu__01+(1/2)*DEV)-(1/4)*gamma*X__2^2*sigma__v^2-(1/4)*gamma*X__1^2*sigma__v^2+(1/2)*gamma*X__2*X__1*sigma__v^2+theta*(lambda__1*(X__1+delta__1-theta)+lambda__2*(X__2+delta__2-theta)-lambda__3*(X__3+delta__3+theta))

 (4)
> 

collect(with_theta, theta);

(-lambda__1-lambda__2-lambda__3)*theta^2+(-X__3*lambda__3+X__2*lambda__2+X__1*lambda__1+lambda__1*(X__1+delta__1)+lambda__2*(X__2+delta__2)-lambda__3*(X__3+delta__3))*theta+X__3*(-X__3*lambda__3-delta__3*lambda__3+DEV)+X__2*(-X__2*lambda__2-delta__2*lambda__2-nu__02)+X__1*(-X__1*lambda__1-delta__1*lambda__1-nu__01)+X__2*(nu__02+(1/2)*DEV)+X__1*(nu__01+(1/2)*DEV)-(1/4)*gamma*X__2^2*sigma__v^2-(1/4)*gamma*X__1^2*sigma__v^2+(1/2)*gamma*X__2*X__1*sigma__v^2

 (5)
> 

solve(wo_theta > with_theta, theta) assuming 0 < gamma, 0 < nu__02, 0 < nu__01, 0 < sigma__v, delta__1::real, delta__2::real, delta__3::real, theta::real;
> 

solve(with_theta < wo_theta, theta);

Warning, solve may be ignoring assumptions on the input variables.

  

Warning, solutions may have been lost

  
> 

difference_term := (-lambda__1 - lambda__2 - lambda__3)*theta^2 + (X__1*lambda__1 + X__2*lambda__2 - X__3*lambda__3 + lambda__1*(X__1 + delta__1) + lambda__2*(X__2 + delta__2) - lambda__3*(X__3 + delta__3))*theta;

(-lambda__1-lambda__2-lambda__3)*theta^2+(-X__3*lambda__3+X__2*lambda__2+X__1*lambda__1+lambda__1*(X__1+delta__1)+lambda__2*(X__2+delta__2)-lambda__3*(X__3+delta__3))*theta

 (6)
> 

# I would expect such difference_term in theta to be always < 0, i.e., for any theta different from 0)
# (Note that lambda_1, lambda_2, and lambda_3 are always > 0, while theta, the three X and the three delta can be positive or negative. In other words, it suffices to show that the linear term in theta is always negative...)
solve(difference_term<0);

Warning, solve may be ignoring assumptions on the input variables.

  

{X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, X__1 < -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1, theta < 0, lambda__1 < 0, lambda__2 < 0, lambda__3 < 0}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, X__2 < (1/2)*(2*X__3*lambda__3+lambda__2*theta+lambda__3*theta-delta__2*lambda__2+delta__3*lambda__3)/lambda__2, theta < 0, lambda__2 < 0, lambda__3 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < lambda__1, theta < 0, lambda__2 < 0, lambda__3 < 0, -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1 < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, X__1 < (1/2)*(2*X__3*lambda__3+lambda__1*theta+lambda__3*theta-delta__1*lambda__1+delta__3*lambda__3)/lambda__1, theta < 0, lambda__1 < 0, lambda__3 < 0}, {X__1 = X__1, X__2 = X__2, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, lambda__2 = 0, theta < 0, lambda__3 < 0, -(1/2)*delta__3-(1/2)*theta < X__3}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, 0 < lambda__1, theta < 0, lambda__3 < 0, (1/2)*(2*X__3*lambda__3+lambda__1*theta+lambda__3*theta-delta__1*lambda__1+delta__3*lambda__3)/lambda__1 < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < lambda__2, X__1 < -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1, theta < 0, lambda__1 < 0, lambda__3 < 0}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, 0 < lambda__2, theta < 0, lambda__3 < 0, (1/2)*(2*X__3*lambda__3+lambda__2*theta+lambda__3*theta-delta__2*lambda__2+delta__3*lambda__3)/lambda__2 < X__2}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < lambda__1, 0 < lambda__2, theta < 0, lambda__3 < 0, -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1 < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__3 = 0, X__1 < -(1/2)*(2*X__2*lambda__2-lambda__1*theta-lambda__2*theta+delta__1*lambda__1+delta__2*lambda__2)/lambda__1, theta < 0, lambda__1 < 0, lambda__2 < 0}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, lambda__3 = 0, X__2 < -(1/2)*delta__2+(1/2)*theta, theta < 0, lambda__2 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__3 = 0, 0 < lambda__1, theta < 0, lambda__2 < 0, -(1/2)*(2*X__2*lambda__2-lambda__1*theta-lambda__2*theta+delta__1*lambda__1+delta__2*lambda__2)/lambda__1 < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, lambda__3 = 0, X__1 < -(1/2)*delta__1+(1/2)*theta, theta < 0, lambda__1 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, lambda__3 = 0, 0 < lambda__1, theta < 0, -(1/2)*delta__1+(1/2)*theta < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__3 = 0, 0 < lambda__2, X__1 < -(1/2)*(2*X__2*lambda__2-lambda__1*theta-lambda__2*theta+delta__1*lambda__1+delta__2*lambda__2)/lambda__1, theta < 0, lambda__1 < 0}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, lambda__3 = 0, 0 < lambda__2, theta < 0, -(1/2)*delta__2+(1/2)*theta < X__2}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__3 = 0, 0 < lambda__1, 0 < lambda__2, theta < 0, -(1/2)*(2*X__2*lambda__2-lambda__1*theta-lambda__2*theta+delta__1*lambda__1+delta__2*lambda__2)/lambda__1 < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < lambda__3, X__1 < -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1, theta < 0, lambda__1 < 0, lambda__2 < 0}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, 0 < lambda__3, X__2 < (1/2)*(2*X__3*lambda__3+lambda__2*theta+lambda__3*theta-delta__2*lambda__2+delta__3*lambda__3)/lambda__2, theta < 0, lambda__2 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < lambda__1, 0 < lambda__3, theta < 0, lambda__2 < 0, -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1 < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, 0 < lambda__3, X__1 < (1/2)*(2*X__3*lambda__3+lambda__1*theta+lambda__3*theta-delta__1*lambda__1+delta__3*lambda__3)/lambda__1, theta < 0, lambda__1 < 0}, {X__1 = X__1, X__2 = X__2, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, lambda__2 = 0, 0 < lambda__3, X__3 < -(1/2)*delta__3-(1/2)*theta, theta < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, 0 < lambda__1, 0 < lambda__3, theta < 0, (1/2)*(2*X__3*lambda__3+lambda__1*theta+lambda__3*theta-delta__1*lambda__1+delta__3*lambda__3)/lambda__1 < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < lambda__2, 0 < lambda__3, X__1 < -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1, theta < 0, lambda__1 < 0}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, 0 < lambda__2, 0 < lambda__3, theta < 0, (1/2)*(2*X__3*lambda__3+lambda__2*theta+lambda__3*theta-delta__2*lambda__2+delta__3*lambda__3)/lambda__2 < X__2}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < lambda__1, 0 < lambda__2, 0 < lambda__3, theta < 0, -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1 < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < theta, lambda__1 < 0, lambda__2 < 0, lambda__3 < 0, -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1 < X__1}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, 0 < theta, lambda__2 < 0, lambda__3 < 0, (1/2)*(2*X__3*lambda__3+lambda__2*theta+lambda__3*theta-delta__2*lambda__2+delta__3*lambda__3)/lambda__2 < X__2}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < theta, 0 < lambda__1, X__1 < -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1, lambda__2 < 0, lambda__3 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, 0 < theta, lambda__1 < 0, lambda__3 < 0, (1/2)*(2*X__3*lambda__3+lambda__1*theta+lambda__3*theta-delta__1*lambda__1+delta__3*lambda__3)/lambda__1 < X__1}, {X__1 = X__1, X__2 = X__2, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, lambda__2 = 0, 0 < theta, X__3 < -(1/2)*delta__3-(1/2)*theta, lambda__3 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, 0 < theta, 0 < lambda__1, X__1 < (1/2)*(2*X__3*lambda__3+lambda__1*theta+lambda__3*theta-delta__1*lambda__1+delta__3*lambda__3)/lambda__1, lambda__3 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < theta, 0 < lambda__2, lambda__1 < 0, lambda__3 < 0, -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1 < X__1}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, 0 < theta, 0 < lambda__2, X__2 < (1/2)*(2*X__3*lambda__3+lambda__2*theta+lambda__3*theta-delta__2*lambda__2+delta__3*lambda__3)/lambda__2, lambda__3 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < theta, 0 < lambda__1, 0 < lambda__2, X__1 < -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1, lambda__3 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__3 = 0, 0 < theta, lambda__1 < 0, lambda__2 < 0, -(1/2)*(2*X__2*lambda__2-lambda__1*theta-lambda__2*theta+delta__1*lambda__1+delta__2*lambda__2)/lambda__1 < X__1}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, lambda__3 = 0, 0 < theta, lambda__2 < 0, -(1/2)*delta__2+(1/2)*theta < X__2}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__3 = 0, 0 < theta, 0 < lambda__1, X__1 < -(1/2)*(2*X__2*lambda__2-lambda__1*theta-lambda__2*theta+delta__1*lambda__1+delta__2*lambda__2)/lambda__1, lambda__2 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, lambda__3 = 0, 0 < theta, lambda__1 < 0, -(1/2)*delta__1+(1/2)*theta < X__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, lambda__3 = 0, 0 < theta, 0 < lambda__1, X__1 < -(1/2)*delta__1+(1/2)*theta}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__3 = 0, 0 < theta, 0 < lambda__2, lambda__1 < 0, -(1/2)*(2*X__2*lambda__2-lambda__1*theta-lambda__2*theta+delta__1*lambda__1+delta__2*lambda__2)/lambda__1 < X__1}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, lambda__3 = 0, 0 < theta, 0 < lambda__2, X__2 < -(1/2)*delta__2+(1/2)*theta}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__3 = 0, 0 < theta, 0 < lambda__1, 0 < lambda__2, X__1 < -(1/2)*(2*X__2*lambda__2-lambda__1*theta-lambda__2*theta+delta__1*lambda__1+delta__2*lambda__2)/lambda__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < theta, 0 < lambda__3, lambda__1 < 0, lambda__2 < 0, -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1 < X__1}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, 0 < theta, 0 < lambda__3, lambda__2 < 0, (1/2)*(2*X__3*lambda__3+lambda__2*theta+lambda__3*theta-delta__2*lambda__2+delta__3*lambda__3)/lambda__2 < X__2}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < theta, 0 < lambda__1, 0 < lambda__3, X__1 < -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1, lambda__2 < 0}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, 0 < theta, 0 < lambda__3, lambda__1 < 0, (1/2)*(2*X__3*lambda__3+lambda__1*theta+lambda__3*theta-delta__1*lambda__1+delta__3*lambda__3)/lambda__1 < X__1}, {X__1 = X__1, X__2 = X__2, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, lambda__2 = 0, 0 < theta, 0 < lambda__3, -(1/2)*delta__3-(1/2)*theta < X__3}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__2 = 0, 0 < theta, 0 < lambda__1, 0 < lambda__3, X__1 < (1/2)*(2*X__3*lambda__3+lambda__1*theta+lambda__3*theta-delta__1*lambda__1+delta__3*lambda__3)/lambda__1}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < theta, 0 < lambda__2, 0 < lambda__3, lambda__1 < 0, -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1 < X__1}, {X__1 = X__1, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, lambda__1 = 0, 0 < theta, 0 < lambda__2, 0 < lambda__3, X__2 < (1/2)*(2*X__3*lambda__3+lambda__2*theta+lambda__3*theta-delta__2*lambda__2+delta__3*lambda__3)/lambda__2}, {X__2 = X__2, X__3 = X__3, delta__1 = delta__1, delta__2 = delta__2, delta__3 = delta__3, 0 < theta, 0 < lambda__1, 0 < lambda__2, 0 < lambda__3, X__1 < -(1/2)*(2*X__2*lambda__2-2*X__3*lambda__3-lambda__1*theta-lambda__2*theta-lambda__3*theta+delta__1*lambda__1+delta__2*lambda__2-delta__3*lambda__3)/lambda__1}

 (7)
 

NULL

Download inequality.mw

How to non-dimensionalize this 10th-degr...

MaPal93 175 Maple
rootof allvalues duplicate-question
March 20 2024
1 29

Details here: non-dimensionalization.mw. Thanks!

In short, I want to plot all the ten roots of a 10th-degree polynomial in _Z with 4 or 5 primitive parameters. To do so, I need to find that combination of parameters that allows me to transform/scale/non-dimensionalize the original polynomial in a polynomial in _Z with just (1) one parameter or at maximum (2) two parameters. In the case of (1), I can plot its roots in a standard 2D plot against the single parameter. In the case of (2), I can plot the roots in a standard 3D plot against the two parameters.

Best quality when manually exporting plo...

MaPal93 175 Maple
export plotting png
March 19 2024
1 1

I want to export plots as PNG but found in the past that when using commands to automate the process (and explicitly controlling for image quality) some symbolic notation on the axes or in the plots themselves are translated to 1D.

Anyway, regardless of the reason just provided, I have a preference for exporting plots as PNG manually rather than automatically in some of my scripts. How to do this while ensuring the best quality? By default, manual exports into PNG have quite bad quality. 

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