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Question: get the error order of the estimated polynomial containing a lot of trigonometric functions

Question:get the error order of the estimated polynomial containing a lot of trigonometric functions

Aixleft math 80 Maple
determinant taylor
July 07 2024
0

Hi all guys, when i am doing error analysis but I meet with an problem. I get the trace and determinant of one matrix which consists a lot trigonometric functions. I wanna get the approximation error order of trace and determinant (Like tr=2+O(v^6),det=1+O(v^6)). But I use Taylor expansion and series, it displays can't compute the series. How to know the other ways to get the error order of it? Thanks all !phase_error_try.mw

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restart
> 

c[2] := 1/2+(1/10)*sqrt(5); c[3] := 1/2-(1/10)*sqrt(5)

1/2+(1/10)*5^(1/2)

  

1/2-(1/10)*5^(1/2)

 (1)
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with(LinearAlgebra)
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``
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A := Matrix([[0, 0, 0], [-(cos((1/10)*(5+sqrt(5))*v)-1)/v^2, 0, 0], [0, -(cos((1/10)*(-5+sqrt(5))*v)-1)/(cos((1/10)*(5+sqrt(5))*v)*v^2), 0]])
> 

C := Matrix([0, 1/2+(1/10)*sqrt(5), 1/2-(1/10)*sqrt(5)])

Matrix(%id = 36893490461606184468)

 (2)
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e := Matrix([[1], [1], [1]])

Matrix(%id = 36893490461606180252)

 (3)
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E := Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]])

Matrix(%id = 36893490461606177116)

 (4)
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G := Matrix([[0], [10*sin((1/10)*(5+sqrt(5))*v)/((5+sqrt(5))*v)], [(10*(sin((1/10)*(-5+sqrt(5))*v)*cos((1/10)*(5+sqrt(5))*v)+sin((1/10)*(5+sqrt(5))*v)*cos((1/10)*(-5+sqrt(5))*v)-sin((1/10)*(5+sqrt(5))*v)))/(v*cos((1/10)*(5+sqrt(5))*v)*(-5+sqrt(5)))]])
> 

b := Matrix([1/24, (-sin((1/10)*v*(-5+sqrt(5)))*v^3+12*cos((1/10)*v*(-5+sqrt(5)))*v^2+24*cos((1/10)*v*(-5+sqrt(5)))*cos(v)-24*sin((1/10)*v*(-5+sqrt(5)))*sin(v)+24*sin((1/10)*v*(-5+sqrt(5)))*v-24*cos((1/10)*v*(-5+sqrt(5))))/(24*v^3*(cos((1/10)*v*(-5+sqrt(5)))*sin((1/10)*v*(5+sqrt(5)))+sin((1/10)*v*(-5+sqrt(5)))*cos((1/10)*v*(5+sqrt(5))))), -(sin((1/10)*v*(5+sqrt(5)))*v^3+12*cos((1/10)*v*(5+sqrt(5)))*v^2+24*cos(v)*cos((1/10)*v*(5+sqrt(5)))+24*sin(v)*sin((1/10)*v*(5+sqrt(5)))-24*v*sin((1/10)*v*(5+sqrt(5)))-24*cos((1/10)*v*(5+sqrt(5))))/(24*v^3*(cos((1/10)*v*(-5+sqrt(5)))*sin((1/10)*v*(5+sqrt(5)))+sin((1/10)*v*(-5+sqrt(5)))*cos((1/10)*v*(5+sqrt(5)))))])
> 

bp := Matrix([1/12, -(sin((1/10)*v*(-5+sqrt(5)))*v^2+12*cos((1/10)*v*(-5+sqrt(5)))*sin(v)-12*cos((1/10)*v*(-5+sqrt(5)))*v+12*cos(v)*sin((1/10)*v*(-5+sqrt(5)))-12*sin((1/10)*v*(-5+sqrt(5))))/(12*v^2*(cos((1/10)*v*(-5+sqrt(5)))*sin((1/10)*v*(5+sqrt(5)))+sin((1/10)*v*(-5+sqrt(5)))*cos((1/10)*v*(5+sqrt(5))))), -(sin((1/10)*v*(5+sqrt(5)))*v^2+12*cos(v)*sin((1/10)*v*(5+sqrt(5)))-12*cos((1/10)*v*(5+sqrt(5)))*sin(v)+12*cos((1/10)*v*(5+sqrt(5)))*v-12*sin((1/10)*v*(5+sqrt(5))))/(12*v^2*(cos((1/10)*v*(-5+sqrt(5)))*sin((1/10)*v*(5+sqrt(5)))+sin((1/10)*v*(-5+sqrt(5)))*cos((1/10)*v*(5+sqrt(5)))))])
> 

L1 := 1/simplify(E+v^2.A); N1 := simplify(1-(1/2)*v^2+v^4*(b.L1.G.C.e)); N11 := (Typesetting[delayDotProduct](sin((1/10)*v*(5+sqrt(5)))*((v^3-24*v+24*sin(v))*sin((1/10)*v*(5+sqrt(5)))+12*cos((1/10)*v*(5+sqrt(5)))*(v^2+2*cos(v)-2))*(-5+sqrt(5)), v^2.((cos((1/10)*v*(-5+sqrt(5)))-1)*sec((1/10)*v*(5+sqrt(5)))/v^2), true)+((cos((1/10)*v*(-5+sqrt(5)))-1)*(v^3-24*v+24*sin(v))*(5+sqrt(5))*tan((1/10)*v*(5+sqrt(5)))+(96*v^2+240*cos(v)-192)*cos((1/10)*v*(-5+sqrt(5)))+2*sqrt(5)*(v^3-24*v+24*sin(v))*sin((1/10)*v*(-5+sqrt(5)))-(12*(v^2+2*cos(v)-2))*(5+sqrt(5)))*sin((1/10)*v*(5+sqrt(5)))+12*cos((1/10)*v*(5+sqrt(5)))*sin((1/10)*v*(-5+sqrt(5)))*(-6+(v^2+2*cos(v)-2)*sqrt(5)+3*v^2+10*cos(v)))/(48*sin((1/10)*v*(-5+sqrt(5)))*cos((1/10)*v*(5+sqrt(5)))+48*sin((1/10)*v*(5+sqrt(5)))*cos((1/10)*v*(-5+sqrt(5))))

Matrix(%id = 36893490461639877084)

 (5)
> 

N2 := simplify(1-v^2*b.L1.e); N22 := (Typesetting[delayDotProduct](((12*v^2+24*cos(v)-24)*cos((1/10)*(5+sqrt(5))*v)+(v^3-24*v+24*sin(v))*sin((1/10)*(5+sqrt(5))*v))*(v^2.((cos((1/10)*(5+sqrt(5))*v)-1)/v^2)+1), v^2.((cos((1/10)*(-5+sqrt(5))*v)-1)*sec((1/10)*(5+sqrt(5))*v)/v^2), true)+Typesetting[delayDotProduct]((-12*v^2-24*cos(v)+24)*cos((1/10)*(-5+sqrt(5))*v)+sin((1/10)*(-5+sqrt(5))*v)*(v^3-24*v+24*sin(v)), v^2.((cos((1/10)*(5+sqrt(5))*v)-1)/v^2), true)+((-v^3+24*v)*sin((1/10)*(-5+sqrt(5))*v)+12*v^2+24*cos(v)-24)*cos((1/10)*(5+sqrt(5))*v)+((-v^3+24*v)*cos((1/10)*(-5+sqrt(5))*v)+v^3-24*v+24*sin(v))*sin((1/10)*(5+sqrt(5))*v)+(-12*v^2-24*cos(v)+24)*cos((1/10)*(-5+sqrt(5))*v)+sin((1/10)*(-5+sqrt(5))*v)*(v^3-24*v+24*sin(v)))/(24*v*(sin((1/10)*(-5+sqrt(5))*v)*cos((1/10)*(5+sqrt(5))*v)+sin((1/10)*(5+sqrt(5))*v)*cos((1/10)*(-5+sqrt(5))*v)))

Matrix(%id = 36893490461606200972)

 (6)
> 

N3 := simplify(-v^2+v^4*bp.L1.G.C.e); N33 := v*(Typesetting[delayDotProduct](((v^2+12*cos(v)-12)*sin((1/10)*(5+sqrt(5))*v)+12*cos((1/10)*(5+sqrt(5))*v)*(v-sin(v)))*sin((1/10)*(5+sqrt(5))*v)*(-5+sqrt(5)), v^2.((cos((1/10)*(-5+sqrt(5))*v)-1)*sec((1/10)*(5+sqrt(5))*v)/v^2), true)+((cos((1/10)*(-5+sqrt(5))*v)-1)*(v^2+12*cos(v)-12)*(5+sqrt(5))*tan((1/10)*(5+sqrt(5))*v)+(96*v-120*sin(v))*cos((1/10)*(-5+sqrt(5))*v)+2*sqrt(5)*(v^2+12*cos(v)-12)*sin((1/10)*(-5+sqrt(5))*v)-(12*(5+sqrt(5)))*(v-sin(v)))*sin((1/10)*(5+sqrt(5))*v)+12*cos((1/10)*(5+sqrt(5))*v)*sin((1/10)*(-5+sqrt(5))*v)*((v-sin(v))*sqrt(5)+3*v-5*sin(v)))/(24*cos((1/10)*(5+sqrt(5))*v)*sin((1/10)*(-5+sqrt(5))*v)+24*sin((1/10)*(5+sqrt(5))*v)*cos((1/10)*(-5+sqrt(5))*v))

Matrix(%id = 36893490461733603676)

 (7)
> 

N4 := simplify(1-v^2*bp.L1.e); N44 := (Typesetting[delayDotProduct](((v^2+12*cos(v)-12)*sin((1/10)*(5+sqrt(5))*v)+12*cos((1/10)*(5+sqrt(5))*v)*(v-sin(v)))*(v^2.((cos((1/10)*(5+sqrt(5))*v)-1)/v^2)+1), v^2.((cos((1/10)*(-5+sqrt(5))*v)-1)*sec((1/10)*(5+sqrt(5))*v)/v^2), true)+Typesetting[delayDotProduct]((v^2+12*cos(v)-12)*sin((1/10)*(-5+sqrt(5))*v)-12*cos((1/10)*(-5+sqrt(5))*v)*(v-sin(v)), v^2.((cos((1/10)*(5+sqrt(5))*v)-1)/v^2), true)+(-cos((1/10)*(-5+sqrt(5))*v)*v^2+v^2+12*cos(v)+12*cos((1/10)*(-5+sqrt(5))*v)-12)*sin((1/10)*(5+sqrt(5))*v)+(-sin((1/10)*(-5+sqrt(5))*v)*v^2+12*v-12*sin(v)+12*sin((1/10)*(-5+sqrt(5))*v))*cos((1/10)*(5+sqrt(5))*v)+(v^2+12*cos(v)-12)*sin((1/10)*(-5+sqrt(5))*v)-12*cos((1/10)*(-5+sqrt(5))*v)*(v-sin(v)))/(12*sin((1/10)*(-5+sqrt(5))*v)*cos((1/10)*(5+sqrt(5))*v)+12*sin((1/10)*(5+sqrt(5))*v)*cos((1/10)*(-5+sqrt(5))*v))

Matrix(%id = 36893490461606185188)

 (8)
> 

tr := N11+N44
> 

det := N11*N44-N22*N33
> 

expand(det, v, 10)

Warning,  computation interrupted

  

` `

 (9)

NULL

NULL


 

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