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Question: Numerical noise

Question:Numerical noise

kh2n 220 Maple
fsolve
February 15 2011
1



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restart:
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Digits:=60:
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variphi:=0.38:R1:=0.0009:R2:=8.75:E1:=1:E2:=1:R4:=177.6:C:=1.96:k:=4:H:=7:Pec_i:=-120:Phi:=1:Q:=1.9:R:=20:
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A:=(Q*(R*S+1-S)*(H+1/(1-exp(Pec_i)))*exp(C*Pec_i*S/(k*(R*S+1-S)))-H)*(1-exp(Pec_i*(1-S)/(R*S+1-S)))=1;

(1)
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eqn2:=Student:-Calculus1:-Roots(subs(H=7,R=20,C=1.96,k=4,Pec_i=-120,Q=1.9,A),S);

(2)
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S0:=.3592598898850950896270594340253413222287;

(3)
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F:=1/(R*S0+1-S0);

(4)
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w_0:=1/k*Pec_i*F*C;

(5)
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w_1:=Pec_i*F;

(6)
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M_1:=evalf(Q*Pec_i/k*(H+1/(1-exp(Pec_i))));

(7)
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M_2:=M_1*exp(w_0*S0);

(8)
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M_3:=w_1/(1-exp(w_1*(S0-1)));

(9)
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M_4:=M_3*exp(w_1*(S0-1));

(10)
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R3l0:=-(-cosh(1/2*S0*(w_0^2)^(1/2))*(w_0^2)^(1/2)+sinh(1/2*S0*(w_0^2)^(1/2))*w_0)*(S0*R2*R1+1-S0)*w_1*((M_2*k*w_0+1/2*M_4*w_1)*C*sinh(1/2*(-1+S0)*(C^2*w_1^2)^(1/2)/C)-1/2*(C^2*w_1^2)^(1/2)*cosh(1/2*(-1+S0)*(C^2*w_1^2)^(1/2)/C)*M_4)*w_0/((-(1/2*M_3*C*(2+w_1*(-1+S0))*w_0*exp(w_1*(-1+S0))+R1*(C*M_1*(S0*w_0+1)*exp(S0*w_0)-w_0*H)*k*w_1)*(w_0^2)^(1/2)*cosh(1/2*S0*(w_0^2)^(1/2))+(1/2*M_3*C*(2+w_1*(-1+S0))*w_0*exp(w_1*(-1+S0))+R1*(C*M_1*(S0*w_0+1)*exp(S0*w_0)-w_0*H)*k*w_1)*w_0*sinh(1/2*S0*(w_0^2)^(1/2))+(w_0^2)^(1/2)*k*M_1*exp(1/2*S0*w_0)*w_1*C*R1)*sinh(1/2*(-1+S0)*(C^2*w_1^2)^(1/2)/C)-1/2*M_3*w_0*(C^2*w_1^2)^(1/2)*cosh(1/2*(-1+S0)*(C^2*w_1^2)^(1/2)/C)*exp(w_1*(-1+S0))*(-cosh(1/2*S0*(w_0^2)^(1/2))*(w_0^2)^(1/2)+sinh(1/2*S0*(w_0^2)^(1/2))*w_0)*(-1+S0))/(R1-1)/R2;

(11)
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R3linf := -1.*Pec_i/k/(R*S0+1-S0)*C*(Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0+exp(Pec_i/(R*S0+1-S0)*(S0-1.))*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*R2*R1+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*R2*R1-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*H*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*H*R2*R1*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*R2*R1*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R2*R1*R*S0*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*H-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*exp(Pec_i/(R*S0+1-S0)*(S0-1.))*R2*R1+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R2*R1*S0*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*H*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*R2*R1*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*H*R2*R1+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R2*R1*R*S0+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*H*R2*R1*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*R2*R1-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R2*R1*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*R2*R1*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*H-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*H*R2*R1-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R*S0*H*R2*R1*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*H*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*H-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*H*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+exp(Pec_i/(R*S0+1-S0)*(S0-1.))*R2*R1*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R2*R1+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*S0*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*H*R2*R1-1.*Q*exp(1/k*Pec_i/(R*S0+1-S0)*C*S0)*R2*R1*S0)/R2/R1/H/(R1-1.-1.*R1*exp(Pec_i/(R*S0+1-S0)*(S0-1.))-1.*R1*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))+R1*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))*exp(Pec_i/(R*S0+1-S0)*(S0-1.))+exp(Pec_i/(R*S0+1-S0)*(S0-1.))+exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0))-1.*exp(Pec_i/(R*S0+1-S0)*(S0-1.))*exp(Pec_i/(R*S0+1-S0)*(R*S0+1.-1.*S0)));

(12)
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R3:=-38.16;

(13)

 

 
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A:=variphi*sigma*Phi = -2*(((1/4*(cosh(l)-sinh(l))*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*w_0*(-l*w_0+E1*sigma)*(R1-1)*C^2*(l*C*w_1-sigma*E2*R1*k)*(l*w_0+E1*sigma)*M_3*R2*(-1/2*w_1+l)*exp(S0*(w_1-l)-w_1+2*l)+(1/4*(1/2*w_1+l)*(cosh(l)-sinh(l))*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*w_0*(-l*w_0+E1*sigma)*(R1-1)*C^2*(l*w_0+E1*sigma)*M_3*R2*exp(S0*(w_1+l)-w_1)+(-1/2*(l^2-1/2*l*w_0+E1*sigma+1/2*w_0^2)*(-l*w_0+E1*sigma)*(R1-1)*(l*R3*R2*cosh(l*(-1+S0))*R1+variphi*sinh(l*(-1+S0))*sigma)*C*M_1*k*exp(-(-w_0+l)*S0)+((-1/2*l*(R1-1)*C*R3*M_1*(l^2+1/2*l*w_0+E1*sigma+1/2*w_0^2)*k*exp((w_0+l)*S0)+(-l*w_0+E1*sigma)*(-1/2*H*k*R3*l*w_0*(R1-1)*cosh(l*S0)+(M_2*k*sigma*E1+1/2*M_2*w_0^2*k+l^2*M_2*k+1/4*w_1*M_4*w_0)*C*sinh(l*S0)))*R1*R2*cosh(l*(-1+S0))-(1/2*(R1-1)*sigma*variphi*C*M_1*(l^2+1/2*l*w_0+E1*sigma+1/2*w_0^2)*k*exp((w_0+l)*S0)+((M_2*k*sigma*E1+1/2*M_2*w_0^2*k+l^2*M_2*k+1/4*w_1*M_4*w_0)*C+1/2*H*k*sigma*variphi*w_0*(R1-1))*(-l*w_0+E1*sigma)*cosh(l*S0))*sinh(l*(-1+S0)))*(l*w_0+E1*sigma))*(l*C*w_1-sigma*E2*R1*k))*(l*C*w_1+sigma*E2*R1*k))*sinh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))-1/2*(w_0^2+4*E1*sigma+4*l^2)^(1/2)*((1/2*(cosh(l)-sinh(l))*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*(-l*w_0+E1*sigma)*(R1-1)*C^2*(l*C*w_1-sigma*E2*R1*k)*(l*w_0+E1*sigma)*M_3*R2*(-1/2*w_1+l)*exp(S0*(w_1-l)-w_1+2*l)+(l*C*w_1+sigma*E2*R1*k)*(1/2*(1/2*w_1+l)*(cosh(l)-sinh(l))*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*(-l*w_0+E1*sigma)*(R1-1)*C^2*(l*w_0+E1*sigma)*M_3*R2*exp(S0*(w_1+l)-w_1)+(1/2*C*k*M_1*(-w_0+l)*(R1-1)*(l*R3*R2*cosh(l*(-1+S0))*R1+variphi*sinh(l*(-1+S0))*sigma)*(-l*w_0+E1*sigma)*exp(-(-w_0+l)*S0)+((-1/2*C*k*R3*M_1*l*(w_0+l)*(R1-1)*exp((w_0+l)*S0)+(-l*w_0+E1*sigma)*(-H*k*R3*l*(R1-1)*cosh(l*S0)+C*(M_2*k*w_0+1/2*M_4*w_1)*sinh(l*S0)))*R1*R2*cosh(l*(-1+S0))-sinh(l*(-1+S0))*(1/2*C*k*M_1*sigma*variphi*(w_0+l)*(R1-1)*exp((w_0+l)*S0)+(-l*w_0+E1*sigma)*cosh(l*S0)*((M_2*k*w_0+1/2*M_4*w_1)*C+H*k*sigma*variphi*(R1-1))))*(l*w_0+E1*sigma))*(l*C*w_1-sigma*E2*R1*k)))*cosh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))+C*k*M_1*w_0*exp(1/2*S0*w_0)*(R1-1)*(l*C*w_1-sigma*E2*R1*k)*(l*C*w_1+sigma*E2*R1*k)*(l*R3*R2*cosh(l*(-1+S0))*R1+variphi*sinh(l*(-1+S0))*sigma)*(E1*sigma+l^2)))*sinh(1/2*(-1+S0)*((w_1^2+4*l^2)*C^2+4*C*sigma*E2*R1*k)^(1/2)/C)-1/8*(-cosh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))*(w_0^2+4*E1*sigma+4*l^2)^(1/2)+sinh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))*w_0)*((w_1^2+4*l^2)*C^2+4*C*sigma*E2*R1*k)^(1/2)*(-l*w_0+E1*sigma)*(l*w_0+E1*sigma)*((-C*M_3*R2*(cosh(l)-sinh(l))*(R1-1)*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*(l*C*w_1-sigma*E2*R1*k)*exp(S0*(w_1-l)-w_1+2*l)+(C*M_3*R2*(cosh(l)-sinh(l))*(R1-1)*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)*exp(S0*(w_1+l)-w_1)+2*M_4*(R1*R2*cosh(l*(-1+S0))*sinh(l*S0)-cosh(l*S0)*sinh(l*(-1+S0)))*(l*C*w_1-sigma*E2*R1*k))*(l*C*w_1+sigma*E2*R1*k))*cosh(1/2*(-1+S0)*((w_1^2+4*l^2)*C^2+4*C*sigma*E2*R1*k)^(1/2)/C)-2*C*k*E2*M_3*R1*sigma*R2*exp(1/2*w_1*(-1+S0)+l)*(cosh(l)-sinh(l))*(R1-1)*(R3*cosh(l*S0)*l+sinh(l*S0)*variphi*sigma)))*Phi/H/k/(l^2*C^2*w_1^2-sigma^2*E2^2*R1^2*k^2)/sinh(1/2*(-1+S0)*((w_1^2+4*l^2)*C^2+4*C*sigma*E2*R1*k)^(1/2)/C)/(R1*R2*cosh(l*(-1+S0))*sinh(l*S0)-cosh(l*S0)*sinh(l*(-1+S0)))/(-cosh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))*(w_0^2+4*E1*sigma+4*l^2)^(1/2)+sinh(1/2*S0*(w_0^2+4*E1*sigma+4*l^2)^(1/2))*w_0)/(-l^2*w_0^2+E1^2*sigma^2):for K from 3.50 by 0.01  to 3.60 do lprint(K, simplify(evalf(fsolve(subs(l=K, A),sigma=-350..5))));od;

3.50, -314.923173164654389867747414398949541923417339307110151060742

3.51, -314.811475122542643304511070165322730779049424307219678486016

3.52, -314.700526967354360817888716698520304562223129549082529139317

3.53, -314.590328376299839649582493242960614863043232135537722776600

3.54, -34.7441833384194630408351197809864124879575282271021278198878

3.55, -35.8905928733776333650866949498015784433129695957897798769347

3.56, -36.7567808965451547118490034402149052470577056267360062869435

3.57, -37.4850257881683914377805659257111370719963613051793057432780

3.58, -38.1270326868227905058572639481915465207214918052207621870875

3.59, -38.7085747177680034072069478913077078347878801119938487508561

3.60, -39.2447080559625876155022825602420878122589586940687333921707

  

In the above data list we can see that due to numerical noise there is a big jump. I am trying to find the solution in the interval [1, 10]. But due to multiple solutions I am unable to control noise. I have also tried to adjust the range (sigma=-350..5 ).  



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