Combining arbitrary order global Pad approximation of the Mittag-Leffler function with its addition formula for a significant accuracy boost
Author
Richard Herrmann
Title
Combining arbitrary order global Pad approximation of the Mittag-Leffler function with its addition formula for a significant accuracy boost
Description
Padé approximant of the Mittag-Leffler function using Solve
Category
Academic Articles & Supplements
Keywords
Fractional Calculus, Mittag-Leffler function
URL
http://www.notebookarchive.org/2024-10-0xt10fv/
DOI
https://notebookarchive.org/2024-10-0xt10fv
Date Added
2024-10-02
Date Last Modified
2024-10-02
File Size
95.81 kilobytes
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Rights
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Supplemental notebook to R. Herrmann, “Combining arbitrary order global Padé approximation of the Mittag-Leffler function with its addition formula for a significant accuracy boost”, arXiv:2408.10257 [physics.gen-ph]. https://doi.org/10.48550/arXiv.2408.10257
Combining arbitrary order global Padé approximation of the Mittag-Leffler function with its addition formula for a significant accuracy boost
Combining arbitrary order global Padé approximation of the Mittag-Leffler function with its addition formula for a significant accuracy boost
Richard Herrmann
gigaHedron, r.herrmann@fractionalcalculus.org
October 2024.
Introduction
Introduction
The combination of the global Padé approximation of the Mittag-Leffler function with its addition formula for the case α < 1 yields significantly higher accuracy results for a given arbitrary order n. We present a solution in terms of a Mathematica notebook to determine the general structure of the system of linear equations to be solved.
Program pade1.nb
Program pade1.nb
start clean
start clean
In[]:=
Clear[alp,bet,vars]
order of PadeApproximant
order of PadeApproximant
In[]:=
n=2
Out[]=
2
asymptotics
asymptotics
In[]:=
a[x_]:=+Gamma[bet-alp]x*Sum[(-x)^k/Gamma[bet+alpk],{k,0,n+1}];b[x_]:=-Gamma[bet-alp]x*Sum[(-x)^(-k)/Gamma[bet-alpk],{k,1,n}];
collect parameters
collect parameters
In[]:=
ps=Symbol["p"<>ToString[#]]&/@Range[0,n-1];qs=Symbol["q"<>ToString[#]]&/@Range[0,n-1];vars=Append[ps,qs]//Flattenxs=(x^#)&/@Range[0,n-1];
Out[]=
{p0,p1,q0,q1}
Pade polynomials p/q
Pade polynomials p/q
In[]:=
p=Total[psxs]+x^n;q=Total[qsxs]+x^n;
collect conditions
collect conditions
In[]:=
ca=CoefficientList[p-qa[x],x];cb=CoefficientList[p/x^n-q/x^nb[x],1/x];
collect equations
collect equations
In[]:=
condsa=(ca[[#]]==0)&/@Range[n+1]condsb=(cb[[#]]==0)&/@Range[2,n]conds=Append[condsa,condsb]//Flatten
Out[]=
p00,p1-0,1-+0
q0Gamma[-alp+bet]
Gamma[bet]
q1Gamma[-alp+bet]
Gamma[bet]
q0Gamma[-alp+bet]
Gamma[alp+bet]
Out[]=
p1-q1+0
Gamma[-alp+bet]
Gamma[-2alp+bet]
Out[]=
p00,p1-0,1-+0,p1-q1+0
q0Gamma[-alp+bet]
Gamma[bet]
q1Gamma[-alp+bet]
Gamma[bet]
q0Gamma[-alp+bet]
Gamma[alp+bet]
Gamma[-alp+bet]
Gamma[-2alp+bet]
solve system of inhomogenous linear equations
solve system of inhomogenous linear equations
In[]:=
result=Solve[conds,vars];padeApproximant=p/q/x/Gamma[bet-alp]/.result
Out[]=
-xGamma[-alp+bet]++(Gamma[bet](Gamma[bet]Gamma[-2alp+bet]-)Gamma[alp+bet])(Gamma[-2alp+bet]Gamma[-alp+bet](-+Gamma[-alp+bet]Gamma[alp+bet]))
2
x
x(Gamma[bet]Gamma[-2alp+bet]-)Gamma[alp+bet]
2
Gamma[-alp+bet]
Gamma[-2alp+bet](-Gamma[-alp+bet]Gamma[alp+bet])
2
Gamma[bet]
2
x
x(Gamma[-alp+bet]-Gamma[bet]Gamma[-2alp+bet]Gamma[alp+bet])
2
Gamma[bet]
Gamma[-2alp+bet](-Gamma[-alp+bet]Gamma[alp+bet])
2
Gamma[bet]
2
Gamma[-alp+bet]
2
Gamma[bet]
unit test
unit test
In[]:=
alp=0.5bet=1.501ug=10^-3;og=10^3;fig1=Plot[{padeApproximant,MittagLefflerE[alp,bet,-x]},{x,ug,og}];fig2=LogLogPlot[1-padeApproximant/MittagLefflerE[alp,bet,-x],{x,ug,og}];fig3=Plot[1-padeApproximant/MittagLefflerE[alp,bet,-x],{x,1,2}];GraphicsRow[{fig1,fig2,fig3},ImageSizeFull]
Out[]=
0.5
Out[]=
1.501
Out[]=
Cite this as: Richard Herrmann, "Combining arbitrary order global Pad approximation of the Mittag-Leffler function with its addition formula for a significant accuracy boost" from the Notebook Archive (2024), https://notebookarchive.org/2024-10-0xt10fv
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