Code covered by the BSD License

Highlights fromMittag-Leffler function

from Mittag-Leffler function by Igor Podlubny
Calculates the Mittag-Leffler function with desired accuracy.

[e]=mlf(alf,bet,c,fi)
```function [e]=mlf(alf,bet,c,fi)
%
% MLF -- Mittag-Leffler function.
%           MLF (alpha,beta,Z,P) is the Mittag-Leffler function E_{alpha,beta}(Z)
%           evaluated with accuracy 10^(-P) for each element of Z.
%           alpha and beta are scalars, P is integer, Z can be a vector or
%           a two-dimensional array. The output is of the same size as Z.
% (C) 2001-2012 Igor Podlubny, Martin Kacenak
% Last update: 2012-09-07

[cRows, cCols] = size(c);
c=m2v(c);

if nargin<4 , fi=6;		end
if nargin<3 || alf<=0 || fi<=0
else
[r,s]=size(c); [r1,s1]=size(alf); [r2,s2]=size(bet);
mx=max([r,s]); mx1=max([r1,s1]); mx2=max([r2,s2]);
if (r>1 && s>1) || (r1>1 && s1>1) || (r2>1 && s2>1) || (mx1>1 && mx2>1)
sprintf('wrong number of input parameters')
else
if mx1>mx2 , mxx=mx1; e=zeros(mx,mx1);
else mxx=mx2; e=zeros(mx,mx2);end;
for i1= 1:mx
for i2=1:mxx

if r>s , z=c(i1,1); else z=c(1,i1); end
if mx1>mx2 , if r1>s1 , alfa=alf(i2,1); else alfa=alf(1,i2);end, beta=bet;
else if r2>s2 ,beta=bet(i2,1); else beta=bet(1,i2); end, alfa=alf; end
if beta<0 , rc=(-2*log(10^(-fi)*pi/(6*(abs(beta)+2)*(2*abs(beta))^(abs(beta)))))^alfa;
else  rc=(-2*log(10^(-fi)*pi/6))^alfa; end
r0=max([1,2*abs(z),rc]);
if (alfa==1 && beta==1)
e(i1,i2)=exp(z);
else
if (alfa<1 && abs(z)<=1) || ( (1<=alfa && alfa <2) && abs(z)<=floor(20/(2.1-alfa)^(5.5-2*alfa))) || (alfa>=2 && abs(z)<=50)
oldsum=0;
k=0;
while (alfa*k+beta)<=0
k=k+1;
end
newsum=z^k/gamma(alfa*k+beta);
while newsum~=oldsum
oldsum=newsum;
k=k+1;
term=z^k/gamma(alfa*k+beta);
newsum=newsum+term;
k=k+1;
term=z^k/gamma(alfa*k+beta);
newsum=newsum+term;
end
e(i1,i2)=newsum;
else
if (alfa<=1 && abs(z)<=fix(5*alfa+10))
if ((abs(angle(z))>pi*alfa) && (abs(abs(angle(z))-(pi*alfa))>10^(-fi)))
if beta<=1
e(i1,i2)=rombint('K',0,r0,fi,alfa,beta,z);
else
eps=1;
e(i1,i2)=rombint('K',eps,r0,fi,alfa,beta,z)+ ...
rombint('P',-pi*alfa,pi*alfa,fi,alfa,beta,z,eps);
end
elseif (abs(angle(z))<pi*alfa && abs(abs(angle(z))-(pi*alfa))>10^(-fi))
if beta<=1
e(i1,i2)=rombint('K',0,r0,fi,alfa,beta,z)+ ...
(z^((1-beta)/alfa))*(exp(z^(1/alfa))/alfa);
else
eps=abs(z)/2;
e(i1,i2)=rombint('K',eps,r0,fi,alfa,beta,z)+ ...
rombint('P',-pi*alfa,pi*alfa,fi,alfa,beta,z,eps)+ ...
(z^((1-beta)/alfa))*(exp(z^(1/alfa))/alfa);
end
else
eps=abs(z)+0.5;
e(i1,i2)=rombint('K',eps,r0,fi,alfa,beta,z)+ ...
rombint('P',-pi*alfa,pi*alfa,fi,alfa,beta,z,eps);
end
else
if alfa<=1
if (abs(angle(z))<(pi*alfa/2+min(pi,pi*alfa))/2)
% alfa
newsum=(z^((1-beta)/alfa))*exp(z^(1/alfa))/alfa;
for k=1:floor(fi/log10(abs(z)))
newsum=newsum-((z^(-k))/gamma(beta-alfa*k));
% k
end
e(i1,i2)=newsum;
else
newsum=0;
for k=1:floor(fi/log10(abs(z)))
newsum=newsum-((z^-k)/gamma(beta-alfa*k));
end
e(i1,i2)=newsum;
end
else
if alfa>=2
m=floor(alfa/2);
sum=0;
for h=0:m
zn=(z^(1/(m+1)))*exp((2*pi*1i*h)/(m+1));
sum=sum+mlf(alfa/(m+1),beta,zn,fi);
end
e(i1,i2)=(1/(m+1))*sum;
else
e(i1,i2)=(mlf(alfa/2,beta,z^(1/2),fi)+mlf(alfa/2,beta,-z^(1/2),fi))/2;
end
end
end
end
end
end
end
end
end

if isreal(c)
e = real(e);
end

e = v2m(e,cRows,cCols);

function [res]=rombint(funfcn,a,b,order,varargin)
if nargin<4 ,order=6; end
if nargin<3
Warning ('Error in input format')
else
rom=zeros(2,order);
h=b-a;
rom(1,1)=h*(feval(funfcn,a,varargin{:})+feval(funfcn,b,varargin{:}))/2;

ipower=1;
for i= 2:order
sum=0;
for j=1:ipower
sum=sum+feval(funfcn,(a+h*(j-0.5)),varargin{:});
end
rom(2,1)=(rom(1,1)+h*sum)/2;
for k=1:i-1
rom(2,k+1)=((4^k)*rom(2,k)-rom(1,k))/((4^k)-1);
end

for j=0:i-1
rom(1,j+1)=rom(2,j+1);
end
ipower=ipower*2;
h=h/2;
end
res=rom(1,order);
end

function res=K(r,alfa,beta,z)
res=r.^((1-beta)/alfa).*exp(-r.^(1/alfa)).*(r*sin(pi*(1-beta))-...
z*sin(pi*(1-beta+alfa)))/(pi*alfa*(r.^2-2*r*z*cos(pi*alfa)+z.^2));

function res=P(r,alfa,beta,z,eps)
w=(eps^(1/alfa))*sin(r/alfa)+r*(1+(1-beta)/alfa);
res=((eps^(1+(1-beta)/alfa))/(2*pi*alfa))*((exp((eps^(1/alfa))*cos(r/alfa)).*...
(cos(w)+1i*sin(w))))/(eps*exp(1i*r)-z);

function A = v2m(V, M, N)
if numel(V)==M*N,
A = reshape(V, [N, M]);
A = A' ;
else
warning('Wrong dimensions of the output in V2M.')
end

function V = m2v(A)
M = A'; V = M(:);

```

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