function?y?=?LaguerreGen（varargin）
%LaguerreGen?calculates?the?generalized?Laguerre?polynomial?L{n?alpha}
%
%?This?function?computes?the?generalized?Laguerre?polynomial?L{nalpha}.?
%?If?no?alpha?is?supplied?alpha?is?set?to?zero?and?this?function?
%?calculates?the?“normal“?Laguerre?polynomial.
%
%?Input:
%??-?n?=?nonnegative?integer?as?degree?level
%??-?alpha?>=?-1?real?number?（input?is?optional）
%
%?The?output?is?formated?as?a?polynomial?vector?of?degree?（n+1）?
%?corresponding?to?MatLab?norms?（that?is?the?highest?coefficient?is?the?
%?first?element）.
%
%?Possible?usage:
%??-?polyval（LaguerreGen（n?alpha）?x）?evaluates?L{n?alpha}（x）
%??-?roots（LaguerreGen（n?alpha））?calculates?roots?of?L{n?alpha}


%?Calculation?is?done?recursively?using?matrix?operations?for?very?fast
%?execution?time.?The?formula?is?taken?from?Szeg??Orthogonal?Polynomials?
%?1958?formula?（5.1.10）


%???Author:?Matthias.Trampisch@rub.de
%???Date:?16.08.2007
%???Version?1.2
?

%%?======================================================================?%
%??????????????set?default?parameters?and?rename?input
%?=======================================================================?%
if?（nargin?==?1）????????%only?one?parameter?“n“?supplied
????????n?=?varargin{1};
????????alpha?=?0;??????%set?defaul?value?for?alpha
elseif?（nargin?==?2）????%at?least?two?parameters?supplied
????????n?=?varargin{1};
????????alpha?=?varargin{2};
end;

%%?======================================================================?%
%????????????????????error?checking?of?input?parameters
%?=======================================================================?%
if?（nargin?==?0）?||?（nargin?>?2）?||?（n~=abs（round（n）））?||?（alpha<-1）
????????error（‘n?must?be?integer?and?（optional）?alpha?>=?-1‘）;
end;

%%?======================================================================?%
%????????Recursive?calculation?of?generalized?Laguerre?polynomial
%?=======================================================================?%
L=zeros（n+1）;??????????%reserve?memory?for?faster?storage
switch?n
????case?0
????????L（1:）=1;
????otherwise???????????%n>1?so?we?need?to?do?recursion
????????L（1:）=[zeros（1n）?1];
????????L（2:）=[zeros（1?n-1）?-1?（alpha+1）];
????????for?i=3:n+1
????????????A1?=?1/（i-1）?*?（conv（[zeros（1?n-1）?-1?（2*（i-1）+alpha-1）]?L（i-1:）））;
????????????A2?=?1/（i-1）?*?（conv（[zeros（1?n）?（（i-1）+alpha-1）]?L（i-2:）））;
????????????B1=A1（length（A1）-n:1:length（A1））;
????????????B2=A2（length（A2）-n:1:length（A2））;
????????????L（i:）=B1-B2;????%?i-th?row?corresponds?to?L{i-1?alpha}
????????end;
end;

%%?======================================================================?%
%???????????????????????????????Define?output
%?=======================================================================?%
y=L（n+1:）;??%last?row?is?the?gen.?Laguerre?polynomial?L{n?alpha}

?屬性????????????大小?????日期????時間???名稱
-----------?---------??----------?-----??----
?????文件????????2846??2015-06-22?20:57??LaguerreGen.m