function?[phi?lam?Xmean?nbasis]=POD（Xcenergy）
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%?This?function?calculates?the?POD?basis?vectors?
%
%?Function?inputs:
%?X???????:??（n?x?nsnap）?snapshot?matrix
%????????????n?=?number?of?states?in?full?space
%????????????nsnap?=?number?of?snapshots
%?cenergy?:??how?much?energy?of?the?ensemble?you?want?
%????????????to?capture?i.e.?cenergy?=?99.9?（percent）
%?
%?Function?outputs:
%?phi?????:???（n?x?nsnap）?matrix?containing?POD?basis?vectors
%?lam?????:???（nsnap?x?1）?vector?containing?POD?eigenvalues
%?Xmean???:???（n?x?1）?the?mean?of?the?ensemble?X
%?nbasis??:???number?of?POD?basis?vectors?you?should?use
%?????????????to?capture?cenergy?（percent）?of?the?ensemble
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%?calculate?the?mean?of?the?snapshots
nsnap?=?size（X2）;
Xmean?=?sum（X2）/nsnap;
%?Obtain?new?snapshot?ensemble?with?zero?mean
for?i=1:nsnap
????X1（:i）?=?X（:i）-Xmean;
end
%???METHOD?OF?SNAPSHOTS
%?calculate?the?empirical?correlation?matrix?C
C?=?X1‘*X1/nsnap;
%?Calculate?the?POD?basis
[evectorCevalueC]?=?eig（C）;
phi?=?X1?*?evectorC;
%?Normalize?the?POD?basis
for?i=1:nsnap
????phi（:i）?=?phi（:i）/norm（phi（:i）2）;
end
%?return?the?POD?eigenvalues?in?a?vector
lam?=?diag（evalueC）;
%?Rearrange?POD?eigenvalues?vectors?in?descending?order.
%?Note?that?the?correlation?matrix?C?is?symmetric?so?SVD?and?EIG
%?will?give?the?same?evectorC?and?evalueC?but?they?are?already?in
%?descending?order?and?hence?we?don‘t?need?to?rearrange?evectorC?and?evalueC
%?if?SVD?is?used
lam?=?rot90（lam2）;
phi?=?fliplr（phi）;

%%%???????Find?the?number?of?POD?basis?vectors?capturing?cenergy?（percent）?of?energy???%%%%

%?total?energy
tenergy?=?sum（lam）;
energy?=?0.;
nbasis?=?0;
i?=?1;
while?（（（energy/tenergy）*100）?????energy?=?energy?+?lam（i）;
????i?=?i+1;
end
nbasis?=?i;
%?plot?eigenvalues?corresponding?to?nbasis?vectors
plot（lam（1:nbasis）/tenergy/100‘*‘）
xlabel（‘Number?of?POD?eigenvalues‘）
ylabel（‘Energy?captured‘）
????????

?屬性????????????大小?????日期????時間???名稱
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?????文件???????2024??2004-10-29?17:36??POD.m

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