function f=galerkin2cheb(g,typ)
%this function transforms from the galerkin basis to the chebyshev basis
% typ = integer
% 1) Dirichlet/Dirichlet Mason Galerkin
% 2) Dirichlet/Dirichlet Julien Galerkin
% 3) Dirichlet/Dirichlet Trefethen Galerkin
% 4) u=u'=0
% 5) u=u''=0
% 6) Neumann/Neumann
% 7) u'[-1]-u[-1]/-1=u'[1]-u[1]==0 %stress free angular velocity
% 8) u[1]=u[-1]=u''[1]+u'[1]-u[1]=u''[-1]-u'[-1]-u[-1]=0 % stress free velocity potential
% 9) u'[1]=u'[-1]=u'''[1]=u'''[-1]==0 %Neumann/Neumann and third deriv
% 9) u[1]=u[-1]=u''''[1]=u''''[-1]==0 %Dirichlet/Dirichlet and fourth deriv
% % Matrix Method ~ O(N^2)
% switch typ
% case 1 %Mason
% A=diag(ones(length(g)+2,1));
% for j = 3:2:length(g)+1
% A(1,j)=-1;
% A(2,j+1)=-1;
% end
% if (mod(length(g+2),2)==0)
% A(2,end)=-1;
% else
% A(1,end)=-1;
% end
% f=A*[0;0;g];
% case 2 %Julien
% A=diag(ones(length(g)+2,1))-diag(ones(length(g),1),2);
% f=A*[0;0;g];
% case 3 %Trefethen
% A=diag(ones(length(g)+2,1))-diag(2*ones(length(g),1),2)+diag(ones(length(g)-2,1),4);
% A(1,3)=-1; A(2,4)=-1;% Correct for TO and T1 mode
% f=A*[0;0;g];
% end
%Direct Solve ~ O(N)
switch typ
case 1 %Mason: u[1]=u[-1]==0
a0=0;a1=0;
for j = 1:2:length(g)-1
a0=-1*g(j)+a0;
a1=-1*g(j+1)+a1;
end
if mod(length(g),2)==1
a0=-1*g(end)+a0;
end
f=[a0;a1;g];
case 2 %Julien: u[1]=u[-1]==0
f=zeros(length(g)+2,1);
f(end)=g(end);
f(end-1)=g(end-1);
for j=length(g):-1:3
f(j)=g(j-2)-g(j);
end
f(1)=-g(1);
f(2)=-g(2);
case 3 %Trefethen: u[1]=u[-1]==0
f=zeros(length(g)+2,1);
f(end)=g(end);
f(end-1)=g(end-1);
f(end-2)=g(end-2)-2*g(end);
f(end-3)=g(end-3)-2*g(end-1);
for j=length(g)-2:-1:3
f(j)=g(j-2)-2*g(j)+g(j+2);
end
f(1)=-g(1)+g(3);
f(2)=-g(2)+g(4);
case 4 %u[1]=u[-1]=u'[1]=u'[-1]=0
L=length(g);
D=ones(L+4,1);
L2=[];
for k=0:L+1;
L2=[L2,-2*(k+2)/(k+3)];
end
L4=[];
for k=0:L-1;
L4=[L4,(k+1)/(k+3)];
end
A=sparse(diag(D))+sparse(diag(L2,-2))+sparse(diag(L4,-4));
f=A*[g;0;0;0;0];
case 5 %u[1]=u[-1]=u''[1]=u''[-1]=0
L=length(g);
D=ones(L+4,1);
L2=[];
for k=0:L+1;
L2=[L2,-(1+((k+1)*(2*k^2+4*k+3)/((k+3)*(2*k^2+12*k+19))))];
end
L4=[];
for k=0:L-1;
L4=[L4,((k+1)*(2*k^2+4*k+3)/((k+3)*(2*k^2+12*k+19)))];
end
A=sparse(diag(D))+sparse(diag(L2,-2))+sparse(diag(L4,-4));
f=A*[g;0;0;0;0];
case 6 %u'[1]=u'[-1]==0
L=length(g);
D=[1];
for k=1:L+1;
D=[D,(k+2)^2/(2*k^2+4*k+4)];
end
L2=[0];
for k=1:L-1;
L2=[L2,-((k)^2)/(2*k^2+4*k+4)];
end
A=sparse(diag(D))+sparse(diag(L2,-2));
f=A*[g;0;0];
case 7 %u'[-1]-u[-1]/-1=u'[1]-u[1]==0
L=length(g);
D=[];
for k=0:L+1;
D=[D,((k+2)^2-1)/(2*(k+1)^2)];
end
L2=[];
for k=0:L-1;
L2=[L2,-((k^2-1))/(2*(k+1)^2)];
end
A=sparse(diag(D))+sparse(diag(L2,-2));
f=A*[g;0;0];
case 8 %u[1]=u[-1]=u''[1]+u'[1]-u[1]=u''[-1]-u'[-1]-u[-1]=0
L=length(g);
D=ones(L+4,1);
L2=[];
for k=0:L+1;
L2=[L2,-2*(k+2)*(k^2+4*k+9)/((k+3)*(k^2+6*k+11))];
end
L4=[];
for k=0:L-1;
L4=[L4,(k+1)*(k^2+2*k+3)/((k+3)*(k^2+6*k+11))];
end
A=sparse(diag(D))+sparse(diag(L2,-2))+sparse(diag(L4,-4));
f=A*[g;0;0;0;0];
case 9 %u'[1]=u'[-1]=u'''[1]=u'''[-1]==0 %Neumann/Neumann and third deriv
L=length(g);
D=ones(L+4,1);
L2=[];
for j=0:L+1;
L2=[L2,(-2*j^2*(11 + 2*j*(4 + j)))/((2 + j)*(3 + j)*...
(15 + 2*j*(6 + j)))];
end
L4=[];
for j=0:L-1;
L4=[L4,(j^2*(1 + j)*(-1 + 2*j*(2 + j)))/ ...
((3 + j)*(4 + j)^2*(15 + 2*j*(6 + j)))];
end
A=sparse(diag(D))+sparse(diag(L2,-2))+sparse(diag(L4,-4));
f=A*[g;0;0;0;0];
case 10 %u[1]=u[-1]=u''''[1]=u''''[-1]==0 %Dirichlet/Dirchlet and fourth deriv
L=length(g);
D=ones(L+4,1);
D(1)=1;
L2=[];
for j=0:L+1;
L2=[L2,(-2*(210 + j*(4 + j)*(53 + 2*j*(4 + j))))/ ...
((4 + j)*(5 + j)*(21 + 2*j*(6 + j)))];
end
L4=[];
for j=0:L-1;
L4=[L4,(j*(-5 + j*(1 + 2*j*(1 + j))))/...
((4 + j)*(5 + j)*(21 + 2*j*(6 + j)))];
end
A=sparse(diag(D))+sparse(diag(L2,-2))+sparse(diag(L4,-4));
f=A*[g;0;0;0;0];
end
