SVM Poly Kernal Machine Training?

Hello. Training Poly Kernel SVM. How do I know when machine trained? I was expecting to use a quadratic solver (quadprog, SMO?) but not sure how and where to insert? Any ideas? Prefer not to use fitcsvm or similar library svm function since I want to see how and where SVM employs quad functions (I will use quad solver though). Thx--AR

%%Poly Kernel Trainer, 2 & 5 Digits

    x=[train2(:,1:64);train5(:,1:64)]; % 2,5 digit samples, 1 sample per row, 0s and 1s for attributes

    [m,n] = size(x); %[302 64] 

    Y=[train2(:,65);train5(:,65)];% Labels for #2 & #5 digits, [302,1]

    d=3;

    alpha=0.0156;

    beta=0;

    xT=x'; % xT=x(j), x=x(i); used for inner product w/o going to higher dim.

    K = (alpha * (x*xT)+ beta).^d; % Polynomial Machine, [302,302]

    %%SV= 0.5( ∑i ∑j alpha(i)alpha(j)y(i)y(j)K(x(i),x(j))) - ∑i alpha(i)%%
    %%where ∑i alpha(i)=0 and 0<=alpha(i)<C
    %%did not employ this equation yet, where??, how?? %%