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We study numerically the ODE/IM correspondence for untwisted affine Liealgebras associated with simple Lie algebras including exceptional type. Weconsider the linear problem obtained from the massless limit of that of themodified affine Toda field equation. We found that the Q-functions inintegrable models are expressed as the inner product of the solution of thedual linear problem and the subdominant solution of the linear problem. UsingCheng's algorithm to obtain the solution of the linear problem, we candetermine efficiently the zeros of the Q-function, which is known to providethe solutions of the Bethe ansatz equations. We calculate the zerosnumerically, which are shown to agree with the results from the Non-LinearIntegral Equations for simply-laced affine Lie algebras including theexceptional type. By the folding procedure of the Dynkin diagrams ofsimply-laced Lie algebras, we also find the correspondence for the linearproblem of the non-simply-laced affine Lie algebras.

ODE/IM correspondence for affine Lie algebras: A numerical approach