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I can write the 1D TISE as $$\psi''(x) = -k^2(x) \psi(x),$$ where $$k^2(x) = \dfrac{2m}{\hbar^2} \left[ E - V(x) \right].$$ Since the TISE is an eigenvalue equation, I do not know the value of E beforehand, and I have to guess it. So, I have to make two guesses — ##\psi'(x = 0)## and ##E##. Now, suppose I have been given ##\psi(x = 0)## and ##\psi(x = x_N)##, and, of course, I know the form of ##V(x)##. After one iteration from ##x = 0 \text{ to } x_N##, I find that the numerically computed ##\psi(x_N)## doesn't quite match the given value. Now, which one do I change — ##\psi'(x = 0)## or ##E##?