Numerical Methods in Computational Finance. Daniel J. Duffy

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The operator H is called the Hamiltonian operator, and it determines the time variation of the system. We are now interested in approximating Equation (5.24) using finite difference schemes. To this end, the explicit Euler FTCS scheme is given by:

(5.25)

      while the implicit Euler BTCS scheme is given by:

      or

(5.26)

      or

Scheme (5.25) is unstable, while (5.26) is stable. However, neither scheme is unitary in the sense of the original problem; that is, the total probability of finding the particle somewhere is 1:

      (5.27)

      A remedy for this is to use the Cayley form (this is essentially the Crank–Nicolson scheme):

      or

(5.28)

      This scheme is unitary; you can check this by a bit of arithmetic using complex arithmetic.

      The solution of (5.24) is:

      (5.29)

and the solution of (5.28) is:

(5.30)

We can see that (5.30) is the (1,1) Padé approximant to the exponential function. Its absolute value is 1.

5.8 SUMMARY AND CONCLUSIONS

      Matrix theory is too important to be ignored or given short shrift in any book on numerical analysis and its applications. For this reason, we gave a reasonably detailed exposition of matrix theory as a companion to the other chapters in this book (and it could possibly be a companion to other books).