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Algebraic Simplification Techniques in Richard Fateman's CS 282 Lecture 13

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Discover advanced algebraic simplification methods taught by Richard Fateman in CS 282 Lecture 13, including principles of rational expressions, variable manipulations, and derivative computations.

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Algebraic Simplification Techniques in Richard Fateman's CS 282 Lecture 13

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  1. Algebraic Simplification, Yet more Lecture 13 Richard Fateman CS 282 Lecture 13

  2. Richardson’s class R2 • rationals and p • variable x • +, *, / • log(u), exp(u), composition • no sin or abs • This has a zero equivalence algorithm if we can tell if a constant is zero. Richard Fateman CS 282 Lecture 13

  3. Consider an ordering of subexpressions • Suppose y is a subexpression of z, then z is more complex than z. Not much more structure is needed. • Let y be most complex subexpression and suppose it is log(u) for some u. • Express F=anyn + ... a1y + a0 • Is an, a simpler expression, zero? Richard Fateman CS 282 Lecture 13

  4. an is zero • Let F1=an-1yn-1+ ..., a simpler expression. Test it for zero. Richard Fateman CS 282 Lecture 13

  5. an is not zero • Let F1=F/an = yn+ ...+ a0/an, ok, since an is not zero. • Take the derivative wrt x of F1, which looks like F2=ny’yn-1+ ...+ (ana0’-a0an’)/an2. This expression is simpler because the derivative of log(u) is du/u. If F2=0 then F1 is a constant. Since it is a constant, test to see if it is zero. Richard Fateman CS 282 Lecture 13

  6. What if y is not log(u)? • Let y be most complex subexpression and suppose it is exp(u) for some u. • Express F=anyn + ... a1y + a0 • Is a0, a simpler expression, zero? • If a0 is not zero, divide through by it to get F1=anyn+ ...+a1y+1. This might not be simpler, but compute its derivative, F2=bnyn+..+b1y+0. Now we can divide through by y, and get something like bnyn-1+ ...+b1 which IS simpler. Note that exp(u) is assumed non-zero since u is supposedly defined and not -1 . Is F2 constant? Richard Fateman CS 282 Lecture 13

  7. What if a0 is zero? • divide through by y to get F1=anyn-1+ a1, a simpler expression. Since F=a0*F1, if F1 is zero, so is F. Richard Fateman CS 282 Lecture 13

  8. Generalizations and speculations • What if the only thing we need to add more functions to this list is to have a sufficiently simple defining equation, • df/dx = f defines exponential • df/dx = 1/x defines log • The speculation is that all the properties of special functions of physics can be derived “automatically” by a computer system which could then use these properties to build a simplifier. (Not just zero-equivalence test as given here.) Richard Fateman CS 282 Lecture 13

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