why is integration harder than differentiation

Discussion in 'Math Research' started by Zig, Feb 16, 2004.

  1. If your question is "how can the inverse of differentiation be so much
    harder than differentiation itself", then I can say that it happens all
    the time in mathematics that an operation is easier to apply than invert.
    For example, multiplying prime numbers is easier than factoring a number
    into primes. This example is more relevant than you might suppose.
    Factoring a number into primes is similar to factoring a polynomial;
    integrating a rational function requires factoring its denominator.

    To be sure, there are fast algorithms to numerically factor polynomials.
    But that dodges your question, because there are also fast algorithms
    to numerically integrate any analytic function. Your question
    is really about exact integration.
     
    Greg Kuperberg, Feb 19, 2004
    #21
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  2. Readers may be interested in my new paper in which I fight the
    curse of dimensionality for numerical integration in high dimensions:

    http://front.math.ucdavis.edu/math.NA/0402047

    Granted, it's impossible to completely defeat the curse. I derive
    good methods for functions that are well-approximated by low-degree
    polynomials.
     
    Greg Kuperberg, Feb 19, 2004
    #22
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  3. Zig

    Mitch Harris Guest

    The key terms are: integration in finite terms, Risch's algorithm, Liouville's theorem.
    The authors associated with it are Liouville, Ostrowski, Ritt, Risch, and Rosenlicht.

    The main references I know of are

    Robert Risch, The Problem of Integration in Finite Terms, Trans. of AMS, v139 (1969) p167-189.
    (the main results)

    and

    Max Rosenlicht, Integration in Finite Terms, Am. Math. Monthly, v79 (1972) p936-972.
    (good exposition with classic examples (like int e^(-x^2)))
     
    Mitch Harris, Feb 19, 2004
    #23
  4. ...

    This answer seems right to me.

    And (to spell it out more explicitly) we find the situation
    puzzling mainly because we don't take ease of differentiation
    as our criterion for selecting this class. Rather, we generate
    this class "naturally" by combining a small number of elementary
    functions via algebraic ops (multiplication and division being
    the pesky ones) and by other kinds of functional composition.
    The chain rule makes differentiation a piece of cake. On the
    other hand, to apply the chain rule in reverse we first have to
    solve a factorization problem, and that's where the difficulty
    lies.
     
    Russell Blackadar, Feb 19, 2004
    #24
  5. Zig

    tchow Guest

    These are certain classical references and the keywords Mitch listed are the
    right ones. However, to dispel a common misconception, let me mention that
    the general problem of integration in finite terms is not completely solved.
    See Bronstein's book, that I mentioned in my other article in this thread,
    for more information.
     
    tchow, Feb 19, 2004
    #25
  6. Zig

    Herman Rubin Guest

    Even in one dimension, there are lots of cases of
    numerical integration where the integrand is not
    well-approximated by even high degree polynomials.

    One such example encountered is \int (F(x+c))^n dF(x),
    where dF(x) = exp(-x^2/2)/sqrt(2*\pi) dx. For moderate
    values of n, using 32-point Gauss-Hermite integration,
    accurate for integrating all polynomials of degree 63
    with respect to dF, the results are not too good.
     
    Herman Rubin, Feb 19, 2004
    #26
  7. Zig

    gowan Guest

    A couple more ideas ... First, integration and differentiation are
    equally easy for polynomials and among continuous functions on an
    interval, say, polynomials are uniformly dense. That suggests that
    the problem comes from trying to work with specific algebraic
    representations of functions. Also, there are plenty of functions not
    easily definable in closed form (maybe most functions are of this
    sort) for which differentiation is as difficult as integration unless
    a power series is available. So the question really seems to be about
    functions built up from basic algebraic operations and composition,
    including the standard transcendental functions, too. You are asking
    why integration isn't well related to the basic operations for
    building up these functions.
     
    gowan, Feb 20, 2004
    #27
  8. I should study the literature of this and your reference to your book
    is certainly helpful. However, I suspect that cubature which is exact
    for polynomials is a very reasonable limiting case of information-based
    complexity. The extra generality from not taking the limit may or may
    not be important, depending on the application.

    To be precise, in your article "Information-based complexity and
    information-based optimization" with Traub on your home page, you
    discuss both worst-case and average-case error of sampling methods.
    Your example of average-case error assumes a function f on the interval
    [0,1] whose rth derivative is a Brownian path. This amounts to assuming
    a spectral decay rate for f. I.e. if f were a function on the circle
    instead of the interval, you would say that the Fourier components are
    i.i.d Gaussian with some a decay rate for their standard deviations.

    In high dimensions you have to assume an abrupt decay for the problem
    to be tractable. It is then very easy to construct natural-looking
    spectral assumptions that limit to the polynomial cubature problem.
    For example, you might very reasonably expand f in Hermite polynomials
    times a Gaussian. Assuming a decay rate for this expansion seems very
    similar to me to polynomial cubature with a Gaussian weight function.
    This is one of the main cases of the cubature problem, and the results
    in my paper apply to it.
     
    Greg Kuperberg, Feb 20, 2004
    #28
  9. Yes, because the integrand (without its Gaussian weight) is
    approximately a step function. If you take the vector space spanned by
    this integrand for different c, then the information-based complexity of
    integration is high, because the integrand can take abrupt steps.
    Any numerical method is going to be bad for this class. I'm more
    interested in cases that information-based complexity doesn't show to
    be impossible.
     
    Greg Kuperberg, Feb 20, 2004
    #29
  10. Zig

    Herman Rubin Guest

    This is not a typical integral arising in obtaining
    statistical procedures, but is in evaluating them.
    The integral is wanted for "small" c, small is a
    function of n.

    A more important integral, which is needed for
    asymptotic (already an approximation) purposes,
    is the probability that

    max (B(t) - L(B)(t)) > q,

    where B is the Brownian bridge, and L is a linear
    functional from B to smooth functions. The probability
    exceeds the maximum of the probabilities over t, which is
    the normal tail integral from hq to infinity. This is
    approximately exp(-(hq)^2/2)/(hq*sqrt(2\pi)). If L=0, it
    is known that h=2 and the precise answer just drops the
    denominator. For other L, the h can be computed, and it
    is known that the correction should be of a similar form,
    but is not that simple. Even a decent first-order
    approximation for large q would be quite useful. The
    methods used for L=0 cannot be used here. Note that B is
    continuous but not differentiable, and we are interested
    in small probabilities, while L(B) is differentiable.
     
    Herman Rubin, Feb 21, 2004
    #30
  11. Differentiation is a breaking up procedure, but integration puts it
    all together. A quote from Sam Rayburn:" Any donkey can kick down a
    barn, but it takes a good carpenter to build one." [Absolutely no
    offence meant for the first activity though.]

    A differential equation links/relates the two. Performing integration
    into an analytical form is as difficult as the complexity of this
    relationship. y''+ y' = 1 may be easily solved, but log(y''+ y')^
    sin(x y') = 1 is much more difficult to solve than simply writing out
    such a differential relationship.

    I too had often wondered why there is no proper mathematical
    definition of "degree of hardness" of solvability of differential
    equations, accenting on relational complexity. All we have is
    categorization into ordinary, partial, non-linear etc. The arrival of
    numerical machine solvers like the Runge-Kutta or implicit functions
    has rendered such an enquiry perhaps needless... once and for all ?
     
    Narasimham G.L., Feb 24, 2004
    #31
  12. Zig

    Axel Vogt Guest

    Is there an example available to look at for computing a cumulative
    multivariate normal distribtuion?
     
    Axel Vogt, Feb 22, 2005
    #32
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