lnfact ============================================== Purpose ---------------- Computes the natural log of the factorial function and can be used to compute log gamma. Format ---------------- .. function:: y = lnfact(x) :param x: all elements must be positive. :type x: NxK matrix or N-dimensional array :return y: containing the natural log of the factorial of each of the elements in *x*. :rtype y: NxK matrix Examples ---------------- :: x = { 100, 500, 1000 }; y = lnfact(x); :: 363.73938 y = 2611.3305 5912.1282 Remarks ------- For integer *x*, this is (approximately) ``ln(x!)``. However, the computation is done using a formula, and the function is defined for noninteger *x*. In most formulae in which the factorial operator appears, it is possible to avoid computing the factorial directly, and to use :func:`lnfact` instead. The advantage of this is that lnfact does not have the overflow problems that the factorial (``!``) operator has. For :math:`x > 1`, this function has at least 6 digit accuracy, for :math:`x > 4` it has at least 9 digit accuracy, and for :math:`x > 10` it has at least 12 digit accuracy. For :math:`0 < x < 1`, accuracy is not known completely but is probably at least 6 digits. Sometimes log gamma is required instead of log factorial. These functions are related by: :: lngamma(x) = lnfact(x - 1); Technical Notes --------------- For :math:`x > 1`, Stirling's formula is used. For :math:`0 < x <= 1`, ``ln(gamma(x+1))`` is used. Source ------ lnfact.src .. seealso:: Functions :func:`gamma`