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1.1     ! takayama    1: % $OpenXM$
        !             2: \name{hgm.Rhgm}
        !             3: \alias{hgm.Rhgm}
        !             4: %- Also NEED an '\alias' for EACH other topic documented here.
        !             5: \title{
        !             6:     The function hgm.Rhgm performs the holonomic gradient method (HGM)
        !             7:    for a given Pfaffian system and an initial value vector.
        !             8: }
        !             9: \description{
        !            10:    The function hgm.Rhgm performs the holonomic gradient method (HGM)
        !            11:    for a given Pfaffian system and an initial value vector
        !            12:    with the deSolve package in R.
        !            13: }
        !            14: \usage{
        !            15:  hgm.Rhgm(th0, G0, th1, dG.fun, times=NULL, fn.params=NULL)
        !            16: }
        !            17: %- maybe also 'usage' for other objects documented here.
        !            18: \arguments{
        !            19:   \item{th0}{ A d-dimensional vector which is an initial point of the parameter vector th (theta).}
        !            20:   \item{G0}{
        !            21:     A r-dimensional vector which is the initial value of the vector G
        !            22:     of the normalizing constant and its derivatives.
        !            23:   }
        !            24:   \item{th1}{
        !            25:     A d-dimensional vector which is the target point of th.
        !            26:   }
        !            27:   \item{dG.fun}{
        !            28:      dG.fun is the ``right hand sides'' of the Pfaffian system.
        !            29:      It is a d*r-dimensional array.
        !            30:   }
        !            31:   \item{times}{a vector; times in [0,1] at which explicit estimates for G are desired.
        !            32:          If time = NULL, the set {0,1} is used, and only the final value is returned.
        !            33:   }
        !            34:   \item{fn.params}{
        !            35:   fn.params: a list of parameters passed to the function dG.fun.
        !            36:          If fn.params = NULL, no parameter is passed to dG.fun.
        !            37:   }
        !            38: }
        !            39: \details{
        !            40:   The function hgm.Rhgm computes the value of a holonomic function
        !            41:   at a given point, using HGM.
        !            42:   This is a ``Step 3'' function (see the reference below),
        !            43:   which can be used for an arbitrary input, in the HGM framework.
        !            44:   Efficient ``Step 3'' functions are given for some distributions
        !            45:   in this package.
        !            46:
        !            47:   The Pfaffian system assumed is
        !            48:     d G_j / d th_i = (dG.fun(th, G))_{i,j}
        !            49:
        !            50:   The inputs of hgm.Rhgm are the initial point th0, initial value G0, final point th1,
        !            51:   and Pfaffian system dG.fun. The output is the final value G1.
        !            52:
        !            53:   If the argument `times' is specified, the function returns a matrix,
        !            54:   where the first column denotes time, the following d-vector denotes th,
        !            55:   and the remaining r-vector denotes G.
        !            56: %  \code{\link[RCurl]{postForm}}.
        !            57: }
        !            58: \value{
        !            59: The output is the value of G at th1. The first element of G is the normalizing
        !            60: constant.
        !            61: }
        !            62: \references{
        !            63: \url{http://www.math.kobe-u.ac.jp/OpenXM/Math/hgm/ref-hgm.html}
        !            64: }
        !            65: \author{
        !            66: Tomonari Sei
        !            67: }
        !            68: \note{
        !            69: %%  ~~further notes~~
        !            70: }
        !            71:
        !            72: %% ~Make other sections like Warning with \section{Warning }{....} ~
        !            73:
        !            74: \seealso{
        !            75: %%\code{\link{oxm.matrix_r2tfb}}
        !            76: }
        !            77: \examples{
        !            78: # Example 1.
        !            79: # A demo program; von Mises--Fisher on S^{3-1}
        !            80:
        !            81: G.exact = function(th){  # exact value by built-in function
        !            82:   c( sinh(th[1])/th[1], cosh(th[1])/th[1] - sinh(th[1])/th[1]^2 )
        !            83: }
        !            84:
        !            85: dG.fun = function(th, G, fn.params=NULL){  # Pfaffian
        !            86:   dG = array(0, c(1, 2))
        !            87:   sh = G[1] * th[1]
        !            88:   ch = G[2] * th[1] + G[1]
        !            89:   dG[1,1] = G[2] # Pfaffian eq's
        !            90:   dG[1,2] = sh/th[1] - 2*ch/th[1]^2 + 2*sh/th[1]^3
        !            91:   dG
        !            92: }
        !            93:
        !            94: th0 = 0.5
        !            95: th1 = 15
        !            96:
        !            97: G0 = G.exact(th0)
        !            98: G0
        !            99:
        !           100: G1 = hgm.Rhgm(th0, G0, th1, dG.fun)  # HGM
        !           101: G1
        !           102:
        !           103: G1.exact = G.exact(th1)
        !           104: G1.exact
        !           105:
        !           106: #
        !           107: # Example 2.
        !           108: #
        !           109: hgm.Rhgm.demo1()
        !           110:
        !           111: }
        !           112: % Add one or more standard keywords, see file 'KEYWORDS' in the
        !           113: % R documentation directory.
        !           114: \keyword{ Normalization constant }
        !           115: \keyword{ Holonomic gradient method }
        !           116: \keyword{ HGM }
        !           117:
        !           118: