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version 1.1, 2016/08/24 08:10:37 version 1.2, 2016/08/25 03:13:54
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 %comment $OpenXM$  %comment $OpenXM: OpenXM/src/asir-contrib/packages/doc/n_wishartd/n_wishartd-en.texi,v 1.1 2016/08/24 08:10:37 noro Exp $
 %comment --- おまじない ---  %comment --- おまじない ---
 \input ../../../../asir-doc/texinfo  \input ../../../../asir-doc/texinfo
 @iftex  @iftex
Line 160  is specified by @var{[s,s]}.
Line 160  is specified by @var{[s,s]}.
   
 @table @t  @table @t
 @item n_wishartd.message(@var{onoff})  @item n_wishartd.message(@var{onoff})
 計算中のメッセージ出力をon/off する.  starts/stops displaying messages during computation.
 @end table  @end table
   
 @table @var  @table @var
 @item onoff  @item onoff
 @var{onoff=1} のときメッセージを出力し, @var{onoff=0} のときしない.  Start displaying messages if @var{onoff}=1.
   Stop displaying messages if @var{onoff}=0.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item 計算中のメッセージ出力を on/off する. デフォルトは off である.  @item This function starts/stops displaying messages during computation.
   The default value is set to 0.
 @end itemize  @end itemize
   
 @node Numerical comptation of restricted function,,, n_wishartd.rr  @node Numerical comptation of restricted function,,, n_wishartd.rr
Line 186  is specified by @var{[s,s]}.
Line 188  is specified by @var{[s,s]}.
 @findex n_wishartd.prob_by_hgm  @findex n_wishartd.prob_by_hgm
   
 @table @t  @table @t
 @item n_wishartd.prrob_by_hgm(@var{m},@var{n},@var{[p1,p2,...]},@var{[s1,s2,...]},@var{t}[|@var{options}])  @item n_wishartd.prob_by_hgm(@var{m},@var{n},@var{[p1,p2,...]},@var{[s1,s2,...]},@var{t}[|@var{options}])
 HGM により重複固有値を持つ共分散行列に対する Wishart 行列の最大固有値の  computes the value of the distribution function of the largest eigenvalue of a Wishart matrix.
 分布関数の値を計算する.  
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 @item m  @item m
 変数の個数  The number of variables.
 @item n  @item n
 自由度  The degrees of freedom.
 @item [p1,p2,...]  @item [p1,p2,...,pk]
 重複固有値の個数のリスト  A list of the multiplicities of repeated eigenvalues.
 @item [s1,s2,...]  @item [s1,s2,...,sk]
 各重複固有値  A list of repeated eigenvalues.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item  @item
 固有値 @var{si} を @var{pi} 個もつ対角行列を共分散行列とする Wishart 行  Let @var{l1} be the largest eigenvalue of a Wishart matrix.
 列の最大固有値 @var{l1}の分布関数の値 @var{Pr[l1<t]} を計算する.  Let @var{Pr[l1<t]} be the distribution function of @var{l1}.
   The function @code{n_wishartd.prob_by_hgm} computes the value of the distribution function by using HGM
   for a covariance matrix which has repeated eigenvalues @var{si} with multiplicity @var{pi} (@var{i=1,...,k}).
   
 @item ステップ数を指定したルンゲ=クッタ法を, ステップ数を 2 倍しながら  @item This function repeats a Runge-Kutta method for the Pfaffian system by doubling the step size
 一つ前の計算結果との相対誤差が @var{eps} (デフォルトで @var{10^(-4)})  until the relative error between the current result and the previous result is less than @var{eps},
 になるまで繰り返す.  The default value of @var{eps} is @var{10^(-4)}.
 @item  @item
 @var{eq} オプション指定がない場合, @var{[p1,p2,...]} で指定される対角領  If an option @var{eq} is not set,
 域に制限した微分方程式系を計算する. 指定がある場合, オプションとして指  a system of PDES satisfied by 1F1 on the diagonal region specified by @var{[p1,p2,...]} is computed.
 定されたリストをチェックなしに制限した方程式と見なして計算する.  If an option @var{eq} is set, the list specified by @var{eq} is regarded as the correct system of PDEs.
 @item @var{eps}オプションが指定された場合, 指定された値を @var{eps} として計算する.  @item If an option @var{eps} is set, the value is used as @var{eps}.
 @item @var{td} オプションが指定された場合, 初期ベクトル計算のためのべき級数を @var{td} で  @item If an option @var{td} is set, the truncated power series solution for computing the initial vector
 指定された全次数まで計算する (デフォルトは100).  is computed up to the total degree specified by @var{td}. The default value is 100.
 @item @var{rk} オプションが指定された場合, 指定された次数のルンゲ=クッタ法を用いる.  @item If an option @var{rk} is set, it is regarded as the order of a Runge-Kutta method. The default vaule is 5.
 許される値は 4 または 5, でデフォルトは 5である.  @item It is recommended to use this function only when @var{k<=2} where @var{k} is the number of diagonal blocks because of
 @item べき級数解の計算の困難さ, およびパフィアン行列の計算の困難さから, ブロック数が 2 以下の場合にのみ  the difficulty of the truncated power series solution and the difficulty of computation of the Pfaffian matrices.
 実用性がある.  
 @end itemize  @end itemize
   
 @example  @example
Line 253  Step=2560000
Line 255  Step=2560000
   
 @table @t  @table @t
 @item n_wishartd.prrob_by_ps(@var{m},@var{n},@var{[p1,p2,...]},@var{[s1,s2,...]},@var{t}[|@var{options}])  @item n_wishartd.prrob_by_ps(@var{m},@var{n},@var{[p1,p2,...]},@var{[s1,s2,...]},@var{t}[|@var{options}])
 べき級数に重複固有値を持つ共分散行列に対する Wishart 行列の最大固有値の  computes the value of the distribution function of the largest eigenvalue of a Wishart matrix.
 分布関数の値を計算する.  
 @end table  @end table
   
 @table @var  @table @var
   @item return
 @item m  @item m
 変数の個数  The number of variables.
 @item n  @item n
 自由度  The degrees of freedom.
 @item [p1,p2,...]  @item [p1,p2,...,pk]
 重複固有値の個数のリスト  A list of the multiplicities of repeated eigenvalues.
 @item [s1,s2,...]  @item [s1,s2,...,sk]
 各重複固有値  A list of repeated eigenvalues.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item  @item
 直前の値との相対誤差が @var{eps} (デフォルト値は @var{10^(-4)}) 以下に  This function compute a truncated power series solution up to a total degree
 なるまで, べき級数を全次数ごとに計算する. その値から分布関数の値を計算  where the relative error between the current value and the previous value at the desired point
 して返す.  is less than @var{eps}. The default value of @var{eps} is @var{10^(-4)}.
 @item @var{eps}オプションが指定された場合, 指定された値を @var{eps} として計算する.  The value of the distribution function is computed by using this power series.
 @var{eq} オプション指定がない場合, @var{[p1,p2,...]} で指定される対角領  @item If an option @var{eps} is set, the value is used as @var{eps}.
 域に制限した微分方程式系を計算する. 指定がある場合, オプションとして指  @item
 定されたリストをチェックなしに制限した方程式と見なして計算する.  If an option @var{eq} is not set,
 @item @var{t} の値が小さい場合にのみ実用的に用いることができる.  a system of PDES satisfied by 1F1 on the diagonal region specified by @var{[p1,p2,...]} is computed.
   If an option @var{eq} is set, the list specified by @var{eq} is regarded as the correct system of PDEs.
   @item It is recommened to use this function when @var{t} is small.
 @end itemize  @end itemize
   
 @example  @example
Line 298  Step=2560000
Line 302  Step=2560000
   
 @table @t  @table @t
 @item n_wishartd.ps(@var{z},@var{v},@var{td})  @item n_wishartd.ps(@var{z},@var{v},@var{td})
 微分方程式系のべき級数解を指定された全次数まで計算する.  computes a truncated power series solution up to the total degree @var{td}.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 多項式リスト  A list of polynomial
   
 @item z  @item z
 differential operators with partial fraction coefficientsのリスト  A list of differential operators with partial fraction coefficients.
 @item v  @item v
 変数リスト  A list of variables.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item  @item
 結果は @var{[p,pd]} なるリストで, @var{p} は @var{td} 次まで求めたべき級数解, @var{pd} は  The result is a list @var{[p,pd]} where
 @var{p} の @var{td} 次部分である.  @var{p} is a truncated power series solution up to the total degree @var{td}
 @item @var{z} は, @var{v} に指定される変数以外のパラメタを含んではいけない.  and @var{pd} is the @var{td} homogeneous part of @var{p}.
   @item @var{z} cannot contain parameters other than the variables in @var{v}.
 @end itemize  @end itemize
   
 @example  @example
Line 335  differential operators with partial fraction coefficie
Line 339  differential operators with partial fraction coefficie
 @section Differential operators with partial fraction coefficients  @section Differential operators with partial fraction coefficients
   
 @menu  @menu
 * Expression of partial fractions::  * Representation of partial fractions::
 * Expression of differential operators with partial fraction coefficients::  * Representation of differential operators with partial fraction coefficients::
 * Operations on differential operators with partial fraction coefficients::  * Operations on differential operators with partial fraction coefficients::
 @end menu  @end menu
   
 @node Expression of partial fractions,,, Differential operators with partial fraction coefficients  @node Representation of partial fractions,,, Differential operators with partial fraction coefficients
 @subsection Expression of partial fractions  @subsection Representation of partial fractions
   
 matrix 1F1 が満たす微分方程式の係数は @var{1/yi}, @var{1/(yi-yj)} の定  The coefficients of the PDE satisfied by the matrix 1F1 are written
 数倍の和として書かれている. さらに, ロピタル則を用いた対角領域への制限  as a sum of @var{1/yi} and @var{1/(yi-yj)} multiplied by constants.
 アルゴリズムの結果も同様に部分分数の和として書ける.  Furthermore the result of diagonalization by l'Hopital rule
   can also be written as a sum of partial fractions.
   
 @itemize @bullet  @itemize @bullet
 @item  @item
 分母に現れる @var{yi0^n0(yi1-yj1)^n1(yi2-yj2)^n2...(yik-yjk)^nk} は  A product @var{yi0^n0(yi1-yj1)^n1(yi2-yj2)^n2...(yik-yjk)^nk}
 @var{[[yi0,n0],[yi1-yj1,n1],...,[yik-yjk,nk]]} なる形のリストとして表現  in the denominator of a fraction is represented as a list @var{[[yi0,n0],[yi1-yj1,n1],...,[yik-yjk,nk]]},
 される. ここで, 各因子 @var{yi-yj} は @var{i>j} を満たし, さらに因子は  Where each @var{yi-yj} satisfies @var{i>j} and the factors are sorted according to an ordering.
 ある一定の順序で整列される.  
 @item  @item
 @var{f} を上のようなべき積とし, @var{c} を定数とするとき, 単項式にあた  Let @var{f} be a power sum as above and @var{c} a constant.
 る @var{c/f} は @var{[c,f]} で表現される.  @var{f=[]} の場合, 分母が 1  Then a monomial @var{c/f} is represented by a list は @var{[c,f]}.
 であることを意味する.  @var{f=[]} means that the denominator is 1.
 @item  @item
 最後に, @var{c1/f1+...+ck/fk} は @var{[[c1,f1],...,[ck,fk]]} と表現され  Finally @var{c1/f1+...+ck/fk} is represented as a list @var{[[c1,f1],...,[ck,fk]]},
 る. ここでも, 各項はある一定の順序で整列される.  where terms are sorted according to an ordering.
 @item  @item
 部分分数を通分して簡約した結果, 0 になることもあることに注意する.  We note that it is possible that a partial fraction is reduced to 0.
 @end itemize  @end itemize
   
 @node Expression of differential operators with partial fraction coefficients,,, Differential operators with partial fraction coefficients  @node Representation of differential operators with partial fraction coefficients,,, Differential operators with partial fraction coefficients
 @subsection Expression of differential operators with partial fraction coefficients  @subsection Representation of differential operators with partial fraction coefficients
   
 前節の部分分数を用いて, それらを係数とする微分作用素が表現できる.  By using partial fractions explained in the previous section,
 @var{f1,...,fk} をExpression of partial fractions, @var{d1,...,dk} を分散表現単項式 (現  differential operators with partial fraction coefficients are represented.
 在設定されている項順序で @var{d1>...>dk}) とするとき, 微分作用素  Let @var{f1,...,fk} be partial fractions and @var{d1,...,dk} distributed monomials such that
 @var{f1*d1+...+fk*dk} が@var{[f1,d1],...[fk,dk]]}で表現される.  @var{d1>...>dk}) with respected to the current monomial ordering.
   Then a differential operator @var{f1*d1+...+fk*dk} is represented as a list @var{[f1,d1],...[fk,dk]]}.
   
 @node Operations on differential operators with partial fraction coefficients,,, Differential operators with partial fraction coefficients  @node Operations on differential operators with partial fraction coefficients,,, Differential operators with partial fraction coefficients
 @subsection Operations on differential operators with partial fraction coefficients  @subsection Operations on differential operators with partial fraction coefficients
Line 393  matrix 1F1 が満たす微分方程式の係数は @var{1/yi}, @var
Line 398  matrix 1F1 が満たす微分方程式の係数は @var{1/yi}, @var
   
 @table @var  @table @var
 @item m  @item m
 自然数  A natural number.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item @var{m} 変数の計算環境をセットする. 変数は @var{y0,y1,...,ym}, @var{dy0,...,dym}  @item This function sets a @var{m}-variate computational enviroment. The variables are @var{y0,y1,...,ym} and @var{dy0,...,dym},
 で @var{y0, dy0} は中間結果の計算のためのダミー変数である.  where @var{y0, dy0} are dummy variables for intermediate computation.
 @end itemize  @end itemize
   
 @node n_wishartd.addpf,,, Operations on differential operators with partial fraction coefficients  @node n_wishartd.addpf,,, Operations on differential operators with partial fraction coefficients
Line 410  matrix 1F1 が満たす微分方程式の係数は @var{1/yi}, @var
Line 415  matrix 1F1 が満たす微分方程式の係数は @var{1/yi}, @var
   
 @table @var  @table @var
 @item return  @item return
 differential operators with partial fraction coefficients  A differential operator with partial fraction coefficients.
 @item p1, p2  @item p1, p2
 differential operators with partial fraction coefficients  Differential operators with partial fraction coefficients.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item 微分作用素 @var{p1}, @var{p2} の和を求める.  @item This function computes the sum of differential operators @var{p1} and @var{p2}.
 @end itemize  @end itemize
   
 @node n_wishartd.mulcpf,,, Operations on differential operators with partial fraction coefficients  @node n_wishartd.mulcpf,,, Operations on differential operators with partial fraction coefficients
Line 428  differential operators with partial fraction coefficie
Line 433  differential operators with partial fraction coefficie
   
 @table @var  @table @var
 @item return  @item return
 differential operators with partial fraction coefficients  A differential operator with partial fraction coefficients.
 @item c  @item c
 部分分数  A partial fraction.
 @item p  @item p
 differential operators with partial fraction coefficients  Differential operators with partial fraction coefficients.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item 部分分数 @var{c} と微分作用素 @var{p} の積を計算する.  @item This function computes the product of a partial fraction @var{c} and a differential operator @var{p}.
 @end itemize  @end itemize
   
 @node n_wishartd.mulpf,,, Operations on differential operators with partial fraction coefficients  @node n_wishartd.mulpf,,, Operations on differential operators with partial fraction coefficients
Line 448  differential operators with partial fraction coefficie
Line 453  differential operators with partial fraction coefficie
   
 @table @var  @table @var
 @item return  @item return
 differential operators with partial fraction coefficients  A differential operator with partial fraction coefficients.
 @item p1, p2  @item p1, p2
 differential operators with partial fraction coefficients  Differential operators with partial fraction coefficients.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item 微分作用素 @var{p1}, @var{p2} の積を計算する.  @item This function computes the product of differential operators @var{p1} and @var{p2}.
 @end itemize  @end itemize
   
 @node n_wishartd.muldpf,,, Operations on differential operators with partial fraction coefficients  @node n_wishartd.muldpf,,, Operations on differential operators with partial fraction coefficients
Line 466  differential operators with partial fraction coefficie
Line 471  differential operators with partial fraction coefficie
   
 @table @var  @table @var
 @item return  @item return
 differential operators with partial fraction coefficients  A differential operator with partial fraction coefficients.
 @item y  @item y
 変数  A variable.
 @item p  @item p
 differential operators with partial fraction coefficients  A differential operator with partial fraction coefficients.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item 変数 @var{y} に対し, 微分作用素 @var{dy} と @var{p} の微分作用素としての  @item
 積を計算する.  This function computes the product of the differential operator @var{dy} corresponding to a variable @var{y} and @var{p}.
 @end itemize  @end itemize
   
 @example  @example
Line 497  differential operators with partial fraction coefficie
Line 502  differential operators with partial fraction coefficie
   
 @node rk_ratmat,,, Experimental implementation of Runge-Kutta methods  @node rk_ratmat,,, Experimental implementation of Runge-Kutta methods
   
 @code{n_wishartd.ps_by_hgm} では, パフィアン行列を計算したあと, 与えられたステップ数で  In the function @code{n_wishartd.ps_by_hgm}, after computing the Pfaffian matrices for
 Runge-Kutta 法を実行して近似解の値を計算する組み込み関数 @code{rk_ratmat} を実行している.  the sytem of PDEs on a diagonal region, it executes a built-in function
 この関数を, 値が与えられた精度で安定するまでステップ数を2倍しながら繰り返して実行する.  @code{rk_ratmat} which computes an approximate solution of the Pfaffian system
 @code{rk_ratmat} 自体, ある程度汎用性があるので, ここでその使用法を解説する.  by Runge-Kutta method for a spcified step size.
   This function is repeated until the result gets stabilized, by doubling the step size.
   @code{rk_ratmat} can be used as a general-purpose Runge-Kutta driver and we explain how to use it.
   
 @subsection @code{rk_ratmat}  @subsection @code{rk_ratmat}
 @findex rk_ratmat  @findex rk_ratmat
   
 @table @t  @table @t
 @item rk_ratmat(@var{rk45},@var{num},@var{den},@var{x0},@var{x1},@var{s},@var{f0})  @item rk_ratmat(@var{rk45},@var{num},@var{den},@var{x0},@var{x1},@var{s},@var{f0})
 有理関数係数のベクトル値一階線形常微分方程式系を Runge-Kutta 法で解く  solves a system of linear ODEs with rational function coefficients.
 @end table  @end table
   
 @table @var  @table @var
 @item return  @item return
 実数のリスト  A list of real numbers.
   
 @item rk45  @item rk45
 4 または 5  4 or 5.
 @item num  @item num
 定数行列の配列  An array of constant matrices.
 @item den  @item den
 多項式  A polynomial.
 @item x0, x1  @item x0, x1
 実数  Real numbers.
 @item s  @item s
 自然数  A natural number.
 @item f0  @item f0
 実ベクトル  A real vector.
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
 @item  @item
 配列 @var{num} のサイズを @var{k} とするとき,  Let @var{k} be the size of an array @var{num}.
 @var{P(x)=1/den(num[0]+num[1]x+...+num[k-1]x^(k-1))} に対し @var{dF/dx = P(x)F}, @var{F(x0)=f0} を  The function @code{rk_ratmat} solves  an initial value problem
 Runge-Kutta 法で解く.  @var{dF/dx = P(x)F}, @var{F(x0)=f0} for @var{P(x)=1/den(num[0]+num[1]x+...+num[k-1]x^(k-1))} by a Runge-Kutta method.
 @item  @item
 @var{rk45} が 4 のとき 4 次 Runge-Kutta, 5 のとき 5 次 Runge-Kutta アルゴリズムを実行する.  @var{rk45} specifies the order of a Runge-Kutta method. Adaptive methods are not implemented.
 実験的実装のため, adaptive アルゴリズムは実装されていない.  
 @item  @item
 @var{s} はステップ数で, 刻み幅は@var{(x1-x0)/s} である.  The step size is specified by @var{s}. The step width is @var{(x1-x0)/s}.
 @item  @item
 @var{f0} がサイズ@var{n} のとき, @var{num} の各成分は @var{n} 次正方行列である.  If the size of @var{f0} is @var{n},  each component of @var{num} is a square matrix of size @var{n}.
 @item  @item
 結果は, 長さ @var{s} の実数リスト @var{[rs,...,r1]} で, @var{ri} は @var{i} ステップ目に計算された  The result is a list of real numbers @var{[r1,...,rs]} of length @var{s}.
 解ベクトルの第0成分である. 次のステップに進む前に解ベクトルを @var{ri} で割るので, 最終的に  @var{ri} is the 0-th component of the solution vector after the step @var{i}.
 解 @var{F(x1)} の第 0 成分が @var{rs*r(s-1)*...*r1} となる.  Before going to the next step the solution vector is divided by @var{ri}.
 @item 方程式が線形なので, Runge-Kutta の各ステップも線形となることを利用し,  Therefore the 0-th component of the final solution vector [var{F(x1)} is equal to @var{rs*r(s-1)*...*r1}.
 第0成分を1に正規化することで, 途中の解の成分が倍精度浮動小数の  @item Since the ODE is linear, each step of Runge-Kutta method is also linear.
 範囲に収まることを期待している. 初期ベクトル @var{f0} の成分が倍精度浮動小数に収まらない場合  This enables us to apply a normalization such that the 0-th
 は, @var{f0} を正規化してから @code{rk_ratmat} を実行し, 前項の結果に @code{f0} の第 0 成分をかければ  component of each intermediate solution vector is set to 1.  By
 よい.  applying this normalization we expect that all the components of
   intermediate solution vectors can be represented by the format of
   double precision floating point number.
   If there exist some components in the initial vector @var{f0}, we apply this normalization
   to @var{f0}. After applying @code{rk_ratmat} we multiply the result for the normalized @var{f0} and the 0-th component
   of the original @var{f0} to get the desired result.
 @end itemize  @end itemize
   
 @example  @example
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