===================================================================
RCS file: /home/cvs/OpenXM/src/asir-doc/parts/builtin/upoly.texi,v
retrieving revision 1.1.1.1
retrieving revision 1.2
diff -u -p -r1.1.1.1 -r1.2
--- OpenXM/src/asir-doc/parts/builtin/upoly.texi	1999/12/08 05:47:44	1.1.1.1
+++ OpenXM/src/asir-doc/parts/builtin/upoly.texi	1999/12/21 02:47:34	1.2
@@ -1,5 +1,12 @@
+@comment $OpenXM$
+\BJP
 @node 一変数多項式の演算,,, 組み込み函数
 @section 一変数多項式の演算
+\E
+\BEG
+@node Univariate polynomials,,, Built-in Function
+@section Univariate polynomials
+\E
 
 @menu
 * umul umul_ff usquare usquare_ff utmul utmul_ff::
@@ -10,7 +17,8 @@
 * udiv urem urembymul urembymul_precomp ugcd::
 @end menu
 
-@node umul umul_ff usquare usquare_ff utmul utmul_ff,,, 一変数多項式の演算
+\JP @node umul umul_ff usquare usquare_ff utmul utmul_ff,,, 一変数多項式の演算
+\EG @node umul umul_ff usquare usquare_ff utmul utmul_ff,,, Univariate polynomials
 @subsection @code{umul}, @code{umul_ff}, @code{usquare}, @code{usquare_ff}, @code{utmul}, @code{utmul_ff}
 @findex umul
 @findex umul_ff
@@ -22,25 +30,32 @@
 @table @t
 @item umul(@var{p1},@var{p2}) 
 @itemx umul_ff(@var{p1},@var{p2}) 
-:: 一変数多項式の高速乗算
+\JP :: 一変数多項式の高速乗算
+\EG :: Fast multiplication of univariate polynomials
 @item usquare(@var{p1})
 @itemx usquare_ff(@var{p1})
-:: 一変数多項式の高速 2 乗算
+\JP :: 一変数多項式の高速 2 乗算
+\EG :: Fast squaring of a univariate polynomial
 @item utmul(@var{p1},@var{p2},@var{d}) 
 @itemx utmul_ff(@var{p1},@var{p2},@var{d}) 
-:: 一変数多項式の高速乗算 (打ち切り次数指定)
+\JP :: 一変数多項式の高速乗算 (打ち切り次数指定)
+\EG :: Fast multiplication of univariate polynomials with truncation
 @end table
 
 @table @var
 @item return
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @item p1 p2
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @item d
-非負整数
+\JP 非負整数
+\EG non-negative integer
 @end table
 
 @itemize @bullet
+\BJP
 @item
 一変数多項式の乗算を, 次数に応じて決まるアルゴリズムを用いて
 高速に行う. 
@@ -51,7 +66,7 @@
 @item
 @code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} は,
 係数を有限体の元と見なして, 有限体上の多項式として
-積を求める. ただし, 引数が全く有限体の元を含まない場合には, 
+積を求める. ただし, 引数の係数が整数の場合, 
 整数係数の多項式を返す場合もあるので, これらを呼び出した結果
 が有限体係数であることを保証するためには
 あらかじめ @code{simp_ff()} で係数を有限体の元に変換しておくとよい. 
@@ -66,13 +81,53 @@ GF(2^n) 係数の多項式を引数に取れない. 
 @item
 いずれも, @code{set_upkara()} (@code{utmul}, @code{utmul_ff} については
 @code{set_uptkara()}) で返される値以下の次数に対しては通常の筆算
-形式の方法, @code{set_uufft()} で返される値以下の次数に対しては Karatsuba
+形式の方法, @code{set_upfft()} で返される値以下の次数に対しては Karatsuba
 法, それ以上では FFTおよび中国剰余定理が用いられる. すなわち, 
 整数に対する FFT ではなく, 十分多くの 1 ワード以内の法 @var{mi} を用意し, 
 @var{p1}, @var{p2} の係数を @var{mi} で割った余りとしたものの積を, 
 FFT で計算し, 最後に中国剰余定理で合成する. その際, 有限体版の関数に
 おいては, 最後に基礎体を表す法で各係数の剰余を計算するが, ここでは
-Shoup によるトリックを用いて高速化してある. 
+Shoup によるトリック @code{[Shoup]} を用いて高速化してある. 
+\E
+\BEG
+@item
+These functions compute products of univariate polynomials
+by selecting an appropriate algorithm depending on the degrees
+of inputs.
+@item
+@code{umul()}, @code{usquare()}, @code{utmul()} 
+compute products over the integers.
+Coefficients in GF(p) are regarded as non-negative integers
+less than p.
+@item
+@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()}
+compute products over a finite field. However, if some of
+the coefficients of the inputs are integral,
+the result may be an integral polynomial. So if one wants
+to assure that the result is a polynomial over the finite field,
+apply @code{simp_ff()} to the inputs.
+@item
+@code{umul_ff()}, @code{usquare_ff()}, @code{utmul_ff()} 
+cannot take polynomials over GF(2^n) as their inputs.
+@item
+@code{umul()}, @code{umul_ff()} produce @var{p1*p2}.
+@code{usquare()}, @code{usquare_ff()} produce @var{p1^2}.
+@code{utmul()}, @code{utmul_ff()} produce @var{p1*p2 mod v^(d+1)},
+where @var{v} is the variable of @var{p1}, @var{p2}.
+@item
+If the degrees of the inputs are less than or equal to the
+value returned by @code{set_upkara()} (@code{set_uptkara()} for
+@code{utmul}, @code{utmul_ff}), usual pencil and paper method is
+used. If the degrees of the inputs are less than or equall to 
+the value returned by @code{set_upfft()}, Karatsuba algorithm
+is used. If the degrees of the inputs exceed it, a combination
+of FFT and Chinese remainder theorem is used.
+First of all sufficiently many primes @var{mi} 
+within 1 machine word are prepared.
+Then @var{p1*p2 mod mi} is computed by FFT for each @var{mi}.
+Finally they are combined by Chinese remainder theorem.
+The functions over finite fields use an improvement by V. Shoup @code{[Shoup]}.
+\E
 @end itemize
 
 @example
@@ -101,12 +156,14 @@ Shoup によるトリックを用いて高速化してある. 
 @end example
 
 @table @t
-@item 参照
+\JP @item 参照
+\EG @item References
 @fref{set_upkara set_uptkara set_upfft},
 @fref{kmul ksquare ktmul}.
 @end table
 
-@node kmul ksquare ktmul,,, 一変数多項式の演算
+\JP @node kmul ksquare ktmul,,, 一変数多項式の演算
+\EG @node kmul ksquare ktmul,,, Univariate polynomials
 @subsection @code{kmul}, @code{ksquare}, @code{ktmul}
 @findex kmul
 @findex ksquare
@@ -114,23 +171,30 @@ Shoup によるトリックを用いて高速化してある. 
 
 @table @t
 @item kmul(@var{p1},@var{p2}) 
-:: 一変数多項式の高速乗算
+\JP :: 一変数多項式の高速乗算
+\EG :: Fast multiplication of univariate polynomials
 @item ksquare(@var{p1})
-:: 一変数多項式の高速 2 乗算
+\JP :: 一変数多項式の高速 2 乗算
+\EG :: Fast squaring of a univariate polynomial
 @item ktmul(@var{p1},@var{p2},@var{d}) 
-:: 一変数多項式の高速乗算 (打ち切り次数指定)
+\JP :: 一変数多項式の高速乗算 (打ち切り次数指定)
+\EG :: Fast multiplication of univariate polynomials with truncation
 @end table
 
 @table @var
 @item return
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @item p1 p2
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @item d
-非負整数
+\JP 非負整数
+\EG non-negative integer
 @end table
 
 @itemize @bullet
+\BJP
 @item
 一変数多項式の乗算を Karatsuba 法で行う. 
 @item
@@ -138,6 +202,15 @@ Shoup によるトリックを用いて高速化してある. 
 FFT を用いた高速化は行わない. 
 @item
 GF(2^n) 係数の多項式にも用いることができる. 
+\E
+\BEG
+These functions compute products of univariate polynomials by Karatsuba
+algorithm.
+@item
+These functions do not apply FFT for large degree inputs.
+@item
+These functions can compute products over GF(2^n).
+\E
 @end itemize
 
 @example
@@ -153,7 +226,8 @@ return to toplevel
 [37] kmul(A,B)$
 @end example
 
-@node set_upkara set_uptkara set_upfft,,, 一変数多項式の演算
+\JP @node set_upkara set_uptkara set_upfft,,, 一変数多項式の演算
+\EG @node set_upkara set_uptkara set_upfft,,, Univariate polynomials
 @subsection @code{set_upkara}, @code{set_uptkara}, @code{set_upfft}
 @findex set_upkara
 @findex set_uptkara
@@ -163,17 +237,24 @@ return to toplevel
 @item set_upkara([@var{threshold}])
 @itemx set_uptkara([@var{threshold}])
 @itemx set_upfft([@var{threshold}])
-:: 1 変数多項式の積演算における N^2 , Karatsuba, FFT アルゴリズムの切替えの閾値
+\JP :: 1 変数多項式の積演算における N^2 , Karatsuba, FFT アルゴリズムの切替えの閾値
+\BEG
+:: Set thresholds in the selection of an algorithm from N^2, Karatsuba,
+FFT algorithms for univariate polynomial multiplication.
+\E
 @end table
 
 @table @var
 @item return
-設定されている値
+\JP 設定されている値
+\EG value currently set
 @item threshold
-非負整数
+\JP 非負整数
+\EG non-negative integer
 @end table
 
 @itemize @bullet
+\BJP
 @item
 いずれも, 一変数多項式の積の計算における, アルゴリズム切替えの閾値を
 設定する. 
@@ -183,15 +264,29 @@ return to toplevel
 される. この切替えの次数を設定する. 
 @item
 詳細は, それぞれの積関数の項を参照のこと. 
+\E
+\BEG
+@item
+These functions set thresholds in the selection of an algorithm from N^2, 
+Karatsuba, FFT algorithms for univariate polynomial multiplication.
+@item
+Products of univariate polynomials are computed by N^2, Karatsuba,
+FFT algorithms. The algorithm selection is done according to the degrees of
+input polynomials and the thresholds.
+@item
+See the description of each function for details.
+\E
 @end itemize
 
 @table @t
-@item 参照
+\JP @item 参照
+\EG @item References
 @fref{kmul ksquare ktmul},
 @fref{umul umul_ff usquare usquare_ff utmul utmul_ff}.
 @end table
 
-@node utrunc udecomp ureverse,,, 一変数多項式の演算
+\JP @node utrunc udecomp ureverse,,, 一変数多項式の演算
+\EG @node utrunc udecomp ureverse,,, Univariate polynomials
 @subsection @code{utrunc}, @code{udecomp}, @code{ureverse}
 @findex utrunc
 @findex udecomp
@@ -201,19 +296,24 @@ return to toplevel
 @item utrunc(@var{p},@var{d}) 
 @itemx udecomp(@var{p},@var{d}) 
 @itemx ureverse(@var{p})
-:: 多項式に対する操作
+\JP :: 多項式に対する操作
+\EG :: Operations on polynomials
 @end table
 
 @table @var
 @item return
-一変数多項式あるいは一変数多項式のリスト
+\JP 一変数多項式あるいは一変数多項式のリスト
+\EG univariate polynomial or list of univariate polynomials
 @item p
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @item d
-非負整数
+\JP 非負整数
+\EG non-negative integer
 @end table
 
 @itemize @bullet
+\BJP
 @item
 @var{p} の変数を x とする. このとき @var{p} = @var{p1}+x^(d+1)@var{p2}
 (@var{p1} の次数は @var{d} 以下) と分解できる. @code{utrunc()} は
@@ -221,6 +321,19 @@ return to toplevel
 @item
 @var{p} の次数を @var{e} とし, @var{i} 次の係数を @var{p[i]} とすれば, 
 @code{ureverse()} は @var{p[e]}+@var{p[e-1]}x+... を返す. 
+\E
+\BEG
+@item
+Let @var{x} be the variable of @var{p}. Then @var{p} can be decomposed
+as @var{p} = @var{p1}+x^(d+1)@var{p2}, where the degree of @var{p1} 
+is less than or equal to @var{d}.
+Under the decomposition, @code{utrunc()} returns 
+@var{p1} and  @code{udecomp()} returns [@var{p1},@var{p2}].
+@item
+Let @var{e} be the degree of @var{p} and @var{p[i]} the coefficient
+of @var{p} at degree @var{i}. Then 
+@code{ureverse()} returns @var{p[e]}+@var{p[e-1]}x+....
+\E
 @end itemize
 
 @example
@@ -233,11 +346,13 @@ return to toplevel
 @end example
 
 @table @t
-@item 参照
+\JP @item 参照
+\EG @item References
 @fref{udiv urem urembymul urembymul_precomp ugcd}.
 @end table
 
-@node uinv_as_power_series ureverse_inv_as_power_series,,, 一変数多項式の演算
+\JP @node uinv_as_power_series ureverse_inv_as_power_series,,, 一変数多項式の演算
+\EG @node uinv_as_power_series ureverse_inv_as_power_series,,, Univariate polynomials
 @subsection @code{uinv_as_power_series}, @code{ureverse_inv_as_power_series}
 @findex uinv_as_power_series
 @findex ureverse_inv_as_power_series
@@ -245,19 +360,24 @@ return to toplevel
 @table @t
 @item uinv_as_power_series(@var{p},@var{d}) 
 @itemx ureverse_inv_as_power_series(@var{p},@var{d})
-:: 多項式を冪級数とみて, 逆元計算
+\JP :: 多項式を冪級数とみて, 逆元計算
+\EG :: Computes the truncated inverse as a power series.
 @end table
 
 @table @var
 @item return
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @item p
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @item d
-非負整数
+\JP 非負整数
+\EG non-negative integer
 @end table
 
 @itemize @bullet
+\BJP
 @item
 @code{uinv_as_power_series(@var{p},@var{d})} は, 定数項が 0 でない
 多項式 @var{p} に対し, @var{p}@var{r}-1 の最低次数が @var{d}+1
@@ -268,6 +388,23 @@ return to toplevel
 に対して @code{uinv_as_power_series(@var{p1},@var{d})} を計算する. 
 @item
 @code{rembymul_precomp()} の引数として用いる場合, @code{ureverse_inv_as_power_series()} の結果をそのまま用いることができる. 
+\E
+\BEG
+@item
+For a polynomial @var{p} with a non zero constant term,
+@code{uinv_as_power_series(@var{p},@var{d})} computes
+a polynomial @var{r} whose degree is at most @var{d}
+such that @var{p*r = 1 mod x^(d+1)}, where @var{x} is the variable
+of @var{p}.
+@item
+Let @var{e} be the degree of @var{p}.
+@code{ureverse_inv_as_power_series(@var{p},@var{d})} computes
+@code{uinv_as_power_series(@var{p1},@var{d})} for
+@var{p1}=@code{ureverse(@var{p},@var{e})}.
+@item
+The output of @code{ureverse_inv_as_power_series()} can be used
+as the input of @code{rembymul_precomp()}.
+\E
 @end itemize
 
 @example
@@ -286,12 +423,14 @@ x^10+x^9
 @end example
 
 @table @t
-@item 参照
+\JP @item 参照
+\EG @item References
 @fref{utrunc udecomp ureverse},
 @fref{udiv urem urembymul urembymul_precomp ugcd}.
 @end table
 
-@node udiv urem urembymul urembymul_precomp ugcd,,, 一変数多項式の演算
+\JP @node udiv urem urembymul urembymul_precomp ugcd,,, 一変数多項式の演算
+\EG @node udiv urem urembymul urembymul_precomp ugcd,,, Univariate polynomials
 @subsection @code{udiv}, @code{urem}, @code{urembymul}, @code{urembymul_precomp}, @code{ugcd}
 @findex udiv
 @findex urem
@@ -305,17 +444,21 @@ x^10+x^9
 @item urembymul(@var{p1},@var{p2}) 
 @item urembymul_precomp(@var{p1},@var{p2},@var{inv}) 
 @item ugcd(@var{p1},@var{p2}) 
-:: 多項式に対する操作
+\JP :: 一変数多項式の除算, GCD
+\EG :: Division and GCD for univariate polynomials.
 @end table
 
 @table @var
 @item return
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @item p1,p2,inv
-一変数多項式
+\JP 一変数多項式
+\EG univariate polynomial
 @end table
 
 @itemize @bullet
+\BJP
 @item
 一変数多項式 @var{p1}, @var{p2} に対し, 
 @code{udiv} は商, @code{urem}, @code{urembymul} は剰余, 
@@ -324,8 +467,29 @@ x^10+x^9
 @code{urembymul} は, @var{p2} による剰余計算を, @var{p2} の
 冪級数としての逆元計算および, 乗算 2 回に置き換えたもので, 
 次数が大きい場合に有効である. 
-@item @code{urembymul_precomp} は, 固定された多項式による剰余
+@item
+@code{urembymul_precomp} は, 固定された多項式による剰余
 計算を多数行う場合などに効果を発揮する. 
+第 3 引数は, あらかじめ @code{ureverse_inv_as_power_series()} に
+より計算しておく. 
+\E
+\BEG
+@item
+For univariate polynomials @var{p1} and @var{p2},
+there exist polynomials @var{q} and @var{r} such that
+@var{p1=q*p2+r} and the degree of @var{r} is less than that of @var{p2}.
+Then @code{udiv} returns @var{q}, @code{urem} and @code{urembymul} return
+@var{r}. @code{ugcd} returns the polynomial GCD of @var{p1} and @var{p2}.
+These functions are specially tuned up for dense univariate polynomials.
+In @code{urembymul} the division by @var{p2} is replaced with 
+the inverse computation of @var{p2} as a power series and
+two polynomial multiplications. It speeds up the computation
+when the degrees of inputs are large.
+@item
+@code{urembymul_precomp} is efficient when one repeats divisions
+by a fixed polynomial.
+One has to compute the third argument by @code{ureverse_inv_as_power_series()}.
+\E
 @end itemize
 
 @example
@@ -348,6 +512,7 @@ x^10+x^9
 @end example
 
 @table @t
-@item 参照
+\JP @item 参照
+\EG @item References
 @fref{uinv_as_power_series ureverse_inv_as_power_series}.
 @end table