Учебники, методички / Квадратурные формулы / metqua

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\define\de{\sc ЋЇаҐ¤Ґ«Ґ­ЁҐ}
\define\ex{\sc ЏаЁ¬Ґа }
\define\so{\sc ђҐиҐ­ЁҐ}
\define\re{\sc ‡ ¬Ґз ­ЁҐ}

\document
\footline={\hfil}
\head ѓ®бг¤ аб⢥­­л© Є®¬ЁвҐв ђ®ббЁ©бЄ®© ”Ґ¤Ґа жЁЁ\
Ї® ўлб襬㠮Ўа §®ў ­Ёо\\
\\
Њ®бЄ®ўбЄЁ© Ј®бг¤ аб⢥­­л©  ўЁ жЁ®­­л©\\
вҐе­®«®ЈЁзҐбЄЁ© г­ЁўҐабЁвҐв Ё¬.\ Љ.~ќ.~–Ё®«Є®ўбЄ®Ј®\endhead
\bigskip
\bigskip
\bigskip
\head Љ дҐ¤а  ``‚лби п ¬ вҐ¬ вЁЄ ''\endhead
\vskip120truept

\topmatter
\title Љ‚Ђ„ђЂ’“ђЌ›… ”ЋђЊ“‹›\endtitle
\endtopmatter
\head ЊҐв®¤ЁзҐбЄЁҐ гЄ § ­Ёп Ї® Єгабг ``—Ёб«Ґ­­лҐ ¬Ґв®¤л''\endhead
\vskip95truept
\head\hfil ‘®бв ўЁвҐ«м: ЋбЁЇҐ­Є® Љ.~ћ.\endhead
\vskip170truept
\head Њ®бЄў  1995\endhead

\newpage
\footline={\hss\tenrm-- \folio\ --\hss}

\head ‚ўҐ¤Ґ­ЁҐ\endhead

…б«Ё $f(x)$ --- ­ҐЇаҐалў­ п ­  ®в१ЄҐ $[a,b]$ дг­ЄжЁп Ё $F(x)$ --- ҐҐ
ЇҐаў®®Ўа §­ п, в® Ї® д®а¬г«Ґ Ќмов®­ --‹Ґ©Ў­Ёж 
$$\int_a^bf(x)\,dx=F(a)-F(b).$$
Ћ¤­ Є® з бв® ЇҐаў®®Ўа §­ п ­Ґ ¬®¦Ґв Ўлвм ўла ¦Ґ­  зҐаҐ§ н«Ґ¬Ґ­в а­лҐ дг­ЄжЁЁ
Ё«Ё пў«пҐвбп б«ЁиЄ®¬ б«®¦­®©. Ќ ЇаЁ¬Ґа, ЇаЁ ўлзЁб«Ґ­ЁЁ Ё­вҐЈа «®ў
$$\int_0^1e^{-x^2}\,dx,\qquad\int_0^1\frac{dx}{(1+x^2)^{10}}$$
ў ЇҐаў®¬ б«гз Ґ ¬л бв «ЄЁў Ґ¬бп б ⥬, зв® ЇҐаў®®Ўа §­ п ¤«п $e^{-x^2}$ ­Ґ
ўла ¦ Ґвбп зҐаҐ§ н«Ґ¬Ґ­в а­лҐ дг­ЄжЁЁ,   ў® ўв®а®¬ --- б ⥬, зв®
ЇҐаў®®Ўа §­ п ¤«п дг­ЄжЁЁ $(1+x^2)^{-10}$ пў«пҐвбп б«ЁиЄ®¬ Ја®¬®§¤ЄЁ¬
ўла ¦Ґ­ЁҐ¬.

Ља®¬Ґ в®Ј®, ­  Їа ЄвЁЄҐ Ї®¤л­вҐЈа «м­ п дг­ЄжЁп $f(x)$ ®Ўлз­® Ўлў Ґв § ¤ ­  ў
¤ЁбЄаҐв­®¬ зЁб«Ґ в®зҐЄ. ‚ н⮬ б«гз Ґ ЇҐаў®®Ўа §­ п $F(x)$ ў®®ЎйҐ ­Ґ ¬®¦Ґв
Ўлвм ­ ©¤Ґ­  в®з­®. ’Ґ¬ б ¬л¬ ў®§­ЁЄ Ґв § ¤ з  ЇаЁЎ«Ё¦Ґ­­®Ј® ўлзЁб«Ґ­Ёп
®ЇаҐ¤Ґ«Ґ­­®Ј® Ё­вҐЈа «  ®в дг­ЄжЁЁ Ї® Ё­д®а¬ жЁЁ ® §­ зҐ­Ёпе нв®© дг­ЄжЁЁ ў
­ҐЄ®в®а®© бЁб⥬Ґ в®зҐЄ. ’ Є®Ј® த  д®а¬г«л ­ §лў овбп {\it Єў ¤а вга­л¬Ё
д®а¬г« ¬Ё}. ‚ ¤ ­­®¬ Ї®б®ЎЁЁ а бб¬ ваЁў овбп ®б­®ў­лҐ ¬Ґв®¤л Ї®бв஥­Ёп
Їа®б⥩иЁе Єў ¤а вга­ле д®а¬г« Ё ®жҐ­ЄЁ Ёе Ї®ЈаҐи­®б⥩.

\head 1. ”®а¬г«  Їаאַ㣮«м­ЁЄ®ў\endhead

ђ бᬮваЁ¬ § ¤ зг ЇаЁЎ«Ё¦Ґ­­®Ј® ўлзЁб«Ґ­Ёп Ё­вҐЈа « 
$$\int_{-h/2}^{h/2}f(x)\,dx$$
Ї® §­ зҐ­Ёо $f(0)=f_0$. …бвҐб⢥­­® бзЁв вм, зв® дг­ЄжЁп ўбо¤г ЇаЁЎ«Ё¦Ґ­­®
а ў­  $f_0$, Ё § ¬Ґ­Ёвм ўлзЁб«Ґ­ЁҐ Ё­вҐЈа «  ®в Ёб室­®© дг­ЄжЁЁ ­ 
ўлзЁб«Ґ­ЁҐ Ё­вҐЈа «  ®в Ї®бв®п­­®© $f_0$. ’ ЄЁ¬ ®Ўа §®¬, ¬л ЇаЁе®¤Ё¬ Є
Їа®б⥩襩 Єў ¤а вга­®© д®а¬г«Ґ
$$\int_{-h/2}^{h/2}f(x)\,dx\approx hf_0,$$
­ §лў Ґ¬®© {\it д®а¬г«®© Їаאַ㣮«м­ЁЄ®ў}. ѓҐ®¬ҐваЁзҐбЄЁ© б¬лб« д®а¬г«л
Їаאַ㣮«м­ЁЄ®ў § Є«оз Ґвбп ў ⮬, зв® Ї«®й ¤м Ї®¤ Ја дЁЄ®¬ дг­ЄжЁЁ $y=f(x)$
§ ¬Ґ­пҐвбп ­  Ї«®й ¤м Їаאַ㣮«м­ЁЄ  б ўлб®в®©, а ў­®© $f(0)$.

ЋжҐ­Ё¬ Ї®ЈаҐи­®бвм д®а¬г«л Їаאַ㣮«м­ЁЄ®ў ў ЇаҐ¤Ї®«®¦Ґ­ЁЁ, зв® г дг­ЄжЁЁ $f(
x)$ бгйҐбвўгҐв ­ҐЇаҐалў­ п ўв®а п Їа®Ё§ў®¤­ п. Џ® д®а¬г«Ґ ’Ґ©«®а  Ё¬ҐҐ¬
$$f(x)=f_0+f_0'x+\frac{f''(\xi)}{2!}x^2,$$
Ј¤Ґ $f_0'=f'(0)$. Ћвбо¤ 
$$\int_{-h/2}^{h/2}f(x)\,dx=hf_0+\int_{-h/2}^{h/2}\frac{f''(\xi)}{2!}x^2\,dx.
$$
Џ®«®¦Ё¬
$$M_2=\max_{x\in[-h/2,h/2]}|f''(x)|.$$
’®Ј¤ 
$$\left|\int_{-h/2}^{h/2}f(x)\,dx-hf_0\right|\le\frac{M_2}2\int_{-h/2}^{h/2}x
^2\,dx=\frac{M_2}{24}h^3.\tag1.1$$

\head 2. “б«®¦­Ґ­­ п д®а¬г«  Їаאַ㣮«м­ЁЄ®ў\endhead

Џгбвм Ё¬ҐҐвбп ®в१®Є $[a,b]$ Ё вॡгҐвбп ЇаЁЎ«Ё¦Ґ­­® ўлзЁб«Ёвм
$$\int_a^bf(x)\,dx.$$
ђ §®ЎмҐ¬ ®в१®Є $[a,b]$ ­  $N$ а ў­ле з б⥩ в®зЄ ¬Ё
$$x_i=a+ih,\quad i=0,1,\dots,N,\qquad h=\frac{b-a}N.$$
Ќ  Є ¦¤®¬ Ё§ ®в१Є®ў $[x_i,x_{i+1}]$, $i=0,\dots,N-1$, ўлзЁб«Ё¬ §­ зҐ­ЁҐ
дг­ЄжЁЁ ў б।­Ґ© в®зЄҐ $x_{i+1/2}=a+(i+1/2)h$ Ё ЇаЁ¬Ґ­Ё¬ д®а¬г«г
Їаאַ㣮«м­ЁЄ®ў
$$\int_{x_i}^{x_{i+1}}f(x)\,dx\approx hf_{i+1/2},\tag2.1$$
Ј¤Ґ $f_{i+1/2}=f(x_{i+1/2})$.

Џ®бЄ®«мЄг
$$\int_a^bf(x)\,dx=\sum_{i=0}^{N-1}\int_{x_i}^{x_{i+1}}f(x)\,dx,$$
в®, б«®¦Ёў ЇаЁЎ«Ё¦Ґ­­лҐ а ўҐ­бвў  \thetag{2.1}, Ї®«гзЁ¬
$$\int_a^bf(x)\,dx\approx h(f_{1/2}+f_{3/2}+\dots+f_{N-1/2}).\tag2.2$$
”®а¬г«  \thetag{2.2} ­ §лў Ґвбп {\it гб«®¦­Ґ­­®© д®а¬г«®© Їаאַ㣮«м­ЁЄ®ў\/}
(з бв® Ё¬Ґ­­® нвг д®а¬г«г ­ §лў ов {\it д®а¬г«®© Їаאַ㣮«м­ЁЄ®ў\/}).

„«п ®жҐ­ЄЁ Ї®ЈаҐи­®бвЁ д®а¬г«л \thetag{2.2} ЇаҐ¤Ї®«®¦Ё¬, зв® дг­ЄжЁп $f(x)$
Ё¬ҐҐв ­ҐЇаҐалў­го ўв®аго Їа®Ё§ў®¤­го ­  ®в१ЄҐ $[a,b]$ Ё
$$M_2=\max_{x\in[a,b]}|f''(x)|.$$
“зЁвлў п ®жҐ­Єг \thetag{1.1}, Ё¬ҐҐ¬
$$\biggl|\int_a^bf(x)\,dx-h\sum_{i=0}^{N-1}f_{i+1/2}\biggr|\le\sum_{i=0}^{N-1
}\frac{M_2}{24}h^3=\frac{M_2}{24}(b-a)h^2.$$
‚ бЁ«г а ўҐ­бвў  $Nh=b-a$ ¬®¦­® ЇҐаҐЇЁб вм нв㠮業Єг б«Ґ¤гойЁ¬ ®Ўа §®¬
$$\biggl|\int_a^bf(x)\,dx-h\sum_{i=0}^{N-1}f_{i+1/2}\biggr|\le\frac{M_2(b-a)
^3}{24N^2}.$$

\head 3. €бЇ®«м§®ў ­ЁҐ Ё­вҐаЇ®«пжЁ®­­®Ј® ¬­®Ј®з«Ґ­  ‹ Ја ­¦  ¤«п Ї®бв஥­Ёп
Єў ¤а вга­ле д®а¬г« \endhead

Ћ¤­Ё¬ Ё§ ®ЎйЁе ЇаЁҐ¬®ў Ї®бв஥­Ёп Єў ¤а вга­ле д®а¬г« пў«пҐвбп § ¬Ґ­ 
дг­ЄжЁЁ, § ¤ ­­®© ­  ®в१ЄҐ $[a,b]$, ­ҐЄ®в®а®© Ў®«ҐҐ Їа®бв®© Ё ў в® ¦Ґ ўаҐ¬п
Ў«Ё§Є®© Є Ёб室­®© дг­ЄжЁҐ©. Ќ ЇаЁ¬Ґа, Ґб«Ё $f(x)$ Ё§ўҐбв­  ў ­ҐЄ®в®але
в®зЄ е ®в१Є  $[a,b]$ \ $x_0,x_1,\dots,x_n$, в® ¬®¦­® § ¬Ґ­Ёвм ҐҐ ­ 
Ё­вҐаЇ®«пжЁ®­­л© ¬­®Ј®з«Ґ­ ‹ Ја ­¦  $L_n(x)$ Ё Ї®«®¦Ёвм
$$\int_a^bf(x)\,dx\approx\int_a^bL_n(x)\,dx.$$

„«п Ё­вҐаЇ®«пжЁ®­­®Ј® ¬­®Ј®з«Ґ­  ‹ Ја ­¦  Ё¬ҐҐв ¬Ґбв® а ўҐ­бвў® (б¬.\ [7,
бва.\ 9])
$$L_n(x)=\sum_{i=0}^nl_{ni}(x)f_i,$$
Ј¤Ґ
$$l_{ni}(x)=\frac{(x-x_0)\dots(x-x_{i-1})(x-x_{i+1})\dots(x-x_n)}{(x_i-x_0)
\dots(x_i-x_{i-1})(x_i-x_{i+1})\dots(x_i-x_n)},\qquad f_i=f(x_i).$$
’ ЄЁ¬ ®Ўа §®¬, Ї®«гз Ґ¬ Єў ¤а вга­го д®а¬г«г
$$\int_a^bf(x)\,dx\approx\sum_{i=0}^nA_if_i,\tag3.1$$
ў Є®в®а®©
$$A_i=\int_a^bl_{ni}(x)\,dx.\tag3.2$$
\remark\re …б«Ё $f(x)$ --- ¬­®Ј®з«Ґ­ б⥯Ґ­Ё $k\le n$, в® ў бЁ«г
Ґ¤Ё­б⢥­­®бвЁ Ё­вҐаЇ®«пжЁ®­­®Ј® ¬­®Ј®з«Ґ­  ‹ Ја ­¦  $f(x)\equiv L_n(x)$. ’Ґ¬
б ¬л¬ Єў ¤а вга­ п д®а¬г«  \thetag{3.1} в®з­  ­  ¬­®Ј®з«Ґ­ е б⥯Ґ­Ё $k\le n
$. ‚ з бв­®бвЁ, ЇаЁ ўбҐе $k=0,1,\dots,n$
$$\int_a^bx^k\,dx=\sum_{i=0}^nA_ix_i^k.\tag3.3$$
\endremark

\example{\ex3.1} Џ®бва®Ёвм Єў ¤а вга­го д®а¬г«г ўЁ¤ 
$$\int_{-1}^1f(x)\,dx\approx A_0f(-1/2)+A_1f(0)+A_2f(1/4).$$
\endexample

\demo\so Њ®¦­® Ўл«® Ўл ўлзЁб«пвм $A_0$, $A_1$ Ё $A_2$ Ї® д®а¬г« ¬ \thetag{3.2
}. Ќ ЇаЁ¬Ґа,
$$A_0=\int_{-1}^1\frac{x(x-1/4)}{3/8}\,dx=\frac83\left.\left(\frac{x^3}3-
\frac{x^2}8\right)\right|_{-1}^1=\frac{16}9.$$
Њл ¦Ґ ў®бЇ®«м§гҐ¬бп а ўҐ­бвў ¬Ё \thetag{3.3}. €¬ҐҐ¬
$$\align\int_{-1}^1\,dx&=A_0+A_1+A_2,\\
\int_{-1}^1x\,dx&=A_0\left(-\frac12\right)+A_2\frac14,\\
\int_{-1}^1x^2\,dx&=A_0\left(-\frac12\right)^2+A_2\left(\frac14\right)^2.
\endalign$$
’ ЄЁ¬ ®Ўа §®¬, Ї®«гз Ґ¬ б«Ґ¤гойго бЁб⥬г
$$\left\{\alignedat3&A_0&+A_1&+&A_2&=2,\\
-\frac12&A_0&&+&\frac14A_2&=0,\\
\frac14&A_0&&+&\frac1{16}A_2&=\frac23.\endalignedat\right.$$
ђҐи п нвг бЁб⥬г, ­ е®¤Ё¬
$$A_0=\frac{16}9,\qquad A_1=-\frac{10}3,\qquad A_2=\frac{32}9.$$
‘«Ґ¤®ў вҐ«м­®, ЁбЄ®¬ п Єў ¤а вга­ п д®а¬г«  Ё¬ҐҐв ўЁ¤
$$\int_{-1}^1f(x)\,dx\approx \frac{16}9f(-1/2)-\frac{10}3f(0)+\frac{32}9f(1/4
).$$
\enddemo

\head 4. Љў ¤а вга­лҐ д®а¬г«л Ќмов®­ --Љ®вҐб \endhead

„«п ЇаЁЎ«Ё¦Ґ­­®Ј® ўлзЁб«Ґ­Ёп Ё­вҐЈа « 
$$\int_a^bf(x)\,dx$$
а §®ЎмҐ¬ ®в१®Є $[a,b]$ ­  $n$ а ў­ле з б⥩ в®зЄ ¬Ё
$$x_i=x_0+ih,\quad i=0,1,\dots,n,\qquad h=\frac{b-a}n$$
($x_0=a$, $x_n=b$). ‡ ¬Ґ­Ё¬ дг­ЄжЁо $f(x)$ ­  ҐҐ Ё­вҐаЇ®«пжЁ®­­л© ¬­®Ј®з«Ґ­
‹ Ја ­¦ , Ї®бв஥­­л© Ї® §­ зҐ­Ёп¬ нв®© дг­ЄжЁЁ ў в®зЄ е $x_0,x_1,\dots,x_n$.
„«п а ў­®®вбв®пйЁе 㧫®ў Ё­вҐаЇ®«пжЁЁ (б¬.\ [7, бва.\ 12])
$$l_{ni}(x)=\frac{(-1)^{n-i}}{i!(n-i)!}q(q-1)\dots(q-i+1)(q-i-1)\dots(q-n),$$
Ј¤Ґ
$$q=\frac{x-x_0}h.\tag4.1$$
‘¤Ґ« ў § ¬Ґ­г ЇҐаҐ¬Ґ­­®© \thetag{4.1} Ё гзЁвлў п, зв® $dx=h\,dq$, Ї®«гзЁ¬
Єў ¤а вга­го д®а¬г«г \thetag{3.1}, ў Є®в®а®©
$$A_i=h\frac{(-1)^{n-i}}{i!(n-i)!}\int_0^nq(q-1)\dots(q-i+1)(q-i-1)\dots(q-n)
\,dq.$$

Џ®«®¦Ё¬
$$H_i=\frac1n\frac{(-1)^{n-i}}{i!(n-i)!}\int_0^nq(q-1)\dots(q-i+1)(q-i-1)
\dots(q-n)\,dq.\tag4.2$$
’ Є Є Є $h=\dfrac{b-a}n$, в® $A_i=(b-a)H_i$. ’Ґ¬ б ¬л¬ Ї®«гз Ґ¬ Єў ¤а вга­го
д®а¬г«г
$$\int_a^bf(x)\,dx\approx(b-a)\sum_{i=o}^nH_if_i,\tag4.3$$
­ §лў Ґ¬го {\it Єў ¤а вга­®© д®а¬г«®© Ќмов®­ --Љ®вҐб }. ‚Ґ«ЁзЁ­л \thetag{4.2}
­ §лў овбп {\it Є®нддЁжЁҐ­в ¬Ё Љ®вҐб }. „«п ­Ёе бгйҐбвўгов в Ў«Ёжл.

Ћв¬ҐвЁ¬ ­ҐЄ®в®алҐ Їа®б⥩訥 бў®©бвў  Є®нддЁжЁҐ­в®ў Љ®вҐб . Џ®бЄ®«мЄг
Єў ¤а вга­ п д®а¬г«  \thetag{4.3} в®з­  ¤«п «оЎ®Ј® ¬­®Ј®з«Ґ­  б⥯Ґ­Ё $k\le n
$, в® ®­ , ў з бв­®бвЁ в®з­  ¤«п $f(x)\equiv1$. Џ®¤бв ўЁў нвг дг­ЄжЁо ў
\thetag{4.3}, ­ е®¤Ё¬
$$\sum_{i=0}^nH_i=1.\tag4.4$$
Ља®¬Ґ в®Ј®, ­ҐЇ®б।б⢥­­® Ё§ \thetag{4.2} ўл⥪ ов а ўҐ­бвў 
$$H_{n-i}=H_i,\quad i=0,1,\dots,n-1.\tag4.5$$

\head 5. ”®а¬г«  ва ЇҐжЁ©\endhead

ђ бᬮваЁ¬ Єў ¤а вга­го д®а¬г«г Ќмов®­ --Љ®вҐб  ЇаЁ $n=1$. ‚ н⮬ б«гз Ґ $h=b
-a$,   б ¬  д®а¬г«  Ё¬ҐҐв ўЁ¤
$$\int_a^bf(x)\,dx\approx h(H_0f_0+H_1f_1).$$
Џ®«м§гпбм бў®©бвў ¬Ё \thetag{4.4} Ё \thetag{4.5}, ­ е®¤Ё¬
$$\gather H_0+H_1=1,\\
H_0=H_1.\endgather$$
‘«Ґ¤®ў вҐ«м­®, $H_0=H_1=1/2$.

€в Є, Ї®«г祭  д®а¬г« 
$$\int_a^bf(x)\,dx\approx h\frac{f_0+f_1}2,$$
­ §лў Ґ¬ п {\it д®а¬г«®© ва ЇҐжЁ©}.

ЋжҐ­Ё¬ Ї®ЈаҐи­®бвм д®а¬г«л ва ЇҐжЁ©. „«п Ї®ЈаҐи­®бвЁ ЇаЁЎ«Ё¦Ґ­Ёп дг­ЄжЁЁ
¬­®Ј®з«Ґ­®¬ ‹ Ја ­¦  ЇҐаў®© б⥯Ґ­Ё Ё¬ҐҐв ¬Ґбв® а ўҐ­бвў® (б¬.\ [7, бва.\ 10,
12])
$$f(x)-L_1(x)=\frac{f''(\xi)}2h^2q(q-1),$$
Ј¤Ґ $\xi$ ­ҐЄ®в®а п в®зЄ  Ё§ Ё­вҐаў «  $(a,b)$,   $q$ ®ЇаҐ¤Ґ«Ґ­® а ўҐ­бвў®¬
\thetag{4.1}. ’ ЄЁ¬ ®Ўа §®¬, Ё¬ҐҐ¬
$$\multline\quad\left|\int_a^bf(x)\,dx-h\frac{f_0+f_1}2\right|=\left|\int_a^b
(f(x)-L_1(x))\,dx\right|\\
=\left|\int_0^1\frac{f''(\xi)}2h^3q(q-1)\,dq\right|\le\frac{M_2}2h^3\int_0^1q
(1-q)\,dq=\frac{M_2}{12}h^3.\quad\endmultline$$

Ђ­ «®ЈЁз­® гб«®¦­Ґ­­®© д®а¬г«Ґ Їаאַ㣮«м­ЁЄ®ў ¬®¦­® Ї®бва®Ёвм гб«®¦­Ґ­­го
д®а¬г«г ва ЇҐжЁ©, а §ЎЁў ўҐбм ®в१®Є $[a,b]$ ­  $n$ а ў­ле з б⥩ Ё ЇаЁ¬Ґ­Ёў
­  Є ¦¤®© Ё§ з б⥩ д®а¬г«г ва ЇҐжЁ©. ’®Ј¤  $h=\dfrac{b-a}n$,  
ᮮ⢥вбвўгой п Єў ¤а вга­ п д®а¬г«  Ё¬ҐҐв ўЁ¤
$$\int_a^bf(x)\,dx\approx h\left(\frac{f_0}2+f_1+\dots+f_{n-1}+\frac{f_n}2
\right).$$
ЏаЁ н⮬ ¤«п Ї®ЈаҐи­®бвЁ нв®© Єў ¤а вга­®© д®а¬г«л (¬л ®Ў®§­ з Ґ¬ ҐҐ зҐаҐ§ $R
_t$) Ўг¤гв бЇа ўҐ¤«Ёўл ­Ґа ўҐ­бвў 
$$R_t\le n\frac{M_2}{12}h^3=\frac{M_2}{12}(b-a)h^2=\frac{M_2(b-a)^3}{12n^2}.
\tag5.1$$

\example{\ex5.1} ‚ бЄ®«мЄЁе в®зЄ е ­ ¤® ўлзЁб«Ёвм дг­ЄжЁо $f(x)=e^{x^2}$,
зв®Ўл ЇаЁ ўлзЁб«Ґ­ЁЁ Ё­вҐЈа « 
$$\int_0^1e^{x^2}\,dx$$
Ї® гб«®¦­Ґ­­®© д®а¬г«Ґ ва ЇҐжЁ© Ї®ЈаҐи­®бвм ­Ґ ЇаҐў®б室Ё«  $10^{-5}$.
\endexample

\demo\so Џ®бЄ®«мЄг $f''(x)=(2+4x^2)e^{x^2}$, в® $M_2=6e$. €§ \thetag{5.1}
ўл⥪ Ґв, зв® ¤«п ¤®бвЁ¦Ґ­Ёп вॡ㥬®© в®з­®бвЁ ¤®бв в®з­® ўлЇ®«­Ґ­Ёп
­Ґа ўҐ­бвў 
$$\frac{M_2}{12n^2}\le10^{-5}.$$
Ћвбо¤  $n\ge100\sqrt{5e}=368,6\dots\,\,$. —Ёб«® ўлзЁб«Ґ­Ё© дг­ЄжЁЁ ЇаЁ
а §ЎЁҐ­ЁЁ ®в१Є  ­  $n$ з б⥩ а ў­® $n+1$, Ї®н⮬㠤«п ўлзЁб«Ґ­Ёп
а бб¬ ваЁў Ґ¬®Ј® Ё­вҐЈа «  б § ¤ ­­®© в®з­®бвмо Ї®вॡгҐвбп ­ е®¤Ёвм §­ зҐ­Ёп
дг­ЄжЁЁ ў 370 в®зЄ е.
\enddemo


\head 6. ”®а¬г«  ‘Ё¬Їб®­ \endhead

Џгбвм ⥯Ґам $n=2$, в.Ґ.\ $x_0=a$, $x_1=a+h$, $x_2=b$,   $h=\dfrac{b-a}2$.
€§ \thetag{4.2} Ё¬ҐҐ¬
$$H_0=\frac14\int_0^2(q-1)(q-2)\,dq=\frac14\left.\left(\frac{q^3}3-\frac{3q^2
}2+2q\right)\right|_0^2=\frac16.$$
’®Ј¤  ў бЁ«г \thetag{4.5} $H_2=H_0=1/6$,   Ё§ \thetag{4.4} ўл⥪ Ґв, зв® $H_1
=2/3$. €в Є, ¬л Ї®«гзЁ«Ё Єў ¤а вга­го д®а¬г«г
$$\int_a^bf(x)\,dx\approx\frac h3(f_0+4f_1+f_2),$$
Є®в®а п ­ §лў Ґвбп {\it д®а¬г«®© ‘Ё¬Їб®­ }. ѓҐ®¬ҐваЁзҐбЄЁ© б¬лб« нв®© д®а¬г«л
§ Є«оз Ґвбп ў ⮬, зв® ў¬Ґбв® Ё­вҐЈа «  ®в Ёб室­®© дг­ЄжЁЁ ўлзЁб«пҐвбп
Ё­вҐЈа « ®в Ї а Ў®«л, Їа®е®¤п饩 зҐаҐ§ §­ зҐ­Ёп дг­ЄжЁЁ ў в®зЄ е $a$, $\dfrac
{a+b}2$ Ё $b$.

ЋжҐ­Ё¬ Ї®ЈаҐи­®бвм д®а¬г«л ‘Ё¬Їб®­ . ЃҐ§ ®Ја ­ЁзҐ­Ёп ®Ўй­®бвЁ ¬®¦­® бзЁв вм,
зв® ­ з «® Є®®а¤Ё­ в ўлЎа ­® ў в®зЄҐ $\dfrac{a+b}2$. ’Ґ¬ б ¬л¬ Ўг¤Ґ¬
®жҐ­Ёў вм Єў ¤а вга­го д®а¬г«г
$$\int_{-h}^hf(x)\,dx\approx\frac h3(f(-h)+4f(0)+f(h)).\tag6.1$$

ЏаҐ¦¤Ґ ўбҐЈ® ®в¬ҐвЁ¬, зв® д®а¬г«  ‘Ё¬Їб®­  в®з­  ­Ґ в®«мЄ® ­  ¬­®Ј®з«Ґ­ е
ўв®а®© б⥯Ґ­Ё, Є Є б«Ґ¤гҐв Ё§ § ¬Ґз ­Ёп ў Ї.\ 3, ­® Ё ­  ¬­®Ј®з«Ґ­ е ваҐв쥩
б⥯Ґ­Ё. „«п в®Ј®, зв®Ўл гЎҐ¤Ёвмбп ў н⮬, ¤®бв в®з­® Їа®ўҐаЁвм, зв® д®а¬г« 
\thetag{6.1} в®з­  ¤«п $f(x)=x^3$.

Џгбвм $c$ --- Їа®Ё§ў®«м­ п в®зЄ  Ё§ Ё­вҐаў «  $(0,h)$. ђ бᬮваЁ¬
Ё­вҐаЇ®«пжЁ®­­л© ¬­®Ј®з«Ґ­ ‹ Ја ­¦  $L_3(x)$, Ё­вҐаЇ®«ЁагойЁ© дг­ЄжЁо $f(x)$
ў в®зЄ е $-h$, $0$, $c$ Ё $h$. €¬ҐҐ¬
$$\int_{-h}^hL_3(x)\,dx=\frac h3(L_3(-h)+4L_3(0)+L_3(h))=\frac h3(f(-h)+4f(0)
+f(h)).$$
Џ®н⮬㠤«п Ї®ЈаҐи­®бвЁ д®а¬г«л ‘Ё¬Їб®­ , Є®в®аго ¬л ®Ў®§­ зЁ¬ зҐаҐ§ $R$,
Ўг¤Ґ¬ Ё¬Ґвм
$$\multline\quad R=\biggl|\int_{-h}^hf(x)\,dx-\frac h3(f(-h)+4f(0)+f(h))
\biggr|\\
=\biggl|\int_{-h}^hf(x)\,dx-\int_{-h}^hL_3(x)\,dx\biggr|=\biggl|\int_{-h}^h(f
(x)-L_3(x))\,dx\biggr|\\
\le\int_{-h}^h|f(x)-L_3(x)|\,dx.\quad\endmultline$$
Џ®«®¦Ё¬
$$M_4=\max_{x\in[-h,h]}|f^{(4)}(x)|.$$
’®Ј¤  ¤«п Ї®ЈаҐи­®бвЁ ЇаЁЎ«Ё¦Ґ­Ёп дг­ЄжЁЁ $f(x)$ ¬­®Ј®з«Ґ­®¬ ‹ Ја ­¦ 
бЇа ўҐ¤«Ёў  б«Ґ¤гой п ®жҐ­Є  (б¬.\ [7, бва.\ 11])
$$|f(x)-L_3(x)|\le\frac{M_4}{4!}|(x+h)x(x-c)(x-h)|.$$
’Ґ¬ б ¬л¬ ¤«п «оЎ®Ј® $c\in(0,h)$
$$R\le\frac{M_4}{24}\int_{-h}^h|(x+h)x(x-c)(x-h)|\,dx.$$
ЏҐаҐе®¤п Є ЇаҐ¤Ґ«г ў н⮬ ­Ґа ўҐ­б⢥ ЇаЁ $c\to0$, Ї®«гз Ґ¬
$$R\le\frac{M_4}{24}\int_{-h}^hx^2(h^2-x^2)\,dx=\frac{M_4}{24}\left.\left(h^2
\frac{x^3}3-\frac{x^5}5\right)\right|_{-h}^h=\frac{M_4}{90}h^5.\tag6.2$$

‚뢥¤Ґ¬ ⥯Ґам гб«®¦­Ґ­­го д®а¬г«г ‘Ё¬Їб®­ . ђ §¤Ґ«Ё¬ ®в१®Є $[a,b]$ ­ 
зҐв­®Ґ зЁб«® ®в१Є®ў $2n$
$$x_i=a+ih,\quad i=0,1,\dots,2n,\qquad h=\frac{b-a}{2n}.$$
Ќ  Є ¦¤®¬ Ё§ ®в१Є®ў $[x_{2i},x_{2i+2}]$ ЇаЁ¬Ґ­Ё¬ д®а¬г«г ‘Ё¬Їб®­ 
$$\int_{x_{2i}}^{x_{2i+2}}f(x)\,dx\approx\frac h3(f_{2i}+4f_{2i+1}+f_{2i+2}).
$$
‘«®¦Ёў нвЁ а ўҐ­бвў , Ї®«гзЁ¬ гб«®¦­Ґ­­го д®а¬г«г ‘Ё¬Їб®­ 
$$\int_a^bf(x)\,dx\approx\frac h3\biggl(f_0+f_{2n}+4\sum_{i=1}^nf_{2i-1}+2
\sum_{i=1}^{n-1}f_{2i}\biggr).$$

“зЁвлў п ®жҐ­Єг \thetag{6.2}, ¤«п Ї®ЈаҐи­®бвЁ гб«®¦­Ґ­­®© д®а¬г«л ‘Ё¬Їб®­ ,
Є®в®аго ¬л ®Ў®§­ зЁ¬ зҐаҐ§ $R_s$, Ўг¤Ґ¬ Ё¬Ґвм
$$R_s\le n\frac{M_4}{90}h^5=\frac{M_4}{180}(b-a)h^4,$$
Ј¤Ґ
$$M_4=\max_{x\in[a,b]}|f^{(4)}(x)|.$$
Џ®бЄ®«мЄг $h=\dfrac{b-a}{2n}$, в® ®жҐ­Є  ўҐ«ЁзЁ­л $R_s$ ¬®¦Ґв Ўлвм ¤ ­  зҐаҐ§
$n$
$$R_s\le\frac{M_4(b-a)^5}{2880n^4}.$$

\example{\ex6.1} ђ бᬮваҐвм ЇаЁ¬Ґа 5.1 ¤«п д®а¬г«л ‘Ё¬Їб®­ .
\endexample

\demo\so €¬ҐҐ¬ $f^{(4)}(x)=(12+48x^2+16x^4)e^{x^2}$. ’Ґ¬ б ¬л¬ $M_4=76e$. „«п
¤®бвЁ¦Ґ­Ёп вॡ㥬®© в®з­®бвЁ ¤®бв в®з­® ўлЇ®«­Ґ­Ёп ­Ґа ўҐ­бвў 
$$\frac{M_4}{2880n^4}\le10^{-5}.$$
Ћвбо¤ 
$$n\ge10\root4\of{\frac{19e}{72}}=9,2\dots\,\,.$$
ЏаЁ ЁбЇ®«м§®ў ­ЁЁ д®а¬г«л ‘Ё¬Їб®­  ®в१®Є а §ЎЁў Ґвбп ­  $2n$ з б⥩.
Џ®н⮬㠤«п ўлзЁб«Ґ­Ёп а бб¬ ваЁў Ґ¬®Ј® Ё­вҐЈа «  б § ¤ ­­®© в®з­®бвмо
Ї®вॡгҐвбп ­ е®¤Ёвм §­ зҐ­Ёп дг­ЄжЁЁ ў 21 в®зЄҐ (­ Ї®¬­Ё¬, зв® ў  ­ «®ЈЁз­®¬
ЇаЁ¬ҐаҐ ¤«п д®а¬г«л ва ЇҐжЁ© ЇаЁ в®© ¦Ґ в®з­®бвЁ вॡ®ў «Ёбм ўлзЁб«Ґ­Ёп ў 370
в®зЄ е).
\enddemo

\head 7. ѓ« ў­ п з бвм Ї®ЈаҐи­®бвЁ Єў ¤а вга­ле д®а¬г«\endhead

„«п гб«®¦­Ґ­­®© д®а¬г«л Їаאַ㣮«м­ЁЄ®ў Ўл«® ¤®Є § ­® б«Ґ¤го饥 а ўҐ­бвў®
$$\int_a^bf(x)\,dx=h\sum_{i=0}^{n-1}f_{i+1/2}+r_1,\tag7.1$$
Ј¤Ґ $h=\dfrac{b-a}n$,  
$$|r_1|\le\frac{M_2}{24}(b-a)h^2.$$
Њл Ї®«гзЁ¬ Ў®«ҐҐ в®з­го ®жҐ­Єг ў ЇаҐ¤Ї®«®¦Ґ­ЁЁ, зв® дг­ЄжЁп $f(x)$ Ё¬ҐҐв
­ҐЇаҐалў­го зҐвўҐавго Їа®Ё§ў®¤­го.

ђ бᬮваЁ¬ б­ з «  б«гз ©, Є®Ј¤  $[a,b]=[-h/2,h/2]$. Џ® д®а¬г«Ґ ’Ґ©«®а 
$$f(x)=f_0+f'_0x+\frac{f_0''}{2!}x^2+\frac{f_0'''}{3!}x^3+\frac{f^{(4)}(\xi)}
{4!}x^4,\tag7.2$$
Ј¤Ґ $f^{(i)}_0=f^{(i)}(0)$, $i=0,1,2,3$,   $\xi$ --- ­ҐЄ®в®а п в®зЄ  Ё§
Ё­вҐаў «  $(-h/2,h/2)$. €­вҐЈаЁагп \thetag{7.2}, Ї®«гз Ґ¬ а ўҐ­бвў®
$$\int_{-h/2}^{h/2}f(x)\,dx=f_0h+\frac{f_0''}{24}h^3+r_2,$$
ў Є®в®а®¬
$$r_2=\int_{-h/2}^{h/2}\frac{f^{(4)}(\xi)}{4!}x^4\,dx.$$
„«п ўҐ«ЁзЁ­л $r_2$ бЇа ўҐ¤«Ёў  б«Ґ¤гой п ®жҐ­Є 
$$|r_2|\le\frac{M_4}{4!}\int_{-h/2}^{h/2}x^4\,dx=\frac{M_4}{1920}h^5.$$
’ ЄЁ¬ ®Ўа §®¬, ¤«п гб«®¦­Ґ­­®© д®а¬г«л Їаאַ㣮«м­ЁЄ®ў Ё¬ҐҐ¬
$$\int_a^bf(x)\,dx=h\sum_{i=0}^{n-1}f_{i+1/2}+\frac{h^3}{24}\sum_{i=0}^{n-1}f
''_{i+1/2}+r_3,\tag7.3$$
Ј¤Ґ
$$|r_3|\le n\frac{M_4}{1920}h^5=\frac{M_4}{1920}(b-a)h^4.$$
ЏаЁ¬Ґ­пп \thetag{7.1} ¤«п ўв®а®© Їа®Ё§ў®¤­®©, ­ е®¤Ё¬
$$\int_a^bf''(x)\,dx=h\sum_{i=0}^{n-1}f''_{i+1/2}+r_4,$$
ЇаЁ н⮬
$$|r_4|\le\frac{M_4}{24}(b-a)h^2.$$
‘«Ґ¤®ў вҐ«м­®,
$$h\sum_{i=0}^{n-1}f''_{i+1/2}=\int_a^bf''(x)\,dx-r_4.$$
Џ®¤бв ў«пп нв® ўла ¦Ґ­ЁҐ ў \thetag{7.3}, Ї®«гз Ґ¬
$$\int_a^bf(x)\,dx=h\sum_{i=0}^{n-1}f_{i+1/2}+\frac{h^2}{24}\int_a^bf''(x)\,d
x+r_5,$$
Ј¤Ґ
$$|r_5|=\Bigl|-\frac{h^2}{24}r_4+r_3\Bigr|\le\frac{M_4}{24^2}(b-a)h^4+\frac{M
_4}{1920}(b-a)h^4=\frac{13M_4}{5760}(b-a)h^4.$$

\definition\de ѓ®ў®апв, зв® $\varphi(h)=O(h^k)$ (зЁв Ґвбп ``® Ў®«м讥''),
Ґб«Ё бгйҐбвўгҐв в Є п Ї®бв®п­­ п $C>0$, ¤«п Є®в®а®©
$$|\varphi(h)|\le Ch^k.$$
\enddefinition

’Ґ¬ б ¬л¬ ¬л ¤®Є § «Ё а ўҐ­бвў®
$$\int_a^bf(x)\,dx=h\sum_{i=0}^{n-1}f_{i+1/2}+ch^2+O(h^4),\tag7.4$$
ў Є®в®а®¬
$$c=\frac1{24}\int_a^bf''(x)\,dx.$$
Џ®«®¦Ё¬
$$I=\int_a^bf(x)\,dx,\qquad I_h^r=h\sum_{i=0}^{n-1}f_{i+1/2}.$$
’®Ј¤  а ўҐ­бвў® \thetag{7.4} § ЇЁиҐвбп ў ўЁ¤Ґ
$$I=I_h^r+ch^2+O(h^4).$$
‚Ґ«ЁзЁ­  $ch^2$ ­ §лў Ґвбп {\it Ј« ў­®© з бвмо\/} Ї®ЈаҐи­®бвЁ д®а¬г«л
Їаאַ㣮«м­ЁЄ®ў.

Ђ­ «®ЈЁз­лҐ а ўҐ­бвў  ¬®¦­® Ї®«гзЁвм ¤«п д®а¬г« ва ЇҐжЁ© Ё ‘Ё¬Їб®­ 
$$\align I&=I_h^t+c_1h^2+O(h^4),\\
I&=I_h^s+c_2h^4+O(h^6)\endalign$$
(ў Ї®б«Ґ¤­Ґ¬ б«гз Ґ ­ ¤® вॡ®ў вм, зв®Ўл дг­ЄжЁп $f(x)$ Ё¬Ґ«  ­ҐЇаҐалў­го
иҐбвго Їа®Ё§ў®¤­го).

\head 8. Џа ўЁ«® ђг­ЈҐ Їа ЄвЁзҐбЄ®© ®жҐ­ЄЁ Ї®ЈаҐи­®бвЁ\endhead

Џгбвм $z$ --- ­ҐЁ§ўҐбв­®Ґ в®з­®Ґ §­ зҐ­ЁҐ ­ҐЄ®в®а®© ўҐ«ЁзЁ­л, $z_h$ ---
Ё§ўҐбв­®Ґ ҐҐ ЇаЁЎ«Ё¦Ґ­­®Ґ §­ зҐ­ЁҐ, § ўЁбп饥 ®в Ї®«®¦ЁвҐ«м­®Ј® Ї а ¬Ґва 
$h$, Є®в®ал© ¬®¦Ґв ЇаЁ­Ё¬ вм бЄ®«м гЈ®¤­® ¬ «лҐ §­ зҐ­Ёп.

ЏаҐ¤Ї®«®¦Ё¬, зв® гбв ­®ў«Ґ­  бўп§м ¬Ґ¦¤г в®з­л¬ Ё ЇаЁЎ«Ё¦Ґ­­л¬ §­ зҐ­Ёп¬Ё
$$z=z_h+ch^k+O(h^{k+m}),\tag8.1$$
Ј¤Ґ $c$ --- ­ҐЁ§ўҐбв­ п ­Ґ § ўЁбпй п ®в $h$ Ї®бв®п­­ п. ’®Ј¤ 
$$z=z_{h/2}+c\left(\frac h2\right)^k+O(h^{k+m}),\tag8.2$$
в Є Є Є ¤«п «оЎ®© Ї®бв®п­­®© $C$ \ $O((Ch)^n)=O(h^n)$.
‚лзЁв п Ё§ \thetag{8.1} а ўҐ­бвў® \thetag{8.2}, Ўг¤Ґ¬ Ё¬Ґвм
$$z_{h/2}-z_h=c\left(\frac h2\right)^k(2^k-1)+O(h^{k+m}).\tag8.3$$
Ћвбо¤ 
$$c\left(\frac h2\right)^k=\frac{z_{h/2}-z_h}{2^k-1}+O(h^{k+m}).$$
‘«Ґ¤®ў вҐ«м­®, ЇаЁ $c\ne0$ ўҐ«ЁзЁ­  $\dfrac{z_{h/2}-z_h}{2^k-1}$ ®в«Ёз Ґвбп
®в Ј« ў­®Ј® з«Ґ­  Ї®ЈаҐи­®бвЁ $z-z_{h/2}$ ­  $O(h^{k+m})$, в.Ґ.
$$z-z_{h/2}=\frac{z_{h/2}-z_h}{2^k-1}+O(h^{k+m}). \tag8.4$$

’Ґ¬ б ¬л¬ ЇаЁ $c\ne0$ ®жҐ­Ёвм Ї®ЈаҐи­®бвм ¬®¦­® в Є:
$$z-z_{h/2}\approx\frac{z_{h/2}-z_h}{2^k-1}.$$
’ Є®© бЇ®б®Ў ®жҐ­ЄЁ Ї®ЈаҐи­®бвЁ ­ §лў Ґвбп {\it Їа ўЁ«®¬ ђг­ЈҐ}.

\remark\re Ќ  Їа ЄвЁЄҐ бзЁв Ґвбп, зв® гб«®ўЁҐ $c\ne0$ ўлЇ®«­Ґ­®, Ґб«Ё
$$\left|2^k\frac{z_{h/2}-z_h}{z_h-z_{2h}}-1\right|<0,1.\tag8.5$$
’®«мЄ® ў н⮬ б«гз Ґ ४®¬Ґ­¤гҐвбп ЇаЁ¬Ґ­Ґ­ЁҐ Їа ўЁ«  ђг­ЈҐ
\endremark

Џ®пб­Ё¬ гб«®ўЁҐ \thetag{8.5}. €§ \thetag{8.3} б«Ґ¤гҐв, зв®
$$\align z_{h/2}-z_h&=c\left(\frac h2\right)^k(2^k-1)+O(h^{k+m}),\\
z_h-z_{2h}&=ch^k(2^k-1)+O(h^{k+m}).\endalign$$
Џ®н⮬г
$$\frac{z_{h/2}-z_h}{z_h-z_{2h}}\approx\frac1{2^k}.$$
’ ЄЁ¬ ®Ўа §®¬,
$$2^k\frac{z_{h/2}-z_h}{z_h-z_{2h}}\approx1.$$

\head 9. “в®з­Ґ­ЁҐ ЇаЁЎ«Ё¦Ґ­­®Ј® аҐиҐ­Ёп Ї® ђЁз а¤б®­г \endhead

Џ®«®¦Ё¬
$$z_h^*=\frac{2^kz_{h/2}-z_h}{2^k-1}.$$
’®Ј¤  Ё§ \thetag{8.4} Ї®«гз Ґ¬
$$z=z_{h/2}+\frac{z_{h/2}-z_h}{2^k-1}+O(h^{k+m})=z_h^*+O(h^{k+m}).$$
ЏаЁ $c\ne0$ \ $z-z_{h/2}$ Ё¬ҐҐв $k$-л© Ї®а冷Є ¬ «®бвЁ,   $z-z_h^*$ --- $(k+m
)$-л© Ї®а冷Є ¬ «®бвЁ, в.Ґ.\ $z_h^*$ --- Ў®«ҐҐ в®з­®Ґ ЇаЁЎ«Ё¦Ґ­ЁҐ. Ћ­® ­®бЁв
­ §ў ­ЁҐ {\it гв®з­Ґ­ЁҐ Ї® ђЁз а¤б®­г}.

’ ЄЁ¬ ®Ўа §®¬, Ґб«Ё ўлзЁб«пҐвбп ®ЇаҐ¤Ґ«Ґ­­л© Ё­вҐЈа « Ї® гб«®¦­Ґ­­л¬ д®а¬г« ¬
Їаאַ㣮«м­ЁЄ®ў Ё«Ё ва ЇҐжЁ©, в® $k=2$, Ё ¬л ¬®¦Ґ¬ ®жҐ­Ёвм ЇаЁЎ«Ё¦Ґ­­®
Ї®ЈаҐи­®бвм Ї® Їа ўЁ«г ђг­ЈҐ
$$I-I_{h/2}\approx\frac{I_{h/2}-I_h}3.$$
Ља®¬Ґ в®Ј®, ¬®¦­® ­ ©вЁ гв®з­Ґ­ЁҐ Ї® ђЁз а¤б®­г
$$I_h^*=\frac{4I_{h/2}-I_h}3.$$

„«п д®а¬г«л ‘Ё¬Їб®­  $k=4$, Ё ЇаЁЎ«Ё¦Ґ­­ п ®жҐ­Є  Ї®ЈаҐи­®бвЁ Ё¬ҐҐв ўЁ¤
$$I-I^s_{h/2}\approx\frac{I_{h/2}^s-I_h^s}{15}.$$
„«п гв®з­Ґ­Ёп Ї® ђЁз а¤б®­г Ё¬ҐҐ¬
$${I_h^s}^*=\frac{16I_{h/2}^s-I_h^s}{15}.$$

\remark\re ”®а¬г«л ва ЇҐжЁ© Ё ‘Ё¬Їб®­  㤮Ў­л ⥬, зв® ЇаЁ ЇҐаҐе®¤Ґ ®в $h$ Є
$h/2$ ўбҐ ўлзЁб«Ґ­­лҐ а ­ҐҐ §­ зҐ­Ёп дг­ЄжЁ© ЁбЇ®«м§говбп ў ­®ў®©
Єў ¤а вга­®© д®а¬г«Ґ.
\endremark

Џа ўЁ«® ђг­ЈҐ Ё гв®з­Ґ­ЁҐ Ї® ђЁз а¤б®­г ¬®¦­® ЇаЁ¬Ґ­пвм Ё ¤«п ¤агЈЁе § ¤ з
ЇаЁЎ«Ё¦Ґ­­®Ј® ўлзЁб«Ґ­Ёп. ђ бᬮваЁ¬ ў Є зҐб⢥ ЇаЁ¬Ґа  зЁб«Ґ­­®Ґ
¤ЁддҐаҐ­жЁа®ў ­ЁҐ.

Џгбвм Ё§ўҐбв­л §­ зҐ­Ёп ­ҐЄ®в®а®© ¤®бв в®з­® Ј« ¤Є®© дг­ЄжЁЁ $f(x)$ ў в®зЄ е
$x_i=x_0+ih$, $i=0,\pm1$. Џ®«®¦Ё¬ $f_i=f(x_i)$, $i=0,\pm1$. •®а®и® Ё§ўҐбв­ 
д®а¬г«  зЁб«Ґ­­®Ј® ¤ЁддҐаҐ­жЁа®ў ­Ёп (б¬., ­ ЇаЁ¬Ґа, [7, бва.\ 26])
$$f_0''\approx\frac{f_1-2f_0+f_{-1}}{h^2}=f''_{0h}.$$
Ќ ©¤Ґ¬ Ј« ў­го з бвм Ї®ЈаҐи­®бвЁ ў н⮬ ¬Ґв®¤Ґ. Џ® д®а¬г«Ґ ’Ґ©«®а  Ё¬ҐҐ¬
$$f_{\pm1}=f_0\pm f_0'h+\frac{f_0''}{2!}h^2\pm\frac{f_0'''}{3!}h^3+\frac{f_0^
{(4)}}{4!}h^4\pm\frac{f_0^{(5)}}{5!}+\frac{f^{(6)}(\xi_{\pm1})}{6!}h^6,$$
Ј¤Ґ $f_0^{(i)}=f^{(i)}(0)$, $i=0,1,\dots,5$, $\xi_1\in(0,h)$,   $\xi_{-1}\in(
-h,0)$.
Џ®«®¦Ё¬
$$r=-\frac{f^{(6)}(\xi_1)+f^{(6)}(\xi_{-1})}{6!}h^4.$$
’®Ј¤ 
$$f_1+f_{-1}=2f_0+f_0''h^2+\frac{f_0^{(4)}}{12}h^4-rh^2.$$
’Ґ¬ б ¬л¬
$$f_0''=\frac{f_1-2f_0+f_{-1}}{h^2}-\frac{f_0^{(4)}}{12}h^2+r.\tag9.1$$
Џ®бЄ®«мЄг
$$|r|\le\frac{2M_6}{6!}h^4,$$
в® а ўҐ­бвў® \thetag{9.1} ¬®¦Ґв Ўлвм § ЇЁб ­® ў ўЁ¤Ґ
$$f_0''=f''_{0h}+ch^2+O(h^4),$$
Ј¤Ґ $c=-f_0^{(4)}/12$. ‘«Ґ¤®ў вҐ«м­®, Ї® Їа ўЁ«г ђг­ЈҐ
$$f_0''-f''_{0,h/2}\approx\frac{f''_{0,h/2}-f''_{0h}}3,$$
  ¤«п гв®з­Ґ­Ёп Ї® ђЁз а¤б®­г бЇа ўҐ¤«Ёў® а ўҐ­бвў®
$${f''_{0h}}^*=\frac{4f''_{0,h/2}-f''_{0h}}3.$$


\head \endhead
\head \endhead
\topmatter
\title ‹ЁвҐа вга \endtitle
\endtopmatter
\frenchspacing
\roster
\item"1." Ѓ еў «®ў Ќ.‘. {\it —Ёб«Ґ­­лҐ ¬Ґв®¤л}. Њ.: Ќ гЄ , 1973.
\item"2." Ѓ еў «®ў Ќ.‘., †Ё¤Є®ў Ќ.Џ., Љ®ЎҐ«мЄ®ў ѓ.Њ. {\it —Ёб«Ґ­­лҐ ¬Ґв®¤л}.
Њ.: Ќ гЄ , 1987.
\item"3." ЃҐаҐ§Ё­ €.‘., †Ё¤Є®ў Ќ.Џ. {\it ЊҐв®¤л ўлзЁб«Ґ­Ё©}. Њ.: Ќ гЄ , 1966.
’.1; ”Ё§¬ вЈЁ§, 1962. ’.2.
\item"4." ‚®«Є®ў ….Ђ. {\it —Ёб«Ґ­­лҐ ¬Ґв®¤л}. Њ.: Ќ гЄ , 1982.
\item"5." „Ґ¬Ё¤®ўЁз Ѓ.Џ., Њ а®­ €.Ђ. {\it Ћб­®ўл ўлзЁб«ЁвҐ«м­®© ¬ вҐ¬ вЁЄЁ}.
Њ.: Ќ гЄ , 1966.
\item"6." Љ «ЁвЄЁ­ Ќ.Ќ. {\it —Ёб«Ґ­­лҐ ¬Ґв®¤л}. Њ.: Ќ гЄ , 1978.
\item"7." ЋбЁЇҐ­Є® Љ.ћ. {\it  ЂЇЇа®ЄбЁ¬ жЁп дг­ЄжЁ© ¬­®Ј®з«Ґ­ ¬Ё Ё зЁб«Ґ­­®Ґ
¤ЁддҐаҐ­жЁа®ў ­ЁҐ}: ЊҐв®¤ЁзҐбЄЁҐ  гЄ § ­Ёп  Ї® Єгабг ``—Ёб«Ґ­­лҐ ¬Ґв®¤л'';
ЊѓЂ’“. Њ., 1994.
\endroster

\newpage
\topmatter
\title ЋЈ« ў«Ґ­ЁҐ\endtitle
\endtopmatter
\bigskip
\toc
\title ‚ўҐ¤Ґ­ЁҐ\page{\rm3}\endtitle
\head 1. ”®а¬г«  Їаאַ㣮«м­ЁЄ®ў\page{3}\endhead
\head 2. “б«®¦­Ґ­­ п д®а¬г«  Їаאַ㣮«м­ЁЄ®ў\page{4}\endhead
\head 3. €бЇ®«м§®ў ­ЁҐ Ё­вҐаЇ®«пжЁ®­­®Ј® ¬­®Ј®з«Ґ­  ‹ Ја ­¦  ¤«п Ї®бв஥­Ёп
Єў ¤а вга­ле д®а¬г«\page{5}\endhead
\head 4. Љў ¤а вга­лҐ д®а¬г«л Ќмов®­ --Љ®вҐб \page{7}\endhead
\head 5. ”®а¬г«  ва ЇҐжЁ©\page{9}\endhead
\head 6. ”®а¬г«  ‘Ё¬Їб®­ \page{10}\endhead
\head 7. ѓ« ў­ п з бвм Ї®ЈаҐи­®бвЁ Єў ¤а вга­ле д®а¬г«\page{13}\endhead
\head 8. Џа ўЁ«® ђг­ЈҐ Їа ЄвЁзҐбЄ®© ®жҐ­ЄЁ Ї®ЈаҐи­®бвЁ\page{15}\endhead
\head 9. “в®з­Ґ­ЁҐ ЇаЁЎ«Ё¦Ґ­­®Ј® аҐиҐ­Ёп Ї® ђЁз а¤б®­г\page{17}\endhead
\title ‹ЁвҐа вга \page{\rm19}\endtitle
\endtoc

\newpage
\footline={\hfil}
\
\vskip80truept
\head ЋбЁЇҐ­Є® Љ®­бв ­вЁ­ ћа쥢Ёз\endhead
\vskip60truept
\head Љў ¤а вга­лҐ д®а¬г«л\endhead
\bigskip
\bigskip
\head ЊҐв®¤ЁзҐбЄЁҐ гЄ § ­Ёп Ї® Єгабг ``—Ёб«Ґ­­лҐ ¬Ґв®¤л''\endhead
\vskip300truept
\leftline{\bf ђҐ¤ Єв®а Њ. Ђ. ‘®Є®«®ў }
\vskip30truept
\line{Џ®¤Ї. ў ЇҐз вм \phantom{ 77.77.7777}\hfil ЋЎкҐ¬ \phantom {77} Ї.«.
\hfil ’Ёа ¦ 75 нЄ§.\hfil ‡ Є § N \phantom{7777}}
\vskip2truept
\hrule
\vskip2truept
\leftline{ђ®в ЇаЁ­в ЊѓЂ’“. ЃҐа­ЁЄ®ўбЄ п ­ Ў., 14}

\enddocument