Математическая обработка экспериментальных данных нейтронного рассеяния в физике низких энергий. Злоказов В.Б. - 41 стр.

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МГУ | Москва

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g(1) = f(1), g(m) = f(m)
f(i) = c(i)f(i + 1) + h(i), c(1) = 0, h(1) = y(0), i = 1, ..., m − 1,
c(i), h(i), i > 1 f(i)
f(i + 1) − 2g(i) + c(i − 1)f(i) + h(i − 1) − βw(i)f(i) = −βw(i)y(i).
c(i), h(i)
c(i) = 1/(2 + βw(i) − c(i − 1)), h(i) = c(i)(βw(i)f(i) + h(i − 1)),
c(1) = 0, h(1) = y(1)
f(i)
y(i)
f(i)
m−1
X
i=2
f
00
(i)
2
e(i) + α
m
X
i=1
w(i)(y(i) − f(i))
2
f(1) = y(1), f(m) = y(m), f
00
(1) = 0, f
00
(m) = 0.
F (f
00
(t)) = f
00
(t)
2
e(t) e(t) w(t)
e(i) = 1/(1 + y
0
(i)
2
) w(i) = 1/σ(i)
2
w(i) = 1
e(i + 1)d
00
(i + 1) − 2e(i)d
00
(i) + e(i − 1)d
00
(i − 1) + α · w(i)f(i) = α · w(i)y(i).
d(i) = f
00
(i)
f(1) = y(1); f(m) = y(m); f
00
(1) = 0; f
00
(m) = 0;
P (i) = A
i
· P (i + 1) + B(i)
P (i) f(i), f
00
(i)) A
i
2·2 B(i)
A
i
(1, 1) = u
i
(1, 1); A
i
(1, 2) = u
i
(1, 2);
A
i
(2, 1) = u
i
(2, 1); A
i
(2, 2) = u
i
(2, 2);
A
1
(1, 1) = A
1
(1, 2) = A
1
(2, 1) = A
1
(2, 2) = 0;
B
1
(i) = u
i
(1, 1)B
1
(i − 1) − u
i
(1, 2)(α · w(i)f(i) − e(i − 1)B
2
(i − 1))/e(i + 1);
ïðè g(1) = f (1), g(m) = f (m).
Ýòè óðàâíåíèÿ ìîãóò áûòü ðåøåíû óñòîé÷èâûì ñïîñîáîì ñëåäóþùèì îáðàçîì.
Ïðåäñòàâèì

            f (i) = c(i)f (i + 1) + h(i),       c(1) = 0, h(1) = y(0), i = 1, ..., m − 1,

ãäå êîýôôèöèåíòû c(i), h(i), i > 1 ÿâëÿþòñÿ ïîêà íåèçâåñòíûìè. Ïîäñòàâèâ f (i) â
óðàâíåíèå Ýéëåðà, ìû ïîëó÷èì

           f (i + 1) − 2g(i) + c(i − 1)f (i) + h(i − 1) − βw(i)f (i) = −βw(i)y(i).

Îòñþäà ìû ïîëó÷àåì ôîðìóëó äëÿ âû÷èñëåíèÿ êîýôôèöèåíòîâ c(i), h(i):

            c(i) = 1/(2 + βw(i) − c(i − 1)), h(i) = c(i)(βw(i)f (i) + h(i − 1)),

ïðè íà÷àëüíûõ c(1) = 0, h(1) =                          y(1), èñïîëüçóÿ êîòîðûå, ìû íàõîäèì
ïîñëåäîâàòåëüíî íóæíîå ðåøåíèå f (i).

    Ñëåäóþùàÿ âàæíàÿ çàäà÷à - ñãëàæèâàíèå ôóíêöèé. Äëÿ çàäàííîé ôóíêöèè y(i)
åå ñãëàæåííûé àíàëîã f (i) íàõîäèòñÿ èç óñëîâèÿ ìèíèìóìà âûðàæåíèÿ
                            m−1
                            X                          m
                                                       X
                                  f 00 (i)2 e(i) + α         w(i)(y(i) − f (i))2               (21)
                            i=2                        i=1

ïðè ãðàíè÷íûõ óñëîâèÿõ f (1) = y(1), f (m) = y(m), f 00 (1) = 0, f 00 (m) = 0.
Ò.å. â äàííîì ñëó÷àå êðèòåðèé F (f 00 (t)) = f 00 (t)2 e(t); e(t) è w(t) - òåì èëè èíûì
ñïîñîáîì âûáðàííûå âåñà. Íàïðèìåð, e(i) = 1/(1 + y 0 (i)2 ), à w(i) = 1/σ(i)2 , èëè
w(i) = 1.
Óðàâíåíèå Ýéëåðà äëÿ ìèíèìèçàöèè (21) âûãëÿäèò ñëîæíåå, òàê êàê ïðèâîäèò
ê 5-òî÷å÷íûì ðàçíîñòíûì ñõåìàì. Äëÿ åãî ðåøåíèÿ ïðèõîäèòñÿ ïðèáåãàòü óæå ê
ìàòðè÷íîé 'ïðîãîíêå'. Óðàâíåíèÿ äëÿ ìèíèìèçàöèè (21) âûãëÿäÿò òàê

    e(i + 1)d00 (i + 1) − 2e(i)d00 (i) + e(i − 1)d00 (i − 1) + α · w(i)f (i) = α · w(i)y(i).   (22)

ãäå d(i) = f 00 (i). Ãðàíè÷íûå óñëîâèÿ èìåþò âèä

                     f (1) = y(1); f (m) = y(m); f 00 (1) = 0; f 00 (m) = 0;

Ïîñòðîèì ôîðìóëû äëÿ ìàòðè÷íîãî àíàëîãà ìåòîäà 'ïðîãîíêè' äëÿ ðåøåíèÿ ýòèõ
óðàâíåíèé. Ìû èìååì
                        P (i) = Ai · P (i + 1) + B(i)                  (23)
ãäå P (i) ÿâëÿåòñÿ âåêòîðîì (f (i), f 00 (i)), à Ai - ìàòðèöåé ðàçìåðà 2·2, è B(i) - âåêòîðîì
ðàçìåðà 2. Ìû èìååì

                            Ai (1, 1) = ui (1, 1); Ai (1, 2) = ui (1, 2);

                            Ai (2, 1) = ui (2, 1); Ai (2, 2) = ui (2, 2);
                        A1 (1, 1) = A1 (1, 2) = A1 (2, 1) = A1 (2, 2) = 0;
      B1 (i) = ui (1, 1)B1 (i − 1) − ui (1, 2)(α · w(i)f (i) − e(i − 1)B2 (i − 1))/e(i + 1);

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