Лабораторные работы по численным методам. Пак Т.В. - 8 стр.

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R
n,k
(x)
•
min R
n,k
< R
n,k
(x) < max R
n,k
R
n,k
(x
m
) = L
(k)
n
(x
m
) − f
(k)
(x
m
).
b
R
a
f (x)dx
I
6m
= 0, 3h(y
0
+ 5y
1
+ y
2
+ 6y
3
+ y
4
+ 5y
5
+ 2y
6
+ . . .
+5y
6m−5
+ y
6m−4
+ 6y
6m−3
+ y
6m−2
+ 5y
6m−1
+ y
6m
);
I
n
= (b − a)
n
P
k=0
c
k
f (x
k
)
n = 1, c
k
= (1/2, 1/2);
n = 2, c
k
= (1/6, 4/6, 1/6);
n = 3, c
k
= (1/8, 3/8, 3/8, 1/8);
n = 4, c
k
= (7/90, 32/90, 12/90, 32/90, 7/90);
n = 5, c
k
= (19/288, 75/288, 50/288, 50/288, 75/288, 19/288);
n = 6, c
k
= (41/840, 216/840, 27/840, 272/840, 27/840, 216/840, 41/840);
I
n
=
(b−a)
2
n
P
k=1
c
k
f (x
k
) x
k
=
b + a
2
+
b − a
2
t
k
n = 1, t
1
= 0, c
1
= 2;
n = 2, t
1,2
= ±0, 577350, c
1
= c
2
= 1;
ïîëó÷èòü è îöåíèòü ìèíèìàëüíîå è ìàêñèìàëüíîå çíà÷åíèÿ îñòàòî÷íîãî
÷ëåíà Rn,k (x).
• Ïðîâåðèòü, âûïîëíÿåòñÿ ëè íåðàâåíñòâî
                                               (k)
min Rn,k < Rn,k (x) < max Rn,k , Rn,k (xm ) = Ln (xm ) − f (k) (xm ).
   Òàáëèöà 2.
             ê=2                        ê=1
             n =3    n=4 n=5            n =3     n=4 n=5 n=6
    m=0      16      20     25          1        5      10         31
    m=1      17      21     26          2        6      11         32
    m=2      18      22     27          3        7      12         33
    m=3      19      23     28          4        8      13         34
    m=4              24     29                   9      14         35
    m=5                     30                          15         36
    m=6                                                            37


Òåìà 4. Êâàäðàòóðíûå ôîðìóëû
                       Rb
Âû÷èñëèòü èíòåãðàë          f (x)dx
                       a
1.   ïî
     ñîñòàâíîé ôîðìóëå ëåâûõ ïðÿìîóãîëüíèêîâ;
2.   ïî
     ñîñòàâíîé ôîðìóëå ïðàâûõ ïðÿìîóãîëüíèêîâ;
3.   ïî
     ñîñòàâíîé ôîðìóëå öåíòðàëüíûõ ïðÿìîóãîëüíèêîâ;
4.   ïî
     ôîðìóëå òðàïåöèè;
5.   ïî
     ôîðìóëå Ñèìïñîíà;
6.   ïî
     ôîðìóëå Âåääëÿ:
           I6m = 0, 3h(y0 + 5y1 + y2 + 6y3 + y4 + 5y5 + 2y6 + . . .
          +5y6m−5 + y6m−4 + 6y6m−3 + y6m−2 + 5y6m−1 + y6m );
                                             P
                                             n
ïî ôîðìóëå Íüþòîíà - Êîòåñà: In = (b − a)       ck f (xk )
                                                       k=0
7. n = 1, ck = (1/2, 1/2);
8. n = 2, ck = (1/6, 4/6, 1/6);
9. n = 3, ck = (1/8, 3/8, 3/8, 1/8);
10. n = 4, ck = (7/90, 32/90, 12/90, 32/90, 7/90);
11. n = 5, ck = (19/288, 75/288, 50/288, 50/288, 75/288, 19/288);
12. n = 6, ck = (41/840, 216/840, 27/840, 272/840, 27/840, 216/840, 41/840);

                              (b−a)   P
                                      n                            b+a b−a
ïî ôîðìóëå Ãàóññà: In =         2           ck f (xk ), ãäå xk =      +    tk
                                      k=1                           2   2
13. n = 1,   t1 = 0, c1 = 2;
14. n = 2,   t1,2 = ±0, 577350,       c1 = c2 = 1;

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