Воеводин Вл. В. Численные методы, параллельные вычисления и информационные технологии
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(m, n)
u(x) =
m
X
i=0
u
i
x
i
, v(x) =
n
X
i=0
v
i
x
i
f(x)v(x) − u(x) = O(x
m+n+1
)
v(0) = 1.
f(x) −
u(x)
v(x)
= O(x
m+n+1
).
n
X
j=0
a
i−j
v
j
= 0, m + 1 6 i 6 m + n,
a
m+1
a
m
... a
m−n+1
a
m+2
a
m+1
... a
m−n
... ... ... ...
a
m+n
a
m+n−1
... a
m
v
0
v
1
...
v
n
= 0.
v
0
= 1
a
m
... a
m−n+1
... ... ...
a
m+n−1
... a
m
v
1
...
v
n
= −
a
m+1
...
a
m+n
.
m = n = 3
a
3
a
2
a
1
a
4
a
3
a
2
a
5
a
4
a
3
v
1
v
2
v
3
= −
a
4
a
5
a
6
.
a
1
a
2
a
3
a
2
a
3
a
4
a
3
a
4
a
5
v
3
v
2
v
1
= −
a
4
a
5
a
6
.
a
m−n+1
... a
m
... ... ...
a
m
... a
m+n−1
v
n
...
v
1
= −
a
m+1
...
a
m+n
,
A
mn
v
n
= −a
mn
.
A
mn
f(x)
a
m
n i < 0
a
i
= 0 a
mn
[A
mn
, a
mn
]
(m, n)
a
mn
∈ A
mn
,
A
mn
= [A
mn
, a
mn
].
A = [a
i+j−1
], 1 6 i, j < ∞,
A
k
k
n
1
< n
2
< ...
A
−1
n
k
A
−1
n
k+1
p =
n
k
q = n
k+1
A
p
a
1
... a
p
... ... ...
a
p
... a
2p−1
s
1
...
s
p
=
a
p+1
...
a
2p
p
p
p+1
s
1
, ... , s
p
p + 1
p+1
r(p) > p
p + 1
n
k+1
= r(n
k
).
p = n
k
r = r(p)
a
1
... a
p
a
p+1
... ... ... ...
a
p
... a
2p−1
a
2p
... ... ... ...
a
r−1
... a
r+p−2
a
r+p−1
a
r
... a
r+p−1
a
r+p
−s
1
...
−s
p
1
=
0
...
0
...
0
γ
, γ 6= 0.
A
r
−s
1
1
... −s
1
1
... ... ...
...
−s
p
... ...
1
1 −s
p
...
1 ... ...
... ...
... −s
p
1
=
=
a
1
a
2
... a
p
a
2
a
3
... a
p+1
... ... ... ...
a
p
a
p+1
... a
2p−1
γ a
p+1
a
p+2
... a
2p
... ...
... ... ... ...
γ ... ...
... ... ... ...
γ ... ... ...
a
r
a
r+1
... a
r+p−1
.
p = n
k
A
p
A
r
γ 6= 0
n
k
6 n 6 n
k+1
A
n
= {n − n
k
, n
k+1
− n}.
A
n
{n − n
k
, n
k+1
− n}
^
A
n
A
n
^
A
n
=
a
1
... a
n
a
n+1
... ... ... ...
a
n
... a
2n−1
a
2n
.
n
k
6 n < n
k+1
A
n
=
^
A
n
n − n
k
< n
k+1
− n.
^
A
n
p = n
k
q = n
k+1
p 6 n = p + i < q p + 1
A
q−1
s
1
, ..., s
p
p+1
A
p+1
p + 2i < q,
A
p+2i
p + 1 + i
^
A
p+i
p s
1
, ..., s
p
p + 2i < q n − p < q − n
^
A
n
A
n
A
n
A
n+1
n − p > q − p.
A
n
= n − {n − p, q − n} = 2n − q,
A
n+1
= n + 1 − {n + 1 − p, q − n − 1} = 2n − q + 2.
A
n+1
= A
n
+ 2,
n
k
6 n < n
k+1
A
n+1
− A
n
6 1.
A n A
ij
n−1 i j A
ik;jl
n − 2 A
i k
j l
i < k j < l
A A
ik;jl
= A
ij
A
kl
− A
il
A
kj
.
i = j = n − 1 k = l = n B = A
ik;jl
A
A =
B v q
u c d
p g h
.
B
u p B
I 0 0
−uB
−1
1 0
−pB
−1
0 1
B v q
u c d
p g h
=
B v q
0 c
1
d
1
0 g
1
h
1
,
h
1
= h−pB
−1
q, c
1
= c−uB
−1
v, g
1
= g−pB
−1
v, d
1
= d−uB
−1
q.
A
ij
A
kl
− A
il
A
kj
= ( B)
2
(h
1
c
1
−g
1
d
1
) = B A.
B
B B
ε
B
ε
B
A =
[a
i+j
]
f(x) =
∞
P
i=0
a
i
x
i
i < 0 a
i
= 0 A
mn
n
a
m
C = [c
mn
] A
c
mn
= A
mn
, 0 6 m, n < ∞.
c
m0
= 1
c
m,n+1
c
m,n−1
= c
m+1,n
c
m−1,n
− c
2
mn
.
A
m,n+1
C A
C
A
a
0
6= 0 c
0n
6= 0
n > 1
c
m0
= 1 m > 0.
C
c
m,n−1
= c
m+1,n
= 0
c
m,n+1
= c
m+1,n
= 0 c
mn
= 0
0 ∗
∗ 0
,
∗ 0
0 ∗
c
m0
6= 0
c
0n
6= 0 C
c
mn
6= 0
c
m+r,n+r
6= 0
c
m+r,n+r−1
= 0
c
m+r−1,n+r
= 0
A
mn
A
m+r,n+r
A
m+r,n+r
A
m+i,n+i
0 < i < r
n + 1 A
m+r,n+r
c
m+1,n
6= 0 c
m+r+1,n+r
6= 0
A
m+1,n
A
m+1+r,n+r
n + 2
A
m+r,n+r
n+1 n+2
A
m+r,n+r
c
m+r,n+r
= A
m+r,n+r
= 0
c
m+r,n+r−1
6= 0
c
m+r,n+r+1
6= 0
c
m+r,n+r
= 0 A
m,n+1
A
m+r,n+r+1
n + 2
A
m+r,n+r+1
n + 2
A
m+r,n+r
n+1
A
m+r,n+r
c
m+r−1,n+r
6= 0
c
m+r,n+r
6= 0, c
m+r,n+r−1
6= 0, c
m+r−1,n+r
6= 0.
c
m+r,n+r
c
mn
c
m+r,n+r
6= 0 r > 0
(m, n)
f(x)
c
mn
= A
mn
6= 0
c
mn
= 0
c
mn
= 0
C c
kl
6=
0
c
k+r,l+r
(m, n) k+l <
m + n < k + l + r
c
st
c
s+p,t+p
h > 0 m = s+h
n = t + h A
st
A
s+p,t+p
A
s+i,t+i
0 < i < p
(t + h) − t < (t + p) − (t + h) ⇔ h < p/2.