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4
Rauan Kaldybaev
Piecewise series solutions to ODEs
Rauan Kaldybaev
Posted
1 month ago
428 Views
|
1 Reply
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Piecewise series solutions to ODEs
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u
r
e
d
a
s
2
2
2
x
(
x
+
3
)
y
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+
7
x
(
x
+
3
)
y
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-
3
y
,
d
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e
s
n
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t
e
x
c
e
e
d
-
1
2
1
0
.
T
h
e
s
o
l
u
t
i
o
n
o
n
l
y
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k
6
9
m
i
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c
o
n
d
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t
o
c
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p
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e
.
O
u
t
[
]
=
A
p
p
r
o
x
i
m
a
t
e
s
o
l
u
t
i
o
n
t
o
-
3
y
[
x
]
+
7
x
(
3
+
x
)
′
y
[
x
]
+
2
2
x
2
(
3
+
x
)
′
′
y
[
x
]
=
0
T
h
e
f
u
n
c
t
i
o
n
s
p
r
e
s
e
n
t
e
d
i
n
t
h
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e
s
s
a
y
c
a
n
a
l
s
o
b
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e
d
t
o
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o
l
v
e
O
D
E
s
o
f
h
i
g
h
e
r
o
r
d
e
r
,
o
r
O
D
E
s
t
h
a
t
d
o
n
’
t
h
a
v
e
a
s
i
n
g
u
l
a
r
p
o
i
n
t
.
B
e
l
o
w
,
f
o
r
e
x
a
m
p
l
e
,
a
r
e
t
h
e
f
o
u
r
l
i
n
e
a
r
l
y
i
n
d
e
p
e
n
d
e
n
t
s
o
l
u
t
i
o
n
s
t
o
(
4
)
x
[
t
]
+
x
[
t
]
=
0
:
I
n
[
]
:
=
p
o
w
e
r
S
o
l
n
s
=
f
r
o
b
e
n
i
u
s
D
S
o
l
v
e
[
x
'
'
'
'
[
t
]
+
x
[
t
]
0
,
x
,
t
,
0
,
4
0
]
;
P
l
o
t
[
#
,
{
t
,
-
6
,
6
}
,
I
m
a
g
e
S
i
z
e
S
m
a
l
l
]
&
/
@
p
o
w
e
r
S
o
l
n
s
O
u
t
[
]
=
,
,
,
Introduction
G
i
v
e
n
a
l
i
n
e
a
r
h
o
m
o
g
e
n
o
u
s
O
D
E
(
o
r
d
i
n
a
r
y
d
i
f
f
e
r
e
n
t
i
a
l
e
q
u
a
t
i
o
n
)
w
i
t
h
p
o
l
y
n
o
m
i
a
l
c
o
e
f
f
i
c
i
e
n
t
s
,
w
e
c
a
n
a
l
w
a
y
s
w
r
i
t
e
i
t
i
n
t
h
e
f
o
r
m
M
∑
m
=
0
Z
∑
p
=
0
C
m
p
m
(
x
-
x
0
)
p
d
u
p
d
x
=
0
(
1
)
w
h
e
r
e
x
0
i
s
s
o
m
e
(
a
n
y
)
n
u
m
b
e
r
a
n
d
C
i
s
a
m
a
t
r
i
x
w
i
t
h
c
o
n
s
t
a
n
t
e
n
t
r
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s
.
T
o
s
o
l
v
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e
q
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a
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g
t
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m
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t
h
o
d
o
f
F
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b
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n
i
u
s
n
e
a
r
x
=
x
0
,
a
s
s
u
m
e
s
o
l
u
t
i
o
n
o
f
t
h
e
f
o
r
m
u
[
t
]
=
∞
∑
n
=
0
a
n
r
+
n
t
(
2
)
w
h
e
r
e
t
≡
x
-
x
0
.
S
u
b
s
t
i
t
u
t
i
n
g
t
h
i
s
i
n
t
o
t
h
e
e
q
u
a
t
i
o
n
(
1
)
y
i
e
l
d
s
0
=
∞
∑
k
=
-
Z
r
+
k
t
k
+
Z
∑
n
=
M
a
x
[
0
,
k
-
M
]
L
k
n
a
n
(
3
)
w
h
e
r
e
t
h
e
f
o
r
m
u
l
a
f
o
r
L
k
n
i
s
g
i
v
e
n
b
e
l
o
w
.
T
o
s
a
t
i
s
f
y
t
h
e
e
q
u
a
t
i
o
n
,
t
h
e
c
o
e
f
f
i
c
i
e
n
t
s
b
e
f
o
r
e
a
l
l
p
o
w
e
r
s
o
f
t
m
u
s
t
b
e
z
e
r
o
:
k
+
Z
∑
n
=
M
a
x
[
0
,
k
-
M
]
L
k
n
a
n
=
0
-
Z
≤
k
(
4
)
T
h
i
s
g
i
v
e
s
u
s
s
a
s
e
t
o
f
e
q
u
a
t
i
o
n
s
t
h
a
t
a
l
l
o
w
s
u
s
t
o
d
e
t
e
r
m
i
n
e
t
h
e
c
o
n
s
t
a
n
t
r
a
n
d
c
o
e
f
f
i
c
i
e
n
t
s
a
n
i
n
(
2
)
.
I
n
a
s
e
n
s
e
,
t
h
e
m
a
t
r
i
x
C
e
n
c
a
p
s
u
l
a
t
e
s
a
l
l
t
h
e
r
e
l
e
v
a
n
t
i
n
f
o
r
m
a
t
i
o
n
a
b
o
u
t
t
h
e
O
D
E
(
1
)
t
h
a
t
w
e
n
e
e
d
t
o
c
o
m
p
u
t
e
a
s
o
l
u
t
i
o
n
.
L
k
n
≡
M
i
n
[
Z
,
M
+
n
-
k
]
∑
p
=
M
a
x
[
0
,
n
-
k
]
C
k
-
n
+
p
,
p
n
Π
i
=
n
+
1
-
p
(
r
+
i
)
(
5
)
Code to extract the matrix
C
T
o
o
b
t
a
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a
F
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:
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[
]
:
=
C
l
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a
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A
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l
[
c
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a
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V
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T
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]
;
P
r
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e
c
t
[
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;
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{
i
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d
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b
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x
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}
;
c
h
a
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[
y
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[
x
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2
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y
[
x
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,
y
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x
,
0
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O
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t
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a
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f
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y
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[
t
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y
[
x
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2
y
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[
x
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i
n
t
o
a
l
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s
t
o
f
t
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m
s
{
y
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[
x
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,
y
[
x
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,
2
y
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[
x
]
}
.
T
h
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m
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a
n
i
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g
o
f
t
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a
t
y
'
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[
x
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+
y
[
x
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2
y
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[
x
]
0
.
I
n
[
]
:
=
C
l
e
a
r
A
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l
[
e
x
t
r
a
c
t
T
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r
m
s
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;
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x
t
r
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c
t
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[
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q
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q
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:
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p
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[
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,
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x
p
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[
1
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x
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[
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;
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x
t
r
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c
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s
[
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y
[
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y
[
t
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2
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[
x
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=
{
y
[
t
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,
2
′
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[
x
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,
′
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[
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E
a
c
h
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r
m
h
a
s
t
h
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f
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m
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[
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:
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[
c
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O
u
t
[
]
=
{
{
1
,
0
,
0
}
,
{
2
,
0
,
1
}
,
{
1
,
0
,
2
}
}
T
h
e
o
u
t
p
u
t
t
e
l
l
s
u
s
t
h
a
t
t
h
e
r
e
a
r
e
3
t
e
r
m
s
,
a
n
d
t
h
e
i
r
n
u
m
b
e
r
s
{
,
m
,
p
}
a
r
e
{
1
,
0
,
0
}
,
{
2
,
0
,
1
}
,
a
n
d
{
1
,
0
,
2
}
,
r
e
s
p
e
c
t
i
v
e
l
y
,
w
h
i
c
h
c
o
r
r
e
s
p
o
n
d
s
t
o
t
h
e
O
D
E
y
[
t
]
+
2
y
'
[
t
]
+
y
'
'
[
t
]
=
0
.
N
o
w
,
w
e
c
a
n
w
r
i
t
e
a
f
u
n
c
t
i
o
n
t
o
c
o
m
p
u
t
e
t
h
e
m
a
t
r
i
x
C
:
I
n
[
]
:
=
C
l
e
a
r
A
l
l
[
e
x
t
r
a
c
t
C
m
a
t
r
i
x
]
;
e
x
t
r
a
c
t
C
m
a
t
r
i
x
[
e
q
n
_
E
q
u
a
l
,
d
e
p
v
a
r
_
,
i
n
d
v
a
r
_
,
x
0
_
]
:
=
B
l
o
c
k
[
t
e
r
m
s
,
m
p
l
i
s
t
,
C
m
a
t
r
i
x
,
M
,
Z
,
t
e
r
m
s
=
e
x
t
r
a
c
t
T
e
r
m
s
@
c
h
a
n
g
e
V
a
r
s
T
o
T
[
e
q
n
,
d
e
p
v
a
r
,
i
n
d
v
a
r
,
x
0
]
;
m
p
l
i
s
t
=
m
p
f
o
r
m
[
#
,
d
e
p
v
a
r
]
&
/
@
t
e
r
m
s
;
I
f
[
M
e
m
b
e
r
Q
[
m
p
l
i
s
t
,
$
F
a
i
l
e
d
]
,
{
$
F
a
i
l
e
d
,
$
F
a
i
l
e
d
,
$
F
a
i
l
e
d
}
,
M
=
M
a
x
[
m
p
l
i
s
t
[
[
A
l
l
,
2
]
]
]
;
Z
=
M
a
x
[
m
p
l
i
s
t
[
[
A
l
l
,
3
]
]
]
;
C
m
a
t
r
i
x
=
C
o
n
s
t
a
n
t
A
r
r
a
y
[
0
,
{
1
+
M
,
1
+
Z
}
]
;
(
C
m
a
t
r
i
x
[
[
1
+
#
[
[
2
]
]
,
1
+
#
[
[
3
]
]
]
]
+
=
#
[
[
1
]
]
)
&
/
@
m
p
l
i
s
t
;
{
M
,
Z
,
C
m
a
t
r
i
x
}
]
]
;
H
e
r
e
i
s
w
h
a
t
t
h
e
f
u
n
c
t
i
o
n
o
u
t
p
u
t
s
f
o
r
t
h
e
e
q
u
a
t
i
o
n
3
x
y
'
'
[
x
]
+
y
[
x
]
0
.
T
h
e
o
u
t
p
u
t
i
s
o
f
t
h
e
f
o
r
m
M
,
P
,
C
,
w
h
e
r
e
M
,
P
a
r
e
t
h
e
d
i
m
e
n
s
i
o
n
s
o
f
t
h
e
r
e
c
t
a
n
g
u
l
a
r
M
P
m
a
t
r
i
x
C
:
e
x
t
r
a
c
t
C
m
a
t
r
i
x
[
3
x
y
'
'
[
x
]
+
y
[
x
]
0
,
y
,
x
,
0
]
O
u
t
[
]
=
{
1
,
2
,
{
{
1
,
0
,
0
}
,
{
0
,
0
,
3
}
}
}
M
a
t
r
i
x
F
o
r
m
[
{
{
1
,
0
,
0
}
,
{
0
,
0
,
3
}
}
]
O
u
t
[
]
/
/
M
a
t
r
i
x
F
o
r
m
=
1
0
0
0
0
3
T
h
e
m
a
t
r
i
x
h
a
s
t
w
o
n
o
n
z
e
r
o
e
n
t
r
i
e
s
.
T
h
e
f
i
r
s
t
o
n
e
,
C
0
0
,
c
o
r
r
e
s
p
o
n
d
s
t
o
t
h
e
t
e
r
m
y
[
x
]
,
a
n
d
t
h
e
o
t
h
e
r
o
n
e
,
C
1
2
,
c
o
r
r
e
s
p
o
n
d
s
t
o
3
x
y
'
'
[
x
]
.
Formula for the Frobenius solution
The recursion relation
F
o
r
M
≤
k
,
t
h
e
e
q
u
a
t
i
o
n
(
4
)
b
e
c
o
m
e
s
k
+
Z
∑
n
=
k
-
M
L
k
n
a
n
=
0
(
6
)
G
i
v
e
n
t
h
a
t
L
k
n
w
o
u
l
d
o
f
t
e
n
b
e
z
e
r
o
f
o
r
n
=
k
+
Z
o
r
o
t
h
e
r
h
i
g
h
v
a
l
u
e
s
,
w
e
c
a
n
f
u
r
t
h
e
r
s
i
m
p
l
i
f
y
t
h
e
a
b
o
v
e
e
q
u
a
t
i
o
n
b
y
w
r
i
t
i
n
g
k
-
M
+
R
∑
n
=
k
-
M
L
k
n
a
n
=
0
w
h
e
r
e
R
≤
Z
+
M
i
s
t
h
e
h
i
g
h
e
s
t
n
u
m
b
e
r
u
n
d
e
r
t
h
e
c
o
n
d
i
t
i
o
n
t
h
a
t
L
k
n
i
s
n
o
n
z
e
r
o
f
o
r
t
h
e
u
p
p
e
r
l
i
m
i
t
o
f
s
u
m
m
a
t
i
o
n
n
=
k
-
M
+
R
.
F
r
o
m
h
e
r
e
,
w
e
o
b
t
a
i
n
t
h
e
r
e
c
u
r
s
i
o
n
r
e
l
a
t
i
o
n
t
h
a
t
t
e
l
l
s
u
s
a
n
i
n
t
e
r
m
s
o
f
t
h
e
p
r
e
v
i
o
u
s
c
o
e
f
f
i
c
i
e
n
t
s
:
a
n
=
-
1
L
n
+
M
-
R
,
n
n
-
1
∑
k
=
n
-
R
L
n
+
M
-
R
,
k
a
k
(
7
)
W
e
d
o
n
’
t
h
a
v
e
t
o
w
o
r
r
y
a
b
o
u
t
L
n
+
M
-
R
,
n
b
e
i
n
g
i
d
e
n
t
i
c
a
l
l
y
z
e
r
o
a
n
y
m
o
r
e
.
N
o
w
,
w
e
n
e
e
d
t
o
a
s
k
t
h
e
q
u
e
s
t
i
o
n
o
f
h
o
w
t
o
f
i
n
d
R
.
T
h
e
n
u
m
b
e
r
R
i
s
d
e
f
i
n
e
d
b
y
t
h
e
c
o
n
d
i
t
i
o
n
t
h
a
t
L
k
+
M
-
R
,
k
i
s
n
o
n
z
e
r
o
.
T
h
e
f
o
r
m
u
l
a
(
5
)
t
e
l
l
s
u
s
t
h
a
t
L
k
+
M
-
R
,
k
=
M
i
n
[
Z
,
R
]
∑
p
=
M
a
x
[
0
,
R
-
M
]
C
M
-
R
+
p
,
p
k
Π
i
=
k
+
1
-
p
(
r
+
i
)
I
n
o
r
d
e
r
f
o
r
t
h
i
s
e
x
p
r
e
s
s
i
o
n
t
o
b
e
z
e
r
o
i
n
d
e
p
e
n
d
e
n
t
l
y
o
f
r
,
a
l
l
c
o
n
s
t
a
n
t
s
C
m
u
s
t
b
e
z
e
r
o
.
T
h
e
n
,
R
i
s
t
h
e
h
i
g
h
e
s
t
n
u
m
b
e
r
s
u
c
h
t
h
a
t
a
t
l
e
a
s
t
o
n
e
o
f
t
h
e
c
o
e
f
f
i
c
i
e
n
t
s
C
M
-
R
+
p
,
p
i
s
n
o
n
z
e
r
o
,
w
h
e
r
e
p
g
o
e
s
f
r
o
m
M
a
x
[
0
,
R
-
M
]
t
o
M
i
n
[
Z
,
R
]
.
O
n
e
w
a
y
t
o
c
o
m
p
u
t
e
R
i
s
t
h
e
f
o
l
l
o
w
i
n
g
.
S
t
e
p
1
:
s
t
a
r
t
w
i
t
h
R
=
Z
+
M
.
S
t
e
p
2
:
c
h
e
c
k
i
f
t
h
e
C
’
s
a
r
e
a
l
l
z
e
r
o
.
S
t
e
p
3
:
I
f
t
r
u
e
c
h
a
n
g
e
R
t
o
R
-
1
a
n
d
r
e
t
u
r
n
t
o
s
t
e
p
2
;
o
t
h
e
r
w
i
s
e
,
w
h
i
c
h
i
s
i
f
a
n
y
o
f
t
h
e
C
’
s
i
s
n
o
n
z
e
r
o
,
s
t
o
p
.
B
e
c
a
u
s
e
w
e
s
t
a
r
t
f
r
o
m
t
h
e
h
i
g
h
e
s
t
p
o
s
s
i
b
l
e
v
a
l
u
e
o
f
R
a
n
d
g
o
d
o
w
n
i
n
s
t
e
p
s
o
f
1
,
t
h
e
a
l
g
o
r
i
t
h
m
i
s
g
o
i
n
g
t
o
r
e
t
u
r
n
t
h
e
c
o
r
r
e
c
t
R
a
n
d
n
o
t
s
o
m
e
o
t
h
e
r
R
w
r
o
n
g
t
h
a
t
p
a
s
s
e
s
t
h
e
c
r
i
t
e
r
i
o
n
L
k
+
M
-
R
,
k
≠
0
b
u
t
i
s
n
’
t
t
h
e
h
i
g
h
e
s
t
n
u
m
b
e
r
t
o
d
o
s
o
.
I
n
[
]
:
=
C
l
e
a
r
A
l
l
[
c
o
m
p
u
t
e
L
s
m
a
t
r
i
x
]
;
c
o
m
p
u
t
e
L
s
m
a
t
r
i
x
[
M
_
,
Z
_
,
C
m
a
t
r
i
x
_
]
:
=
T
a
b
l
e
[
S
u
m
[
C
m
a
t
r
i
x
[
[
1
+
k
-
n
+
p
,
1
+
p
]
]
P
r
o
d
u
c
t
[
r
+
i
,
{
i
,
n
+
1
-
p
,
n
}
]
,
{
p
,
M
a
x
[
0
,
n
-
k
]
,
M
i
n
[
Z
,
M
+
n
-
k
]
}
]
,
{
k
,
-
Z
,
M
-
1
}
,
{
n
,
0
,
Z
+
M
-
1
}
]
I
n
[
]
:
=
C
l
e
a
r
A
l
l
[
f
i
n
d
R
e
c
u
r
s
i
o
n
R
e
l
n
]
;
f
i
n
d
R
e
c
u
r
s
i
o
n
R
e
l
n
[
M
_
,
Z
_
,
C
m
a
t
r
i
x
_
,
L
_
]
:
=
B
l
o
c
k
{
R
}
,
R
=
Z
+
M
;
W
h
i
l
e
[
A
l
l
T
r
u
e
[
(
C
m
a
t
r
i
x
[
[
1
+
M
-
R
+
#
,
1
+
#
]
]
0
)
&
/
@
R
a
n
g
e
[
M
a
x
[
0
,
R
-
M
]
,
M
i
n
[
Z
,
R
]
]
,
#
&
]
,
R
-
-
;
]
;
R
,
a
[
n
]
1
L
[
n
+
M
-
R
,
n
]
S
u
m
[
E
x
p
a
n
d
[
-
L
[
n
+
M
-
R
,
k
]
]
a
[
k
]
,
{
k
,
n
-
R
,
n
-
1
}
]
;
The indicial equation
F
o
r
-
Z
≤
k
<
M
,
t
h
e
e
q
u
a
t
i
o
n
(
4
)
c
a
n
b
e
w
r
i
t
t
e
n
a
s
k
+
Z
∑
n
=
0
L
k
n
a
n
=
0
-
Z
≤
k
<
M
(
8
)
I
f
w
e
u
s
e
t
h
e
f
a
c
t
t
h
a
t
L
k
n
=
0
f
o
r
n
>
k
,
w
e
c
a
n
r
e
a
d
(
8
)
a
s
a
m
a
t
r
i
x
e
q
u
a
t
i
o
n
.
T
h
e
Z
+
M
d
i
m
e
n
s
i
o
n
a
l
v
e
c
t
o
r
a
l
i
e
s
i
n
t
h
e
n
u
l
l
s
p
a
c
e
o
f
t
h
e
(
Z
+
M
)
(
Z
+
M
)
m
a
t
r
i
x
L
s
,
w
h
e
r
e
L
s
i
s
a
“
s
l
i
c
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