.

681.3
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Abstract. The way of a parallel implementation of the least squares method with information processing at the level of complex data structures in the processing elements including a scalar multiplier is proposed and described. It is solving the problems of mathematical physics in a single computational (technology) flow oriented method. A description is considered on the basis of specific example in comparison with the known solution. It is useful for specialists of mathematics and developers of methods and structures of special processor for solving problems of mathematical physics and other problems.
Keywords: polynomial, integration and differentiation of splines, discrepancy, parallel processing, complex data structures, scalar multiplier, single computational (technology) flow.
1.
, () . (), . (GPU) (CPU), nVIDIA AMD CUDA STREAM, .
[1] () , ( ) () () Ax = b, . , , (). (),
.., 2012
ISSN 1028-9763. , 2012, 4
. (), , , () [], (). . . () , , .
, , () , (), (), (), (). , , [1], .
[2, 3] , , , , .
. . ,
.
2.
[4] :
e = L(u) - p, (1)
F =<,>=< L(u) - p, L(u) - p >. (2)

u=Z (3)
F a.:
dF/da. = 0, i = 1,2,...,N. (4)

dF / = / . <,>=/ da {< L(Z af X L(Z akfk) > -2 < L(Z akk), P) > + < A P >b (5)
, L - :
2 < L(ZaKfk),() > -2 < L((pi),p >= 0 (6)

< L(Z aKfK-p), ,) >= 0. (7)
, [4], (), [1] .

L(u) - p = 2u / dx2 + u + x = 0. (8)

= ( -2)1 + (2 -3)2, (9)

= + (-2 + -2)1 + (2 -6 + 2 -3)2. (10)
1, 2,

| (-2 + - 2)^ = [ ]1 = 0, (11)
0
1
|(2 -6 + 2 - 3)^ = []2 = 0. (11)
0
(11), (11 )
2.
202 101 101 1532
}-ї (12)

1 = 0,192, 2 = 0,165. (13)
:
1
| (-2 + - 2) = 0,
0
1
+ , (-2 + - 2) + (2 - 6 + 2 - 3))(-2 + - 2'
|( + 1 (-2 + -2) + 2(2-6 + 2 -3))(-2 + -2 )^ =
0
= ((-2 + 2 - 3) + 1(4 - 4 + 52 - 23 + 4) + 2(-4 + 14 - 102 + 93 - 24 + 5)) = =|(-2 + 3/3 - 4/4) + 1(4- 22 + 5/33 4/2 + 5/5) + 2(-4 + 72 - 103/3+
+ 94/4 - 25 /5 + /6)||1 = (-1 +1/3 -1/4) + (4 - 2 + 5/3 -1/2 +1/5) +
+ 2 (-4 + 7 -10/3 + 9/4 - 2/5 +1/6) = - 55/60 + 1202/60 + 2(101/60)=0, (14)
202 + 1012 = 55.
(11 ) :
1 1
|(2 -6 + 2 -) = |(2 -62 + 3 -4) + 1(-4 +14 - 102 + 93 -24 + 5 ) +
0
+ 2(4-24 + 402 -163 + 134 -25 + 6^ ==-19/2 + 1101/60 + 2393/105=0, (15) 01/60 + 2393/105 = 19/2.
(14) (15) :
0
1( ) =(-2 + - 2),
/2() = (2-6 + 2 -3), /3 () = (-2 + 2 - 3), /4() = (2 - 62 + 3 - 4)

|(/ () + 1/1()/1() + 2/1()/())^ = 0 ,
0
1
| (/ () + / ()/ 2 () + 2/2 ()/2 () = 0 .
.
(16)-(19) :
() * [1] =

/1 () * /1 ( ) = [,] * [*] = [,] * '() *
"- 2 "- 2' " 4'
1 -2 1 - 4
-1 1 -2 * -1 = 5
0 -1 1 -2 0 - 2
_ 0 0 -1 1 - 2. _ 0. 1






/ () * /*() *
- 2
1 -2
-1 1 -2
0 -1 1 -2
0 0 -1 1
0 0 0 -1
2
1
2
" 2' ' - 4' 0
- 6 14 1
1 = -10 2
-1 9 3
0 - 2 4
_ 0. _ 1^ 5
(16)
(17)
(18)
(19)
(20) (21)
"- 2 1 -1 0 0' "- 2 "- 2'
- 2 1 -1 0 1 -2 1
- 2 1 -1 * [/ = -1 1 -2 * /1() = -1
-2 1 0 -1 1 - 2 0
- 2 _ 0 0 -1 1 - 2. _ 0.





(22)
(23)
(24)
0
>
/2 * /2 (-* ) * [ 2) *
2 2 4 0
- 6 2 - 6 - 24 1
1 -6 2 1 40 2
* = * 3
-1 1 -6 2 -1 -16
0 -1 1 -6 2 0 13 4
0 0 -1 1 -6 2 0 - 2 ^
0 0 0 -1 1 - 6 2 0 1 6
* /2*
(25)
. , () 1, 2, ..
= 1 + 2 +1.
, , . []
[ ] =
0 0 0 0 0
1 0 0 0 0
0 1/2 0 0 0
0 0 1/3 0 0
0 0 0 1/4 0
0 0 0 0 1/5
.
(26)


|/1 () = ^ |[1 ] = [] * [*] -
'0 0 0 0 0 ..." ' 4"
1 0 0 0 0 ... - 4
0 1/2 0 0 0 ... * 5
0 0 1/3 0 0 ... - 2
0 0 0 1/4 0 ... 1
0 0 0 0 1/5 ..._ _ 0_
=
0
-
0
4 1 2
-2 *
*
5/3 3
-2/4 4
_ 1/5. 5

= /1* () ^ [1]1 1 ^ 1[] *[4 - 4 + 52 - 23 + 4] {} =
= 1 []*[4-4 5-21] {} = [0 4-2 5/3 -2/4 1/5] {} |0
(27)
0
0
1
0
.
[ * (*)* /2 ().
1
[2]1 = |2 [1]2 = 2[]*[*]2 = 2[][-4 14 -10/3 9/4 -2/5 1/6]{} =
= 2[0 - 47 -10/39/4 - 2/51/6] = 2
0
-4
7
-10/3
9/4
-2/5
1/6







* /1*( )
(28)
[3 ] * /3* () :
[3]1 = |[3 ] = [] [3] = [] * [-2 1 -1] {} =[0 0 -1 1/3 -1/4] {}
1
* /3*().
" 0' 0
0 1
-1 2
1/3 3
_-1/4 4
(29)
0
(11) :
1 1
|/ () = [4]2 = |[4 ] = [] * [4] * [] * [0 2 -6 1 -1] {} = [][^] =
0
0 0 1 -2 1/4 -1/5] 5 0
0 1




* /*2().
(30)
|/1() /2() = [ 1]2 = |1[1]2 = []1[*]2 = [-4 14 -10 9 -21 ] {} =
0
1
0
0
0
0
=,
0 - 4 7
-10/3 9/4 - 2/5 1/6







* /1,2(.).
(31)
0

| 2/2 ()* /2 () = [2 ]2 = 2 " = [] 2 [4 -24 40 -16 13 -2 1] {}
0
4
-12
40/3
-4
13/5
-1/3
1/7.



* /2*2().
(32)
0
(11), (11 ), (23)-(29), 1, 2 .
:
[ ]1 = ,[ 1] [] + 2[ 2] [] + [ 3] [],
[]2 = 1[ 1 ] [] + 2[2 ]2 [] + [3] []
(33)
, , , ( = 1):
1[1] [] = 1[04 - 25/3 - 2/41/ 5] * [01 2345]|10:
= 1 (0 * 1+4 * 1 -2 * 1+5/2 * 1 -2/4 * 1+1/5 * 1)= 1 * 101/30
2[2] [] = 2 [0 -4 7 -10/3 9/4 -2/5 1/6]* [∞ 1 2 3 4 5 ] = 2 (0 * 1 -4 * 1 7 * 1 -10/3 * 1 9/4 * 1 -2/5 * 1 1/6 * 1)= 2 * 101/60,
0 1 2 3 4 5
[ 3] []=[0 0 -1 1/3 -1/4] * [ 1 2 " 4 3 ] =(0 * 1 0 * 1 -1 * 1 + 1/3 * 1 -1/4 * 1)= -11/12. [ ]2 ( = 1):
(34)
(35)
(36)
[ ]=[0 0 1 -2 1/4 -1/5] * [ ∞ 1 2 3 4 5 ]
1
0
1

2

3

4

7


=(0 * 1+0 * 1+1 * 1 -2 * 1+1/4 * 1 -1/5 * 1)=-19/20, (37)
1[2]2[] = 1 [0 -4 7-10/3 9/4 -2/5 1/6]* [∞ 1 2 3 4 5 6 ]
= 1 (0 * 1 -4 * 1 7 * 1 -10/3 * 1 9/4 * 1 -2/5 * 1 1/6 * 1)= 1 * 101/60, (38)
2[2]2[] = 2[0 4 -12 40/3 -4 13/5 -1/3 1/7]* [∞ 1 2 3 4 5 ] |'0 =
= 2 (0 * 1 4* 1 -12* 1 40/3 * 1 -4* 1+13/5 * 1 -1/3 * 1+1/7* 1)= 2 * 131/35. (39)

'202/60 101/60' * 11 = 55/601
_101/60 393/105] {2} {19/20}

(40)
3,36667 1,683331 * | 11 0,916671
{1,68333 3,74286[2= { 0,95 }.

'3,36667 1,68333 | 0,91667'
0 9,76671 1,65537
(42)
= 0,18753, 2 = 0,16949, (43)
ui
. = { (1- {)(+ 2 1) (44)
{ =0,25; 2 = 0,5; 3 = 0,75:
1 = 0,25 * 0,75(0,18754+0,16984 * 0,25)=0,043125, (44)
2 = 0,5 * 0,5(0,18754+0,16984 * 0,5)=0,068115, (44)
3 = 0,75 * 0,25(0,18754+0,16984 * 0,75)=0,05905. (44)
: 1 =0,044014; 2 =0,069747; 3 =0,060056.
. , {
. :
= () - , (45)
^ =< (() - )2, > , (45)
- , = 1,..., ;
(45) {-
< {) - }* { ()/ {}, >= 0, (46)
(46)
< {1(2, ї') - } *('), , >= 0, , = 1,2,...,N.
. (8)(10) [4] (44)
= + (-2 + -2)1 + (2-6 + 2 -3)2
= + (1)1 + ( 2)2
3- 1 = 1/4 ,
1 =
" (1)1 ( 2)1" , 1 1
(1)2 ( 2)2 * 1= 2
_ (1)3 (2)3 _ 1 2 J 3
10 = 11
.0 = 1 1
+ 2 [2-6 1-1] * 1 = 1/4
1 = 1/4
2 = 1/16
2 = 1/16 1
1 3 = 1/64
( ) , ,, , = 1, 2, 3.
1 = 1 +(-2+ 1 - 2) 1 +(2-6 1 + ^ - ^) 2 ^
*1/4+ 1 [-2 1-1]:
=1/4+ 1 (-2+1 / 4-1)+ 2 (2-6/2 1/16-1/64)= 1/4+(-29/16 1+35/64 2). (50 ) 2 3:
2=1/2+(-2 1/4-1/4) 1+(2 -6/2 1/4 -1/8) 2=1/2+(-7/4 1-7/8 2), 3 =3/4+(-29/16 1-151/64 2).
1 2:
(48)
(49)
(50)
(50)
(50)
(1)2 (1)3 1 * (2>1 * (1>1 (1)1 (1>31 * - 1 0'
(2)2 (2)3 (1>2 (2)2 1(2)1 (22 (2)3 - 2 > 0
)3 (2)3 - 3 0
(51)
, =(-2+ , - ^^ ) * (-2+ , - 2 )=(4-4 , +5 _1 -2 _1 + _1 )*
-2 " "-2" 0 " 4
1 -2 1 1 -4
-1 1 -2 * -1 2 = 5
0 -1 1 -2 0 3 -2
0 0 -1 1 -2 0 4 1
2 =(4-4 +5 2 -2 3 + 4 )
(52)
0


3

4

( (1 )1)2 == * 1=[4 -4 5 -2 1]
0 = 1 1 = 1/4 2 =1/16 3 = 1/64 4 = 1/256
=(4 *1 -4 *1/4+5 * 1/16 -2 * 1/64+1 * 1/256)=841/256= (29 /16 *29/16).
(53)
( (1 )2)2 = 2 = 2 * 2 =[4 -4 5 -2 1]:
0 = 1

( (1 )3)2 =3
2 = 1/4 3 = 1/8 _4 =1/16_
=(4 *1 -4 * 1/2+5 * 1/4 -2 * 1/8+1 * 1/16)=49/16=(7/4 * 7/4).
V = 1
1 =3/ 4 2 = 9/16 3 = 27/64 4 = 81/256
3 * 3
:[4 -4 5 -2 1]:
(53)
= -841/256=(-29/16 * -29/16). (53)
( (2 )1)2 - .
1' "-29/16 35/64 '
2 = -7/4 -7/8 *
3 -29/16 -151/64


-1/4
-1/2
-3/4
(54)

29/16
-7/4
-29/16
36/64 ' -7/8 -151/64


35/64 -7/8
-1/4' 0'
-29/16
* -1/2 > < 0
-151/64
-3/4 0
(55)
29/16=1,8125; 7/4 =1,75; 35/64 =0,5469; 7/8 =0,875; 151/64=2,3594,
2 2 2
(-1,8125) +(1,75) +(1,8125) =9,6334,
2 2 2
(0,5469) +(0,875) +(2,3594) =6,6314,
(0,5469) * (-1,8125)+(0,875 * 1,75)+(2,3594 * 1,8125)=4,8164,
(-1,8125 * 0,5469)+(1,75 * 0,875)+(1,8125 * 2,3594)=4,8164.
(55)
9,6334 4,8164 4,8164 6,6314


2,68751 2,0704
(56)

*
9,334 4,81^ I 2, 875~
0 -40,852 | -7,0009 ,
:
a2 =0,1722, a1 =0,1928. (5)

U = xt (1- xt )(al + a2xt X (57)
u, = 0,25 * 0,5(0,1928+0,1722 * 0,25)=0,0442, u2 =0,5 * 0,5(0,1928+0,1722 * 0,5)=0,097, u3 = 0,75 * 0,25(0,1928+0,1722 * 0,75)=0,003.
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1. PM .
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4. , , Host , .

1. - / .. - , 1989. - 20 . - (. / . - . B.M. ; 89-57).
2. .. / .. // M . - 2012. - 1. - C. 28 - 35.
3. .. M , , . / .. // M . - 2012. - 2. - C. 17 - 28.
4. . M / . , . ; . . - .: , 1979. - 24 .
08.08.2012