1. . ..................................................................... 3
2. ............................................................ 4
2.1 ................................................................ 4
2.2. ............................................................. 4
2.3. .................................................................... 5
2.4. .......................................................... 5
3. - ............................................................. 8
3.1. ............................................................... 8
3.2. ............................................................. 9
3.3. .................................................................. 10
3.4. ......................................................... 11
4. ......................................................................... 13
5. ................................................................... 15
6. .................................................................................................... 18
........................................................... 19
1. ......................................................... 20
2. .................................................................... 22
. .
, , , (). , , ( , , , . .) , . , , .
, :
A .
:
1.
2.
3.
4.
. , , .., .. [3]. : .. , .. [1], Y.Saad Iterative Method for Sparse Linear System [5]. , CG (Conjugate Gradient ) MinRES (Minimal Residual ). [5] , .
|
|
A = AT >0.
(1)
, xk, k - , (1). , zk=xk - rk Azk=rk.
, (2) <=> (3)
t k +l || rk +1|| || rk ||. (2) .
t k +l. (3)
, (4) => (5)
. . rk , zk=xk - . (5) ,
(6)
, rk +1|| ,
(7)
, k - (k +1)- . xk rk=Axk - f (7) t k +l. (3) xk+ 1.
(3), (7) , t.
, (8)
. (9)
w k =B -1 rk (k +1)- . , zk=xk - x , . .
. (10)
, B = BT >0. (8), t k +l ||w k +1|| B =(B w k +1,w k +1)1/2 w k. B = .
t k +l. (10)
.
,
. (11)
B w k= rk Bvk=A w k, vk = B -1 w k ( (9)).
. (12)
(2) t k +l || zk +1|| A zk, zk +1 =xk +1- x. zk
,
.
, ,
.
zk=xk - x ( ), , Azk=rk=Axk - f, t k +l
. (13)