w10 b, Inżynierskie, Semestr III, Metody obliczeniowe, Wyklady

>
restart:
Metoda Jacobiego
+ −=
−+ + =
20 4 3 19
3 0 2 3
xxx
1
2
3
x
1
x x
2
3
4
xx x
1
+ + =
3
2
20
3
70
>
n:=3;
:=
n
3
>
A:=Array([[20,4,-3],[-3,40,2],[4,3,20]]);

20 4 -3
-3 40 2
43 0

A
:=


>
b:=Array([19,83,70]);
:=
b
[
19 83 70
, ,
]
>
x:=Array(1..n):
>
x[1],x[2],x[3]:=0.,0.,0.;
x
1
, ,
x
2
x
3
:=
0. 0. 0.
1

n

∑
x
( )
k
+
= −
b a x
()
k
,
i
=
1, 2,..., ,
n k
=
0,1, 2,...
i

i
i j j

a


j
=
≠
1
ii


ji
>
for k to 15 do
x:=[seq(1/A[i,i]*(b[i]-add(A[i,j]*x[j],j=1..n)+A[i,i]*x[i]),i=1.
.n)]:
end do;
0.9500000000 2.075000000 3.500000000
:=
x
[
,
,
]
1.060000000 1.971250000 2.998750000
,
,
]
:=
x
[
1.005562500 2.004562500 2.992312500
,
,
]
:=
x
[
0.9979343750 2.000801562 2.998203125
,
,
]
0.9995701560 1.999934922 3.000292890
:=
x
[
,
,
]
1.000056949 1.999953117 3.000095730
,
,
]
:=
x
[
1.000023736 1.999999485 2.999995642
,
,
]
:=
x
[
0.9999994495 2.000001998 2.999995330
,
,
]
:=
x
[
0.9999989000 2.000000192 2.999999810
,
,
]
:=
x
[
0.9999999330 1.999999927 3.000000191
,
,
]


,,


:=
x
[
:=
x
[
:=
x
[
1.000000043 1.999999986 3.000000024
,
,
]
:=
x
[
1.000000006 2.000000002 2.999999994
,
,
]
:=
x
[
0.9999999985 2.000000001 2.999999998
,
,
]
:=
x
[
0.9999999995 2.000000000 3.000000000
:=
x
[
,
,
]
1.000000000 2.000000000 3.000000000
,
,
]
>
x:='x':
Inny sposób - bez znajomości wzorów
+−=
−+ + =
20 4 3 19
3 0 2 3
x x x
1
2
3
x
1
x x
2
3
43 0 0
1
++ =
2
3
1

n

∑
x
= −
b a x i
,
=
1, 2,...,
n
i

i
i j j

a


j
=
≠
1
ii


ji
>
for i to n do
r||i:=x[i]=1/A[i,i]*(b[i]-add(A[i,j]*x[j],j=1..n)+A[i,i]*x[i]);
end do;
r1
=
:=
x
1
19
20
− +
1
5
x
2
3
20
x
3
r2
=
:=
x
2
83
40
+ −
3
40
x
1
1
20
x
3
r3
=
:=
x
3
7
2
− −
1
5
x
1
3
20
x
2
>
x[1],x[2],x[3]:=0.,0.,0.;
x
1
, ,
x
3
:=
0. 0. 0.
0.9500000000 2.075000000 3.500000000
>
x[1],x[2],x[3]:=0.,0.,0.;
,
,
]
x
1
, ,
x
3
:=
0. 0. 0.
>
for k to 15 do
x:=[seq(rhs(r||i),i=1..n)];
end do;
:=
x
[
0.9500000000 2.075000000 3.500000000
:=
x
[
,
,
]
1.060000000 1.971250000 2.998750000
,
,
]
:=
x
[
1.005562500 2.004562500 2.992312500
,
,
]
:=
x
[
0.9979343750 2.000801563 2.998203125
,
,
]
:=
x
[
0.9995701562 1.999934922 3.000292891
:=
x
[
,
,
]
1.000056949 1.999953117 3.000095731
,
,
]
xx x


x
2
,,
>
x:=[seq(rhs(r||i),i=1..n)];
:=
x
[
x
2
,,
x
[
:=
1.000023736 1.999999484 2.999995642
,
,
]
x
[
0.9999994495 2.000001998 2.999995330
,
,
]
x
[
0.9999988999 2.000000192 2.999999810
,
,
]
x
[
0.9999999331 1.999999926 3.000000191
,
,
]
x
[
:=
1.000000043 1.999999985 3.000000024
,
,
]
x
[
:=
1.000000007 2.000000002 2.999999993
,
,
]
x
[
0.9999999986 2.000000001 2.999999999
,
,
]
x
[
0.9999999996 2.000000000 3.000000000
,
,
]
x
[
:=
1.000000000 2.000000000 3.000000000
,
,
]
>
x:='x':
Metoda Gaussa-Seidla
>
x[1],x[2],x[3]:=0.,0.,0.;
x
1
, ,
x
2
x
3
:=
0. 0. 0.
1

∑∑
−
1
n

x
()
k
+
= −
b a x
()
k
+
−
a x
()
k
,
i
=
1, 2,..., ,
n k
=
0,1, 2,...


i
a
i
i j j
i j j


ii
j
=
1
j i
= +
1
>
for k to 8 do
for i to n do
x[i]:=1/A[i,i]*(b[i]-add(A[i,j]*x[j],j=1..i-1)-add(A[i,j]*x[j],j
=i+1..n));
end do;
[seq(x[j],j=1..n)];
end do;
[
0.9500000000 2.146250000 2.988062500
,
,
]
[
0.9689593750 1.998268828 3.006467801
[
,
,
]
1.001316404 1.999775340 2.999770418
,
,
]
[
1.000010494 2.000012266 2.999996062
,
,
]
[
0.9999969560 1.999999969 3.000000614
[
,
,
]
1.000000098 1.999999976 2.999999984
,
,
]
[
1.000000002 2.000000001 3.000000000
,
,
]
[
1.000000000 2.000000000 3.000000000
,
,
]
>
x:='x':
Inny sposób - bez znajomości wzorów
1

∑
n

x
= −
b a x i
,
=
1, 2,...,
n
i

i
i j j

a


ii
j
=
≠
1


ji
>
for i to n do r||i end do;
:=
:=
:=
:=
:=
,,
i


x
1
=
19
20
− +
1
5
x
2
3
20
x
3
x
2
=
83
40
+ −
3
40
x
1
1
20
x
3
x
3
=
7
2
− −
1
5
x
1
3
20
x
2
>
x[1],x[2],x[3]:=0.,0.,0.;
x
1
, ,
x
2
x
3
:=
0. 0. 0.
,,
>
x[1]:=rhs(r1);
x
1
:=
0.9500000000
>
x[2]:=rhs(r2);
x
2
:=
2.146250000
>
x[3]:=rhs(r3);
x
3
:=
2.988062500
>
x[1],x[2],x[3]:=0.,0.,0.;
x
1
, ,
x
2
x
3
:=
0. 0. 0.
,,
>
for k to 9 do
for i to n do
x[i]:=rhs(r||i);
end do;
[seq(x[j],j=1..n)];
end do;
[
0.9500000000 2.146250000 2.988062500
,
,
]
[
0.9689593750 1.998268828 3.006467801
[
,
,
]
1.001316405 1.999775340 2.999770418
,
,
]
[
1.000010495 2.000012266 2.999996061
,
,
]
[
0.9999969560 1.999999969 3.000000614
[
,
,
]
1.000000098 1.999999976 2.999999984
,
,
]
[
1.000000002 2.000000001 3.000000000
,
,
]
[
0.9999999998 2.000000000 3.000000000
[
,
,
]
1.000000000 2.000000000 3.000000000
,
,
]
>
>

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