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Numeeriset menetelmät (2017)
Harjoitus 2/viikko 5
1. Ratkaise Gaussin eliminointimenetelmällä yhtälöryhmä

x1 + x 2 + x3 + x4
=1




 x1 + x2 + 2x3 + 3x4 = 2

−x1 + 2x3 + x4
=1




3x1 + 2x2 − x3
=1
2. (a) Muodosta LU-hajotelma matriisille


2 7 5
A = 6 20 10 .
4 3 0
>
(b) Ratkaise LU-hajotelman avulla yhtälöryhmä Ax = b, missä b = 0 4 1 .
3. Olkoon
A=
1
−2
.
−0.99 1.99
(a) Muodosta matriisin A käänteismatriisi sekä laske A:n kuntoluku joko k · k∞ normia tai k · k1 -normia käyttäen.
(b) Ratkaise yhtälöryhmä
Ax = b + εr,
−1
missä b =
,
1
1
r=
1
ja ε ≥ 0 kuvaa mittausvirheen suuruutta b:ssä.
(c) Laske ratkaisuvektorin x absoluuttinen ja suhteellinen virhe k · k2 -normin suhteen.
4. Laske yhtälöryhmän

−3x + 9y + 4z = −14


5x + y + 2z
= 10


x + 2y − 7z
= −33
ratkaisun kymmenes likiarvo X10 = (x10 , y10 , z10 )>
(a) Jacobin
(b) Gauss-Seidel
menetelmällä kun X0 = (0, 0, 0)> .
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Exercise 2/week 5
1. Solve the following system of equation by using the Gaussian elimination method

x1 + x 2 + x3 + x4
=1




 x1 + x2 + 2x3 + 3x4 = 2

−x1 + 2x3 + x4
=1




3x1 + 2x2 − x3
=1
2. (a) Find the LU decomposition of the matrix


2 7 5
A = 6 20 10 .
4 3 0
>
(b) Solve Ax = b with b = 0 4 1 by using the LU decomposition of A.
3. Let
A=
1
−2
.
−0.99 1.99
(a) Find A−1 and calculate the condition number of A by using either the norm
k · k∞ or k · k1 .
(b) Solve the system of equation
Ax = b + εr,
where b =
−1
,
1
1
r=
1
and ε ≥ 0 is the measurement error in b.
(c) Calculate the absolute and relative errors of x with respect to the norm k · k2 .
4. Find the tenth approximation X10 = (x10 , y10 , z10 )> of the solution of the system of
equation

−3x + 9y + 4z = −14


5x + y + 2z
= 10


x + 2y − 7z
= −33
by using
(a) Jacobi
(b) Gauss-Seidel method
with X0 = (0, 0, 0)> .
2