Použití interpolace, která byla popsána v předchozí
kapitole je v některých případech nevhodné. Metoda nejmenších čtverců
patří mezi metody aproximace funkce, při níž je
f(
x
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@3A72@
funkce a
{
x
i
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadIhadaWgaaWcbaGaamyAaaqabaGccaGG9baaaa@3A2E@
,
i =1,…, n
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaabccacaqG9aGaaGymaiaacYcacqWIMaYscaGGSaGaaeiiaiaad6gacaqGGaaaaa@3DD4@
posloupnost bodů neboli argumentů,
v nichž jsme naměřili hodnoty funkce
f(x)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@394E@
,
které jsou obecně zatíženy chybami a koeficienty se volí z podmínky, aby
součet čtverců rozdílů mezi funkcí
f(x)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@394E@
a její aproximací na konečné množině bodů byl
minimální. Předpokládáme, že chyby v různých bodech jsou na sobě navzájem
nezávislé. Na rozdíl od interpolace nepožadujeme, aby všechny uzlové body byly
navzájem různé. Funkci je tedy zadána tabulkou:
|
x
i
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3804@
|
x
0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIWaaabeaaaaa@37D0@
|
x
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIXaaabeaaaaa@37D1@
|
x
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaaIYaaabeaaaaa@37D2@
|
…
|
x
n−1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaaaa@39B1@
|
x
n
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGUbaabeaaaaa@3809@
|
(1)
|
|
y
i
=f(
x
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadAgacaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaacMcaaaa@3D7B@
|
y
0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIWaaabeaaaaa@37D1@
|
y
1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIXaaabeaaaaa@37D2@
|
y
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaaIYaaabeaaaaa@37D3@
|
…
|
y
n−1
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGUbGaeyOeI0IaaGymaaqabaaaaa@39B2@
|
y
n
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGUbaabeaaaaa@380A@
|
Hledáme mnohočlen
m
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@36FF@
-tého stupně, kde
m < n
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiaad2gacaqGGaGaeyipaWJaaeiiaiaad6gaaaa@3ADF@
:
P
m
(x)=
a
m
x
m
+
a
m
x
m−1
+…
a
1
x +
a
0
=
∑
j=0
m
a
j
x
j
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@5DF7@
takový, aby součet čtverců odchylek mnohočlenu
P
m
(x)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBaaaleaacaWGTbaabeaakiaabccacaGGOaGaamiEaiaacMcaaaa@3B03@
funkce
f(x)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@394E@
v uzlových bodech
S=
∑
i=0
n
(
y
i
−
∑
j=0
m
a
j
x
i
j
)
2
=
∑
i=0
n
∑
j=0
m
(
y
i
−
a
j
x
i
j
)
2
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=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@654B@
byl nejmenší. Funkce
S (
a
0
,
a
1
, …,
a
m
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaabccacaGGOaGaamyyamaaBaaaleaacaaIWaaabeaakiaacYcacaqGGaGaamyyamaaBaaaleaacaaIXaaabeaakiaacYcacaqGGaGaeSOjGSKaaiilaiaabccacaWGHbWaaSbaaSqaaiaad2gaaeqaaOGaaiykaaaa@43B7@
má extrém (minimum) tam, kde všechny
∂S
∂
a
k
=0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacqGHciITcaWGtbaabaGaeyOaIyRaamyyamaaBaaaleaacaWGRbaabeaaaaGccqGH9aqpcaaIWaaaaa@3D8D@
(k = 0, 1, …, m)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadUgacaqGGaGaaeypaiaabccacaaIWaGaaiilaiaabccacaaIXaGaaiilaiaabccacqWIMaYscaGGSaGaaeiiaiaad2gacaGGPaaaaa@41DE@
2
∑
i=0
n
[
∑
j=0
m
(
a
j
x
i
j
−
y
i
)
]
x
i
k
=0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaadiaabaWaaabCaeaacaaMc8+aamqaaeaadaaeWbqaaiaacIcacaWGHbWaaSbaaSqaaiaadQgaaeqaaOGaamiEamaaDaaaleaacaWGPbaabaGaamOAaaaakiabgkHiTiaadMhadaWgaaWcbaGaamyAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdaakiaawUfaaaWcbaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdGccaGGPaaacaGLDbaacaWG4bWaa0baaSqaaiaadMgaaeaacaWGRbaaaOGaeyypa0JaaGimaaaa@5468@
odtud
∑
i=0
n
y
i
x
i
k
=
∑
i=0
n
∑
j=0
m
x
i
j+k
a
j
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaamiEamaaDaaaleaacaWGPbaabaGaam4Aaaaakiabg2da9maaqahabaaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoakmaaqahabaGaamiEamaaDaaaleaacaWGPbaabaGaamOAaiabgUcaRiaadUgaaaGccaWGHbWaaSbaaSqaaiaadQgaaeqaaaqaaiaadQgacqGH9aqpcaaIWaaabaGaamyBaaqdcqGHris5aaWcbaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdaaaa@54D3@
pro
k = 0, 1, …, m
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiaabccacaqG9aGaaeiiaiaaicdacaGGSaGaaeiiaiaaigdacaGGSaGaaeiiaiablAciljaacYcacaqGGaGaamyBaaaa@4085@
;
je to soustava normálních rovnic. Upravíme ji na tvar:
∑
i=0
n
y
i
x
i
k
=
∑
j=0
m
a
j
∑
i=0
n
x
i
j+k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaamiEamaaDaaaleaacaWGPbaabaGaam4Aaaaakiabg2da9maaqahabaGaamyyamaaBaaaleaacaWGQbaabeaaaeaacaWGQbGaeyypa0JaaGimaaqaaiaad2gaa0GaeyyeIuoakmaaqahabaGaamiEamaaDaaaleaacaWGPbaabaGaamOAaiabgUcaRiaadUgaaaaabaGaamyAaiabg2da9iaaicdaaeaacaWGUbaaniabggHiLdaaleaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoaaaa@54BE@
pro
k=0,…,m
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2da9iaaicdacaGGSaGaeSOjGSKaaiilaiaad2gaaaa@3C31@
a označíme
g
jk
=
∑
i=0
n
x
i
j+k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zamaaBaaaleaacaWGQbGaam4AaaqabaGccqGH9aqpdaaeWbqaaiaadIhadaqhaaWcbaGaamyAaaqaaiaadQgacqGHRaWkcaWGRbaaaaqaaiaadMgacqGH9aqpcaaIWaaabaGaamOBaaqdcqGHris5aaaa@44C5@
a
ρ
k
=
∑
i=0
n
y
i
x
i
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaSbaaSqaaiaadUgaaeqaaOGaeyypa0ZaaabCaeaacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaamiEamaaDaaaleaacaWGPbaabaGaam4AaaaaaeaacaWGPbGaeyypa0JaaGimaaqaaiaad6gaa0GaeyyeIuoaaaa@44FB@
Výsledný tvar soustavy:
∑
j=0
m
g
jk
a
j
=
ρ
k
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacaWGNbWaaSbaaSqaaiaadQgacaWGRbaabeaakiaadggadaWgaaWcbaGaamOAaaqabaaabaGaamOAaiabg2da9iaaicdaaeaacaWGTbaaniabggHiLdGccqGH9aqpcqaHbpGCdaWgaaWcbaGaam4Aaaqabaaaaa@44D3@
Řešením této soustavy získáme hledané koeficienty mnohočlenu
P
m
(x)=
∑
j=0
m
a
j
x
j
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBaaaleaacaWGTbaabeaakiaabccacaGGOaGaamiEaiaacMcacqGH9aqpdaaeWbqaaiaadggadaWgaaWcbaGaamOAaaqabaGccaWG4bWaaWbaaSqabeaacaWGQbaaaaqaaiaadQgacqGH9aqpcaaIWaaabaGaamyBaaqdcqGHris5aaaa@4605@
.
Této aproximace lze pak užít nejen v bodech
{
x
i
}
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaadIhadaWgaaWcbaGaamyAaaqabaGccaGG9baaaa@3A2E@
,
ale také v jiných bodech
x
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaaaa@370A@
pro výpočet funkční hodnoty.
Pomocí maticového počtu lze dokázat, že problém nejmenších
čtverců, a tedy i soustava , má jediné řešení.
Řešení normálních rovnic pro malé hodnoty
m
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaaykW7aaa@388A@
(m<6)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaad2gacqGH8aapcaaI2aGaaiykaaaa@3A1C@
poskytuje velmi dobré aproximace. Avšak čím
větší hodnoty
m
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaaykW7aaa@388A@
zvolíme, tím horší aproximace řešením
normálních rovnic získáme. Volba stupně polynomu
m
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiaaykW7aaa@388A@
(stupeň aproximujícího polynomu), je ve
většině případů založena na znalosti fyzikální podstaty řešeného problému, tedy
na znalosti teoretického očekávaného průběhu.
Příklad 1.
Metodou nejmenších čtverců nalezněte mnohočlen druhého stupně
pro funkci zadanou tabulkou:
|
x
i
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3804@
|
0,549
|
0,805
|
1,129
|
2,439
|
3,159
|
|
y
i
=f(
x
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadAgacaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaacMcaaaa@3D7B@
|
-0,784
|
-0,989
|
-0,981
|
1,073
|
3,659
|
Řešení:
Příklad 2.
Jsou dány empirické hodnoty:
|
x
i
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaaaaa@3804@
|
0,1
|
0,2
|
0,3
|
0,4
|
0,5
|
0,6
|
0,7
|
0,8
|
0,9
|
|
y
i
=f(
x
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBaaaleaacaWGPbaabeaakiabg2da9iaadAgacaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaacMcaaaa@3D7B@
|
5,1234
|
5,3057
|
5,5687
|
5,9378
|
6,4370
|
7,0978
|
7,9493
|
9,0253
|
10,3627
|
Nalezněte nejlepší aproximaci funkce
f(x)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@392F@
V aplikaci Excel získáme výsledek:
y = 0,9988
x
4
+ 2,9899
x
3
+ 2,0172
x
2
+ 0,9920 x + 5,0010
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaabccacqGH9aqpcaqGGaGaaGimaiaacYcacaaI5aGaaGyoaiaaiIdacaaI4aGaaeiiaiaadIhadaahaaWcbeqaaiaaisdaaaGccaqGGaGaey4kaSIaaeiiaiaaikdacaGGSaGaaGyoaiaaiIdacaaI5aGaaGyoaiaabccacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaaeiiaiabgUcaRiaabccacaaIYaGaaiilaiaaicdacaaIXaGaaG4naiaaikdacaqGGaGaamiEamaaCaaaleqabaGaaGOmaaaakiaabccacqGHRaWkcaqGGaGaaGimaiaacYcacaaI5aGaaGyoaiaaikdacaaIWaGaaeiiaiaadIhacaqGGaGaey4kaSIaaeiiaiaaiwdacaGGSaGaaGimaiaaicdacaaIXaGaaGimaaaa@613A@
Pozn. k postupu:
Vložit > Graf >
Graf > Přidat spojnici trendu >
Použijeme-li funkci MatLab: polyfit, pak po zadání dat
x
i
,
y
i
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaakiaacYcacaWG5bWaaSbaaSqaaiaadMgaaeqaaaaa@3AD7@
získáme výsledek:
>> x=[0.1 0.2 0.3
0.4 0.5 0.6 0.7 0.8 0.9];
>> y=[5.1234
5.3057 5.5687 5.9378 6.437 7.0978 7.9493 9.0253 10.3627];
>> p =
polyfit(x,y,4)
p =
0.9988
2.9899 2.0172 0.9920
5.0010
>>
Tedy výsledný polynom je:
y = 0,9988
x
4
+ 2,9899
x
3
+ 2,0172
x
2
+ 0,9920 x + 5,0010
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaabccacqGH9aqpcaqGGaGaaGimaiaacYcacaaI5aGaaGyoaiaaiIdacaaI4aGaaeiiaiaadIhadaahaaWcbeqaaiaaisdaaaGccaqGGaGaey4kaSIaaeiiaiaaikdacaGGSaGaaGyoaiaaiIdacaaI5aGaaGyoaiaabccacaWG4bWaaWbaaSqabeaacaaIZaaaaOGaaeiiaiabgUcaRiaabccacaaIYaGaaiilaiaaicdacaaIXaGaaG4naiaaikdacaqGGaGaamiEamaaCaaaleqabaGaaGOmaaaakiaabccacqGHRaWkcaqGGaGaaGimaiaacYcacaaI5aGaaGyoaiaaikdacaaIWaGaaeiiaiaadIhacaqGGaGaey4kaSIaaeiiaiaaiwdacaGGSaGaaGimaiaaicdacaaIXaGaaGimaaaa@613A@
a je zcela shodný s polynomem získaným z aplikace
Excel