Předpokládáme, že funkce
f(x)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaaaa@394E@
je v intervalu
〈a,b〉
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeUaamyyaiaacYcacaWGIbGaeyOkJepaaa@3C0D@
spojitá; dále v koncových bodech
intervalu
〈a,b〉
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeUaamyyaiaacYcacaWGIbGaeyOkJepaaa@3C0D@
platí
f(a)⋅f(b)<0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGHbGaaiykaiabgwSixlaadAgacaGGOaGaamOyaiaacMcacqGH8aapcaaIWaaaaa@406A@
a rovnice
f(x)=0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bGaaiykaiabg2da9iaaicdaaaa@3B0E@
má v intervalu
〈a,b〉
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeUaamyyaiaacYcacaWGIbGaeyOkJepaaa@3C0D@
právě jeden kořen.
Křivka se nahradíme tětivou AB procházející dvěma krajními body
A,B
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiaacYcacaWGcbaaaa@384A@
na křivce. Tětiva je úsečka (část přímky)
spojující oba body křivky. Body
A=[a,f(a)]
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaiabg2da9iaacUfacaWGHbGaaiilaiaadAgacaGGOaGaamyyaiaacMcacaGGDbaaaa@3E59@
B=[b,f(b)]
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaiabg2da9iaacUfacaWGIbGaaiilaiaadAgacaGGOaGaamOyaiaacMcacaGGDbaaaa@3E5C@
jsou voleny, jak již bylo uvedeno, tak, aby
f(a)⋅f(b)<0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGHbGaaiykaiabgwSixlaadAgacaGGOaGaamOyaiaacMcacqGH8aapcaaIWaaaaa@406A@
.
Uvedený postup představuje nahrazení funkce
y=f(x)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiabg2da9iaadAgacaGGOaGaamiEaiaacMcaaaa@3B52@
v intervalu
〈a,b〉
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeUaamyyaiaacYcacaWGIbGaeyOkJepaaa@3C0D@
lineárním interpolačním mnohočlenem:
P
1
(x)=f(a)+
f(b)−f(a)
b−a
(x−a)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabg2da9iaadAgacaGGOaGaamyyaiaacMcacqGHRaWkdaWcaaqaaiaadAgacaGGOaGaamOyaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaaabaGaamOyaiabgkHiTiaadggaaaGaaiikaiaadIhacqGHsislcaWGHbGaaiykaaaa@4D70@
;
b≠a
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgcMi5kaadggaaaa@39A1@
Průsečík přímky AB s osou x je zpřesnění kořene. Metoda
regula falsi je konvergentní pro všechny spojité funkce. Metoda konverguje
nejvýše lineárně. Obecně se jedná o nestacionární metodu, mnohdy (např. u
konvexních funkcí, kdy je druhá derivace kladná) je stacionární - jeden krajní
bod intervalu je vždy jedním ze dvou bodů užitých v následující iteraci. Tato
metoda je velmi vhodná v případě, že nejsou známy výchozí údaje o poloze
kořenu; konverguje však relativně pomalu. Stačí ale nalézt pouze dva body, ve
kterých má hodnota funkce opačné znaménko a pak tuto metodu lze aplikovat na
zadaný interval. Princip metody je znázorněn na následujícím obrázku.
|
Obr. 1
|
Pro výpočet aproximace kořene s použijeme vztah:
|
s=
a⋅f(b)−b⋅f(a)
f(b)−f(a)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2da9maalaaabaGaamyyaiabgwSixlaadAgacaGGOaGaamOyaiaacMcacqGHsislcaWGIbGaeyyXICTaamOzaiaacIcacaWGHbGaaiykaaqaaiaadAgacaGGOaGaamOyaiaacMcacqGHsislcaWGMbGaaiikaiaadggacaGGPaaaaaaa@4D00@
|
(1)
|
nebo v jiné formě:
|
s=a−
f(a)
f(b)−f(a)
⋅(b−a)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2da9iaadggacqGHsisldaWcaaqaaiaadAgacaGGOaGaamyyaiaacMcaaeaacaWGMbGaaiikaiaadkgacaGGPaGaeyOeI0IaamOzaiaacIcacaWGHbGaaiykaaaacqGHflY1caGGOaGaamOyaiabgkHiTiaadggacaGGPaaaaa@4AB7@
|
(2)
|
Při
f(s) = 0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGZbGaaiykaiaabccacqGH9aqpcaqGGaGaaGimaaaa@3C4F@
nebo
f(s) ≈ 0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGZbGaaiykaiaabccacqGHijYUcaqGGaGaaGimaaaa@3CFA@
výpočet končí. V opačném případě (
f(s) ≠ 0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGZbGaaiykaiaabccacqGHGjsUcaqGGaGaaGimaaaa@3D10@
nebo
f(s) > ε
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGZbGaaiykaiaabccatCvAUfKttLearyqr1ngBPrgaiuaacqWF+aGpcqWFGaairmqr1ngBPrgitLxBI9gBaGGbaiab+v7aLbaa@4747@
,
kde
ε
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqeduuDJXwAKbYu51MyVXgaiuaacqWF1oqzaaa@3CBC@
je „malé“ číslo) postupujeme takto:
bude-li
f(s) f(b) < 0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWGZbGaaiykaiaabccacaWGMbGaaiikaiaadkgacaGGPaGaaeiiaiabgYda8iaabccacaaIWaaaaa@401B@
,
přeznačíme
s
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3705@
na
a
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36F3@
a
s
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3705@
počítáme podle stejného vztahu jako předtím,
tedy s užitím (1) nebo (2). V opačném případě přeznačíme
s
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3705@
na
b
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaaaa@36F4@
a
s
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3705@
počítáme opět podle jednoho z výše
uváděných vztahů.
Pro funkce, pro které je metoda regula falsi stacionární (tj.
nemění-li funkce v intervalu svůj charakter, což znamená, že v tomto
intervalu je první derivace jen kladná nebo jen záporná (
f
′
(x)≠0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOzayaafaGaaiikaiaadIhacaGGPaGaeyiyIKRaaGimaaaa@3BDB@
) a pro druhou derivaci a funkční
hodnoty platí jeden ze čtyř následujících vztahů):
|
|
|
Obr.
2
|
Obr.3
|
|
|
|
Obr.4
|
Obr.5
|
Pak platí jeden ze dvou vzorců:
|
x
i+1
=
x
i
f(b)−bf(
x
i
)
f(b)−f(
x
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaGccqGH9aqpdaWcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccaWGMbGaaiikaiaadkgacaGGPaGaeyOeI0IaamOyaiaadAgacaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaacMcaaeaacaWGMbGaaiikaiaadkgacaGGPaGaeyOeI0IaamOzaiaacIcacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaaaaa@4EE3@
|
x
i
+1=
x
i
f(a)−af(
x
i
)
f(a)−f(
x
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbaabeaakiabgUcaRiaaigdacqGH9aqpdaWcaaqaaiaadIhadaWgaaWcbaGaamyAaaqabaGccaWGMbGaaiikaiaadggacaGGPaGaeyOeI0IaamyyaiaadAgacaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaacMcaaeaacaWGMbGaaiikaiaadggacaGGPaGaeyOeI0IaamOzaiaacIcacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaaaaa@4EE0@
|
|
platí pro
Obr. 2 a Obr. 4
|
platí pro
Obr. 3 a Obr. 5
|
nebo
v jiném tvaru:
|
x
i+1
=
x
i
−
f(
x
i
)
f(b)−f(
x
i
)
(b−
x
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0YaaSaaaeaacaWGMbGaaiikaiaadIhadaWgaaWcbaGaamyAaaqabaGccaGGPaaabaGaamOzaiaacIcacaWGIbGaaiykaiabgkHiTiaadAgacaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaacMcaaaGaaiikaiaadkgacqGHsislcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@501F@
|
x
i+1
=
x
i
−
f(
x
i
)
f(a)−f(
x
i
)
(a−
x
i
)
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBaaaleaacaWGPbGaey4kaSIaaGymaaqabaGccqGH9aqpcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0YaaSaaaeaacaWGMbGaaiikaiaadIhadaWgaaWcbaGaamyAaaqabaGccaGGPaaabaGaamOzaiaacIcacaWGHbGaaiykaiabgkHiTiaadAgacaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaacMcaaaGaaiikaiaadggacqGHsislcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaaaa@501D@
|
|
platí pro
Obr. 2 a Obr. 4
|
platí pro
Obr. 3 a Obr. 5
|
Pravým nebo levým krajním bodem intervalu je po celou dobu
výpočtu bod
(b,f(b))
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadkgacaGGSaGaamOzaiaacIcacaWGIbGaaiykaiaacMcaaaa@3C28@
resp.
(a,f(a))
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaadggacaGGSaGaamOzaiaacIcacaWGHbGaaiykaiaacMcaaaa@3C26@
.
Pro odhad chyby lze s využitím věty o střední hodnotě
získat vztah:
|
x
i
−ξ
|≤
|
f(
x
i
)
|
m
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaeyOeI0IaeqOVdGhacaGLhWUaayjcSdGaeyizIm6aaSaaaeaadaabdaqaaiaadAgacaGGOaGaamiEamaaBaaaleaacaWGPbaabeaakiaacMcaaiaawEa7caGLiWoaaeaacaWGTbaaaaaa@483E@
,
kde
m=
min
x∈〈
a,b
〉
|
f
′
(x)
|
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyBaiabg2da9maaxababaGaciyBaiaacMgacaGGUbaaleaacaWG4bGaeyicI48aaaWaaeaacaWGHbGaaiilaiaadkgaaiaawMYicaGLQmcaaeqaaOWaaqWaaeaaceWGMbGbauaacaGGOaGaamiEaiaacMcaaiaawEa7caGLiWoaaaa@4857@
Pro zastavení výpočtu můžeme používat též podmínku
f(
x
i
)<ε
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacIcacaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaaiykaiabgYda8iabew7aLbaa@3D1D@
Příklad 1.
(
x
2
)
2
−sinx=0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr4rNCHbGeaGqipv0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaadaWcaaqaaiaadIhaaeaacaaIYaaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiabgkHiTiGacohacaGGPbGaaiOBaiaadIhacqGH9aqpcaaIWaaaaa@40D4@
:
Algoritmus:
function
TForm1.F(X:extended): extended;
begin
F:=(x/2)*(x/2)-Sin(x);
end;
procedure TForm1.Button1Click(Sender: TObject);
var A,B,C,EPS,FA,FB,FC: extended;
k:integer;
begin
A:=StrToFloat(edit1.text);
B:=StrToFloat(edit2.text);
EPS:=StrToFloat(edit3.text);
FA:=F(A);
FB:=F(B);
k:=0;
repeat
C:=(A*FB-B*FA)/(FB-FA);
FC:=F(C);
inc(k);
if FC*FB<0 then
begin
A:=C;
FA:=FC
end
else
begin
B:=C;
FB:=FC
end
until (abs(FC)<EPS);
edit4.text:=FloatToStr(C);
Button1.Caption:='Spočteno';
Edit5.text
:=IntToStr(k);
end;
