Rautaki mō te Whakaoti i ngā Whārite Kore-Raina
Ko te whārite kore-rārangi he whārite kāore e hanga i te rārangi tika ina tuhia ki te kauwhata. Ko te tikanga he uaua ake te āhua o ēnei whārite i ngā whārite rārangi, ā, he maha ngā wā kāore e taea te whakaoti mā te tātari mā te whakamahi i ngā tikanga taketake pēnei i te tāpiri, te tango, te whakarea, te wehewehe rānei.
He mea nui te mārama ki te whakaoti rapanga whārite kore-rārangi i roto i ngā mara maha o te pūtaiao, tae atu ki te ahupūngao, te matū, te koiora, te ōhanga, me te hangarau. Ka matapakihia e tēnei tuhinga ētahi rautaki rongonui mō te whakaoti rapanga whārite kore-rārangi, tae atu ki ngā tikanga tau me ngā tikanga tātari.
Pendahuluan
I roto i te nuinga o ngā wā, ka puta ake ngā whārite kore-raina hei tauira mō ngā āhuatanga uaua. Hei tauira, i roto i ngā whanaketanga wai, i ngā tauhohenga matū, i ngā pūnaha ōhanga rānei, he maha ngā wā ka tika ake, ka whai tikanga ake hoki ngā tauira kore-raina. Heoi, nā te uaua o ngā whārite kore-raina ka uaua te whakaoti mā te whakamahi i ngā tikanga māmā, i te taurangi taketake rānei . Nō reira, kua whakawhanakehia ngā tikanga me ngā tikanga maha hei whakatika i tēnei wero.
Tikanga Whakahōu
1. Tikanga Newton-Raphson
Ko te tikanga Newton-Raphson tētahi o ngā tikanga tāruarua rongonui mō te kimi i ngā pūtake o ngā whārite kore-raina. Mō tētahi mahi \( f(x) = 0 \), ka whakamahia e tēnei tikanga he huarahi tāruarua hei whakatau tata i te otinga mā te tātai:
\[ x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)} \]
I konei, ko \( f'(x_n) \) te tuatahi o ngā pānga o te mahi \( f \) i te pūwāhi \( x_n \). He tere, he taupatupatu hoki tēnei tikanga ina whakamahia tata ki ngā pūtake o te otinga, mena kāore te pānga o te mahi e tata ki te kore.
Tauira whakatinanatanga:
1. Kōwhiria te pūwāhi tīmatanga \( x_0 \).
2. Tātaihia \( f(x_0) \) me \( f'(x_0) \).
3. Whakamahia te tātai tāruarua hei whiwhi i te \( x_1 \).
4. Whakahokia ngā taahiraa 2 me te 3 kia tata rā anō te uara o \( x_{n+1} \) ki te pūtake me te manawanui e hiahiatia ana.
Heoi anō, he ngoikoretanga anō tō te tikanga Newton-Raphson, inā koa mēnā ka whiriwhiria e koe tētahi tīmatanga e tawhiti ana i te pūtake tūturu, e tata ana rānei te pānga tuatahi ki te kore.
2. Tikanga Momotu
Ko te tikanga Secant he whakarerekētanga o te tikanga Newton-Raphson kāore e hiahiatia te taupū tuatahi. Ko te tātai tāruarua ko:
\[ x_{n+1} = x_n – \frac{f(x_n)(x_n – x_{n-1})}{f(x_n) – f(x_{n-1})} \]
Ko te painga o tēnei tikanga, kāore e hiahiatia te tatau i ngā tātaitanga, he mea uaua pea tēnei. Heoi, i te nuinga o te wā, he puhoi ake te tūhono o tēnei tikanga i a Newton-Raphson.
3. Tikanga Wehewehe
Ko te tikanga Wehewehe he tikanga taketake e whakamana ana i te huihuinga, engari i te tere o te whakahoutanga puhoi. E whakawhirinaki ana tēnei tikanga ki te Ariā a Bolzano e kī ana mēnā he tonu te mahi (f(x)) i roto i te wā ([a, b]\) me \( f(a) \cdot f(b) < 0 \), kāti he iti rawa te kotahi te pūwāhi \( c \) kei reira \( f(c) = 0 \). Ko ngā mahi:
1. Kōwhiria kia rua ngā tīmatanga \( a \) me \( b \) kia \( f(a) \cdot f(b) < 0 \).
2. Tāutuhia te pūwāhi waenga \( c = \frac{a + b}{2} \).
3. Whakatauhia \( f(c) \).
4. Mena ko \( f(c) = 0 \), ko te \( c \) he pūtake.
5. Mena he 0 te tohu o \( f(c) \neq 0 \), tirohia te tohu o \( f(a) \cdot f(c) \). Mena he kino, whakakapia te \( b \) ki te \( c \); mena he pai, whakakapia te \( a \) ki te \( c \).
6. Whakahokia te tukanga kia iti rawa te wā [a, b].
He tino pūmau tēnei tikanga, ā, ka kitea tonutia ngā pūtake i roto i te wā kua whakaritea, engari ka puhoi pea te huihuinga.
Ngā Tikanga Tātari
Ko ngā tikanga tātari e uru ana ki te whakaaroaro pāngarau hōhonu me te raweketanga taurangi hei kimi otinga mō ngā whārite kore-raina.
1. Te Whakakapinga me te Whakawhitiwhiti
Ka taea te whakangawari i ētahi whārite kore-rārangi mā te whakarite anō i ngā taurangi, mā te whakakapi rānei. Ka taea e ēnei panonitanga taurangi te huri i ngā whārite kore-rārangi ki tētahi āhua e māmā ake ai te whakaoti.
2. Te Whakawehewehe
Ka taea te whakauru i ngā whārite teitei ake ki roto i ngā hua o ngā whārite rārangi, whārite tapawhā rānei. Hei tauira, ka taea te whakangawari i ngā whārite pūronomial kore-rārangi mā te kimi i ō rātou pūtake tauwehe.
3. Raupapa
He āwhina pea te whakamahi i te raupapa Taylor , i te raupapa Fourier rānei, i ētahi wā, ki te whakaoti rapanga, ki te whakatau tata rānei i te otinga o ngā whārite kore-raina. Ko tēnei huarahi ko te whakawhānui i te mahi i roto i te āhua raupapa, kātahi ka tapahia kia tae atu ki tētahi taumata hei whakatutuki i tētahi otinga tata.
Tikanga Whakamātautau
1. Te Pūnaha Ira
He huarahi whakatauira whanaketanga, e ahu mai ana i te arotautanga, ngā rauropi ira hoki, hei whakaoti rapanga kore-raina. Kei roto i tēnei tikanga ko te whiriwhiri, te whakawhiti, me ngā tukanga whakarerekētanga hei kimi i ngā otinga tino pai, tata rānei ki te tino pai.
2. Te Whakakorikori Whakaari
He tikanga arotau te Whakamātao Whakaari e whai ana i te tukanga whakamatao i roto i te mahi whakarewa. He tino whai hua tēnei tikanga mō te kimi i te iti rawa ā-ao o tētahi mahi kore-raina.
Tikanga Whakairoiro
I ētahi wā, mā te tuhi kauwhata i tētahi whārite kore-raina ka taea te whakarato i tētahi tirohanga nui ki te āhua o te otinga. Mā te tuhi i te mahi me te tiro i ngā taunga-x ka āwhina i te mārama ki te whanonga o te otinga.
Tauira tauira
1. Ngā whārite a Kepler
I roto i te hangarau o te rangi, kei roto i ngā ture a Kepler ngā whārite kore-raina e kore e taea te whakaoti tika. He maha ngā wā ka whakamahia te tikanga Newton-Raphson hei whakaoti i ēnei whārite.
2. Te peita kore-Newtonian
I roto i te hangarau wai mō ngā wai ehara i te Newtonian, ka uru atu ki ngā tauira pāngarau ngā whārite kore-raina uaua, ā, he maha ngā wā ka whakatauhia mā te whakamahi i ngā tikanga tau pēnei i te tikanga Runge-Kutta.
Whakamutunga
He wero nui te whakaoti rapanga kore-raina i roto i ngā momo mara. Ko ngā tikanga Newton-Raphson, Secant, me Bisection ētahi o ngā tikanga tau e whakamahia whānuitia ana. Ka tukuna hoki e ngā huarahi tātari me ngā tikanga tauira ngā huarahi maha hei wikitoria i ngā uauatanga o ngā whārite kore-raina. Ko te whiriwhiri i te tikanga tika e whakawhirinaki ana ki te āhua o te whārite me te tika me te whai huatanga e hiahiatia ana mō te otinga.
