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 mahi 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 arapū 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. Ka whakawhirinaki tēnei tikanga ki te Ariā a Bolzano e kī ana mēnā he mahi tonu \( f(x) \) i roto i te wā \([a, b]\) me \( f(a) \cdot f(b) < 0 \), kātahi ka kotahi te pūwāhi \( c \) kei reira \( f(c) = 0 \). Ko ngā mahi: 1. Kōwhiria kia rua ngā pūwāhi tīmatanga \( a \) me \( b \) kia \( f(a) \cdot f(b) < 0 \). 2. Kimihia te pūwāhi \( c = \frac{a + b}{2} \). 3. Kimihia \( f(c) \). 4. Mēnā \( f(c) = 0 \), kātahi ka putake a \( c \). 5. Mēnā \( f(c) \neq 0 \), tirohia te tohu o \( f(a) \cdot f(c) \). Mena he kino, whakakapia a \( b \) ki a \( c \); ki te mea he pai, whakakapia a \( a \) ki a \( c \). 6. Whakahokia te tukanga kia iti rawa te wā [a, b].
He tino pumau tēnei tikanga, ā, ka kitea tonutia ngā pūtake i roto i te wā kua hoatu engari ka puhoi pea te hononga. Ngā Tikanga Tātari Ko ngā tikanga tātari he whakaaro pāngarau hohonu ake me ngā mahi taurangi hei kimi otinga mō ngā whārite kore-rārangi. 1. Te Whakakapinga me te Whakawhiti Ka taea te whakangawari i ētahi whārite kore-rārangi mā te whakarite i ngā taurangi, te whakakapi rānei. Ka taea e ēnei whakawhiti taurangi te huri i te whārite kore-rārangi ki tētahi āhua e ngāwari ake ai te whakaoti. 2. Te Whakawehewehe He maha ngā wā ka taea te wehewehe i ngā whārite tohu-tiketike ki roto i tētahi hua o ngā whārite rārangi, whārite tapawhā rānei. Hei tauira, ka taea te whakangawari i tētahi whārite pūronomial kore-rārangi mā te kimi i ōna pūtake wehewehe. 3. Raupapa Ka taea te āwhina i ētahi wā mā te whakamahi i te raupapa Taylor, i te raupapa Fourier rānei ki te whakaoti, ki te whakatata rānei i te otinga o tētahi whārite kore-rārangi. Ko tēnei huarahi ko te whakawhānui i te mahi i roto i te āhua raupapa, kātahi ka tapahia ki tētahi taumata hei tae atu ki tētahi otinga tata. Ngā Tikanga Whakamātautau 1. Te Aratohu Ira Ko te Aratohu Ira he huarahi arotau whanaketanga me te whaihanga e hangai ana ki te whakaoti rapanga kore-raina. Kei roto i tēnei tikanga te whiriwhiri, te whakawhiti, me ngā tukanga whakarerekētanga hei kimi i ngā otinga tino pai, tata-tino pai rānei. 2. Te Whakamahana Whakaari Ko te Whakamahana Whakaari he tikanga arotau 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 o ngā mahi kore-raina. Ngā Tikanga Whakairoiro I ētahi wā ka taea e te whakatakoto kauwhata i tētahi whārite kore-raina te whakarato i te māramatanga 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 Take 1. Ngā Whārite a Kepler I roto i te miihini rangi, ko ngā ture a Kepler e uru ana ki ngā whārite kore-raina kāore 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 miihini wai mō ngā wai kore-Newtonian, ko ngā tauira pāngarau e uru ana ki ngā whārite kore-raina uaua, ā, he maha ngā wā ka whakatauhia mā te whakamahi i ngā tikanga tau pērā 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 whakatau 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 hei whakaoti.