Ma kēia ʻatikala e hoʻāʻo mākou i ka ʻike makemakika o ChatGPT. E hoʻāʻo mākou e hoʻohana pono i ka naʻauao hana e ʻimi ai i kahi laʻana kūʻē i ka Fundamental Theorem of Algebra, me ka ʻike ʻana he mea kānalua ia e hoʻokuʻu iā mākou i ka Fields Medal.
Inā mākou e nīnau e pili ana i nā aʻa o kahi polynomial o ke degere 3, i kēia hihia ʻoiaʻiʻo, ʻōlelo ʻo ChatGPT e hilinaʻi paha ka hoʻonā analytical i ka polynomial i manaʻo ʻia, no laila ke paipai nei mākou e hoʻohana i ke ʻano helu helu e like me ke ʻano Newton-Raphson.
He hewa i ka helu ʻana i ka derivative
I kēia manawa, ʻaʻole hiki iā mākou ke kānalua i ka hiki ke makemakika o AI, no laila ua hoʻāʻo mākou e hoʻoponopono i ka pilikia o ka loaʻa ʻana o nā aʻa o ka polynomial p(x) = x3 – 3×2 + 4 a i ko mākou kahaha ua hewa ka helu ʻana. o ka derivative , no laila, ʻaʻole pololei ka loaʻa ʻana o nā aʻa. Hoʻihoʻi ia i ka x = 0 ma ke kumu o ka polynomial a noi mākou iā ia e nānā. ʻO ka mea maʻamau, ʻike ia i ke ʻano o kahi hewa akā ʻaʻole ʻike i kahi i loaʻa ai. Ua ʻike mākou aia ka hewa i ka derivative o ka polynomial a ke noi nei mākou ua helu ʻia mai nā aʻa ma o ke ʻano Newton-Raphson. ʻO ka mea kupanaha, hana hou ia i ka hewa helu helu, i kēia manawa ma kahi hana maʻalahi, e like me kā mākou e ʻike ai ma ke kiʻi aʻe.
Ke kuhi hewa
I ka ike ana i ka hewa o ka helu ana, ninau hou aku makou ia ia, me ka hana ana i kekahi hewa hou, no laila, haawi aku makou ia ia i ka huahelu mua o ka Newton-Raphson Method, oia hoi, x₁ = 5/3 a ke noi aku nei makou e hoomau aku i na helu ana, a loaa ka x₁ = 5 /3 ke kumu o ka polynomial. Hoʻoikaika mākou ma ka nīnau hou ʻana inā he kumu ka waiwai 5/3 o ka polynomial, a loaʻa iā mākou kahi pane ʻae. Noi mākou e helu i ka waiwai o ka polynomial ma ia waiwai, a no ka mea ʻokoʻa ka hopena mai ka ʻole, hōʻike mākou ʻaʻole hiki ke lilo i kumu. Hoʻomaopopo ʻo ia a kala mai e like me kā mākou e ʻike ai ma lalo nei.
Manaʻo mākou he pololei ke kumumanaʻo o ke ʻano Newton-Raphson, akā ʻaʻole pololei kāna hoʻohana ʻana, no laila ke hoʻāʻo nei mākou e ʻimi i nā aʻa me ke ʻano ʻē aʻe, e like me ka factorization o ka polynomial.
I kēia hihia, ʻike mākou ʻo nā kumu o ka polynomial p(x) he x = r a me x = 1 ± 2i.
Ke noi ʻia e hōʻoia ʻaʻole ʻole ka waiwai o p(1+2i) a no laila ʻaʻole hiki ke lilo i kumu o kā mākou polynomial, e ʻae hou i ka hewa. I ka hōʻea ʻana i kēia kūlana, hele mākou me kahi ʻike, a haʻi mākou iā ia he kumu maoli ʻo x = – 1 o ka polynomial a ʻo ke koena o nā aʻa e helu. ʻAʻole hiki ke kāhāhā ʻia kāna pane mua, e haʻi ana iā mākou ma waho o x = – 1, ʻo nā kumu ʻē aʻe o ka polynomial p(x)=4 – 3×2 + x3 he x = 1 + 2i a me x = 1 – 2i. . A hiki i ʻehā manawa hāʻawi hou i nā hopena hewa, no laila ʻaʻohe o mākou koho akā e hāʻawi iā ia i kahi aʻa hou. I kēia hihia, ma kahi o ka hāʻawi ʻana, nīnau mākou inā x = 2 ke kumu o kā mākou polynomial. E hoʻoholo no ʻoukou iho i ka pane, a i ʻole, nā helu a ChatGPT e hana ai no ka nānā ʻana ʻaʻole ʻo x = 2 he kumu:
Ma hope o ka nānā hou ʻana i kāu helu ʻana, e hoʻopau me ka wehewehe ʻana ʻo nā kumu o kā mākou polynomial he x = – 1, x = 1, a me x = 2.
E hōʻike mākou iā ʻoe ʻo nā aʻa ʻekolu a pau, e hoʻihoʻi ana i ka pololei, ʻo ia hoʻi, x = 1 ʻaʻole ia he kumu aʻo nā kumu waiwai ʻelua. ʻAʻole mākou e haʻalele a hoʻāʻo e ʻimi i ke kumu ʻekolu o ka polynomial, a ma laila mākou e ʻike ai i ka wehewehe nui loa. ʻO kā mākou wehewehe ʻana: “Ua hoʻokumu ka Theorem Fundamental of Algebra i kēlā me kēia polynomial o ke degere he n aʻa pololei, ʻoiaʻiʻo a paʻakikī. No laila, inā he k kumu maoli ka polynomial o ke degere n, a laila, pono ke loaʻa nā kumu paʻakikī nk. I kēia manawa ua ʻae mākou. Ke hoʻomau nei mākou me kēia: “Ma ke ʻano o ka polynomial p(x) = 4 – 3×2 + x3, ua ʻike mākou he ʻelua mau kumu maoli, x = – 1 a me x = 2. No ka mea he p(x) polynomial o ke degere 3, a laila pono ke loaʻa i ke kumu paʻakikī hou. ʻO kēia kumu paʻakikī he x = 1 ± 2i.” ʻAʻole hiki iā mākou ke haʻalele i ko mākou kahaha a ke manaʻo nei mākou ua makemake wale ʻo ia e hōʻike iā mākou i kekahi o nā aʻa ʻelua, no laila hāʻawi mākou iā ia i kahi manawa hou aʻe, ʻo ka hopena:
No laila, inā pololei mākou, ua loaʻa iā mākou kahi hiʻohiʻona kūʻē i ka Theorem Fundamental of Algebra, he polynomial o ka degere 3 me nā kumu 4. Ke holo nei mākou no ka Medal Fields?
Ua hōʻoia hou ʻo AI ua pololei kāna pane a hiki i ʻelua mau manawa hou, e hōʻike ana i kahi degere 3 polynomial hiki ke loaʻa i nā aʻa 4. Hoʻomaka mākou e ʻimi iā lākou me ka hoʻohana ʻana i ke ʻano Bisection. ʻAe, haʻalele mākou i ka ʻimi ʻana i nā aʻa o kahi degere maʻalahi 3 polynomial. Ke haʻi aloha aku nei mākou me hoʻokahi pila hope loa:
E like me ka hōʻuluʻulu manaʻo, ʻaʻole mākou e ʻōlelo nei ʻo ChatGPT kahi mea akamai Artificial maikaʻi ʻole, mamao loa ia, inā ʻaʻole wale ka mea kūʻē, he AI maikaʻi loa ia, akā i loko o kāna iho, i ka Natural Language Processing, ʻoiai ma ka Matematika aia nō kahi. lōʻihi ka hele ʻana, aʻo. Pono kākou e koʻikoʻi i nā hopena i hoʻihoʻi ʻia e nā ʻenekini iā mākou: ʻaʻole ia he ʻoiaʻiʻo me ka maikaʻi o ka wehewehe ʻana, a me he mea lā ua nalowale ke kanaka nāna e hōʻoia i ko lākou ʻoiaʻiʻo.
NO KA MEA KAKAU
ʻO Íñigo Sarría Martínez De Mendivil
Ua paʻi mua ʻia kēia ʻatikala ma The Conversation.
