Iinkqubo zezilinganiso ezilinganayo ze-algebraic. Iinkqubo ezihambelanayo ezilinganayo zeelgebraic equation

Emuva esikolweni, ngamnye wethu wafunda ukulinganisa kunye, mhlawumbi, inkqubo yokulinganisa. Kodwa a baninzi abantu bayazi ukuba kukho iindlela ezininzi zokuzisombulula. Namhlanje siza kuxubusha ngeenkcukacha zonke iindlela zokusombulula inkqubo yee-algebraic equation equation ezingaphezu kwamabini alinganayo.

Imbali

Kuze kube namhlanje, kwaziwa ukuba ubugcisa bokusombulula ukulingana kunye neenkqubo zabo ezivela kwiBhabhiloni lakudala naseYiputa. Nangona kunjalo, ukulingana kwifomu yabo eqhelekileyo ngathi kubonakala emva kokubonakala komqondiso wokulingana "=", owasungulwa ngo-1556 nguMqulu wezeMathematika weNgesi. Ngendlela, lo mqondiso ukhethwe ngesizathu: uthetha amabini ahlukeneyo alinganayo. Kwaye kuyinyaniso ukuba umzekelo omhle wokulingana awukwazi ukucinga.

Umsunguli wee-alfabhethi zamagama ezingabonakaliyo kunye nezibonakaliso zamadigri ngumfuzulwazi waseFransi uFrancois Viet. Nangona kunjalo, ukutyunjwa kwayo kwahluka kakhulu kunamhlanje. Ngokomzekelo, isikwere senani elingaziwa lichazwe ngetekisi Q (i-Latin "quadratus"), kunye nekhamera yileta C (iCatus "cubus"). Ezi zikhundla ngoku zibonakala zingenakonwaba, kodwa ke yindlela eqondakalayo yokubhala iinkqubo ze-algebraic equation.

Nangona kunjalo, ukungonakali kwezi ndlela zokusombululo kukuba izibalo zithathwa njengezimpembelelo ezintle kuphela. Mhlawumbi oku kubangelwa kukuba ixabiso elibi alizange libe nesicelo esisebenzayo. Nangona kunjalo, izibalo zamatriki zaseItaliya uNiccolo Tartaglia, uGerolamo Cardano kunye noRafael Bombelli ngekhulu le-16 babenokuqala ukuqwalasela iingcambu ezimbi. Ifom yesimanje, indlela eyona nto yokuxazulula ukulinganisa kwe-quadratic (nge-discriminant) yenziwa kuphela ngekhulu le-17 ngenxa yemisebenzi ye-Descartes ne-Newton.

Phakathi kwekhulu le-18, isazi sezibalo seSwitzerland uGabriel Kramer safumana indlela entsha yokulungisa iinkqubo zokulinganisa okulula. Le ndlela yabizwa emva kwayo kwaye kuze kube yimhla siyayisebenzisa. Kodwa siya kuthetha ngeCramamer kwindlela encinane kamva, kodwa ngoku, siza kuxubusha ukulingana okulinganayo kunye neendlela zokuzisombulula ngokwahlukileyo kwinkqubo.

Ukulinganisa okulinganayo

Ukulinganisa okuqhelekileyo kukulingana okulula kunye neziguquko. Zihlelwa njenge-algebraic. Ukulingana okulinganayo kubhalwe kwifom jikelele ngokulandelayo: ₁ * x ₁ + _{2 *} x ₂ + ... _n * x _n = b. Ukumelwa kwabo kule fomu kuyadingeka ukuba kuhlanganiswe iistim kunye neematri.

Iinkqubo zezilinganiso ezilinganayo ze-algebraic

Inkcazo yeli gama yile: kukuqokelela kwamanani alinganayo kunye nesisombululo esifanayo. Njengomthetho, esikolweni, yonke into ixazululwe yiinkqubo ezimbini okanye ezilinganayo ezintathu. Kodwa kukho iinkqubo ezine okanye ngaphezulu. Makhe siqwalasele okokuqala ukuba ubhale phantsi ukwenzela ukuba kwixesha elizayo kwakulungele ukucombulula. Okokuqala, iinkqubo ze-algebraic equation ziza kubonakala zingcono ukuba zonke iinguqu zibhalwa njenge x kunye nesalathisi esifanayo: 1,2,3 njalo njalo. Okwesibini, kuyimfuneko ukuzisa zonke izilinganiso kwifomu ye-canonical: ₁ * x ₁ + _{2 *} x ₂ + ... _n * x _n = b.

Emva kwayo yonke le ntshukumo, sinokuqala ukuxelela indlela yokufumana isisombululo kwiinkqubo zokulingana okulinganayo. Kuninzi kule nto sidinga amatriki.

Matrices

I-matrix yile tafile equkethe imigqa kunye neikholomu, kwaye kwi-intersection yazo zizinto zayo. Oku kunokuba yixabiso okanye iimpawu ezithile. Ngokuqhelekileyo, ukubonisa izixhobo, zibekwe ngaphantsi kweemviwo (umzekelo, i- ₁₁ okanye i- ₂₃ ). Inqaku lokuqala liyinombolo yomqolo, kwaye yesibini yikholam. Iimatriki ezingaphezulu, kunye nakwezinye izixhobo zeemathematika, ungenza imisebenzi eyahlukeneyo. Ngaloo ndlela, unako:

1) Susa kwaye ungeze amanani etafile afanayo.

2) Yandisa i-matrix ngenani okanye kumvolisi.

3) Ukutshintshela: guqula imigqa ye-matrix kwiikholam, kunye neentsika - kwimigca.

4) Yandisa inani leematriki ukuba inani lemiqolo yomnye lilingana nenani lamakholomu enye.

Siza kuxubusha zonke ezo zinto ngokubanzi, kuba ziya kuba luncedo kuthi kwixesha elizayo. Ukukhupha kunye nokongeza kwamatrices kulula. Ekubeni sithatha matrices ngobukhulu obufanayo, iqela ngalinye letafile enye lihambelana nento nganye yomnye. Ngaloo ndlela songeza (shenxisa) ezi zinto zimbini (kubalulekile ukuba zimi kwiindawo ezifanayo kumatrices abo). Xa ukwandisa i-matrix ngenani okanye vector, usandula inxalenye nganye ye-matrix yile nombolo (okanye i-vector). Utshintsho luyinkqubo enomdla kakhulu. Kubonakala kunomdla ngokukhawuleza ukuyijonga ebomini bokwenyani, umzekelo, xa utshintsha ukuqhelaniswa kwethebhulethi okanye ifowuni. Iimpawu kwi desktop ziba matrix, kwaye xa isikhundla sitshintsha, sitshintshwa kwaye siba sikanzi, kodwa sinciphisa ukuphakama.

Masihlalutye inkqubo enjalo, njengokuphindaphinda kwamatriki. Nangona ayifumaneki kakuhle, kuya kuba luncedo ukwazi. Yandisa amanqaku amabini kuphela xa inani leenqwelwana zetafile enye lilingana nenani lemiqolo yomnye. Ngoku sithatha imimiselo yomgca we-matrix kunye neemeko zekholam ehambelanayo enye. Sibaphindaphinda ngeyodwa baze bangeze (oko kukuthi, umzekelo wezinto ezili- ₁₁ kunye ne- ₁₂ b b ₁₂ kunye b ₂₂ ngu: ₁₁ * b ₁₂ + i- ₁₂ * b ₂₂ ). Ngaloo ndlela, inxalenye enye yetafile ifumaneka, kwaye indlela efanayo iyazaliswa ngokuqhubekayo.

Ngoku sinokuqala ukuqwalasela indlela inkqubo yokulingana okulinganayo ixazululwa ngayo.

Indlela yeGauss

Esi sihloko siqala ukuqhutywa esikolweni. Siyazi umgaqo "weenkqubo ezimbini zokulinganisa" kunye kwaye unokuzixazulula. Kuthekani kuthi ukuba inani lamanani alingaphezu kweebini? Indlela yeGauss iza kusinceda kule nto .

Ewe, kulungele ukusebenzisa le ndlela xa senza i-matrix kwi-system. Kodwa awukwazi ukuyitshintsha kwaye uyisombululo kwifomu yayo ecocekileyo.

Ngoko indlela yokulingana kweGauss edibeneyo isombulula njani le ndlela? Ngendlela, nangona le ndlela ibizwa ngokuba nguye, kodwa yafunyanwa kumaxesha amandulo. UGauss ubonisa oku kulandelayo: ukwenza imisebenzi kunye nama-equation, ukuze ekugqibeleni uholele lonke udidi kwifom efana nesinyathelo. Okokuthi, kuyimfuneko ukuba ukusuka phezulu ukuya phezulu (ukuba ulungiswe kakuhle) ukususela kwi-equation yokuqala ukuya ekugqibeleni kuya kuncipha ngenye into engaziwayo. Ngamanye amagama, kufuneka siyenze ukuze sifike, sitsho, ukulinganisa kathathu: kwizinto ezintathu zokuqala ezingaziwa, kwiyesibini - ezimbini, kwiyesithathu enye. Emva koko, ukusuka ekulinganisweni kokugqibela sithola i-yokuqala engaziwayo, sitshintshe ixabiso layo kwisibini okanye ukulinganisa kokuqala, kwaye sifumane iinguqu ezimbini eziseleyo.

Indlela yeCraamer

Ukuqonda le ndlela, kubaluleke kakhulu ukuba ube nezakhono zokongeza, ukukhupha kwamatrices, kunye nokukwazi ukufumana izigqibo. Ngoko ke, ukuba ukwenza kakubi okanye ungazi njani, kuya kufuneka ufunde kwaye usebenzise.

Yiyiphi intsingiselo yale ndlela, kunye nendlela yokwenza ngayo ukuze inkqubo yokulinganisa i-Cramer edibeneyo ifumaneke? Kulula kakhulu. Simele sakhe i-matrix yenani (phantse njalo) i-coefficients of system ye-algebraic equations. Ukwenza oku, thatha kuphela amanani phambi kwezinto ezingaziwa uze uzibeke etafileni ngendlela ebhalwa ngayo kwinkqubo. Ukuba kukho "uphawu" phambi kweenombolo, bhala u-coefficient. Ngoko ke, siqulunqe i-matrix yokuqala yee-coefficients for unknowns, ingabandakanyi manani emva kweempawu ezilinganayo (kungokwemvelo ukuba i-equation kufuneka iyancitshiswe kwifomu ye-canonical, xa iqhosha lesandla sokunene liqule kuphela inani, kwaye ngakwesobunxele yonke into engaziwa ngama coefficients). Emva koko sifuna ukudala amanani matrices ambalwa, omnye kwisigidi ngasinye. Ukwenza oku, faka i-matrix yokuqala ngoku, iikholomu nganye kunye neefowuni zekholomu yamanani emva kwesalathisi esilinganayo. Ngaloo ndlela sifumana matrices ambalwa kwaye sifumane izigqibo zabo.

Emva kokuba sifumane iinqununu, into encinane. Sine-matrix yokuqala, kwaye kukho izibalo ezininzi ezifumanekayo ezihambelanayo neziguquko ezahlukeneyo. Ukufumana isisombululo sesistim, sihlula i-taxi yetafile efunyenweyo kwisigxina setafile yokuqala. Inombolo ekhuphayo yixabiso leminye yezinto eziguqukayo. Ngokufanayo, sifumana yonke into engaziwayo.

Ezinye iindlela

Kukho ezinye iindlela ezininzi zokufumana isisombululo seenkqubo zokulingana okulinganayo. Ngokomzekelo, indlela okuthiwa yiGauss-Jordan, esetyenziselwa ukufumana isisombululo kwinkqubo ye-equation quadratic, idibene nokusetyenziswa kwamatrices. Kukho nendlela yeJacob yokulungisa isistim se-algebraic equation. Yona iyakwazi ukulungelelanisa ikhompyutha kwaye isetyenziswe kwikhompyutha yekhompyutha.

Matyala a

Ngokuqhelekileyo kuvela ukuba inani lezilinganiso lingaphantsi kwenani lezinto eziguqukayo. Emva koko sinokuthi ngokuqinisekileyo ukuba inkqubo leyo ayihambelani (oko kukuthi, ingenazo iingcambu) okanye inani lezisombululo zayo liye lithengelelanise. Ukuba sinesizathu sesibini, ngoko kufuneka sibhale isisombululo esipheleleyo senkqubo yokulinganisa okulinganayo. Iqulethe ubuncinane enye.

Isiphelo

Ngoko safika ekupheleni. Masiqokelele: sihlalutyile ukuba yintoni inkqubo kunye neematriki, kwaye sifunde indlela yokufumana isisombululo jikelele sesistim yokulinganisa okulinganayo. Ukongeza, saqwalasela ezinye iindlela. Sifumene indlela inkqubo yokulinganisa ngokulinganayo ixazululwa ngayo: indlela yeGauss kunye nendlela yeCramer. Sathetha ngeemeko ezinzima kunye nezinye iindlela zokufumana izisombululo.

Enyanisweni, esi sihloko sininzi kakhulu, kwaye ukuba ufuna ukuyiqonda kakuhle, ngoko sincoma ukufunda iincwadi ezizodwa.