❈❤✳✺✿ ❆rr❛② ❝♦♠♣✉t✐♥❣ ❛♥❞ ❝✉r✈❡ ♣❧♦tt✐♥❣
❍❛♥s P❡tt❡r ▲❛♥❣t❛♥❣❡♥✶,✷
❙✐♠✉❧❛ ❘❡s❡❛r❝❤ ▲❛❜♦r❛t♦r②✶ ❯♥✐✈❡rs✐t② ♦❢ ❖s❧♦✱ ❉❡♣t✳ ♦❢ ■♥❢♦r♠❛t✐❝s✷
❆✉❣ ✷✶✱ ✷✵✶✻
- ♦❛❧✿ ❧❡❛r♥ t♦ ✈✐s✉❛❧✐③❡ ❢✉♥❝t✐♦♥s
❲❡ ♥❡❡❞ t♦ ❧❡❛r♥ ❛❜♦✉t ❛ ♥❡✇ ♦❜❥❡❝t✿ ❛rr❛②
❈✉r✈❡s y = f (x) ❛r❡ ✈✐s✉❛❧✐③❡❞ ❜② ❞r❛✇✐♥❣ str❛✐❣❤t ❧✐♥❡s ❜❡t✇❡❡♥ ♣♦✐♥ts ❛❧♦♥❣ t❤❡ ❝✉r✈❡ ▼❡❡❞ t♦ st♦r❡ t❤❡ ❝♦♦r❞✐♥❛t❡s ♦❢ t❤❡ ♣♦✐♥ts ❛❧♦♥❣ t❤❡ ❝✉r✈❡ ✐♥ ❧✐sts ♦r ❛rr❛②s ① ❛♥❞ ② ❆rr❛②s ≈ ❧✐sts✱ ❜✉t ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♠✉❝❤ ♠♦r❡ ❡✣❝✐❡♥t ❚♦ ❝♦♠♣✉t❡ t❤❡ ② ❝♦♦r❞✐♥❛t❡s ✭✐♥ ❛♥ ❛rr❛②✮ ✇❡ ♥❡❡❞ t♦ ❧❡❛r♥ ❛❜♦✉t ❛rr❛② ❝♦♠♣✉t❛t✐♦♥s ♦r ✈❡❝t♦r✐③❛t✐♦♥ ❆rr❛② ❝♦♠♣✉t❛t✐♦♥s ❛r❡ ✉s❡❢✉❧ ❢♦r ♠✉❝❤ ♠♦r❡ t❤❛♥ ♣❧♦tt✐♥❣ ❝✉r✈❡s✦
❚❤❡ ♠✐♥✐♠❛❧ ♥❡❡❞✲t♦✲❦♥♦✇ ❛❜♦✉t ✈❡❝t♦rs
❱❡❝t♦rs ❛r❡ ❦♥♦✇♥ ❢r♦♠ ❤✐❣❤ s❝❤♦♦❧ ♠❛t❤❡♠❛t✐❝s✱ ❡✳❣✳✱ ♣♦✐♥t (x, y) ✐♥ t❤❡ ♣❧❛♥❡✱ ♣♦✐♥t (x, y, z) ✐♥ s♣❛❝❡ ■♥ ❣❡♥❡r❛❧✱ ❛ ✈❡❝t♦r v ✐s ❛♥ n✲t✉♣❧❡ ♦❢ ♥✉♠❜❡rs✿ v = (v✵, . . . , vn−✶) ❱❡❝t♦rs ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❧✐sts✿ vi ✐s st♦r❡❞ ❛s ✈❬✐❪✱ ❜✉t ✇❡ s❤❛❧❧ ✉s❡ ❛rr❛②s ✐♥st❡❛❞
❱❡❝t♦rs ❛♥❞ ❛rr❛②s ❛r❡ ❦❡② ❝♦♥❝❡♣ts ✐♥ t❤✐s ❝❤❛♣t❡r✳ ■t t❛❦❡s s❡♣❛r❛t❡ ♠❛t❤ ❝♦✉rs❡s t♦ ✉♥❞❡rst❛♥❞ ✇❤❛t ✈❡❝t♦rs ❛♥❞ ❛rr❛②s r❡❛❧❧② ❛r❡✱ ❜✉t ✐♥ t❤✐s ❝♦✉rs❡ ✇❡ ♦♥❧② ♥❡❡❞ ❛ s♠❛❧❧ s✉❜s❡t ♦❢ t❤❡ ❝♦♠♣❧❡t❡ st♦r②✳ ❆ ❧❡❛r♥✐♥❣ str❛t❡❣② ♠❛② ❜❡ t♦ ❥✉st st❛rt ✉s✐♥❣ ✈❡❝t♦rs✴❛rr❛②s ✐♥ ♣r♦❣r❛♠s ❛♥❞ ❧❛t❡r✱ ✐❢ ♥❡❝❡ss❛r②✱ ❣♦ ❜❛❝❦ t♦ t❤❡ ♠♦r❡ ♠❛t❤❡♠❛t✐❝❛❧ ❞❡t❛✐❧s ✐♥ t❤❡ ✜rst ♣❛rt ♦❢ ❈❤✳ ✺✳
❚❤❡ ♠✐♥✐♠❛❧ ♥❡❡❞✲t♦✲❦♥♦✇ ❛❜♦✉t ❛rr❛②s
❆rr❛②s ❛r❡ ❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ✈❡❝t♦rs ✇❤❡r❡ ✇❡ ❝❛♥ ❤❛✈❡ ♠✉❧t✐♣❧❡ ✐♥❞✐❝❡s✿ Ai,j✱ Ai,j,k ❊①❛♠♣❧❡✿ t❛❜❧❡ ♦❢ ♥✉♠❜❡rs✱ ♦♥❡ ✐♥❞❡① ❢♦r t❤❡ r♦✇✱ ♦♥❡ ❢♦r t❤❡ ❝♦❧✉♠♥ ✵ ✶✷ −✶ ✺ −✶ −✶ −✶ ✵ ✶✶ ✺ ✺ −✷ A = A✵,✵ · · · A✵,n−✶ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ Am−✶,✵ · · · Am−✶,n−✶ ❚❤❡ ♥♦ ♦❢ ✐♥❞✐❝❡s ✐♥ ❛♥ ❛rr❛② ✐s t❤❡ r❛♥❦ ♦r ♥✉♠❜❡r ♦❢ ❞✐♠❡♥s✐♦♥s ❱❡❝t♦r ❂ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❛rr❛②✱ ♦r r❛♥❦ ✶ ❛rr❛② ■♥ P②t❤♦♥ ❝♦❞❡✱ ✇❡ ✉s❡ ◆✉♠❡r✐❝❛❧ P②t❤♦♥ ❛rr❛②s ✐♥st❡❛❞ ♦❢ ♥❡st❡❞ ❧✐sts t♦ r❡♣r❡s❡♥t ♠❛t❤❡♠❛t✐❝❛❧ ❛rr❛②s ✭❜❡❝❛✉s❡ t❤✐s ✐s ❝♦♠♣✉t❛t✐♦♥❛❧❧② ♠♦r❡ ❡✣❝✐❡♥t✮
❙t♦r✐♥❣ ✭①✱②✮ ♣♦✐♥ts ♦♥ ❛ ❝✉r✈❡ ✐♥ ❧✐sts
❈♦❧❧❡❝t ♣♦✐♥ts ♦♥ ❛ ❢✉♥❝t✐♦♥ ❝✉r✈❡ y = f (x) ✐♥ ❧✐sts✿
❃❃❃ ❞❡❢ ❢✭①✮✿ ✳✳✳ r❡t✉r♥ ①✯✯✸ ✳✳✳ ❃❃❃ ♥ ❂ ✺ ★ ♥♦ ♦❢ ♣♦✐♥ts ❃❃❃ ❞① ❂ ✶✳✵✴✭♥✲✶✮ ★ ① s♣❛❝✐♥❣ ✐♥ ❬✵✱✶❪ ❃❃❃ ①❧✐st ❂ ❬✐✯❞① ❢♦r ✐ ✐♥ r❛♥❣❡✭♥✮❪ ❃❃❃ ②❧✐st ❂ ❬❢✭①✮ ❢♦r ① ✐♥ ①❧✐st❪ ❃❃❃ ♣❛✐rs ❂ ❬❬①✱ ②❪ ❢♦r ①✱ ② ✐♥ ③✐♣✭①❧✐st✱ ②❧✐st✮❪
❚✉r♥ ❧✐sts ✐♥t♦ ◆✉♠❡r✐❝❛❧ P②t❤♦♥ ✭◆✉♠P②✮ ❛rr❛②s✿
❃❃❃ ✐♠♣♦rt ♥✉♠♣② ❛s ♥♣ ★ ♠♦❞✉❧❡ ❢♦r ❛rr❛②s ❃❃❃ ① ❂ ♥♣✳❛rr❛②✭①❧✐st✮ ★ t✉r♥ ❧✐st ①❧✐st ✐♥t♦ ❛rr❛② ❃❃❃ ② ❂ ♥♣✳❛rr❛②✭②❧✐st✮