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❈❤❡❜②s❤❡✈ ■♥t❡r♣♦❧❛t✐♦♥ P♦❧②♥♦♠✐❛❧✲❜❛s❡❞ ❚♦♦❧s ❢♦r ❘✐❣♦r♦✉s ❈♦♠♣✉t✐♥❣

◆✐❝♦❧❛s ❇r✐s❡❜❛rr❡ ▼✐♦❛r❛ ❏♦❧❞❡s ❆♣r✐❧ ✾✱ ✷✵✶✵

✶ ✴ ✸✽

▼♦t✐✈❛t✐♦♥

✷ ✴ ✸✽

▼♦t✐✈❛t✐♦♥

✷ ✴ ✸✽

▼♦t✐✈❛t✐♦♥

✷ ✴ ✸✽

▼♦t✐✈❛t✐♦♥

✷ ✴ ✸✽

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s

❲❤②❄

• ❡t t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ♥♦t ❛♥ ✑❛❧♠♦st✑ ❝♦rr❡❝t ♦♥❡

❇r✐❞❣❡ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s❝✐❡♥t✐✜❝ ❝♦♠♣✉t✐♥❣ ❛♥❞ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s ✲ s♣❡❡❞ ❛♥❞ r❡❧✐❛❜✐❧✐t②

❍♦✇❄

❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ ❞✐s❝r❡t✐③❛t✐♦♥✱ tr✉♥❝❛t✐♦♥ ❡rr♦rs ✐♥ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❈♦♠♣✉t❡ ❡♥❝❧♦s✉r❡s ✐♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s

❲❤❛t❄

✶✳ ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✭■❆✮ ✷✳ ❚❛②❧♦r ♠♦❞❡❧s ✭❚▼✮ ✸✳ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❈▼✮

❲❤❡r❡❄ ❇❡❛♠ P❤②s✐❝s ✭▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✮✱ ▲♦r❡♥t③ ❛ttr❛❝t♦r ✭❲✳ ❚✉❝❦❡r✮✱ ❋❧②s♣❡❝❦ ♣r♦❥❡❝t ✭❘✳ ❩✉♠❦❡❧❧❡r✮

✸ ✴ ✸✽

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s

❲❤②❄

• ❡t t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ♥♦t ❛♥ ✑❛❧♠♦st✑ ❝♦rr❡❝t ♦♥❡

❇r✐❞❣❡ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s❝✐❡♥t✐✜❝ ❝♦♠♣✉t✐♥❣ ❛♥❞ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s ✲ s♣❡❡❞ ❛♥❞ r❡❧✐❛❜✐❧✐t②

❍♦✇❄

❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ ❞✐s❝r❡t✐③❛t✐♦♥✱ tr✉♥❝❛t✐♦♥ ❡rr♦rs ✐♥ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❈♦♠♣✉t❡ ❡♥❝❧♦s✉r❡s ✐♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s

❲❤❛t❄

✶✳ ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✭■❆✮ ✷✳ ❚❛②❧♦r ♠♦❞❡❧s ✭❚▼✮ ✸✳ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❈▼✮

❲❤❡r❡❄ ❇❡❛♠ P❤②s✐❝s ✭▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✮✱ ▲♦r❡♥t③ ❛ttr❛❝t♦r ✭❲✳ ❚✉❝❦❡r✮✱ ❋❧②s♣❡❝❦ ♣r♦❥❡❝t ✭❘✳ ❩✉♠❦❡❧❧❡r✮

✸ ✴ ✸✽

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s

❲❤②❄

• ❡t t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ♥♦t ❛♥ ✑❛❧♠♦st✑ ❝♦rr❡❝t ♦♥❡

❇r✐❞❣❡ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s❝✐❡♥t✐✜❝ ❝♦♠♣✉t✐♥❣ ❛♥❞ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s ✲ s♣❡❡❞ ❛♥❞ r❡❧✐❛❜✐❧✐t②

❍♦✇❄

❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ ❞✐s❝r❡t✐③❛t✐♦♥✱ tr✉♥❝❛t✐♦♥ ❡rr♦rs ✐♥ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❈♦♠♣✉t❡ ❡♥❝❧♦s✉r❡s ✐♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s

❲❤❛t❄

✶✳ ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✭■❆✮ ✷✳ ❚❛②❧♦r ♠♦❞❡❧s ✭❚▼✮ ✸✳ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❈▼✮

❲❤❡r❡❄ ❇❡❛♠ P❤②s✐❝s ✭▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✮✱ ▲♦r❡♥t③ ❛ttr❛❝t♦r ✭❲✳ ❚✉❝❦❡r✮✱ ❋❧②s♣❡❝❦ ♣r♦❥❡❝t ✭❘✳ ❩✉♠❦❡❧❧❡r✮

✸ ✴ ✸✽

❘✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ t♦♦❧s

❲❤②❄

• ❡t t❤❡ ❝♦rr❡❝t ❛♥s✇❡r✱ ♥♦t ❛♥ ✑❛❧♠♦st✑ ❝♦rr❡❝t ♦♥❡

❇r✐❞❣❡ t❤❡ ❣❛♣ ❜❡t✇❡❡♥ s❝✐❡♥t✐✜❝ ❝♦♠♣✉t✐♥❣ ❛♥❞ ♣✉r❡ ♠❛t❤❡♠❛t✐❝s ✲ s♣❡❡❞ ❛♥❞ r❡❧✐❛❜✐❧✐t②

❍♦✇❄

❯s❡ ❋❧♦❛t✐♥❣✲P♦✐♥t ❛s s✉♣♣♦rt ❢♦r ❝♦♠♣✉t❛t✐♦♥s ✭❢❛st✮ ❇♦✉♥❞ r♦✉♥❞♦✛✱ ❞✐s❝r❡t✐③❛t✐♦♥✱ tr✉♥❝❛t✐♦♥ ❡rr♦rs ✐♥ ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠s ❈♦♠♣✉t❡ ❡♥❝❧♦s✉r❡s ✐♥st❡❛❞ ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥s

❲❤❛t❄

✶✳ ■♥t❡r✈❛❧ ❛r✐t❤♠❡t✐❝ ✭■❆✮ ✷✳ ❚❛②❧♦r ♠♦❞❡❧s ✭❚▼✮ ✸✳ ❈❤❡❜②s❤❡✈ ♠♦❞❡❧s ✭❈▼✮

❲❤❡r❡❄ ❇❡❛♠ P❤②s✐❝s ✭▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✮✱ ▲♦r❡♥t③ ❛ttr❛❝t♦r ✭❲✳ ❚✉❝❦❡r✮✱ ❋❧②s♣❡❝❦ ♣r♦❥❡❝t ✭❘✳ ❩✉♠❦❡❧❧❡r✮

✸ ✴ ✸✽

❲❤❛t ❦✐♥❞ ♦❢ ♣r♦❜❧❡♠s ❝❛♥ ✇❡ ✭❈▼✮ ❛❞❞r❡ss ❄

❈✉rr❡♥t❧② ✇❡ ❝♦♥s✐❞❡r ✉♥✐✈❛r✐❛t❡ ❢✉♥❝t✐♦♥s f✱ ✏s✉✣❝✐❡♥t❧② s♠♦♦t❤✑ ♦✈❡r [a, b]✳ Pr❛❝t✐❝❛❧ ❊①❛♠♣❧❡s✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❢✉♥❝t✐♦♥s✿ ✇❤❡r❡ ✐s ❛ ✈❡r② ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ✳ ❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡✿

✹ ✴ ✸✽

❲❤❛t ❦✐♥❞ ♦❢ ♣r♦❜❧❡♠s ❝❛♥ ✇❡ ✭❈▼✮ ❛❞❞r❡ss ❄

❈✉rr❡♥t❧② ✇❡ ❝♦♥s✐❞❡r ✉♥✐✈❛r✐❛t❡ ❢✉♥❝t✐♦♥s f✱ ✏s✉✣❝✐❡♥t❧② s♠♦♦t❤✑ ♦✈❡r [a, b]✳ Pr❛❝t✐❝❛❧ ❊①❛♠♣❧❡s✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❢✉♥❝t✐♦♥s✿ sup

x∈[a, b]

{|f(x) − g(x)|}, ✇❤❡r❡ g ✐s ❛ ✈❡r② ❣♦♦❞ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ f✳ ❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡✿ π =

1

• 4

1 + x2 dx

✹ ✴ ✸✽

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ✱ ❜✉t ■♠

✺ ✴ ✸✽

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs π ∈ [3.1415, 3.1416] ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ✱ ❜✉t ■♠

✺ ✴ ✸✽

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs π ∈ [3.1415, 3.1416] ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ [1, 2] + [−3, 2] = [−2, 4] ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ✱ ❜✉t ■♠

✺ ✴ ✸✽

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs π ∈ [3.1415, 3.1416] ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ [1, 2] + [−3, 2] = [−2, 4] ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ x ∈ [−1, 2], f(x) = x2 + x + 1 F(X) = X2 + X + 1 F([−1, 2]) = [−1, 2]2 + [−1, 2] + [1, 1] F([−1, 2]) = [0, 4] + [−1, 2] + [1, 1] F([−1, 2]) = [0, 7] ✱ ❜✉t ■♠

✺ ✴ ✸✽

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs π ∈ [3.1415, 3.1416] ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ [1, 2] + [−3, 2] = [−2, 4] ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ x ∈ [−1, 2], f(x) = x2 + x + 1 F(X) = X2 + X + 1 F([−1, 2]) = [−1, 2]2 + [−1, 2] + [1, 1] F([−1, 2]) = [0, 4] + [−1, 2] + [1, 1] F([−1, 2]) = [0, 7] x ∈ [−1, 2], f(x) ∈ [0, 7]✱ ❜✉t ■♠(f) = [3/4, 7]

✺ ✴ ✸✽

❲❤❡♥ ■❆ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

✱ ✱ s✳t✳ ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮

✻ ✴ ✸✽

❲❤❡♥ ■❆ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = ex, x ∈ [0, 1]✱ p(x) = 5

i=0 cixi✱ ε(x) = f(x) − p(x)

s✳t✳ ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮

✻ ✴ ✸✽

❲❤❡♥ ■❆ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = ex, x ∈ [0, 1]✱ p(x) = 5

i=0 cixi✱ ε(x) = f(x) − p(x)

s✳t✳ ε∞ = supx∈[a, b]{|ε(x)|} ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮

✻ ✴ ✸✽

❲❤❡♥ ■❆ ❞♦❡s ♥♦t s✉✣❝❡✿ ❈♦♠♣✉t✐♥❣ s✉♣r❡♠✉♠ ♥♦r♠s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs

f(x) = ex, x ∈ [0, 1]✱ p(x) = 5

i=0 cixi✱ ε(x) = f(x) − p(x)

s✳t✳ ε∞ = supx∈[a, b]{|ε(x)|} ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮

✼ ✴ ✸✽

❲❤② ■❆ ❞♦❡s ♥♦t s✉✣❝❡✿ ❖✈❡r❡st✐♠❛t✐♦♥

f(x) = ex, x ∈ [0, 1]✱ p(x) = 5

i=0 cixi✱ ε(x) = f(x) − p(x)

s✳t✳ ε∞ = supx∈[a, b]{|ε(x)|} ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮ ❯s✐♥❣ ■❆✱ ε(x) ∈ [−0.4, 0.4]✱ ❜✉t ε(x)∞ ≃ 1.1295 · 10−6✿

✽ ✴ ✸✽

❲❤② ■❆ ❞♦❡s ♥♦t s✉✣❝❡✿ ❖✈❡r❡st✐♠❛t✐♦♥

❖✈❡r❡st✐♠❛t✐♦♥ ❝❛♥ ❜❡ r❡❞✉❝❡❞ ❜② ✉s✐♥❣ ✐♥t❡r✈❛❧s ♦❢ s♠❛❧❧❡r ✇✐❞t❤✳ ■♥ t❤✐s ❝❛s❡✱ ♦✈❡r [0, 1] ✇❡ ♥❡❡❞ 107 ✐♥t❡r✈❛❧s✦

✾ ✴ ✸✽

❘✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

r❡♣❧❛❝❡❞ ✇✐t❤ ❛ r✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✿ ✲ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❞❡❣r❡❡ ✲ ✐♥t❡r✈❛❧ s✳ t✳ ▼❛✐♥ ♣♦✐♥t ♦❢ t❤✐s t❛❧❦✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❄

✶✵ ✴ ✸✽

❘✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

f r❡♣❧❛❝❡❞ ✇✐t❤ ❛ r✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✿ ✲ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ T ♦❢ ❞❡❣r❡❡ n ✲ ✐♥t❡r✈❛❧ s✳ t✳ ▼❛✐♥ ♣♦✐♥t ♦❢ t❤✐s t❛❧❦✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❄

✶✵ ✴ ✸✽

❘✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

f r❡♣❧❛❝❡❞ ✇✐t❤ ❛ r✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✿ ✲ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ T ♦❢ ❞❡❣r❡❡ n ✲ ✐♥t❡r✈❛❧ ∆ s✳ t✳ f(x) − T(x) ∈ ∆, ∀x ∈ [a, b] ▼❛✐♥ ♣♦✐♥t ♦❢ t❤✐s t❛❧❦✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❄

✶✵ ✴ ✸✽

❘✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s

f r❡♣❧❛❝❡❞ ✇✐t❤ ❛ r✐❣♦r♦✉s ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ✿ (T, ∆) ✲ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ T ♦❢ ❞❡❣r❡❡ n ✲ ✐♥t❡r✈❛❧ ∆ s✳ t✳ f(x) − T(x) ∈ ∆, ∀x ∈ [a, b] ▼❛✐♥ ♣♦✐♥t ♦❢ t❤✐s t❛❧❦✿ ❍♦✇ t♦ ❝♦♠♣✉t❡ (T, ∆) ❄

✶✵ ✴ ✸✽

❚❛②❧♦r ▼♦❞❡❧s ✲ ❍♦✇ ❞♦ ✇❡ ♦❜t❛✐♥ t❤❡♠❄

■❞❡❛✿ ❈♦♥s✐❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ▲❡t ✱ t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ ♦✈❡r ❛r♦✉♥❞ ✳

r❡♠❛✐♥❞❡r

✱ ✱ ❧✐❡s str✐❝t❧② ❜❡t✇❡❡♥ ❛♥❞ ❍♦✇ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❄ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❛♥ ✐♥t❡r✈❛❧ ❡♥❝❧♦s✉r❡ ❢♦r ❄

✶✶ ✴ ✸✽

❚❛②❧♦r ▼♦❞❡❧s ✲ ❍♦✇ ❞♦ ✇❡ ♦❜t❛✐♥ t❤❡♠❄

■❞❡❛✿ ❈♦♥s✐❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ▲❡t n ∈ N✱ n + 1 t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ♦✈❡r [a, b] ❛r♦✉♥❞ x0✳ f(x) =

n

• i=0

f(i)(x0)(x − x0)i i!

• T(x)

+ ∆n(x, ξ)

• r❡♠❛✐♥❞❡r

∆n(x, ξ) = f(n+1)(ξ)(x − x0)n+1 (n + 1)! ✱ x ∈ [a, b]✱ ξ ❧✐❡s str✐❝t❧② ❜❡t✇❡❡♥ x ❛♥❞ x0 ❍♦✇ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ ❄ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❛♥ ✐♥t❡r✈❛❧ ❡♥❝❧♦s✉r❡ ❢♦r ❄

✶✶ ✴ ✸✽

❚❛②❧♦r ▼♦❞❡❧s ✲ ❍♦✇ ❞♦ ✇❡ ♦❜t❛✐♥ t❤❡♠❄

■❞❡❛✿ ❈♦♥s✐❞❡r ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ▲❡t n ∈ N✱ n + 1 t✐♠❡s ❞✐✛❡r❡♥t✐❛❜❧❡ ❢✉♥❝t✐♦♥ f ♦✈❡r [a, b] ❛r♦✉♥❞ x0✳ f(x) =

n

• i=0

f(i)(x0)(x − x0)i i!

• T(x)

+ ∆n(x, ξ)

• r❡♠❛✐♥❞❡r

∆n(x, ξ) = f(n+1)(ξ)(x − x0)n+1 (n + 1)! ✱ x ∈ [a, b]✱ ξ ❧✐❡s str✐❝t❧② ❜❡t✇❡❡♥ x ❛♥❞ x0 ❍♦✇ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡✣❝✐❡♥ts f(i)(x0) i! ♦❢ T(x) ❄ ❍♦✇ t♦ ❝♦♠♣✉t❡ ❛♥ ✐♥t❡r✈❛❧ ❡♥❝❧♦s✉r❡ ∆ ❢♦r ∆n(x, ξ) ❄

✶✶ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ P♦✐♥t ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)(x0) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ ✱ ✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥

❝♦♠♣✉t❡

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ P♦✐♥t ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)(x0) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ ✱ ✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)(0)

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ P♦✐♥t ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)(x0) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)(0)

sin(x) → u = [sin(0), cos(0), − sin(0), − cos(0), sin(0)] cos(x) → v = [cos(0), − sin(0), − cos(0), sin(0), cos(0)]

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ P♦✐♥t ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)(x0) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ f(i)(x0) = i

k=0 uk vi−k

✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)(0)

sin(x) → u = [sin(0), cos(0), − sin(0), − cos(0), sin(0)] cos(x) → v = [cos(0), − sin(0), − cos(0), sin(0), cos(0)]

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ P♦✐♥t ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)(x0) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ f(i)(x0) = i

k=0 uk vi−k

✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)(0)

sin(x) → u = [sin(0), cos(0), − sin(0), − cos(0), sin(0)] cos(x) → v = [cos(0), − sin(0), − cos(0), sin(0), cos(0)] f(x) → [u0 v0, u0 v1+u1 v0, . . . , u0 v4+u1 v3+u2 v2+u3 v1+u4 v0]

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ P♦✐♥t ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)(x0) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ f(i)(x0) = i

k=0 uk vi−k

✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)(0)

sin(x) → u = [sin(0), cos(0), − sin(0), − cos(0), sin(0)] cos(x) → v = [cos(0), − sin(0), − cos(0), sin(0), cos(0)] f(x) → [u0 v0, u0 v1+u1 v0, . . . , u0 v4+u1 v3+u2 v2+u3 v1+u4 v0]

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ P♦✐♥t ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)(x0) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ f(i)(x0) = i

k=0 uk vi−k

✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)(0)

sin(x) → u = [sin(0), cos(0), − sin(0), − cos(0), sin(0)] cos(x) → v = [cos(0), − sin(0), − cos(0), sin(0), cos(0)] f(x) → [u0 v0, u0 v1+u1 v0, . . . , u0 v4+u1 v3+u2 v2+u3 v1+u4 v0]

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ ▲❛r❣❡r ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)([a, b]) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ f(i)(x0) = i

k=0 uk vi−k

✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)(0)

sin(x) → u = [sin(0), cos(0), − sin(0), − cos(0), sin(0)] cos(x) → v = [cos(0), − sin(0), − cos(0), sin(0), cos(0)] f(x) → [u0 v0, u0 v1+u1 v0, . . . , u0 v4+u1 v3+u2 v2+u3 v1+u4 v0]

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ ▲❛r❣❡r ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)([a, b]) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ f(i)([a, b]) = i

k=0 uk vi−k

✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)([0, 1])

sin(x) → u = [sin(0), cos(0), − sin(0), − cos(0), sin(0)] cos(x) → v = [cos(0), − sin(0), − cos(0), sin(0), cos(0)] f(x) → [u0 v0, u0 v1+u1 v0, . . . , u0 v4+u1 v3+u2 v2+u3 v1+u4 v0]

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ ▲❛r❣❡r ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)([a, b]) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ f(i)([a, b]) = i

k=0 uk vi−k

✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)([0, 1])

sin(x) → U = [[0, 0.85], [0.54, 1], [−0.85, 0], [−1, −0.54], [0, 0.85]] cos(x) → U = [[0.54, 1], [−0.85, 0], [−1, −0.55], [0, 0.85], [0.54, 1]] f(x) → [u0 v0, u0 v1+u1 v0, . . . , u0 v4+u1 v3+u2 v2+u3 v1+u4 v0]

✶✷ ✴ ✸✽

❆✉t♦♠❛t✐❝ ❉✐✛❡r❡♥t✐❛t✐♦♥ ✲ ▲❛r❣❡r ✐♥t❡r✈❛❧s

❈♦♠♣✉t❡ f(i)([a, b]) ✲ f r❡♣r❡s❡♥t❡❞ ❛s ❛♥ ❡①♣r❡ss✐♦♥ tr❡❡ ✲ ❙✐♠♣❧❡ ❢♦r♠✉❧❛s ❢♦r ❞❡r✐✈❛t✐✈❡s ♦❢ ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑✿ exp✱ sin✱ ❡t❝✳ ✲ ▲❡✐❜♥✐t③ ❢♦r♠✉❧❛✿ f(i)([a, b]) = i

k=0 uk vi−k

✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ r❡❝✉rs✐✈❡❧② ❛♣♣❧② ♦♣❡r❛t✐♦♥s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ❊①❛♠♣❧❡✿

• ✐✈❡♥ f(x) = sin(x) cos(x), ❝♦♠♣✉t❡ f(4)([0, 1])

sin(x) → U = [[0, 0.85], [0.54, 1], [−0.85, 0], [−1, −0.54], [0, 0.85]] cos(x) → U = [[0.54, 1], [−0.85, 0], [−1, −0.55], [0, 0.85], [0.54, 1]] f(x) → [u0 v0, u0 v1 + u1 v0, . . . , [0, 13.5]] ❇✉t f(4)([0, 1]) = [0, 8]

✶✷ ✴ ✸✽

❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ f ✐s ❛ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥❄

f(x) =

n

• i=0

f(i)(x0)(x − x0)i i!

• T(x)

+ ∆n(x, ξ)

• r❡♠❛✐♥❞❡r

❚❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ∆ ❢♦r ∆n(x, ξ) = f(n+1)(ξ)(x − x0)n+1 (n + 1)! ✱ ξ ∈ [a, b] ❝❛♥ ❜❡ ❧❛r❣❡❧② ♦✈❡r❡st✐♠❛t❡❞✳

✶✸ ✴ ✸✽

❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ f ✐s ❛ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥❄

❚❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ∆ ❢♦r ∆n(x, ξ) = f(n+1)(ξ)(x − x0)n+1 (n + 1)! ✱ ξ ∈ [a, b] ❝❛♥ ❜❡ ❧❛r❣❡❧② ♦✈❡r❡st✐♠❛t❡❞✳ ❊①❛♠♣❧❡✿ f(x) = e1/ cos x✱ ♦✈❡r [0, 1]✱ n = 13✱ x0 = 0.5✳ ❯s✐♥❣ ❆❉✿ ∆ = [−1.93 · 102, 1.35 · 103] ■♥ ❢❛❝t✱ f(x) − T(x) ∈ [0, 4.56 · 10−3]

✶✸ ✴ ✸✽

❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ f ✐s ❛ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥❄

❚❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ∆ ❢♦r ∆n(x, ξ) = f(n+1)(ξ)(x − x0)n+1 (n + 1)! ✱ ξ ∈ [a, b] ❝❛♥ ❜❡ ❧❛r❣❡❧② ♦✈❡r❡st✐♠❛t❡❞✳ ◗✿ ❲❤❛t ❞♦❡s ✐♥✢✉❡♥❝❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ❢♦r t❤❡ r❡♠❛✐♥❞❡r❄ ✲ ❚❤❡ ✇❛② ✇❡ ❝♦♠♣✉t❡ ❛♥ ✐♥t❡r✈❛❧ ❡♥❝❧♦s✉r❡ ❢♦r t❤❡ r❡♠❛✐♥❞❡r ✉s✐♥❣ s✐♠♣❧② t❤✐s ❢♦r♠✉❧❛ ❢♦r ❛♥② ❢✉♥❝t✐♦♥

✶✸ ✴ ✸✽

❲❤❛t ❤❛♣♣❡♥s ✇❤❡♥ f ✐s ❛ ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥❄

❚❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ∆ ❢♦r ∆n(x, ξ) = f(n+1)(ξ)(x − x0)n+1 (n + 1)! ✱ ξ ∈ [a, b] ❝❛♥ ❜❡ ❧❛r❣❡❧② ♦✈❡r❡st✐♠❛t❡❞✳ ◗✿ ❲❤❛t ❞♦❡s ✐♥✢✉❡♥❝❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ❢♦r t❤❡ r❡♠❛✐♥❞❡r❄ ✲ ❚❤❡ ✇❛② ✇❡ ❝♦♠♣✉t❡ ❛♥ ✐♥t❡r✈❛❧ ❡♥❝❧♦s✉r❡ ❢♦r t❤❡ r❡♠❛✐♥❞❡r ✉s✐♥❣ s✐♠♣❧② t❤✐s ❢♦r♠✉❧❛ ❢♦r ❛♥② ❢✉♥❝t✐♦♥

✶✸ ✴ ✸✽

❚❛②❧♦r ▼♦❞❡❧s P❤✐❧♦s♦♣❤②

❋♦r ❜♦✉♥❞✐♥❣ t❤❡ r❡♠❛✐♥❞❡rs✿ ❋♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ✉s❡ ❆❉✳ ❋♦r ✏❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✑✉s❡ ❛ t✇♦✲st❡♣ ♣r♦❝❡❞✉r❡✿ ✲ ❝♦♠♣✉t❡ ♠♦❞❡❧s (T, I) ❢♦r ❛❧❧ ❜❛s✐❝ ❢✉♥❝t✐♦♥s❀ ✲ ❛♣♣❧② ❛❧❣❡❜r❛✐❝ r✉❧❡s ✇✐t❤ t❤❡s❡ ♠♦❞❡❧s✱ ✐♥st❡❛❞ ♦❢ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥s✳

✶✹ ✴ ✸✽

❚❛②❧♦r ▼♦❞❡❧s ■ss✉❡s

❊①❛♠♣❧❡✿ f(x) = arctan(x) ♦✈❡r [−0.9, 0.9] p(x) ✲ ♠✐♥✐♠❛①✱ ❞❡❣r❡❡ 15 ε(x) = p(x) − f(x) ε∞ ≃ 10−8

✶✺ ✴ ✸✽

❚❛②❧♦r ▼♦❞❡❧s ■ss✉❡s

❊①❛♠♣❧❡✿ f(x) = arctan(x) ♦✈❡r [−0.9, 0.9] p(x) ✲ ♠✐♥✐♠❛①✱ ❞❡❣r❡❡ 15 ε(x) = p(x) − f(x) ε∞ ≃ 10−8 ■♥ t❤✐s ❝❛s❡ ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ♥♦t ❣♦♦❞✱ ✇❡ ♥❡❡❞ t❤❡♦r❡t✐❝❛❧❧② ❛ ❚▼ ♦❢ ❞❡❣r❡❡ 120✳ Pr❛❝t✐❝❛❧❧②✱ t❤❡ ❝♦♠♣✉t❡❞ ✐♥t❡r✈❛❧ r❡♠❛✐♥❞❡r ❝❛♥ ♥♦t ❜❡ ♠❛❞❡ s✉✣❝✐❡♥t❧② s♠❛❧❧ ❞✉❡ t♦ ♦✈❡r❡st✐♠❛t✐♦♥ ❈♦♥s❡q✉❡♥❝❡✿ ❘❡♠❛✐♥❞❡r ❜♦✉♥❞s ❛r❡ ✉♥s❛t✐s❢❛❝t♦r② ✐♥ ♦✉r ❝❛s❡✳

✶✺ ✴ ✸✽

❖✉r ❆♣♣r♦❛❝❤ ✲ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s

❇❛s✐❝ ✐❞❡❛✿ ✲ ❯s❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❜❡tt❡r t❤❛♥ ❚❛②❧♦r✿ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧✳ ✲ ❯s❡ t❤❡ t✇♦ st❡♣ ❛♣♣r♦❛❝❤ ❛s ❚❛②❧♦r ▼♦❞❡❧s✿ ❝♦♠♣✉t❡ ♠♦❞❡❧s (P, I) ❢♦r ❜❛s✐❝ ❢✉♥❝t✐♦♥s❀ ❛♣♣❧② ❛❧❣❡❜r❛✐❝ r✉❧❡s ✇✐t❤ t❤❡s❡ ♠♦❞❡❧s✱ ✐♥st❡❛❞ ♦❢ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥s✳ ◆♦t❡✿ ❈❤❡❜❢✉♥ ✲ ✑❈♦♠♣✉t✐♥❣ ◆✉♠❡r✐❝❛❧❧② ✇✐t❤ ❋✉♥❝t✐♦♥s ■♥st❡❛❞ ♦❢ ◆✉♠❜❡rs✏ ✭◆✳ ❚r❡❢❡t❤❡♥ ❡t ❛❧✳✮✿ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❛❧r❡❛❞② ✉s❡❞✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ♥♦t r✐❣♦r♦✉s ✳

✶✻ ✴ ✸✽

❖✉r ❆♣♣r♦❛❝❤ ✲ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s

❇❛s✐❝ ✐❞❡❛✿ ✲ ❯s❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❜❡tt❡r t❤❛♥ ❚❛②❧♦r✿ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧✳ ✲ ❯s❡ t❤❡ t✇♦ st❡♣ ❛♣♣r♦❛❝❤ ❛s ❚❛②❧♦r ▼♦❞❡❧s✿ ❝♦♠♣✉t❡ ♠♦❞❡❧s (P, I) ❢♦r ❜❛s✐❝ ❢✉♥❝t✐♦♥s❀ ❛♣♣❧② ❛❧❣❡❜r❛✐❝ r✉❧❡s ✇✐t❤ t❤❡s❡ ♠♦❞❡❧s✱ ✐♥st❡❛❞ ♦❢ ♦♣❡r❛t✐♦♥s ✇✐t❤ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥s✳ ◆♦t❡✿ ❈❤❡❜❢✉♥ ✲ ✑❈♦♠♣✉t✐♥❣ ◆✉♠❡r✐❝❛❧❧② ✇✐t❤ ❋✉♥❝t✐♦♥s ■♥st❡❛❞ ♦❢ ◆✉♠❜❡rs✏ ✭◆✳ ❚r❡❢❡t❤❡♥ ❡t ❛❧✳✮✿ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧s ❛r❡ ❛❧r❡❛❞② ✉s❡❞✱ ❜✉t t❤❡ ❛♣♣r♦❛❝❤ ✐s ♥♦t r✐❣♦r♦✉s✳

✶✻ ✴ ✸✽

❖✉r ❆♣♣r♦❛❝❤ ✲ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s

❈♦♠♣✉t❡ ♠♦❞❡❧s (P, I) ❢♦r ❜❛s✐❝ ❢✉♥❝t✐♦♥s f✱ ✇❤❡r❡ P ✐s t❤❡ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧ ❍♦✇ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ P❄ ❍♦✇ t♦ ❜♦✉♥❞ t❤❡ r❡♠❛✐♥❞❡r❄ ❲❤❛t ❜❛s✐s ❢♦r r❡♣r❡s❡♥t✐♥❣ P❄ ❲❤❛t ❛r❡ t❤❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✇✐t❤ t❤❡s❡ ♠♦❞❡❧s ❄

✶✼ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❈❤♦✐❝❡ ♦❢ ❇❛s✐s P♦❧②♥♦♠✐❛❧s

❲❡ ✉s❡❞✿ ◆❡✇t♦♥ ❇❛s✐s ❈❤❡❜②s❤❡✈ ❇❛s✐s ✲ ❞✐s❝✉ss❡❞ ✐♥ ✇❤❛t ❢♦❧❧♦✇s ◆♦t❡✿ ❝❤♦✐❝❡ ♦❢ ♦t❤❡r ❜❛s❡s ✐s ♥♦t ❞❡t❛✐❧❡❞ ✐♥ t❤✐s t❛❧❦ ✑▼♦r❛❧ ♣r✐♥❝✐♣❧❡✿ ❯♥❧❡ss ②♦✉✬r❡ r❡❛❧❧②✱ r❡❛❧❧② s✉r❡ t❤❛t ❛♥♦t❤❡r s❡t ♦❢ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐s ❜❡tt❡r✱ ✉s❡ ❈❤❡❜②s❤❡✈ ♣♦❧②♥♦♠✐❛❧s✳✏✱❏✳ P✳ ❇♦②❞

✶✽ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❈❤♦✐❝❡ ♦❢ ❇❛s✐s P♦❧②♥♦♠✐❛❧s

❲❡ ✉s❡❞✿ ◆❡✇t♦♥ ❇❛s✐s ❈❤❡❜②s❤❡✈ ❇❛s✐s ✲ ❞✐s❝✉ss❡❞ ✐♥ ✇❤❛t ❢♦❧❧♦✇s ◆♦t❡✿ ❝❤♦✐❝❡ ♦❢ ♦t❤❡r ❜❛s❡s ✐s ♥♦t ❞❡t❛✐❧❡❞ ✐♥ t❤✐s t❛❧❦ ✑▼♦r❛❧ ♣r✐♥❝✐♣❧❡✿ ❯♥❧❡ss ②♦✉✬r❡ r❡❛❧❧②✱ r❡❛❧❧② s✉r❡ t❤❛t ❛♥♦t❤❡r s❡t ♦❢ ❜❛s✐s ❢✉♥❝t✐♦♥s ✐s ❜❡tt❡r✱ ✉s❡ ❈❤❡❜②s❤❡✈ ♣♦❧②♥♦♠✐❛❧s✳✏✱❏✳ P✳ ❇♦②❞✶

✶❯♥✐✈❡rs✐t② ♦❢ ▼✐❝❤✐❣❛♥✱ ❤tt♣✿✴✴✇✇✇✲♣❡rs♦♥❛❧✳✉♠✐❝❤✳❡❞✉✴∼❥♣❜♦②❞✴ ✶✽ ✴ ✸✽

❈❤❡❜②s❤❡✈ P♦❧②♥♦♠✐❛❧s

❖✈❡r [−1, 1]✱ Tn(x) = cos (n arccos x) , n ≥ 0✳ ✏❈❤❡❜②s❤❡✈ ♥♦❞❡s✑✿ n ❞✐st✐♥❝t r❡❛❧ r♦♦ts ✐♥ [−1, 1] ♦❢ Tn✿ xi = cos

• (i+1/2) π

n

• , i = 0, . . . , n − 1.

✶✾ ✴ ✸✽

■♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧s ✲ ✏❇❛s✐❝✑ ❋✉♥❝t✐♦♥s ❙t❡♣

▲❡t {yi, i = 0, . . . , n} ❜❡ n + 1 ♣♦✐♥ts ✐♥ [−1, 1]✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ♣♦❧②♥♦♠✐❛❧ P ♦❢ ❞❡❣r❡❡ ≤ n s✳t✳ P(yi) = f(yi), ∀i = 0, . . . , n✱ ♦r ✐❢ yi ✐s r❡♣❡❛t❡❞ k t✐♠❡s✱ P (j)(yi) = f(j)(yi), ∀j = 0, . . . , k − 1✳ ❛❧❧ yi ❡q✉❛❧✱ P ✐s t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ f ❛❧❧ yi ❞✐st✐♥❝t✿ ▲❛❣r❛♥❣❡ ✐♥t❡r♣♦❧❛t✐♦♥

✷✵ ✴ ✸✽

■♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧s ✲ ✏❇❛s✐❝✑ ❋✉♥❝t✐♦♥s ❙t❡♣

▲❡t {yi, i = 0, . . . , n} ❜❡ n + 1 ♣♦✐♥ts ✐♥ [−1, 1]✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ♣♦❧②♥♦♠✐❛❧ P ♦❢ ❞❡❣r❡❡ ≤ n s✳t✳ P(yi) = f(yi), ∀i = 0, . . . , n✱ ♦r ✐❢ yi ✐s r❡♣❡❛t❡❞ k t✐♠❡s✱ P (j)(yi) = f(j)(yi), ∀j = 0, . . . , k − 1✳ ❛❧❧ yi ❡q✉❛❧✱ P ✐s t❤❡ ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ f ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r✿ ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) (n + 1)! (x − yi)n+1.

✷✵ ✴ ✸✽

■♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧s ✲ ✏❇❛s✐❝✑ ❋✉♥❝t✐♦♥s ❙t❡♣

▲❡t {yi, i = 0, . . . , n} ❜❡ n + 1 ♣♦✐♥ts ✐♥ [−1, 1]✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ♣♦❧②♥♦♠✐❛❧ P ♦❢ ❞❡❣r❡❡ ≤ n s✳t✳ P(yi) = f(yi), ∀i = 0, . . . , n✱ ♦r ✐❢ yi ✐s r❡♣❡❛t❡❞ k t✐♠❡s✱ P (j)(yi) = f(j)(yi), ∀j = 0, . . . , k − 1✳ ❛❧❧ yi ❞✐st✐♥❝t✿ ▲❛❣r❛♥❣❡ ✐♥t❡r♣♦❧❛t✐♦♥ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r✿ ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) (n + 1)! Wy(x), ✇✐t❤ Wy(x) = n

i=0(x − yi)✳

✷✵ ✴ ✸✽

■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r ✲ ✏❇❛s✐❝✑ ❋✉♥❝t✐♦♥ ❙t❡♣

▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r✿ ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) (n + 1)! Wy(x), ✇✐t❤ Wy(x) = n

i=0(x − yi)✳

◆♦t❡✿ ❖♣t✐♠❛❧ ❝❤♦✐❝❡ ♦❢ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦✐♥ts ✐s t❤❡ ❈❤❡❜②s❤❡✈ ♥♦❞❡s✱ Wx(x) =

1 2n Tn+1 (x)✳

❲❡ s❤♦✉❧❞ ❤❛✈❡ ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦❢ ✐♥ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ r❡♠❛✐♥❞❡r✱ ❝♦♠♣❛r❡❞ t♦ ❚❛②❧♦r r❡♠❛✐♥❞❡r✳ ❳ ❲❡ ✐♥❤❡r✐t ❛❧❧ ✐ss✉❡s r❡❧❛t❡❞ t♦ ♦✈❡r❡st✐♠❛t✐♦♥ ♦❢

✷✶ ✴ ✸✽

■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r ✲ ✏❇❛s✐❝✑ ❋✉♥❝t✐♦♥ ❙t❡♣

▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r✿ ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) (n + 1)! Wy(x), ✇✐t❤ Wy(x) = n

i=0(x − yi)✳

◆♦t❡✿ ❖♣t✐♠❛❧ ❝❤♦✐❝❡ ♦❢ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦✐♥ts ✐s t❤❡ ❈❤❡❜②s❤❡✈ ♥♦❞❡s✱ Wx(x) =

1 2n Tn+1 (x)✳

❲❡ s❤♦✉❧❞ ❤❛✈❡ ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦❢ 2n ✐♥ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ r❡♠❛✐♥❞❡r✱ ❝♦♠♣❛r❡❞ t♦ ❚❛②❧♦r r❡♠❛✐♥❞❡r✳ ❳ ❲❡ ✐♥❤❡r✐t ❛❧❧ ✐ss✉❡s r❡❧❛t❡❞ t♦ ♦✈❡r❡st✐♠❛t✐♦♥ ♦❢ f(n+1)

✷✶ ✴ ✸✽

❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧ ✲ ✏❇❛s✐❝✑ ❋✉♥❝t✐♦♥ ❙t❡♣

P(x) =

n

• i=0

piTi(x) ✐♥t❡r♣♦❧❛t❡s f ❛t xk ∈ [−1, 1]✱ ❈❤❡❜②s❤❡✈ ♥♦❞❡s ♦❢ ♦r❞❡r n + 1✳ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts✿ pi =

n

• k=0

2 n+1f(xk)Ti(xk)✱ i = 0, . . . , n

❘❡♠❛r❦✿ ❈✉rr❡♥t❧②✱ t❤✐s st❡♣ ✐s ♠♦r❡ ❝♦st❧② t❤❛♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❚▼s✳

✷✷ ✴ ✸✽

❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧ ✲ ✏❇❛s✐❝✑ ❋✉♥❝t✐♦♥ ❙t❡♣

P(x) =

n

• i=0

piTi(x) ✐♥t❡r♣♦❧❛t❡s f ❛t xk ∈ [−1, 1]✱ ❈❤❡❜②s❤❡✈ ♥♦❞❡s ♦❢ ♦r❞❡r n + 1✳ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts✿ pi =

n

• k=0

2 n+1f(xk)Ti(xk)✱ i = 0, . . . , n

❘❡♠❛r❦✿ ❈✉rr❡♥t❧②✱ t❤✐s st❡♣ ✐s ♠♦r❡ ❝♦st❧② t❤❛♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❚▼s✳

✷✷ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s

• ✐✈❡♥ t✇♦ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ❢♦r f1 ❛♥❞ f2✱ ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(x) − P1(x) ∈ ∆1 ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ✇❡ ♥❡❡❞ ❛❧❣❡❜r❛✐❝ r✉❧❡s ❢♦r✿ s✳t✳ ✱ ❲❤❡r❡ ✐s✿ ❆❞❞✐t✐♦♥ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❈♦♠♣♦s✐t✐♦♥ ❚❤✐s ✐s t❤❡ ✏❤✐❞❞❡♥ ❞✐✣❝✉❧t ♣❛rt✑ ✐♥ ❞❡s✐❣♥✐♥❣ s✉❝❤ ♠♦❞❡❧s✿ ❤❛s t♦ ❜❡ ❦❡♣t t✐❣❤t✱ ✇❤✐❧❡ ❤❛s t♦ ❜❡ ❝♦♠♣✉t❡❞ ❢❛st✳

✷✸ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s

• ✐✈❡♥ t✇♦ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ❢♦r f1 ❛♥❞ f2✱ ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(x) − P1(x) ∈ ∆1 ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ✇❡ ♥❡❡❞ ❛❧❣❡❜r❛✐❝ r✉❧❡s ❢♦r✿ (P1, ∆1) (P2, ∆2) = (P, ∆) s✳t✳ f1(x) f2(x) − P(x) ∈ ∆✱ ∀x ∈ [a, b] ❲❤❡r❡ ✐s✿ ❆❞❞✐t✐♦♥ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❈♦♠♣♦s✐t✐♦♥ ❚❤✐s ✐s t❤❡ ✏❤✐❞❞❡♥ ❞✐✣❝✉❧t ♣❛rt✑ ✐♥ ❞❡s✐❣♥✐♥❣ s✉❝❤ ♠♦❞❡❧s✿ ❤❛s t♦ ❜❡ ❦❡♣t t✐❣❤t✱ ✇❤✐❧❡ ❤❛s t♦ ❜❡ ❝♦♠♣✉t❡❞ ❢❛st✳

✷✸ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s

• ✐✈❡♥ t✇♦ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ❢♦r f1 ❛♥❞ f2✱ ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(x) − P1(x) ∈ ∆1 ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ✇❡ ♥❡❡❞ ❛❧❣❡❜r❛✐❝ r✉❧❡s ❢♦r✿ (P1, ∆1) (P2, ∆2) = (P, ∆) s✳t✳ f1(x) f2(x) − P(x) ∈ ∆✱ ∀x ∈ [a, b] ❲❤❡r❡ ✐s✿ ❆❞❞✐t✐♦♥ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❈♦♠♣♦s✐t✐♦♥ ❚❤✐s ✐s t❤❡ ✏❤✐❞❞❡♥ ❞✐✣❝✉❧t ♣❛rt✑ ✐♥ ❞❡s✐❣♥✐♥❣ s✉❝❤ ♠♦❞❡❧s✿ ∆ ❤❛s t♦ ❜❡ ❦❡♣t t✐❣❤t✱ ✇❤✐❧❡ (P, ∆) ❤❛s t♦ ❜❡ ❝♦♠♣✉t❡❞ ❢❛st✳

✷✸ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ❆❞❞✐t✐♦♥

• ✐✈❡♥ t✇♦ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ❢♦r f1 ❛♥❞ f2✱ ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(x) − P1(x) ∈ ∆1 ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ❆❞❞✐t✐♦♥ (P1, ∆1) + (P2, ∆2) = (P1 + P2, ∆1 + ∆2).

✷✹ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥

• ✐✈❡♥ t✇♦ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ❢♦r f1 ❛♥❞ f2✱ ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(x) − P1(x) ∈ ∆1 ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❲❡ ♥❡❡❞ ❛❧❣❡❜r❛✐❝ r✉❧❡ ❢♦r✿ (P1, ∆1) · (P2, ∆2) = (P, ∆) s✳t✳ f1(x) · f2(x) − P(x) ∈ ∆✱ ∀x ∈ [a, b] ■♥ ♦✉r ❝❛s❡✱ ❢♦r ❜♦✉♥❞✐♥❣ ✏ s✑✿ ✳

✷✺ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥

• ✐✈❡♥ t✇♦ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ❢♦r f1 ❛♥❞ f2✱ ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(x) − P1(x) ∈ ∆1 ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❲❡ ♥❡❡❞ ❛❧❣❡❜r❛✐❝ r✉❧❡ ❢♦r✿ (P1, ∆1) · (P2, ∆2) = (P, ∆) s✳t✳ f1(x) · f2(x) − P(x) ∈ ∆✱ ∀x ∈ [a, b] f1(x) · f2(x) ∈ P1 · P2 + P2 · ∆1 + P1 · ∆2 + ∆1 · ∆2

• I2

. (P1 · P2)0...n

• P

+ (P1 · P2)n+1...2n

• I1

∆ = I1 + I2 ■♥ ♦✉r ❝❛s❡✱ ❢♦r ❜♦✉♥❞✐♥❣ ✏ s✑✿ ✳

✷✺ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ▼✉❧t✐♣❧✐❝❛t✐♦♥

• ✐✈❡♥ t✇♦ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ❢♦r f1 ❛♥❞ f2✱ ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(x) − P1(x) ∈ ∆1 ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❲❡ ♥❡❡❞ ❛❧❣❡❜r❛✐❝ r✉❧❡ ❢♦r✿ (P1, ∆1) · (P2, ∆2) = (P, ∆) s✳t✳ f1(x) · f2(x) − P(x) ∈ ∆✱ ∀x ∈ [a, b] f1(x) · f2(x) ∈ P1 · P2 + P2 · ∆1 + P1 · ∆2 + ∆1 · ∆2

• I2

. (P1 · P2)0...n

• P

+ (P1 · P2)n+1...2n

• I1

∆ = I1 + I2 ■♥ ♦✉r ❝❛s❡✱ ❢♦r ❜♦✉♥❞✐♥❣ ✏P s✑✿ P = p0 +

n

• i=1

pi · [−1, 1]✳

✷✺ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ❈♦♠♣♦s✐t✐♦♥

• ✐✈❡♥ ❈▼s ❢♦r f1 ♦✈❡r [c, d]✱ ❢♦r f2 ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(y) − P1(y) ∈ ∆1✱ ∀y ∈ [c, d] ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ❘❡♠❛r❦✿ ✐s ❡✈❛❧✉❛t❡❞ ❛t ✳ ❲❡ ♥❡❡❞✿ ✱ ❝❤❡❝❦❡❞ ❜② ❊①tr❛❝t ♣♦❧②♥♦♠✐❛❧ ❛♥❞ r❡♠❛✐♥❞❡r✿ ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ✉s✐♥❣ ♦♥❧② ❛❞❞✐t✐♦♥s ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥s✿ ❈❧❡♥s❤❛✇✬s ❛❧❣♦r✐t❤♠

✷✻ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ❈♦♠♣♦s✐t✐♦♥

• ✐✈❡♥ ❈▼s ❢♦r f1 ♦✈❡r [c, d]✱ ❢♦r f2 ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(y) − P1(y) ∈ ∆1✱ ∀y ∈ [c, d] ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ❘❡♠❛r❦✿ (f1 ◦ f2)(x) ✐s f1 ❡✈❛❧✉❛t❡❞ ❛t y = f2(x)✳ ❲❡ ♥❡❡❞✿ f2([a, b]) ⊆ [c, d]✱ ❝❤❡❝❦❡❞ ❜② P2 + ∆2 ⊆ [c, d] ❊①tr❛❝t ♣♦❧②♥♦♠✐❛❧ ❛♥❞ r❡♠❛✐♥❞❡r✿ ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ✉s✐♥❣ ♦♥❧② ❛❞❞✐t✐♦♥s ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥s✿ ❈❧❡♥s❤❛✇✬s ❛❧❣♦r✐t❤♠

✷✻ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ❈♦♠♣♦s✐t✐♦♥

• ✐✈❡♥ ❈▼s ❢♦r f1 ♦✈❡r [c, d]✱ ❢♦r f2 ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(y) − P1(y) ∈ ∆1✱ ∀y ∈ [c, d] ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ❘❡♠❛r❦✿ (f1 ◦ f2)(x) ✐s f1 ❡✈❛❧✉❛t❡❞ ❛t y = f2(x)✳ ❲❡ ♥❡❡❞✿ f2([a, b]) ⊆ [c, d]✱ ❝❤❡❝❦❡❞ ❜② P2 + ∆2 ⊆ [c, d] f1(y) ∈ P1(y) + ∆1 ❊①tr❛❝t ♣♦❧②♥♦♠✐❛❧ ❛♥❞ r❡♠❛✐♥❞❡r✿ ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ✉s✐♥❣ ♦♥❧② ❛❞❞✐t✐♦♥s ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥s✿ ❈❧❡♥s❤❛✇✬s ❛❧❣♦r✐t❤♠

✷✻ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ❈♦♠♣♦s✐t✐♦♥

• ✐✈❡♥ ❈▼s ❢♦r f1 ♦✈❡r [c, d]✱ ❢♦r f2 ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(y) − P1(y) ∈ ∆1✱ ∀y ∈ [c, d] ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ❘❡♠❛r❦✿ (f1 ◦ f2)(x) ✐s f1 ❡✈❛❧✉❛t❡❞ ❛t y = f2(x)✳ ❲❡ ♥❡❡❞✿ f2([a, b]) ⊆ [c, d]✱ ❝❤❡❝❦❡❞ ❜② P2 + ∆2 ⊆ [c, d] f1(f2(x)) ∈ P1(f2(x)) + ∆1 ❊①tr❛❝t ♣♦❧②♥♦♠✐❛❧ ❛♥❞ r❡♠❛✐♥❞❡r✿ ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ✉s✐♥❣ ♦♥❧② ❛❞❞✐t✐♦♥s ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥s✿ ❈❧❡♥s❤❛✇✬s ❛❧❣♦r✐t❤♠

✷✻ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❖♣❡r❛t✐♦♥s✿ ❈♦♠♣♦s✐t✐♦♥

• ✐✈❡♥ ❈▼s ❢♦r f1 ♦✈❡r [c, d]✱ ❢♦r f2 ♦✈❡r [a, b]✱ ❞❡❣r❡❡ n✿

f1(y) − P1(y) ∈ ∆1✱ ∀y ∈ [c, d] ❛♥❞ f2(x) − P2(x) ∈ ∆2✱ ∀x ∈ [a, b]✳ ❘❡♠❛r❦✿ (f1 ◦ f2)(x) ✐s f1 ❡✈❛❧✉❛t❡❞ ❛t y = f2(x)✳ ❲❡ ♥❡❡❞✿ f2([a, b]) ⊆ [c, d]✱ ❝❤❡❝❦❡❞ ❜② P2 + ∆2 ⊆ [c, d] f1(f2(x)) ∈ P1(P2(x) + ∆2) + ∆1 ❊①tr❛❝t ♣♦❧②♥♦♠✐❛❧ ❛♥❞ r❡♠❛✐♥❞❡r✿ P1 ❝❛♥ ❜❡ ❡✈❛❧✉❛t❡❞ ✉s✐♥❣ ♦♥❧② ❛❞❞✐t✐♦♥s ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥s✿ ❈❧❡♥s❤❛✇✬s ❛❧❣♦r✐t❤♠

✷✻ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❙✉♣r❡♠✉♠ ♥♦r♠ ❡①❛♠♣❧❡

❊①❛♠♣❧❡✿ f(x) = arctan(x) ♦✈❡r [−0.9, 0.9] p(x) ✲ ♠✐♥✐♠❛①✱ ❞❡❣r❡❡ 15 ε(x) = p(x) − f(x) ε∞ ≃ 10−8 ✳

✷✼ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❙✉♣r❡♠✉♠ ♥♦r♠ ❡①❛♠♣❧❡

❊①❛♠♣❧❡✿ f(x) = arctan(x) ♦✈❡r [−0.9, 0.9] p(x) ✲ ♠✐♥✐♠❛①✱ ❞❡❣r❡❡ 15 ε(x) = p(x) − f(x) ε∞ ≃ 10−8 ■♥ t❤✐s ❝❛s❡ ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ♥♦t ❣♦♦❞✱ ✇❡ ♥❡❡❞ t❤❡♦r❡t✐❝❛❧❧② ❛ ❚▼ ♦❢ ❞❡❣r❡❡ 120✳ Pr❛❝t✐❝❛❧❧②✱ t❤❡ ❝♦♠♣✉t❡❞ ✐♥t❡r✈❛❧ r❡♠❛✐♥❞❡r ❝❛♥ ♥♦t ❜❡ ♠❛❞❡ s✉✣❝✐❡♥t❧② s♠❛❧❧ ❞✉❡ t♦ ♦✈❡r❡st✐♠❛t✐♦♥✳ ✳

✷✼ ✴ ✸✽

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s ✲ ❙✉♣r❡♠✉♠ ♥♦r♠ ❡①❛♠♣❧❡

❊①❛♠♣❧❡✿ f(x) = arctan(x) ♦✈❡r [−0.9, 0.9] p(x) ✲ ♠✐♥✐♠❛①✱ ❞❡❣r❡❡ 15 ε(x) = p(x) − f(x) ε∞ ≃ 10−8 ■♥ t❤✐s ❝❛s❡ ❚❛②❧♦r ❛♣♣r♦①✐♠❛t✐♦♥s ❛r❡ ♥♦t ❣♦♦❞✱ ✇❡ ♥❡❡❞ t❤❡♦r❡t✐❝❛❧❧② ❛ ❚▼ ♦❢ ❞❡❣r❡❡ 120✳ Pr❛❝t✐❝❛❧❧②✱ t❤❡ ❝♦♠♣✉t❡❞ ✐♥t❡r✈❛❧ r❡♠❛✐♥❞❡r ❝❛♥ ♥♦t ❜❡ ♠❛❞❡ s✉✣❝✐❡♥t❧② s♠❛❧❧ ❞✉❡ t♦ ♦✈❡r❡st✐♠❛t✐♦♥✳ ❆ ❈▼ ♦❢ ❞❡❣r❡❡ 60 ✇♦r❦s✳

✷✼ ✴ ✸✽

❈▼s ✈s✳ ❚▼s

❖♣❡r❛t✐♦♥s ❝♦♠♣❧❡①✐t②✿ ❆❞❞✐t✐♦♥ ✭ ✮✱ ▼✉❧t✐♣❧✐❝❛t✐♦♥✭ ✮ ❛♥❞ ❈♦♠♣♦s✐t✐♦♥ ✭ ✮ ❤❛✈❡ s✐♠✐❧❛r ❝♦♠♣❧❡①✐t②✳ ❳ ■♥✐t✐❛❧ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❝♦❡✣❝✐❡♥ts ❢♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ✐s s❧♦✇❡r ✇✐t❤ ❈▼s ✭ ✮ ✈s✳ ❚▼s ✭ ✮ ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ r❡♠❛✐♥❞❡r ❜♦✉♥❞s ❢♦r s❡✈❡r❛❧ ❢✉♥❝t✐♦♥s✿

f(x)✱ I✱ n ❈▼ ❊①❛❝t ❜♦✉♥❞ ❚▼ ❊①❛❝t ❜♦✉♥❞ sin(x)✱ [3, 4]✱ 10 1.19 · 10−14 1.13 · 10−14 1.22 · 10−11 1.16 · 10−11 arctan(x)✱ [−0.25, 0.25]✱ 15 7.89 · 10−15 7.95 · 10−17 2.58 · 10−10 3.24 · 10−12 arctan(x)✱ [−0.9, 0.9]✱ 15 5.10 · 10−3 1.76 · 10−8 1.67 · 102 5.70 · 10−3 exp(1/ cos(x))✱ [0, 1]✱ 14 5.22 · 10−7 4.95 · 10−7 9.06 · 10−3 2.59 · 10−3

exp(x) log(2+x) cos(x) ✱ [0, 1]✱ 15

9.11 · 10−9 2.21 · 10−9 1.18 · 10−3 3.38 · 10−5 sin(exp(x))✱[−1, 1]✱ 10 9.47 · 10−5 3.72 · 10−6 2.96 · 10−2 1.55 · 10−3 ✷✽ ✴ ✸✽

❈▼s ✈s✳ ❚▼s

❖♣❡r❛t✐♦♥s ❝♦♠♣❧❡①✐t②✿ ❆❞❞✐t✐♦♥ ✭O(n)✮✱ ▼✉❧t✐♣❧✐❝❛t✐♦♥✭O(n2)✮ ❛♥❞ ❈♦♠♣♦s✐t✐♦♥ ✭O(n3)✮ ❤❛✈❡ s✐♠✐❧❛r ❝♦♠♣❧❡①✐t②✳ ❳ ■♥✐t✐❛❧ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❝♦❡✣❝✐❡♥ts ❢♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ✐s s❧♦✇❡r ✇✐t❤ ❈▼s ✭O(n2)✮ ✈s✳ ❚▼s ✭O(n)✮ ❈♦♠♣❛r✐s♦♥ ❜❡t✇❡❡♥ r❡♠❛✐♥❞❡r ❜♦✉♥❞s ❢♦r s❡✈❡r❛❧ ❢✉♥❝t✐♦♥s✿

f(x)✱ I✱ n ❈▼ ❊①❛❝t ❜♦✉♥❞ ❚▼ ❊①❛❝t ❜♦✉♥❞ sin(x)✱ [3, 4]✱ 10 1.19 · 10−14 1.13 · 10−14 1.22 · 10−11 1.16 · 10−11 arctan(x)✱ [−0.25, 0.25]✱ 15 7.89 · 10−15 7.95 · 10−17 2.58 · 10−10 3.24 · 10−12 arctan(x)✱ [−0.9, 0.9]✱ 15 5.10 · 10−3 1.76 · 10−8 1.67 · 102 5.70 · 10−3 exp(1/ cos(x))✱ [0, 1]✱ 14 5.22 · 10−7 4.95 · 10−7 9.06 · 10−3 2.59 · 10−3

exp(x) log(2+x) cos(x) ✱ [0, 1]✱ 15

9.11 · 10−9 2.21 · 10−9 1.18 · 10−3 3.38 · 10−5 sin(exp(x))✱[−1, 1]✱ 10 9.47 · 10−5 3.72 · 10−6 2.96 · 10−2 1.55 · 10−3 ✷✽ ✴ ✸✽

❲❤❛t ❛❜♦✉t ♦t❤❡r ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s❄

❘❡♠❡③ ✭♠✐♥✐♠❛①✮✿

❳ ▼♦r❡ ❝♦st❧② t♦ ♦❜t❛✐♥ ✭♠♦r❡ ❝♦♠♣❧❡① ♥✉♠❡r✐❝❛❧ ❛❧❣♦r✐t❤♠✮❀ ❳ ❊①✐st❡♥t ❝❧♦s❡ ❢♦r♠✉❧❛ ❢♦r r❡♠❛✐♥❞❡r ❤❛s t❤❡ s❛♠❡ q✉❛❧✐t② ❛s t❤❡ ♦♥❡ ✇❡ ✉s❡✳

✷✾ ✴ ✸✽

◗✉❛❧✐t② ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❝♦♠♣❛r❡❞ t♦ ♠✐♥✐♠❛①

❘❡♠❛r❦✿ ■t ✐s ❦♥♦✇♥ ❬❊❤❧✐❝❤ ✫ ❩❡❧❧❡r✱ ✶✾✻✻❪ t❤❛t ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛♥ts ❛r❡ ✧♥❡❛r✲❜❡st✧✿ ε∞ ≤ (2 + (2/π) log(n)

• Λn

) ε♠✐♥✐♠❛①∞ Λ15 = 3.72... → ✇❡ ❧♦s❡ ❛t ♠♦st 2 ❜✐ts Λ30 = 4.16... → ✇❡ ❧♦s❡ ❛t ♠♦st 3 ❜✐ts Λ100 = 4.93... → ✇❡ ❧♦s❡ ❛t ♠♦st 3 ❜✐ts Λ100000 = 9.32... → ✇❡ ❧♦s❡ ❛t ♠♦st 4 ❜✐ts

✸✵ ✴ ✸✽

◗✉❛❧✐t② ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❝♦♠♣❛r❡❞ t♦ ♠✐♥✐♠❛①

◆♦ f(x)✱ I✱ n ❈▼ ❊①❛❝t ❜♦✉♥❞ ▼✐♥✐♠❛① 1 sin(x)✱ [3, 4]✱ 10 1.19 · 10−14 1.13 · 10−14 1.12 · 10−14 2 arctan(x)✱ [−0.25, 0.25]✱ 15 7.89 · 10−15 7.95 · 10−17 4.03 · 10−17 3 arctan(x)✱ [−0.9, 0.9]✱ 15 5.10 · 10−3 1.76 · 10−8 1.01 · 10−8 4 exp(1/ cos(x))✱ [0, 1]✱ 14 5.22 · 10−7 4.95 · 10−7 3.57 · 10−7 5

exp(x) log(2+x) cos(x) ✱ [0, 1]✱ 15

9.11 · 10−9 2.21 · 10−9 1.72 · 10−9 6 sin(exp(x))✱[−1, 1]✱ 10 9.47 · 10−5 3.72 · 10−6 1.78 · 10−6 ✸✶ ✴ ✸✽

❲❤❛t ❛❜♦✉t ♦t❤❡r ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s❄

❚r✉♥❝❛t❡❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿

✱ ✇❤❡r❡ ♣♦ss✐❜❧❡ s♣❡❡❞✲✉♣✿ r❡❝✉rr❡♥❝❡ ❢♦r♠✉❧❛❡ ❢♦r ❝♦♠♣✉t✐♥❣ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❢♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ❄❄ P♦ss✐❜❧❡ ❧♦ss ✐♥ t❤❡ q✉❛❧✐t② ♦❢ r❡♠❛✐♥❞❡r✳

✸✷ ✴ ✸✽

❲❤❛t ❛❜♦✉t ♦t❤❡r ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥s❄

❚r✉♥❝❛t❡❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s✿

P(x) =

n

• k=0

′akTk(x)✱ ✇❤❡r❡ ak = 2 π 1

• −1

f(x)Tk(x) √ 1−x2 dx

♣♦ss✐❜❧❡ s♣❡❡❞✲✉♣✿ r❡❝✉rr❡♥❝❡ ❢♦r♠✉❧❛❡ ❢♦r ❝♦♠♣✉t✐♥❣ ♣♦❧②♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❢♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ❄❄ P♦ss✐❜❧❡ ❧♦ss ✐♥ t❤❡ q✉❛❧✐t② ♦❢ r❡♠❛✐♥❞❡r✳

✸✷ ✴ ✸✽

❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡

❊①❛♠♣❧❡✿ π =

1

• 4

1 + x2 dx ❈♦♠♣✉t❡ ❛ ❚▼✴❈▼ (P, I) ❢♦r f(x) = 4 1 + x2 ✳

✸✸ ✴ ✸✽

❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡

❊①❛♠♣❧❡✿ π =

1

• 4

1 + x2 dx ❈♦♠♣✉t❡ ❛ ❚▼✴❈▼ (P, I) ❢♦r f(x) = 4 1 + x2 ✳ P(x) + I ≤ f(x) ≤ P(x) + I

✸✸ ✴ ✸✽

❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡

❊①❛♠♣❧❡✿ π =

1

• 4

1 + x2 dx ❈♦♠♣✉t❡ ❛ ❚▼✴❈▼ (P, I) ❢♦r f(x) = 4 1 + x2 ✳

b

• a

(P(x) + I)dx ≤

b

• a

f(x)dx ≤

b

• a

(P(x) + I)dx

✸✸ ✴ ✸✽

❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡

❊①❛♠♣❧❡✿ π =

1

• 4

1 + x2 dx

❖r❞❡r ❙✉❜❞✐✈✳ ❇♦✉♥❞ ❚▼✷ ❇♦✉♥❞ ❈▼ ✺ ✶ ❬ ✸✳✵✷✸✶✽✾✸✸✸✸✸✸✸✱ ✽✳✺✽✵✼✼✽✻✻✻✻✻✻✻ ❪ ❬✸✳✵✾✽✻✾✹✶✶✾✵✶✾✺✱ ✸✳✶✽✺✾✾✻✷✶✹✵✼✹✷❪ ✹ ❬ ✸✳✶✹✶✺✸✻✸✷✷✾✹✶✺✱ ✸✳✶✹✶✻✻✷✾✺✸✻✷✾✷ ❪ ❬✸✳✶✹✶✺✾✵✼✼✶✼✼✻✾✱ ✸✳✶✹✶✺✾✹✸✻✶✵✼✼✷❪ ✶✻ ❬ ✸✳✶✹✶✺✾✷✻✶✵✶✻✶✹✱ ✸✳✶✹✶✺✾✷✻✾✽✵✼✽✻ ❪ ❬✸✳✶✹✶✺✾✷✻✺✸✶✷✻✾✱ ✸✳✶✹✶✺✾✷✻✺✸✾✶✸✶❪ ✶✵ ✶ ❬✲✷✳✶✾✽✹✵✶✵✷✻✻✵✵✻✱ ✸✳✷✶✶✸✾✻✸✶✼✺✷✻✼ ❪ ❬✸✳✶✹✶✶✾✽✶✾✾✹✾✻✾✱ ✸✳✶✹✶✾✾✵✾✾✸✹✺✷✺❪ ✹ ❬ ✸✳✶✹✶✺✾✷✻✺✶✾✺✸✺✱ ✸✳✶✹✶✺✾✷✻✺✹✻✽✼✵ ❪ ❬✸✳✶✹✶✺✾✷✻✺✸✺✽✵✺✱ ✸✳✶✹✶✺✾✷✻✺✸✺✾✾✵❪ ✶✻ ❬ ✸✳✶✹✶✺✾✷✻✺✸✺✽✾✼✱ ✸✳✶✹✶✺✾✷✻✺✸✺✽✾✼ ❪ ❬✸✳✶✹✶✺✾✷✻✺✸✺✽✾✼✾✸✷✱ ✸✳✶✹✶✺✾✷✻✺✸✺✽✾✼✾✸✷❪ ✷❘❡s✉❧ts t❛❦❡♥ ❢r♦♠ ▼✳ ❇❡r③✱ ❑✳ ▼❛❦✐♥♦✱ ✏◆❡✇ ▼❡t❤♦❞s ❢♦r

❍✐❣❤✲❉✐♠❡♥s✐♦♥❛❧ ❱❡r✐✜❡❞ ◗✉❛❞r❛t✉r❡✑✱ ❘❡❧✐❛❜❧❡ ❈♦♠♣✉t✐♥❣ ✺✿✶✸✲✷✷✱ ✶✾✾✾

✸✹ ✴ ✸✽

❈♦♥❝❧✉s✐♦♥

❈▼s ❛r❡ ♣♦t❡♥t✐❛❧❧② ✉s❡❢✉❧ ✐♥ ✈❛r✐♦✉s r✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✿ s♠❛❧❧❡r r❡♠❛✐♥❞❡rs t❤❛♥ ❚▼s✱ ❜✉t r❡q✉✐r❡ ♠♦r❡ ❝♦♠♣✉t✐♥❣ t✐♠❡✳ ❈✉rr❡♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥✿ ♣❛rt✐❛❧❧② ❙♦❧❧②❛ ❛♥❞ ▼❛♣❧❡✳ ❲♦r❦ ✐♥ ♣r♦❣r❡ss✿ ✉s❡ ❈❤❡❜②s❤❡✈ tr✉♥❝❛t❡❞ s❡r✐❡s ✐♥st❡❛❞ ♦❢ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧s✳ ❋✉t✉r❡ ✇♦r❦✿ ❡①t❡♥❞ t♦ ♠✉❧t✐✈❛r✐❛t❡ ❢✉♥❝t✐♦♥s

✸✺ ✴ ✸✽

❍♦✇ t♦ r❡❞✉❝❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ r❡♠❛✐♥❞❡r❄

▼♦♥♦t♦♥✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ r❡♠❛✐♥❞❡r ❝❛♥ ❜❡ ✐♥❢❡r❡❞ ❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s ✸✿ ❊①❛♠♣❧❡✿ ✱ ♦✈❡r ✱ ✱ ✳ ❲✐t❤ ❩✬s r❡♠❛r❦✱

✸❈♦r♦❧❧❛r② ✐♥ ❘✳ ❩✉♠❦❡❧❧❡r✱ ✑●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥ ✐♥ ❚②♣❡ ❚❤❡♦r②✏✱ P❤❉

t❤❡s✐s✱ ♣❛❣❡ ✽✹

✸✻ ✴ ✸✽

❍♦✇ t♦ r❡❞✉❝❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ r❡♠❛✐♥❞❡r❄

▼♦♥♦t♦♥✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ r❡♠❛✐♥❞❡r ❝❛♥ ❜❡ ✐♥❢❡r❡❞ ❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s ✸✿ ■❢ f(n+1)([a, b]) ≥ 0✱ ❛♥❞ T ✐s ❚❛②❧♦r ♣♦❧②♥♦♠✐❛❧ ♦❢ ❞❡❣r❡❡ n ❢♦r f ✱ f − T (x − x0)n+1 ✐s ♠♦♥♦t♦♥✐❝ ♦✈❡r [a, b]✿ t❤❡ r❡♠❛✐♥❞❡r ❝❛♥ ❜❡ ❡①❛❝t❧② ❜♦✉♥❞❡❞ ✉s✐♥❣ t✇♦ ❡✈❛❧✉❛t✐♦♥s ♦❢ f − T ✐♥ t❤❡ ❡♥❞ ♣♦✐♥ts ♦❢ [a, b]✳ ❊①❛♠♣❧❡✿ ✱ ♦✈❡r ✱ ✱ ✳ ❲✐t❤ ❩✬s r❡♠❛r❦✱

✸❈♦r♦❧❧❛r② ✐♥ ❘✳ ❩✉♠❦❡❧❧❡r✱ ✑●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥ ✐♥ ❚②♣❡ ❚❤❡♦r②✏✱ P❤❉

t❤❡s✐s✱ ♣❛❣❡ ✽✹

✸✻ ✴ ✸✽

❍♦✇ t♦ r❡❞✉❝❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ r❡♠❛✐♥❞❡r❄

▼♦♥♦t♦♥✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ r❡♠❛✐♥❞❡r ❝❛♥ ❜❡ ✐♥❢❡r❡❞ ❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s ✸✿ ❊①❛♠♣❧❡✿ f(x) = log(x)✱ ♦✈❡r [0.001, 1.001]✱ n = 13✱ x0 = 0.5✳ ∆n(x, ξ) = −1 ξ14 · (x − 0.5)14 14! ∆ ⊆ [−2.66 · 1031, 2.63 · 10−10] ❲✐t❤ ❩✬s r❡♠❛r❦✱

✸❈♦r♦❧❧❛r② ✐♥ ❘✳ ❩✉♠❦❡❧❧❡r✱ ✑●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥ ✐♥ ❚②♣❡ ❚❤❡♦r②✏✱ P❤❉

t❤❡s✐s✱ ♣❛❣❡ ✽✹

✸✻ ✴ ✸✽

❍♦✇ t♦ r❡❞✉❝❡ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ r❡♠❛✐♥❞❡r❄

▼♦♥♦t♦♥✐❝ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ r❡♠❛✐♥❞❡r ❝❛♥ ❜❡ ✐♥❢❡r❡❞ ❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s ✸✿ ❊①❛♠♣❧❡✿ f(x) = log(x)✱ ♦✈❡r [0.001, 1.001]✱ n = 13✱ x0 = 0.5✳ ∆n(x, ξ) = −1 ξ14 · (x − 0.5)14 14! ∆ ⊆ [−2.66 · 1031, 2.63 · 10−10] ❲✐t❤ ❩✬s r❡♠❛r❦✱ f(x) − T(x) ∈ [−3.06, 7.89 · 10−31]

✸❈♦r♦❧❧❛r② ✐♥ ❘✳ ❩✉♠❦❡❧❧❡r✱ ✑●❧♦❜❛❧ ❖♣t✐♠✐③❛t✐♦♥ ✐♥ ❚②♣❡ ❚❤❡♦r②✏✱ P❤❉

t❤❡s✐s✱ ♣❛❣❡ ✽✹

✸✻ ✴ ✸✽

■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r ✲ ✏❇❛s✐❝✑ ❋✉♥❝t✐♦♥ ❙t❡♣

❲❡ ❝❛♥ ♣r♦✈❡ ✭❛ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ❩✉♠❦❡❧❧❡r✬s r❡♠❛r❦✮✿ ✐❢ f(n+1) ✐s ♠♦♥♦t♦♥✐❝ ♦✈❡r I✱ t❤❡♥ f(x)−P(x)

Wy(x)

✐s ♠♦♥♦t♦♥✐❝ ♦✈❡r I✳ ❚❤❡ r❡♠❛✐♥❞❡r ❝❛♥ ❜❡ ❡①❛❝t❧② ❜♦✉♥❞❡❞ ✉s✐♥❣ t✇♦ ❡✈❛❧✉❛t✐♦♥s ✐♥ t❤❡ ❡♥❞ ♣♦✐♥ts ♦❢ I

✸✼ ✴ ✸✽