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Mar 23, 2024 14 likes •874 views

s r rs t s s rs rts s rsrr r s

slide-1
SLIDE 1

❚♦♦❧s ❢♦r ❘✐❣♦r♦✉s ❈♦♠♣✉t✐♥❣ ✉s✐♥❣ ❈❤❡❜②s❤❡✈ ❙❡r✐❡s ❆♣♣r♦①✐♠❛t✐♦♥s

◆✐❝♦❧❛s ❇r✐s❡❜❛rr❡ ▼✐♦❛r❛ ❏♦❧❞❡s

✶ ✴ ✷✾

slide-2
SLIDE 2

❘✐❣♦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✮

✷ ✴ ✷✾

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SLIDE 3

❘✐❣♦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✮

✷ ✴ ✷✾

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SLIDE 4

❘✐❣♦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✮

✷ ✴ ✷✾

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SLIDE 5

❘✐❣♦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✮

✷ ✴ ✷✾

slide-6
SLIDE 6

❲❤❛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❡✿

✸ ✴ ✷✾

slide-7
SLIDE 7

❲❤❛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

✸ ✴ ✷✾

slide-8
SLIDE 8

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✭♠✉❧t✐♣❧❡ ♣r❡❝✐s✐♦♥ ■❆ ❧✐❜r❛r✐❡s ❡①✐st✱ ❡✳❣✳ ▼P❋■✶✮ ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ✱ ❜✉t ■♠

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾

slide-9
SLIDE 9

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✭♠✉❧t✐♣❧❡ ♣r❡❝✐s✐♦♥ ■❆ ❧✐❜r❛r✐❡s ❡①✐st✱ ❡✳❣✳ ▼P❋■✶✮ π ∈ [3.1415, 3.1416] ■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ❖♣❡r❛t✐♦♥s ❊❣✳ ❘❛♥❣❡ ❜♦✉♥❞✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s ❊❣✳ ✱ ❜✉t ■♠

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾

slide-10
SLIDE 10

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

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

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾

slide-11
SLIDE 11

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✭♠✉❧t✐♣❧❡ ♣r❡❝✐s✐♦♥ ■❆ ❧✐❜r❛r✐❡s ❡①✐st✱ ❡✳❣✳ ▼P❋■✶✮ 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]) = [−1, 6] ✱ ❜✉t ■♠

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾

slide-12
SLIDE 12

■♥t❡r✈❛❧ ❆r✐t❤♠❡t✐❝ ✭■❆✮

❊❛❝❤ ✐♥t❡r✈❛❧ ❂ ♣❛✐r ♦❢ ✢♦❛t✐♥❣✲♣♦✐♥t ♥✉♠❜❡rs ✭♠✉❧t✐♣❧❡ ♣r❡❝✐s✐♦♥ ■❆ ❧✐❜r❛r✐❡s ❡①✐st✱ ❡✳❣✳ ▼P❋■✶✮ 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]) = [−1, 6] x ∈ [−1, 2], f(x) ∈ [−1, 6]✱ ❜✉t ■♠(f) = [3/4, 3]

✶❤tt♣✿✴✴❣❢♦r❣❡✳✐♥r✐❛✳❢r✴♣r♦❥❡❝ts✴♠♣❢✐✴ ✹ ✴ ✷✾

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SLIDE 13

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

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

i=0 cixi✱

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

✺ ✴ ✷✾

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SLIDE 14

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

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

i=0 cixi✱

ε(x) = f(x) − p(x) s✳t✳ ✐s ❛s s♠❛❧❧ ❛s ♣♦ss✐❜❧❡ ✭❘❡♠❡③ ❛❧❣♦r✐t❤♠✮

✺ ✴ ✷✾

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SLIDE 15

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

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

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❤♠✮

✺ ✴ ✷✾

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SLIDE 16

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

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

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

✻ ✴ ✷✾

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SLIDE 17

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

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

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

✻ ✴ ✷✾

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SLIDE 18

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

f(x) = e1/ cos(x), x ∈ [0, 1]✱ p(x) = 10

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) ∈ [−233, 298]✱ ❜✉t ε(x)∞ ≃ 3.8325 · 10−5

✻ ✴ ✷✾

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SLIDE 19

❲❤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✦

✼ ✴ ✷✾

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SLIDE 20

❘✐❣♦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❡ ❄

✽ ✴ ✷✾

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SLIDE 21

❘✐❣♦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❡ ❄

✽ ✴ ✷✾

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SLIDE 22

❘✐❣♦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❡ ❄

✽ ✴ ✷✾

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SLIDE 23

❘✐❣♦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, ∆) ❄

✽ ✴ ✷✾

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SLIDE 24

❚❛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 ❄

✾ ✴ ✷✾

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SLIDE 25

❚❛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 ❄

✾ ✴ ✷✾

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SLIDE 26

❚❛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) ❄

✾ ✴ ✷✾

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SLIDE 27

❆✉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❡

✶✵ ✴ ✷✾

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SLIDE 28

❆✉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)

✶✵ ✴ ✷✾

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SLIDE 29

❆✉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)]

✶✵ ✴ ✷✾

slide-30
SLIDE 30

❆✉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)]

✶✵ ✴ ✷✾

slide-31
SLIDE 31

❆✉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]

✶✵ ✴ ✷✾

slide-32
SLIDE 32

❆✉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]

✶✵ ✴ ✷✾

slide-33
SLIDE 33

❆✉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]

✶✵ ✴ ✷✾

slide-34
SLIDE 34

❆✉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]

✶✵ ✴ ✷✾

slide-35
SLIDE 35

❆✉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]

✶✵ ✴ ✷✾

slide-36
SLIDE 36

❆✉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]

✶✵ ✴ ✷✾

slide-37
SLIDE 37

❆✉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]

✶✵ ✴ ✷✾

slide-38
SLIDE 38

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

❚❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ∆ ❢♦r ∆n(x, ξ) ❝❛♥ ❜❡ ❧❛r❣❡❧② ♦✈❡r❡st✐♠❛t❡❞✳ ❊①❛♠♣❧❡ ✷✿ ✱ ♦✈❡r ✱ ✱ ✳ ❆✉t♦♠❛t✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛✿ ❈❛✉❝❤②✬s ❊st✐♠❛t❡ ❚❛②❧♦r ▼♦❞❡❧s

✷❙✳ ❈❤❡✈✐❧❧❛r❞✱ ❏✳❍❛rr✐s♦♥✱ ▼✳ ❏♦❧❞❡s✱ ❈❤✳ ▲❛✉t❡r✱ ❊✣❝✐❡♥t ❛♥❞ ❛❝❝✉r❛t❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ✉♣♣❡r ❜♦✉♥❞s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs✱ ✷✵✶✵✱ ❘❘▲■P✷✵✶✵✲✷ ✶✶ ✴ ✷✾

slide-39
SLIDE 39

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

❚❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ∆ ❢♦r ∆n(x, ξ) ❝❛♥ ❜❡ ❧❛r❣❡❧② ♦✈❡r❡st✐♠❛t❡❞✳ ❊①❛♠♣❧❡ ✷✿ f(x) = e1/ cos x✱ ♦✈❡r [0, 1]✱ n = 13✱ x0 = 0.5✳ f(x) − T(x) ∈ [0, 4.56 · 10−3] ❆✉t♦♠❛t✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛✿ ❈❛✉❝❤②✬s ❊st✐♠❛t❡ ❚❛②❧♦r ▼♦❞❡❧s

✷❙✳ ❈❤❡✈✐❧❧❛r❞✱ ❏✳❍❛rr✐s♦♥✱ ▼✳ ❏♦❧❞❡s✱ ❈❤✳ ▲❛✉t❡r✱ ❊✣❝✐❡♥t ❛♥❞ ❛❝❝✉r❛t❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ✉♣♣❡r ❜♦✉♥❞s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs✱ ✷✵✶✵✱ ❘❘▲■P✷✵✶✵✲✷ ✶✶ ✴ ✷✾

slide-40
SLIDE 40

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

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

✷❙✳ ❈❤❡✈✐❧❧❛r❞✱ ❏✳❍❛rr✐s♦♥✱ ▼✳ ❏♦❧❞❡s✱ ❈❤✳ ▲❛✉t❡r✱ ❊✣❝✐❡♥t ❛♥❞ ❛❝❝✉r❛t❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ✉♣♣❡r ❜♦✉♥❞s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs✱ ✷✵✶✵✱ ❘❘▲■P✷✵✶✵✲✷ ✶✶ ✴ ✷✾

slide-41
SLIDE 41

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

❚❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ∆ ❢♦r ∆n(x, ξ) ❝❛♥ ❜❡ ❧❛r❣❡❧② ♦✈❡r❡st✐♠❛t❡❞✳ ❊①❛♠♣❧❡ ✷✿ f(x) = e1/ cos x✱ ♦✈❡r [0, 1]✱ n = 13✱ x0 = 0.5✳ f(x) − T(x) ∈ [0, 4.56 · 10−3] ❆✉t♦♠❛t✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛✿ ∆ = [−1.93 · 102, 1.35 · 103] ❈❛✉❝❤②✬s ❊st✐♠❛t❡ ∆ = [−9.17 · 10−2, 9.17 · 10−2] ❚❛②❧♦r ▼♦❞❡❧s

✷❙✳ ❈❤❡✈✐❧❧❛r❞✱ ❏✳❍❛rr✐s♦♥✱ ▼✳ ❏♦❧❞❡s✱ ❈❤✳ ▲❛✉t❡r✱ ❊✣❝✐❡♥t ❛♥❞ ❛❝❝✉r❛t❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ✉♣♣❡r ❜♦✉♥❞s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs✱ ✷✵✶✵✱ ❘❘▲■P✷✵✶✵✲✷ ✶✶ ✴ ✷✾

slide-42
SLIDE 42

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

❚❤❡ ✐♥t❡r✈❛❧ ❜♦✉♥❞ ∆ ❢♦r ∆n(x, ξ) ❝❛♥ ❜❡ ❧❛r❣❡❧② ♦✈❡r❡st✐♠❛t❡❞✳ ❊①❛♠♣❧❡ ✷✿ f(x) = e1/ cos x✱ ♦✈❡r [0, 1]✱ n = 13✱ x0 = 0.5✳ f(x) − T(x) ∈ [0, 4.56 · 10−3] ❆✉t♦♠❛t✐❝ ❞✐✛❡r❡♥t✐❛t✐♦♥ ❛♥❞ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛✿ ∆ = [−1.93 · 102, 1.35 · 103] ❈❛✉❝❤②✬s ❊st✐♠❛t❡ ∆ = [−9.17 · 10−2, 9.17 · 10−2] ❚❛②❧♦r ▼♦❞❡❧s ∆ = [−9.04 · 10−3, 9.06 · 10−3]

✷❙✳ ❈❤❡✈✐❧❧❛r❞✱ ❏✳❍❛rr✐s♦♥✱ ▼✳ ❏♦❧❞❡s✱ ❈❤✳ ▲❛✉t❡r✱ ❊✣❝✐❡♥t ❛♥❞ ❛❝❝✉r❛t❡ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ✉♣♣❡r ❜♦✉♥❞s ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦rs✱ ✷✵✶✵✱ ❘❘▲■P✷✵✶✵✲✷ ✶✶ ✴ ✷✾

slide-43
SLIDE 43

❚❛②❧♦r ▼♦❞❡❧s P❤✐❧♦s♦♣❤②

❋♦r ❜♦✉♥❞✐♥❣ t❤❡ r❡♠❛✐♥❞❡rs✿ ❋♦r ✏❜❛s✐❝ ❢✉♥❝t✐♦♥s✑ ✉s❡ ▲❛❣r❛♥❣❡ ❢♦r♠✉❧❛✳ ❋♦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✳

✶✷ ✴ ✷✾

slide-44
SLIDE 44

❚❛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

✶✸ ✴ ✷✾

slide-45
SLIDE 45

❚❛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❡✳

✶✸ ✴ ✷✾

slide-46
SLIDE 46

❖✉r ❆♣♣r♦❛❝❤ ✲ ❈❤❡❜②s❤❡✈ ▼♦❞❡❧s

❇❛s✐❝ ✐❞❡❛✿ ✲ ❯s❡ ❛ ♣♦❧②♥♦♠✐❛❧ ❛♣♣r♦①✐♠❛t✐♦♥ ❜❡tt❡r t❤❛♥ ❚❛②❧♦r✿ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧✸✳ ❈❤❡❜②s❤❡✈ tr✉♥❝❛t❡❞ s❡r✐❡s✳ ✲ ❯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✳

✸◆✳ ❇r✐s❡❜❛rr❡✱ ▼✳ ❏♦❧❞❡s✱ ❈❤❡❜②s❤❡✈ ✐♥t❡r♣♦❧❛t✐♦♥ ♣♦❧②♥♦♠✐❛❧✲❜❛s❡❞ t♦♦❧s ❢♦r r✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣✳ ■♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ✷✵✶✵ ✐♥t❡r♥❛t✐♦♥❛❧ ❙②♠♣♦s✐✉♠ ♦♥ ❙②♠❜♦❧✐❝ ❛♥❞ ❆❧❣❡❜r❛✐❝ ❈♦♠♣✉t❛t✐♦♥ ✭▼✉♥✐❝❤✱

  • ❡r♠❛♥②✱ ❏✉❧② ✷✺ ✲ ✷✽✱ ✷✵✶✵✮✳ ❆❈▼✱ ◆❡✇ ❨♦r❦✱ ◆❨✱ ✶✹✼✲✶✺✹

✶✹ ✴ ✷✾

slide-47
SLIDE 47

◗✉✐❝❦ ❘❡♠✐♥❞❡r✿ ❈❤❡❜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.

✶✺ ✴ ✷✾

slide-48
SLIDE 48

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

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 ■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐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

✶✻ ✴ ✷✾

slide-49
SLIDE 49

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

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

■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐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

✶✻ ✴ ✷✾

slide-50
SLIDE 50

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

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✳ ❲❡ ❝❛♥ ✉s❡ tr✉♥❝❛t❡❞ ❈❤❡❜②s❤❡✈ s❡r✐❡s ✐♥st❡❛❞✳ ■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐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

✶✻ ✴ ✷✾

slide-51
SLIDE 51

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

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 ■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) (n + 1)! n

i=0(x − xi)✳

❲❡ s❤♦✉❧❞ ❤❛✈❡ ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦❢ ✐♥ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ r❡♠❛✐♥❞❡r✱ ❝♦♠♣❛r❡❞ t♦ ❚❛②❧♦r r❡♠❛✐♥❞❡r✳ ❳ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ ♦✈❡r❡st✐♠❛t✐♦♥ ♦❢ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐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

✶✻ ✴ ✷✾

slide-52
SLIDE 52

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

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 ■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) (n + 1)! n

i=0(x − xi)✳

❲❡ s❤♦✉❧❞ ❤❛✈❡ ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦❢ 2n ✐♥ t❤❡ ✇✐❞t❤ ♦❢ t❤❡ r❡♠❛✐♥❞❡r✱ ❝♦♠♣❛r❡❞ t♦ ❚❛②❧♦r r❡♠❛✐♥❞❡r✳ ❳ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱ ♦✈❡r❡st✐♠❛t✐♦♥ ♦❢ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐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

✶✻ ✴ ✷✾

slide-53
SLIDE 53

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

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 ■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) (n + 1)! n

i=0(x − xi)✳

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

✶✻ ✴ ✷✾

slide-54
SLIDE 54

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

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 ✭❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s✮ ■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ✭❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s✮ ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) (n + 1)! n

i=0(x − xi)✳

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

✶✻ ✴ ✷✾

slide-55
SLIDE 55

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

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 ✭❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s✮ ■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ✭❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s✮ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐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

✶✻ ✴ ✷✾

slide-56
SLIDE 56

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

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 ✭❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s✮ ■♥t❡r♣♦❧❛t✐♦♥ ❊rr♦r✿ ▲❛❣r❛♥❣❡ r❡♠❛✐♥❞❡r ✭❢♦r ✏❜❛s✐❝✑ ❢✉♥❝t✐♦♥s✮ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐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

✶✻ ✴ ✷✾

slide-57
SLIDE 57

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s✿ ✉s✐♥❣ tr✉♥❝❛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✳ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ❚r✉♥❝❛t✐♦♥ ❊rr♦r✿ ❇❡r♥st❡✐♥✲❧✐❦❡ ❢♦r♠✉❧❛ ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐t❤ ♠♦❞❡❧s

✶✼ ✴ ✷✾

slide-58
SLIDE 58

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s✿ ✉s✐♥❣ tr✉♥❝❛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✳ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ✲ r❡❝✉rr❡♥❝❡ ❢♦r♠✉❧❛❡ ❢♦r ❝♦♠♣✉t✐♥❣ ak ❚r✉♥❝❛t✐♦♥ ❊rr♦r✿ ❇❡r♥st❡✐♥✲❧✐❦❡ ❢♦r♠✉❧❛ ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐t❤ ♠♦❞❡❧s

✶✼ ✴ ✷✾

slide-59
SLIDE 59

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s✿ ✉s✐♥❣ tr✉♥❝❛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✳ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ✲ r❡❝✉rr❡♥❝❡ ❢♦r♠✉❧❛❡ ❢♦r ❝♦♠♣✉t✐♥❣ ak ❘❡♠❛r❦✿ ❆s ❢❛st ❛s ❚▼s✳ ❚r✉♥❝❛t✐♦♥ ❊rr♦r✿ ❇❡r♥st❡✐♥✲❧✐❦❡ ❢♦r♠✉❧❛ ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐t❤ ♠♦❞❡❧s

✶✼ ✴ ✷✾

slide-60
SLIDE 60

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s✿ ✉s✐♥❣ tr✉♥❝❛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✳ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ❚r✉♥❝❛t✐♦♥ ❊rr♦r✿ ❇❡r♥st❡✐♥✲❧✐❦❡ ❢♦r♠✉❧❛ ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ∀x ∈ [−1, 1], ∃ξ ∈ [−1, 1] s✳t✳ f(x) − P(x) = f(n+1)(ξ) 2n(n + 1)!✳ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐t❤ ♠♦❞❡❧s

✶✼ ✴ ✷✾

slide-61
SLIDE 61

❈❤❡❜②s❤❡✈ ▼♦❞❡❧s✿ ✉s✐♥❣ tr✉♥❝❛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✳ ❈♦♠♣✉t❛t✐♦♥ ♦❢ t❤❡ ❝♦❡✣❝✐❡♥ts ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ❚r✉♥❝❛t✐♦♥ ❊rr♦r✿ ❇❡r♥st❡✐♥✲❧✐❦❡ ❢♦r♠✉❧❛ ✭❢♦r ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s✮ ✲ ❋♦r ❝♦♠♣♦s✐t❡ ❢✉♥❝t✐♦♥s✱✉s❡ ❛❧❣❡❜r❛✐❝ r✉❧❡s ✭❛❞❞✐t✐♦♥✱ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❝♦♠♣♦s✐t✐♦♥✮ ✇✐t❤ ♠♦❞❡❧s

✶✼ ✴ ✷✾

slide-62
SLIDE 62

❈❤❡❜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).

✶✽ ✴ ✷✾

slide-63
SLIDE 63

❈❤❡❜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✑✿ ✳

✶✾ ✴ ✷✾

slide-64
SLIDE 64

❈❤❡❜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✑✿ ✳

✶✾ ✴ ✷✾

slide-65
SLIDE 65

❈❤❡❜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]✳

✶✾ ✴ ✷✾

slide-66
SLIDE 66

❈❤❡❜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❤♠

✷✵ ✴ ✷✾

slide-67
SLIDE 67

❈❤❡❜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❤♠

✷✵ ✴ ✷✾

slide-68
SLIDE 68

❈❤❡❜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❤♠

✷✵ ✴ ✷✾

slide-69
SLIDE 69

❈❤❡❜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❤♠

✷✵ ✴ ✷✾

slide-70
SLIDE 70

❈❤❡❜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❤♠

✷✵ ✴ ✷✾

slide-71
SLIDE 71

❈❤❡❜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

✷✶ ✴ ✷✾

slide-72
SLIDE 72

❈❤❡❜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✐♦♥✳

✷✶ ✴ ✷✾

slide-73
SLIDE 73

❈❤❡❜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✳

✷✶ ✴ ✷✾

slide-74
SLIDE 74

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

❊①❛♠♣❧❡✿ ε(x) = f(x) − p(x) f(x) = e1/ cos x✱ ♦✈❡r [0, 1]✱ p(x) ✲ ♠✐♥✐♠❛①✱ ❞❡❣r❡❡ 10 ε(x)∞ ≃ 3.8325 · 10−5 ◆❡❡❞✿ ❚▼ ♦❢ ❞❡❣r❡❡ ✳ ❈▼ ♦❢ ❞❡❣r❡❡ ✳

✷✷ ✴ ✷✾

slide-75
SLIDE 75

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

❊①❛♠♣❧❡✿ ε(x) = f(x) − p(x) f(x) = e1/ cos x✱ ♦✈❡r [0, 1]✱ p(x) ✲ ♠✐♥✐♠❛①✱ ❞❡❣r❡❡ 10 ε(x)∞ ≃ 3.8325 · 10−5 ◆❡❡❞✿ ❚▼ ♦❢ ❞❡❣r❡❡ 30✳ ❈▼ ♦❢ ❞❡❣r❡❡ ✳

✷✷ ✴ ✷✾

slide-76
SLIDE 76

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

❊①❛♠♣❧❡✿ ε(x) = f(x) − p(x) f(x) = e1/ cos x✱ ♦✈❡r [0, 1]✱ p(x) ✲ ♠✐♥✐♠❛①✱ ❞❡❣r❡❡ 10 ε(x)∞ ≃ 3.8325 · 10−5 ◆❡❡❞✿ ❚▼ ♦❢ ❞❡❣r❡❡ 30✳ ❈▼ ♦❢ ❞❡❣r❡❡ 13✳

✷✷ ✴ ✷✾

slide-77
SLIDE 77

❈▼s ✈s✳ ❚▼s

❖♣❡r❛t✐♦♥s ❝♦♠♣❧❡①✐t②✿ ❆❞❞✐t✐♦♥ ✭ ✮✱ ▼✉❧t✐♣❧✐❝❛t✐♦♥✭ ✮ ❛♥❞ ❈♦♠♣♦s✐t✐♦♥ ✭ ✮ ❤❛✈❡ s✐♠✐❧❛r ❝♦♠♣❧❡①✐t②✳ ■♥✐t✐❛❧ ❝♦♠♣✉t❛t✐♦♥ ♦❢ ❝♦❡✣❝✐❡♥ts ❢♦r ❛❧❧ ✏❜❛s✐❝✑ ❉✲✜♥✐t❡ ❢✉♥❝t✐♦♥s ✐s s✐♠✐❧❛r ✭ ✮✳ ❈♦♠♣❛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 ✷✸ ✴ ✷✾

slide-78
SLIDE 78

❈▼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❡ ❢✉♥❝t✐♦♥s ✐s s✐♠✐❧❛r ✭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 ✷✸ ✴ ✷✾

slide-79
SLIDE 79

❲❤❛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❡✳

✷✹ ✴ ✷✾

slide-80
SLIDE 80

◗✉❛❧✐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

✷✺ ✴ ✷✾

slide-81
SLIDE 81

◗✉❛❧✐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 ✷✻ ✴ ✷✾

slide-82
SLIDE 82

❘✐❣♦r♦✉s q✉❛❞r❛t✉r❡

❊①❛♠♣❧❡✿ π =

1

  • 4

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

✷✼ ✴ ✷✾

slide-83
SLIDE 83

❘✐❣♦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

✷✼ ✴ ✷✾

slide-84
SLIDE 84

❘✐❣♦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

✷✼ ✴ ✷✾

slide-85
SLIDE 85

❘✐❣♦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✐♥❣ ✺✿✶✸✲✷✷✱ ✶✾✾✾

✷✽ ✴ ✷✾

slide-86
SLIDE 86

❈♦♥❝❧✉s✐♦♥

❈▼s ❛r❡ ♣♦t❡♥t✐❛❧❧② ✉s❡❢✉❧ ✐♥ ✈❛r✐♦✉s r✐❣♦r♦✉s ❝♦♠♣✉t✐♥❣ ❛♣♣❧✐❝❛t✐♦♥s✿ s♠❛❧❧❡r r❡♠❛✐♥❞❡rs t❤❛♥ ❚▼s✱ s✐♠✐❧❛r ❝♦♠♣✉t✐♥❣ t✐♠❡s✳ ❈✉rr❡♥t ✐♠♣❧❡♠❡♥t❛t✐♦♥✿ ✐♥ ▼❛♣❧❡ ✇✇✇✳❡♥s✲❧②♦♥✳❢r✴▲■P✴❆r❡♥❛✐r❡✴❲❛r❡✴❈❤❡❜▼♦❞❡❧s✴✳ ❲♦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

✷✾ ✴ ✷✾