sqr - PowerPoint PPT Presentation

❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡ ▼✐❝❤❛ë❧ ❘❛♦ ❈◆❘❙ ✲ ❊◆❙ ▲②♦♥ ▲■P ✲ éq✉✐♣❡ ▼❈✷ ♠✐❝❤❛❡❧✳r❛♦❅❡♥s✲❧②♦♥✳❢r ❤tt♣✿✴✴♣❡rs♦✳❡♥s✲❧②♦♥✳❢r✴♠✐❝❤❛❡❧✳r❛♦✴ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❦ ✲❛❜❡❧✐❛♥ ❡q✉✐✈❛❧❡♥❝❡ ❚✇♦ ✇♦r❞s ✉ ❛♥❞ ✈ ❛r❡ ❦ ✲❛❜❡❧✐❛♥ ❡q✉✐✈❛❧❡♥t ✐❢✿ ❢♦r ❡✈❡r② ✇ ✇✐t❤ ∣ ✇ ∣ ≤ ❦ ✱ ∣ ✉ ∣ ✇ = ∣ ✈ ∣ ✇ ✳ ❊q✉✐✈❛❧❡♥t❧②✿ ❢♦r ❡✈❡r② ✇ ✇✐t❤ ∣ ✇ ∣ = ❦ ✱ ∣ ✉ ∣ ✇ = ∣ ✈ ∣ ✇ ❛♥❞ ✉ [ ✶ ∶ ❦ − ✶ ] = ✈ [ ✶ ∶ ❦ − ✶ ] ❛♥❞ ✉ [∣ ✉ ∣ − ❦ + ✷ ∶ ∣ ✉ ∣] = ✈ [∣ ✈ ∣ − ❦ + ✷ ∶ ∣ ✈ ∣] ✳ ♣qr ✐s ❛ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡ ✐❢ ♣ ✱ q ✱ r ❛r❡ ♣❛✐r✇✐s❡ ❦ ✲❛❜❡❧✐❛♥ ❡q✉✐✈❛❧❡♥t✳ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❨❡s ✸ ❦ Pr♦❜❧❡♠s ★ ✶✻ Pr♦❜❧❡♠ ❆r❡ ✷ ✲❛❜❡❧✐❛♥ ❝✉❜❡ ❛✈♦✐❞❛❜❧❡ ♦♥ ❛ ❜✐♥❛r② ❛❧♣❤❛❜❡t ❄ Pr♦❜❧❡♠ ❋♦r ✇❤✐❝❤ ❦ ❛r❡ ❦✲❛❜❡❧✐❛♥ sq✉❛r❡s ❛✈♦✐❞❛❜❧❡ ♦♥ ❛ t❡r♥❛r② ❛❧♣❤❛❜❡t ❄ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

Pr♦❜❧❡♠s ★ ✶✻ Pr♦❜❧❡♠ ❆r❡ ✷ ✲❛❜❡❧✐❛♥ ❝✉❜❡ ❛✈♦✐❞❛❜❧❡ ♦♥ ❛ ❜✐♥❛r② ❛❧♣❤❛❜❡t ❄ ⇒ ❨❡s Pr♦❜❧❡♠ ❋♦r ✇❤✐❝❤ ❦ ❛r❡ ❦✲❛❜❡❧✐❛♥ sq✉❛r❡s ❛✈♦✐❞❛❜❧❡ ♦♥ ❛ t❡r♥❛r② ❛❧♣❤❛❜❡t ❄ ⇒ ❦ = ✸ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

✷✲❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡ ❜✐♥❛r② ✇♦r❞ ▲❡t ❤ ∶ Σ ∗ ✸ → Σ ∗ ✷ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✹✼✲✉♥✐❢♦r♠ ♠♦r♣❤✐s♠✳ ⎧ ⎪ ✵ → ✵✵✶✵✵✶✵✶✵✵✶✵✶✶✵✵✶✵✵✶✵✶✵✵✶✵✵✶✶✵✵✶✵✵✶✶✵✵✶✵✶✶✵✶✵✶✶ ⎪ ⎪ ⎪ ❤ ∶ ⎨ ✶ → ✵✵✶✵✵✶✶✵✵✶✵✵✶✶✵✵✶✶✵✶✶✵✵✶✶✵✶✶✵✵✶✵✵✶✶✵✵✶✶✵✶✶✵✶✵✶✶ ⎪ ⎪ ⎪ ⎪ ⎩ ✷ → ✵✵✶✶✵✶✶✵✶✵✶✶✵✶✵✵✶✵✶✶✵✶✵✶✶✵✶✵✵✶✵✶✵✵✶✵✵✶✶✵✶✶✵✶✵✶✶ ❚❤❡♦r❡♠ ❋♦r ❡✈❡r② ❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡ ✇♦r❞ ✇ ∈ Σ ∗ ✸ ✱ ❤ ( ✇ ) ✐s ✷ ✲❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡✳ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ✭✶✮ ψ ( ✇ ) = (∣ ✇ ∣ ✵ , ∣ ✇ ∣ ✶ , ∣ ✇ ∣ ✷ ) ❚ ✳ ψ ′ ( ✇ ) = (∣ ✇ ∣ ✵✵ , ∣ ✇ ∣ ✵✶ , ∣ ✇ ∣ ✶✶ ) ❚ ✳ ▲❡t ▼ s✳t✳ ▼ ψ ( ① ) = ψ ′ ( ❤ ( ① ) ✵ ) ❢♦r ❡✈❡r② ① ∈ Σ ✸ ✳ ⎛ ⎞ ✶✵ ✾ ✺ ⎜ ⎟ ▼ = ✶✻ ✶✹ ✶✼ . ⎝ ⎠ ✺ ✶✵ ✽ ❲❡ ❤❛✈❡ ψ ′ ( ❤ ( ✇ ) ✵ ) = ▼ ψ ( ✇ ) ❢♦r ❡✈❡r② ✇♦r❞ ✇ ∈ Σ ∗ ✸ ✳ ✺✽ ✷ − ✽✸ ⎛ ⎞ ✺✶✼ ✹✼ ✺✶✼ ▼ − ✶ = ⎜ ⎟ ✹✸ − ✺ ✾✵ . ✺✶✼ ✹✼ ✺✶✼ ⎝ ⎠ − ✾✵ ✺ ✹ ✺✶✼ ✹✼ ✺✶✼ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ✭✷✮ ✇ = ❛♣ ′ ❜q ′ ❝r ′ ❞ ✱ ♣ = ♣ ✶ ♣ ✷ ♣ ✸ ✱ q = q ✶ q ✷ q ✸ ✱ r = r ✶ r ✷ r ✸ ✱ ❤ ( ❛ ) = ✉♣ ✶ ✱ ❤ ( ♣ ′ ) = ♣ ✷ ✱ ❤ ( ❜ ) = ♣ ✸ q ✶ ✱ ❤ ( q ′ ) = q ✷ ✱ ❤ ( ❝ ) = q ✸ r ✶ ✱ ❤ ( r ′ ) = r ✷ ✱ ❤ ( ❞ ) = r ✸ ✈ ✳ ●❡♥❡r❛❧ ❝❛s❡✿ ✉ ♣ ✶ ✱ ♣ ✸ ✱ q ✶ ✱ q ✸ ✱ r ✶ ✱ r ✸ ❛♥❞ ✈ ❛r❡ ♥♦t ❡♠♣t②✳ ✭❖t❤❡r ❝❛s❡s ❛r❡ tr❡❛t❡❞ s✐♠✐❧❛r❧②✳✮ ❙✐♥❝❡ ♣qr ✐s ❛ ✷✲❛❜❡❧✐❛♥✲❝✉❜❡✱ t❤❡ ✜rst ❧❡tt❡r ♦❢ ♣ ✶ ✱ q ✶ ✱ r ✶ ❤❛s t❤❡ s❛♠❡ ✜rst ❧❡tt❡r✱ ❛♥❞ ♣ ✸ ✱ q ✸ ✱ r ✸ ❤❛s t❤❡ s❛♠❡ ❧❛st ❧❡tt❡r✳ ❲❡ ❤❛✈❡ ψ ′ ( ♣ ) = ψ ′ ( ♣ ✶ ♣ ✸ ) + ▼ ψ ( ♣ ′ ) ✱ ψ ′ ( q ) = ψ ′ ( q ✶ q ✸ ) + ▼ ψ ( q ′ ) ❛♥❞ ψ ′ ( r ) = ψ ′ ( r ✶ r ✸ ) + ▼ ψ ( r ′ ) ❚❤❡ ❝♦♥❞✐t✐♦♥ ψ ′ ( ♣ ) = ψ ′ ( q ) = ψ ′ ( r ) ❜❡❝♦♠❡✿ ▼ − ✶ ψ ′ ( ♣ ✶ ♣ ✸ ) + ψ ( ♣ ′ ) = ▼ − ✶ ψ ′ ( q ✶ q ✸ ) + ψ ( q ′ ) = ▼ − ✶ ψ ′ ( r ✶ r ✸ ) + ψ ( r ′ ) ✳ ◆♦t❡✿ t❤✐s ❝♦♥❞✐t✐♦♥ ❝❛♥♥♦t ❜❡ ❢✉❧✜❧❧❡❞ ✐❢ ▼ − ✶ ( ψ ′ ( ♣ ✶ ♣ ✸ ) − ψ ′ ( q ✶ q ✸ )) ✱ ♦r ▼ − ✶ ( ψ ′ ( ♣ ✶ ♣ ✸ ) − ψ ′ ( r ✶ r ✸ )) ❛r❡ ♥♦t ✐♥t❡❣❡r ✈❡❝t♦rs✳ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ✭✸✮ ❇② ❝♦♠♣✉t❡r✱ ✇❡ ❝❤❡❝❦ t❤❛t ❢♦r ❡✈❡r② ❛ , ❜ , ❝ , ❞ ∈ Σ ✱ ❛♥❞ ❢♦r ❡✈❡r② ♥♦♥✲❡♠♣t② ✉ , ♣ ✶ , ♣ ✸ , q ✶ , q ✸ , r ✶ , r ✸ , ✈ ∈ Σ ∗ s✉❝❤ t❤❛t✿ ✉♣ ✶ = ❤ ( ❛ ) ✱ ♣ ✸ q ✶ = ❤ ( ❜ ) ✱ q ✸ r ✶ = ❤ ( ❝ ) ✱ r ✸ ✈ = ❤ ( ❞ ) ✱ ♣ ✶ ✱ q ✶ ✱ r ✶ ❤❛✈❡ t❤❡ s❛♠❡ ✜rst ❧❡tt❡r✱ ♣ ✸ ✱ q ✸ ✱ r ✸ ❤❛✈❡ t❤❡ s❛♠❡ ❧❛st ❧❡tt❡r✱ ▼ − ✶ ( ψ ′ ( ♣ ✶ ♣ ✸ ) − ψ ′ ( q ✶ q ✸ )) ❛♥❞ ▼ − ✶ ( ψ ′ ( ♣ ✶ ♣ ✸ ) − ψ ′ ( r ✶ r ✸ )) ❛r❡ ✐♥t❡❣❡r ✈❡❝t♦rs✱ t❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ψ ( ❛♣ ′ ) = ψ ( ❜q ′ ) = ψ ( ❝r ′ ) ✱ t❤❛t ✐s✿ ▼ − ✶ ψ ′ ( ♣ ✶ ♣ ✸ )− ψ ( ❛ ) = ▼ − ✶ ψ ′ ( q ✶ q ✸ )− ψ ( ❜ ) = ▼ − ✶ ψ ′ ( r ✶ r ✸ )− ψ ( ❝ ) , ψ ( ♣ ′ ❜ ) = ψ ( q ′ ❝ ) = ψ ( r ′ ❞ ) ✱ ψ ( ❛♣ ′ ) = ψ ( ❜q ′ ) = ψ ( ❝r ′ ❞ ) ✱ ψ ( ❛♣ ′ ) = ψ ( ❜q ′ ❝ ) = ψ ( r ′ ❞ ) ✳ ■♥ ❛❧❧ ❝❛s❡s✱ ✇ ❤❛s ❛♥ ❛❜❡❧✐❛♥ ❝✉❜❡✳ ❈♦♥tr❛❞✐❝t✐♦♥✳ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

✷✲❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡ ❜✐♥❛r② ✇♦r❞ ✭♥♦♥✲✉♥✐❢♦r♠✮ ▲❡t ❤ ∶ Σ ∗ ✸ → Σ ∗ ✷ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦r♣❤✐s♠✳ ⎧ ⎪ ✵ → ✵✵✶✵✵✶✵✶✵✵✶✶✵✵✶✵✵✶✵✶✵✵✶✵✵✶✶✵✵✶✵✵✶✶✵✵✶✶✵✶✶ ⎪ ⎪ ⎪ ❤ ∶ ⎨ ✶ → ✵✶✵✶✶✵✶✶✵✵✶✶✵✶✶✵✵✶✶✵✵✶✵✵✶✶✵✵✶✶✵✶✶ ⎪ ⎪ ⎪ ⎪ ⎩ ✷ → ✵✶✵✶✶✵✶✵✵✶✵✶✵✵✶✵✶✶✵✵✶✶✵✶✶ ❚❤❡♦r❡♠ ❋♦r ❡✈❡r② ❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡ ✇♦r❞ ✇ ∈ Σ ∗ ✸ ✱ ❤ ( ✇ ) ✐s ✷ ✲❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡✳ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

✸✲❛❜❡❧✐❛♥✲sq✉❛r❡✲❢r❡❡ t❡r♥❛r② ✇♦r❞ ▲❡t ❤ ∶ Σ ∗ ✹ → Σ ∗ ✸ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✷✺✲✉♥✐❢♦r♠ ♠♦r♣❤✐s♠✳ ⎧ ⎪ ⎪ ✵ → ✵✶✵✷✵✶✷✵✷✶✵✶✷✵✶✵✷✵✶✷✶✵✷✶✷ ⎪ ⎪ ⎪ ⎪ ⎪ ✶ → ✵✶✵✷✶✵✶✷✵✶✵✷✶✷✵✶✷✶✵✶✷✵✷✶✷ ❤ ∶ ⎨ ⎪ ⎪ ⎪ ✷ → ✵✶✵✷✶✵✶✷✶✵✷✶✷✵✷✶✵✷✵✶✷✵✷✶✷ ⎪ ⎪ ⎪ ⎪ ⎩ ✸ → ✵✶✷✶✵✷✵✶✷✵✷✶✵✷✵✶✷✶✵✶✷✵✷✶✷ ❚❤❡♦r❡♠ ❋♦r ❡✈❡r② ❛❜❡❧✐❛♥✲sq✉❛r❡✲❢r❡❡ ✇♦r❞ ✇ ∈ Σ ∗ ✹ ✱ ❤ ( ✇ ) ✐s ✸ ✲❛❜❡❧✐❛♥✲sq✉❛r❡✲❢r❡❡✳ ▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦ ✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡