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❉②♥❛♠✐❝ ❡❝♦♥♦♠✐❝s ✐♥ Pr❛❝t✐❝❡

◆✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ✇✐t❤ ▼❛t❧❛❜ ▼♦♥✐❝❛ ❈♦st❛ ❉✐❛s ❛♥❞ ❈♦r♠❛❝ ❖✬❉❡❛

■♥tr♦❞✉❝t✐♦♥

▼❛t❧❛❜

◮ ▼❛t❧❛❜ ✐s ❛ s♦❢t✇❛r❡ ♣❛❝❦❛❣❡ ❛♥❞ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡ ◮ ❲✐❞❡❧② ✉s❡❞ ✐♥ ❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❛♥❞ ✐♥ ❡❝♦♥♦♠✐❝s ✐♥ ❣❡♥❡r❛❧ ◮ Pr♦♣r✐❡t❛r② ❛♥❞ ❡①♣❡♥s✐✈❡

◮ ❚❤♦✉❣❤ ♠♦st ✉♥✐✈❡rs✐t✐❡s ❤❛✈❡ ✐t ❛♥❞ ❛ s✉❜st❛♥t✐❛❧❧② ❞✐s❝♦✉♥t❡❞ st✉❞❡♥t ✈❡rs✐♦♥ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞

◮ ❍❛s ❛ ♥✉♠❜❡r ♦❢ ❛❞❞✐t✐♦♥❛❧ ❵t♦♦❧❜♦①❡s✬ t❤❛t s✉♣♣❧❡♠❡♥t st❛♥❞❛r❞ ❢❡❛t✉r❡s

◮ ❖♣t✐♠✐s❛t✐♦♥ t♦♦❧❜♦① ✐s ❡ss❡♥t✐❛❧ ❢♦r ✇♦r❦ ✐♥ ❞②♥❛♠✐❝ ♣r♦❣r❛♠♠✐♥❣

❆❧t❡r♥❛t✐✈❡s t♦ ▼❛t❧❛❜

◮ ❋♦rtr❛♥✱ ❈✰✰✱ ❈

◮ ▼✉❝❤ ♠✉❝❤ ❢❛st❡r✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ✈❡r② ✐♠♣♦rt❛♥t

◮ ❘

◮ ❋r❡❡✱ ✐♥❝r❡❛s✐♥❣❧② ♣♦♣✉❧❛r ✐♥ ❡❝♦♥♦♠✐❝s

◮ ▼❛t❛ ✭❙t❛t❛✬s ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡✮

◮ ◆❡✇❡r✱ ❧❡ss ✇✐❞❡❧② ✉s❡❞

❈♦❞❡ ❞❡✈❡❧♦♣❡❞ ❢♦r t❤✐s ❝♦✉rs❡

◮ ❚❤❡ ❝♦❞❡ ✇❡ ✇✐❧❧ ❣♦ t❤r♦✉❣❤ ✐s ❞❡s✐❣♥❡❞ t♦ ❜❡ ❡❛s② t♦ ✉♥❞❡rst❛♥❞✱ ♥♦t t♦ ❜❡ ❛s ❢❛st ❛s ♣♦ss✐❜❧❡ ◮ ■t ❞♦❡s ♥♦t ❝♦♠❡ ✇✐t❤ ❛ ✇❛rr❛♥t②✦

◮ ❚❤❡r❡ ♠❛② ❜❡ ♣❛r❛♠❡t❡r s❡ts t❤❛t ❞❡❧✐✈❡r ❛♥ ❡rr♦r ♦r ❛♥ ♦❞❞ r❡s✉❧t

◮ ❨♦✉ ♠❛② ♥♦t ✉♥❞❡rst❛♥❞ ✐t ❛❧❧ ❝♦♠♣❧❡t❡❧② ❛s ✇❡ ③♦♦♠ t❤r♦✉❣❤ ✐t

◮ ■t ✐s ❛♥♥♦t❛t❡❞ ◮ ❚♦ ♣r♦♣❡r❧② ✉♥❞❡rst❛♥❞ ✐t✱ ✐t ✇✐❧❧ ❜❡ ♥❡❝❡ss❛r② t♦ ❣♦ t❤r♦✉❣❤ ✐t ♠♦r❡ ♠❡t✐❝✉❧♦✉s❧② t❤❛♥ ✇❡ ✇✐❧❧ ❤❛✈❡ t✐♠❡ t♦ ❞♦ ❤❡r❡

◮ ❚❤❡ ❡①✐st✐♥❣ ❝♦❞❡ ❞♦❡s ♥♦t ✇♦r❦ ❛s ❛ ❜❧❛❝❦ ❜♦① ✕ ②♦✉ ♥❡❡❞ t♦ ✉♥❞❡rst❛♥❞ ✐t t♦ ❡❞✐t✴❝❤❛♥❣❡ ◮ ❇✉t t❤❡♥ ✐t ❝❛♥ ❜❡ ❣❡♥❡r❛❧✐s❡❞✴s♣❡❝✐❛❧✐s❡❞ t♦ ❞✐✛❡r❡♥t ♣r♦❜❧❡♠s ◮ ❆♥❞ ✐t ✐s s✐♠♣❧❡ t♦ tr❛♥s♣♦s❡ t♦ ♦t❤❡r ♠♦r❡ ❡✣❝✐❡♥t ❧❛♥❣✉❛❣❡s

❈♦❞❡ ❢♦r t❤✐s ❝♦✉rs❡

❲❡ ❤❛✈❡ s✐① ✈❡rs✐♦♥s ♦❢ t❤❡ ❝♦❞❡✱ t❤❛t ❢♦❧❧♦✇ t❤❡ ✐♥❝r❡❛s✐♥❣❧② ♠♦r❡ s♦♣❤✐st✐❝❛t❡❞ ♠♦❞❡❧s t❤❛t ✇✐❧❧ ❜❡ ❞✐s❝✉ss❡❞ ✶✳ ❋✐♥✐t❡ ❝♦♥s✉♠♣t✐♦♥ s❛✈✐♥❣ ❡♥❞♦✇♠❡♥t ♣r♦❜❧❡♠ ✭s♦❧✈❡❞ ❜② ♠❛①✐♠✐s✐♥❣ t❤❡ ❱❛❧✉❡ ❢✉♥❝t✐♦♥✮ ✷✳ ❆❞❞s t♦ ✭✶✮ ♦♣t✐♦♥ t♦ s♦❧✈❡ ✉s✐♥❣ t❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ✸✳ ❆❞❞s t♦ ✭✷✮ ❞❡t❡r♠✐♥✐st✐❝ str❡❛♠ ♦❢ ✐♥❝♦♠❡ ❛♥❞ ❝❛♣❛❝✐t② t♦ ❜♦rr♦✇ ✹✳ ❆❞❞s t♦ ✭✸✮ s✐♠♣❧❡ ✉♥❝❡rt❛✐♥t② ✐♥ ✐♥❝♦♠❡ ✺✳ ❆❞❞s t♦ ✭✹✮ ❛ ❝♦♥✈❡♥t✐♦♥❛❧ st♦❝❤❛st✐❝ ✐♥❝♦♠❡ ♣r♦❝❡ss ✻✳ ❙♦❧✈❡s ✐♥✜♥✐t❡ ❤♦r✐③♦♥ ✈❡rs✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧ ✐♥ ✭✺✮

▼❛t❧❛❜ ❜❛s✐❝s

❋✉♥❝t✐♦♥s

❲❡ ❝❛♥ ❝r❡❛t❡ ❛ s✐♠♣❧❡ ❢✉♥❝t✐♦♥ ❜② ❝r❡❛t✐♥❣ ❛ ✜❧❡ ✇✐t❤ t❤❡ ❢♦❧❧♦✇✐♥❣ t❡①t✿

function [ output ] = myFunction( input ) % This function outputs the twice the square of the input

• utput = 2 * (input^2);

end

❛♥❞ s❛✈✐♥❣ ✐t ❜② t❤❡ ♥❛♠❡ myFunction.m✳ ❲❤❡♥ ✇❡ ♣❧❛❝❡ t❤✐s ✜❧❡ ✐♥ ♦✉r ✇♦r❦✐♥❣ ❞✐r❡❝t♦r②✱ ✇❡ ❝❛♥ t❤❡♥ ❝❛❧❧ t❤❡ ❢✉♥❝t✐♦♥ ❢r♦♠ ▼❛t❧❛❜✿

[y] = myFunction(2);

❚❤✐s r❡t✉r♥s y = 8✳

❙❝r✐♣ts

◮ ❙❝r✐♣ts ❛r❡ ❛❧s♦ ♦r❣❛♥✐s❡ ✐♥ ✜❧❡s ◮ ❚❤❡② ❝♦♥t❛✐♥ ❛ ❧✐sts ♦❢ ❝♦♠♠❛♥❞s ◮ ❯♥❧✐❦❡ ❢✉♥❝t✐♦♥s t❤❡② ❞♦♥✬t st❛rt ✇✐t❤ ❛ ❤❡❛❞❡r ✳ ✳ ✳

◮ function [ output ] = myFunction( input )

◮ ✳ ✳ ✳ ❛♥❞ t❤❡② ❞♦♥✬t ✜♥✐s❤ ✇✐t❤✿

◮ end

◮ ❆ ❦❡② ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ s❝r✐♣ts ❛♥❞ ❢✉♥❝t✐♦♥s ❝♦♥❝❡r♥s ✇❤✐❝❤ ♣❛rts ♦❢ ♠❡♠♦r② t❤❡② ❤❛✈❡ ❛❝❝❡ss t♦

◮ ❋✉♥❝t✐♦♥s ❛❝❝❡♣t ✐♥♣✉ts ❛♥❞ r❡t✉r♥ ♦✉t♣✉ts❀ ♦t❤❡r ✈❛r✐❛❜❧❡s ❛r❡ ✐♥t❡r♥❛❧ t♦ t❤❡ ❢✉♥❝t✐♦♥ ◮ ❙❝r✐♣ts ❞♦ ♥♦t ❛❝❝❡♣t ❛r❣✉♠❡♥ts ❛♥❞ ♦♣❡r❛t❡ ♦♥ ❞❛t❛ ✐♥ t❤❡ ✇♦r❦s♣❛❝❡

▲♦❝❛❧s ❛♥❞ ●❧♦❜❛❧s ■

▲♦❝❛❧ ❛♥❞ ❣❧♦❜❛❧ ✈❛r✐❛❜❧❡s ◮ ▲♦❝❛❧ ✈❛r✐❛❜❧❡s ❛r❡ t❤♦s❡ t❤❛t ❝❛♥ ♦♥❧② ❜❡ ❛❝❝❡ss❡❞ ❜② t❤❡ ✜❧❡ ✐♥ ✇❤✐❝❤ t❤❡② ❛r❡ ✐♥tr♦❞✉❝❡❞ ◮ ●❧♦❜❛❧ ✈❛r✐❛❜❧❡s ❛r❡ t❤♦s❡ t❤❛t ❝❛♥ ❜❡ ❛❝❝❡ss❡❞ ✭❛♥❞ ❝❤❛♥❣❡❞✮ ❜② ❛♥② ✜❧❡ ❑❡② ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ s❝r✐♣ts ❛♥❞ ❢✉♥❝t✐♦♥s✿ ◮ ■❢ s❝r✐♣t ♦r ❢✉♥❝t✐♦♥ a.m ❝❛❧❧s s❝r✐♣t b.m✱ t❤❡♥ b.m ❝❛♥ ❛❝❝❡ss ❛♥② ✈❛r✐❛❜❧❡ t❤❛t a.m ❤❛s ❛❝❝❡ss t♦ ✭✐♥ t❤❡ ✇♦r❦s♣❛❝❡ ♦❢ a.m✮ ◮ ■❢ s❝r✐♣t ♦r ❢✉♥❝t✐♦♥ a.m ❝❛❧❧s ❢✉♥❝t✐♦♥ b.m✱ t❤❡♥ b.m ❤❛s ❛ s❡♣❛r❛t❡ ✇♦rs♣❛❝❡ ❛♥❞ ❝❛♥ ♦♥❧② ❛❝❝❡ss ❣❧♦❜❛❧ ✈❛r✐❛❜❧❡s ❞❡❝❧❛r❡❞ ✐♥ b.m

▲♦❝❛❧s ❛♥❞ ●❧♦❜❛❧s ■■

❚♦ ❞❡❝❧❛r❡ ❛ ✈❛r✐❛❜❧❡ ✭myvar✮ ❛s ❛ ❣❧♦❜❛❧✱ ✐♥ t❤❡ ✜❧❡ ✇❤❡r❡ ✐t ✐s ✜rst ✐♥✐t✐❛❧✐s❡❞ ✐♥❝❧✉❞❡ t❤❡ ❧✐♥❡✿

global myvar

❲❡ ❝❛♥ ❛❝❝❡ss t❤✐s ✈❛r✐❛❜❧❡ ✐♥ ❛♥② ♦t❤❡r s❝r✐♣t ♦r ❢✉♥❝t✐♦♥ ❝❛❧❧❡❞ ❜② t❤✐s ✜❧❡ ❜② ❛❧s♦ ✐♥❝❧✉❞✐♥❣ t❤❡ ❧✐♥❡✿

global myvar

❙❡t✉♣ ♦❢ ❛ ♣r♦❥❡❝t

◮ ■♥ ❛ ♣❛rt✐❝✉❧❛r ❢♦❧❞❡r ②♦✉ ✇✐❧❧ ♥❡❡❞ ❛ ❵♠❛st❡r✬ s❝r✐♣t ✭❝❛❧❧❡❞ ♣❡r❤❛♣s

main.m✮ ❛♥❞ ❛❧❧ t❤❡ s❝r✐♣ts ❛♥❞ ❢✉♥❝t✐♦♥s t❤❛t ✇✐❧❧ ❜❡ ❝❛❧❧❡❞

◮ ❙❡t t❤❡ ✇♦r❦✐♥❣ ❞✐r❡❝t♦r② ✐♥ ▼❛t❧❛❜ ❛s t❤✐s ❢♦❧❞❡r ◮ ❘✉♥ t❤❡ s❝r✐♣t main.m ❜② t②♣✐♥❣ main ❛t t❤❡ ▼❛t❧❛❜ ♣r♦♠♣t ♦r ❜② ❝❧✐❝❦✐♥❣ t❤❡ ❣r❡❡♥ ❵r✉♥✬ ❜✉tt♦♥ ✐♥ t❤❡ ▼❛t❧❛❜ t❡①t ❡❞✐t♦r

❙♦♠❡ ❜❛s✐❝ ❝♦♠♠❛♥❞s

◮ + - * / ^ ❜❛s✐❝ ♦♣❡r❛t♦rs ◮ .* ./ .^ ❡❧❡♠❡♥t✇✐s❡ ♦♣❡r❛t♦rs ◮ ; s✉♣♣r❡ss ♦✉t♣✉t

◮ A = 1; ❛♥❞ A = 1 ❜♦t❤ st♦r❡ s❝❛❧❛r ✶ ✐♥ ❆ ◮ ❚❤❡ ❧❛tt❡r ❛❧s♦ ♣r✐♥ts A=1 t♦ s❝r❡❡♥ ◮ ❨♦✉✬❧❧ ✇❛♥t t♦ t❡r♠✐♥❛t❡ ♠♦st ❝♦♠♠❛♥❞s ✇✐t❤ ;

◮ tic, toc st❛rts ❛♥❞ ✜♥✐s❤❡s ❛ st♦♣✇❛t❝❤ ❛♥❞ ♣r✐♥ts t✐♠❡ t❛❦❡♥

❈r❡❛t✐♥❣ ❛ ♠❛tr✐①

◮ A=[1 2 3; 3 5 6] ❝r❡❛t❡s t❤❡ ♠❛tr✐① A = ✶ ✷ ✸ ✹ ✺ ✻

• ◮ B=zeros(1,3)

❝r❡❛t❡s ❛ ✶ × ✸ ✈❡❝t♦r ♦❢ ③❡r♦s ◮ C=NaN(2,1) ❝r❡❛t❡s ❛ ✷ × ✶ ✈❡❝t♦r ♦❢ ♠✐ss✐♥❣ ✈❛❧✉❡s ◮ X=[A;B] ❝♦♥❝❛t❡♥❛t❡s ✈❡rt✐❝❛❧❧② ♠❛tr✐❝❡s A ❛♥❞ B

◮ X ❤❛s ❞✐♠❡♥s✐♦♥ ✸ × ✸

◮ X=[A,C] ❝♦♥❝❛t❡♥❛t❡s ❤♦r✐③♦♥t❛❧❧② ♠❛tr✐❝❡s A ❛♥❞ C

◮ X ❤❛s ❞✐♠❡♥s✐♦♥ ✷ × ✹

◮ X=repmat(A, (n, m)) r❡♣❧✐❝❛t❡s ♠❛tr✐① ❆ ♥ ❜② ♠ t✐♠❡s

◮ ■❢ ♠❛tr✐① ❆ ❤❛s ❞✐♠❡♥s✐♦♥ j × k✱ X ✇✐❧❧ ❤❛✈❡ ❞✐♠❡♥s✐♦♥ nj × mk

❙❡❧❡❝t✐♥❣ ♦✉t ❡❧❡♠❡♥ts ♦❢ ❛ ♠❛tr✐①

❙❛② ✇❡ ✇❛♥t ❛ ✈❡❝t♦r t❤❛t ❝♦♥t❛✐♥s t❤❡ r♦✇ t ♦❢ ♠❛tr✐① V

Vt = V(t , :) ;

❙❛② ✇❡ ✇❛♥t ❡❧❡♠❡♥ts ✐♥ ❝♦❧✉♠♥s ✸ t♦ ✺ ♦❢ r♦✇ t ♦❢ V

Vt = V(t , 3:5) ;

▲♦♦♣s

❆ ❵❢♦r✬ ❧♦♦♣✿

squares = NaN(10, 1); for ix = 1:1:10 squares(ix) = ix ^ 2; end

❆ ❵✇❤✐❧❡✬ ❧♦♦♣ ♣r♦❞✉❝✐♥❣ t❤❡ s❛♠❡ r❡s✉❧t✿

squares = NaN(10, 1); ix = 1; while ix

≤10

squares(ix) = ix ^ 2; ix = ix + 1; end

■❢ ❜❧♦❝❦s

❆ s✐♠♣❧❡ ❵✐❢✬ ❜❧♦❝❦✿

if x1 < 0 fprintf('x1 is negative') elseif x1 == 0 fprintf('x1 is zero') else fprintf('x1 is positive') end

• ❡♥❡r❛t✐♥❣ ❛ ❣r✐❞ ♦❢ ❡q✉❛❧❧② s♣❛❝❡❞ ✈❛❧✉❡s ✐♥ ▼❛t❧❛❜

grid = linspace(0, 1, 4)

• ❡♥❡r❛t❡s ❛♥❞ st♦r❡s ✐♥ ❛ ✈❡❝t♦r ❛ ❣r✐❞ ✇✐t❤ ♠✐♥✐♠✉♠ ✵✱ ♠❛①✐♠✉♠ ✶ ❛♥❞

❢♦✉r ♣♦✐♥ts s✉❝❤ t❤❛t t❤❡ s♣❛❝❡ ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts ✐s ❡q✉❛❧

grid = 0 0.3333 0.6667 1.0000

• ❡♥❡r❛t✐♥❣ ❛ ❣r✐❞ ♦❢ ✈❛❧✉❡s ♠♦r❡ ❝♦♥❝❡♥tr❛t❡❞ t♦✇❛r❞s ❧♦✇❡r

✈❛❧✉❡s

❉♦ t❤✐s ❜② ❡q✉❛❧✐s✐♥❣ t❤❡ ❧♦❣✲❞✐st❛♥❝❡ ❜❡t✇❡❡♥ ✵ ❛♥❞ s♦♠❡ ✉♣♣❡r ❜♦✉♥❞ ✭❝❛❧❧ ✐t top✮ ❢♦r t❤❡ ❣r✐❞✿

top = 1; loggrid = linspace(log(1),log(1 + top), 4); grid = exp(loggrid)-1;

❘❡t✉r♥s ❛ ❣r✐❞ ✇✐t❤ ♠✐♥✐♠✉♠ ✵✱ ♠❛①✐♠✉♠ top ❛♥❞ ❢♦✉r ♣♦✐♥ts ❛t ✐♥❝r❡❛s✐♥❣ ❞✐st❛♥❝❡✿

grid = 0.2599 0.5874 1.0000

❋✐rst ▼❛t❧❛❜ ♣r♦❣r❛♠ ✕ ❢♦❧❞❡r ✬❝♦❞❡❭✈✶✬ ❙❡t ✉♣ ▼❛t❧❛❜ ❢♦❧❞❡r ❛♥❞ ♣r♦❣r❛♠ t♦ s♦❧✈❡ ❛♥❞ s✐♠✉❧❛t❡ s✐♠♣❧❡ ❝❛❦❡ ❡❛t✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤♦✉t ✉♥❝❡rt❛✐♥t②

❖♣t✐♠✐s❛t✐♦♥

❖♣t✐♠✐s❛t✐♦♥ ✐♥ ▼❛t❧❛❜ ■

◮ Pr♦❜❧❡♠✿ ❋✐♥❞ x s✉❝❤ t❤❛t f (x) ✐s ♠✐♥✐♠✐s❡❞ ◮ ▼❛t❧❛❜ ❤❛s ♠❛♥② ❝♦♠♠❛♥❞s ❛s ♣❛rt ♦❢ ✐ts ♦♣t✐♠✐s❛t✐♦♥ t♦♦❧❜♦①✳ ◮ ❲❡ ✇✐❧❧ ❞✐s❝✉ss fminbnd

[Y, fval] = fminbnd(@(x) f(x), lowerbound, upperbound);

❚❤✐s r❡t✉r♥s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛r❣✉♠❡♥t t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ❢✉♥❝t✐♦♥ ✐♥ Y ❛♥❞ t❤❡ ♠✐♥✐♠✐s❡❞ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ✐♥ fval

[Y, fval] = fminbnd(@(x) (2*(x^2) - x), -100, 100);

❚❤✐s r❡t✉r♥s Y = 0.25 ❛♥❞ fval = -0.125

❖♣t✐♠✐s❛t✐♦♥ ✐♥ ▼❛t❧❛❜ ■■

◮ ❲❤❛t ✇♦✉❧❞ ✇❡ ❞♦ ✐❢ ✇❡ ✇❛♥t❡❞ t♦ ♠❛①✐♠✐s❡ ❛ ❢✉♥❝t✐♦♥ r❛t❤❡r t❤❛♥ ♠✐♥✐♠✐s❡ ✐t❄ ◮ ❲❡ ❝♦✉❧❞ ♠✐♥✐♠✐s❡ t❤❡ ♥❡❣❛t✐✈❡ ♦❢ t❤❛t ❢✉♥❝t✐♦♥

[Y, fval] = fminbnd(@(x) -(2*(x^2) - x), -100, 100);

◮ ❚❤❡ ❛❧❣♦r✐t❤♠ ✉s❡❞ ❜② fminbnd ❛❧t❡r♥❛t❡s ❜❡t✇❡❡♥ t❤❡ ●♦❧❞❡♥ ❙❡❝t✐♦♥ ❙❡❛r❝❤ ❛♥❞ ❙✉❝❝❡ss✐✈❡ P❛r❛❜♦❧✐❝ ■♥t❡r♣♦❧❛t✐♦♥

◮ ❖♣t✐♠✐s❡r ❢♦r ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ◮ ❆s ♦t❤❡r ♥✉♠❡r✐❝❛❧ ♦♣t✐♠✐s❡rs✿ r❡t✉r♥s ❧♦❝❛❧ s♦❧✉t✐♦♥s ◮ ❉♦❡s ♥♦t r❡t✉r♥ ❛ ♠✐♥✐♠✉♠ ♦♥ t❤❡ ❜♦✉♥❞❛r② ♦❢ t❤❡ ♦♣t✐♠✐s✐♥❣ ✐♥t❡r✈❛❧ ✕ ✇✐❧❧ r❡t✉r♥ ❛ ❝❧♦s❡ ❛♣♣r♦①✐♠❛t✐♦♥ ◮ ❙❧♦✇ ❝♦♥✈❡r❣❡♥❝❡ ✐❢ s♦❧✉t✐♦♥ ✐s ♦♥ t❤❡ ❜♦✉♥❞❛r②

❖♣t✐♠✐s❛t✐♦♥ ✐♥ ▼❛t❧❛❜ ■■■

◮ ■♥st❡❛❞ ♦❢ ✇r✐t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ❞✐r❡❝t❧② ✐♥ t❤❡ fminbnd ❝♦♠♠❛♥❞ ❧✐♥❡✱ ✇❡ ❝❛♥ ✇r✐t❡ ✐t ✐♥ ❛ s❡♣❛r❛t❡ ✜❧❡ ◮ ❚❤❡ ❢✉♥❝t✐♦♥ ♠❛② ✐♥❝❧✉❞❡ ❛r❣✉♠❡♥ts ♦t❤❡r t❤❛♥ t❤❡ ♦♥❡ ✇❡ ❛r❡ ♠❛①✐♠✐s✐♥❣ ✇✐t❤ r❡s♣❡❝t t♦ ❋✐rst ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ s❛✈❡ t❤❡ ✜❧❡ ❛s myFunction.m

function [ z ] = myFunction(x, a) z = a*(x^2) - x; end

❚❤❡♥ ✜♥❞ ✐ts ♠✐♥✐♠✉♠ ❢♦r ❛ ❣✐✈❡♥ ✈❛❧✉❡ ♦❢ a

a = 5; [Y, fval] = fminbnd(@(x) myFunction(x, a), -100, 100);

❚❤✐s r❡t✉r♥s Y = 0.1 ❛♥❞ fval = -0.05

❆♣♣r♦①✐♠❛t✐♦♥

❘❡♣r❡s❡♥t✐♥❣ ❛ ❢✉♥❝t✐♦♥ ♥✉♠❡r✐❝❛❧❧②

❍♦✇ ❝❛♥ ✇❡ st♦r❡ ❛ ❢✉♥❝t✐♦♥ t❤❛t ✐s ❞❡✜♥❡❞ ♥✉♠❡r✐❝❛❧❧②❄ ◮ ❚❛❦❡ ❛ s✐♠♣❧❡ ❢✉♥❝t✐♦♥ f (x) = ln(x) ◮ ❈❧♦s❡❞✲❢♦r♠ r❡♣r❡s❡♥t❛t✐♦♥ ❡①✐sts ❜✉t s❛② ✇❡ ✇❛♥t❡❞ t♦ st♦r❡ ✐t ♥✉♠❡r✐❝❛❧❧②❄ ◮ ❲❡ ✇♦✉❧❞ ❞❡✜♥❡ t✇♦ ✈❡❝t♦rs ✭s❛② x ❛♥❞ f ✮ ◮ x ✇✐❧❧ ❝♦♥t❛✐♥ ❛ ♥✉♠❜❡r ♦❢ ♣♦✐♥ts ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ ❢✉♥❝t✐♦♥✱ ❛t ✇❤✐❝❤ ✇❡ ❞❡❝✐❞❡ t♦ ❡✈❛❧✉❛t❡ f (x) ✳ ✳ ✳ ◮ ❛♥❞ f ✇✐❧❧ ❝♦♥t❛✐♥ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤♦s❡ ♣♦✐♥ts       ✵.✶ ✵.✷ ✵.✺ ✶ ✷             −✷.✸ −✶.✻ −✵.✼ ✵.✵ ✵.✼      

❆♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❛ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ▼❛t❧❛❜

◮ ❙✉♣♣♦s❡ ✇❡ ♦♥❧② ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ ❛ ❢✉♥❝t✐♦♥ ♦♥ ❛ ❞✐s❝r❡t❡ s✉❜s❡t ♦❢ ✐ts ❞♦♠❛✐♥ ◮ ❆♥❞ ✇❡ ✇❛♥t t♦ ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t ❛ ♣♦✐♥t x0 ♦♥ t❤❡ ❞♦♠❛✐♥ ❜✉t ♥♦t ✐♥ t❤❛t s✉❜s❡t ◮ ▼❛t❧❛❜ ❤❛s ❛ t♦♦❧ ✭interp1✮ t❤❛t ❛❧❧♦✇s ✉s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ❛t t❤❛t ♣♦✐♥t

approx = interp1(x, f, x0, method);

✇❤❡r❡✿ ◮ x ✐s t❤❡ ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ t❤❡ ♣♦✐♥ts ❛t ✇❤✐❝❤ ✇❡ ❦♥♦✇ f (x) ◮ f ✐s t❤❡ ✈❡❝t♦r ❝♦♥t❛✐♥✐♥❣ t❤❡ ✈❛❧✉❡ ♦❢ f ❛t ❡❛❝❤ ♣♦✐♥t ✐♥ x ◮ x0 ✐s t❤❡ ♣♦✐♥t ❛t ✇❤✐❝❤ ✇❡ ✇❛♥t t♦ ❛♣♣r♦①✐♠❛t❡ f ◮ method s♣❡❝✐✜❡s ✇❤✐❝❤ ❛❧❣♦r✐t❤♠ t♦ ✉s❡

▲✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥

P❧♦t ♦❢ f (x) = ln(x) ✐♥ ✐♥t❡r✈❛❧ (✵, ✷]

▲✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥

❙✉♣♣♦s❡ ✇❡ ♦♥❧② ❦♥♦✇ t❤❡ ✈❛❧✉❡ ♦❢ f (x) ♦♥ ✺ ♣♦✐♥ts

▲✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥

■♥t❡r♣♦❧❛t❡ ❧✐♥❡❛r❧② ❜❡t✇❡❡♥ ✷ s✉❜s❡q✉❡♥t ♣♦✐♥ts

▲✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥

❆♣♣r♦①✐♠❛t❡ ln(x) ❜② ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ✐♥ t❤❡ s♣❛❝❡ ❞❡✜♥❡❞ ❜② t❤❡ ❣r✐❞

▲✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥

▲❡t✬s ❛♣♣r♦①✐♠❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ f (x) = ln(x) ❛t x = ✶.✺ ❣✐✈❡♥ t❤❡ ✈❛❧✉❡s ♦❢ f (x) ❛t t❤❡ ✺ ♣♦✐♥ts ✇❡ ♣r❡✈✐♦✉s❧② ❞❡✜♥❡❞✿       ✵.✶ ✵.✷ ✵.✺ ✶ ✷             −✷.✸ −✶.✻ −✵.✼ ✵.✵ ✵.✼      

x = [0.1, 0.2, 0.5, 1, 2]; f = [-2.3, -1.6, -0.7, 0.0, 0.7]; approx = interp1(x, f, 1.5, 'linear');

❚❤✐s r❡t✉r♥s approx = 0.35 ✭t❤❡ tr✉t❤ ✐s ln(✶.✺) = ✵.✹✵✺✮

❍♦✇ ❣♦♦❞ ✇✐❧❧ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❜❡❄

◮ ❆♣♣r♦①✐♠❛t✐♥❣ ❛ ❢✉♥❝t✐♦♥ ✉s✐♥❣ ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ✇✐❧❧ ❜❡ ❜❡tt❡r✿

◮ ❚❤❡ ❝❧♦s❡r t♦❣❡t❤❡r ❛r❡ t❤❡ ❵❣r✐❞ ♣♦✐♥ts✬✱ ❛t ✇❤✐❝❤ ✇❡ ❦♥♦✇ t❤❡ tr✉❡ ❢✉♥❝t✐♦♥ ✈❛❧✉❡ ◮ ❚❤❡ ❝❧♦s❡r t❤❡ ✉♥❞❡r❧②✐♥❣ tr✉❡ ❢✉♥❝t✐♦♥ t♦ ❧✐♥❡❛r

◮ ■t ❤❡❧♣s t♦ ❝♦♥❝❡♥tr❛t❡ ❣r✐❞ ♣♦✐♥ts ✇❤❡r❡ t❤❡ ❢✉♥❝t✐♦♥ ✐s ♠♦r❡ ♥♦♥✲❧✐♥❡❛r ◮ ❆❧t❡r♥❛t✐✈❡ ❛♣♣r♦①✐♠❛t✐♦♥ t❡❝❤♥✐q✉❡s ❛r❡ ❣❡♥❡r❛❧❧② ♠♦r❡ ❛♣♣r♦♣r✐❛t❡ t♦ ❞❡❛❧ ✇✐t❤ ♥♦♥✲❧✐♥❡❛r✐t✐❡s✿

◮ ▲♦❝❛❧ ♣♦❧②♥♦♠✐❛❧s ✭s♣❧✐♥❡s✮✱ ❝❛♥ ❜❡ s❤❛♣❡ ♣r❡s❡r✈✐♥❣ ◮ ❇✉t t❤❡② ✐♥❝r❡❛s❡ ❝♦♠♣✉t❛t✐♦♥ t✐♠❡

❋✐rst ▼❛t❧❛❜ ♣r♦❣r❛♠ ✕ ❢♦❧❞❡r ✬❝♦❞❡❭✈✶✬ ❇❛❝❦ t♦ t❤❡ ❝♦❞❡ t♦ s♦❧✈❡ ❛♥❞ s✐♠✉❧❛t❡ s✐♠♣❧❡ ❝❛❦❡ ❡❛t✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤♦✉t ✉♥❝❡rt❛✐♥t② ◮ ❯s❡ fminbnd t♦ ♠❛①✐♠✐s❡ t❤❡ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ◮ ❯s❡ interp1 t♦ ✐♥t❡r♣♦❧❛t❡ t❤❡ s♦❧✉t✐♦♥ ♦♥ ❣r✐❞

❘♦♦t✲✜♥❞✐♥❣

❘♦♦t✲✜♥❞✐♥❣ ✐♥ ▼❛t❧❛❜ ■

◮ Pr♦❜❧❡♠✿ ❋✐♥❞ x s✉❝❤ t❤❛t f (x) = ✵ ◮ ❋♦r ♠❛♥② ❢✉♥❝t✐♦♥s f (x)✱ t❤❡r❡ ✐s ♥♦ ❝❧♦s❡❞ ❢♦r♠ s♦❧✉t✐♦♥ ❢♦r x s✉❝❤ t❤❛t f (x) = ✵ ◮ ▼❛t❧❛❜ ❤❛s ❛ ❝♦♠♠❛♥❞ ✭fzero✮ t♦ ❞♦ t❤✐s✿

[Y, fval] = fzero(@(x) f(x), StartingValue)

◮ ❚❤✐s r❡t✉r♥s t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❛r❣✉♠❡♥t t❤❛t ♠✐♥✐♠✐s❡s t❤❡ ❢✉♥❝t✐♦♥ ✐♥ Y ❛♥❞ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ f ❛t Y ✐♥ fval✳ ◮ fval s❤♦✉❧❞ ❜❡ ✈❡r② ❝❧♦s❡ t♦ ✵ ✐❢ t❤❡ ❢✉♥❝t✐♦♥ ❤❛s ✇♦r❦❡❞ ❝♦rr❡❝t❧②

[Y, fval] = fzero(@(x) (2*x - 1), 0.2);

r❡t✉r♥s Y = 0.5 ❛♥❞ fval = 0

❘♦♦t✲✜♥❞✐♥❣ ✉s✐♥❣ ▼❛t❧❛❜ ■■

■♥st❡❛❞ ♦❢ ❣✐✈✐♥❣ ▼❛t❧❛❜ ❛ st❛rt✐♥❣ ✈❛❧✉❡✱ ✐t ✐s ❜❡tt❡r✱ ✐❢ ②♦✉ ❝❛♥✱ t♦ ❣✐✈❡ ✐t ❛ ❜♦✉♥❞ ✇✐t❤✐♥ ✇❤✐❝❤ ②♦✉ ❦♥♦✇ t❤❡ ③❡r♦ ❧✐❡s✿

boundforzero = [0, 1]; [Y] = fzero(@(x) 2*x - 1, boundforzero);

❚❤✐s r❡t✉r♥s Y = 0.5 ❚❤❡ ❛❧❣♦r✐t❤♠ ✉s❡s ❜✐s❡❝t✐♦♥✱ s❡❝❛♥t✱ ❛♥❞ ✐♥✈❡rs❡ q✉❛❞r❛t✐❝ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞s ❛♥❞ ✇❛s ❞❡✈❡❧♦♣❡❞ ❜② ❉❡❦❦❡r ✭✶✾✻✾✮ ❛♥❞ ❇r❡♥t ✭✶✾✼✸✮✳

❘♦♦t✲✜♥❞✐♥❣ ✉s✐♥❣ ▼❛t❧❛❜ ■■■

■♥st❡❛❞ ♦❢ ✇r✐t✐♥❣ t❤❡ ❢✉♥❝t✐♦♥ ❞✐r❡❝t❧② ✇❤❡♥ fzero ✐s ❝❛❧❧❡❞✱ ✇❡ ❝❛♥ ✇r✐t❡ t❤❡ ❢✉♥❝t✐♦♥ s❡♣❛r❛t❡❧② ❛♥❞ ✐♥❝❧✉❞❡ ♦t❤❡r ❛r❣✉♠❡♥ts ❋✐rst ✇r✐t❡ t❤❡ ❢✉♥❝t✐♦♥ ❛♥❞ s❛✈❡ ✐t ✐♥ ✜❧❡ myFunction.m

function [ z ] = myFunction(x, a) z = a*x - 1; end

❚❤❡♥ ❝❛❧❧ ✐t ❛❢t❡r ❞❡✜♥✐♥❣ t❤❡ ♣❛r❛♠❡t❡r a ❛♥❞ t❤❡ st❛rt✐♥❣ ✐♥t❡r✈❛❧

a = 3 boundforzero = [0, 1]; [Y] = fzero(@(x) myFunction(x, a), boundforzero);

❚❤✐s r❡t✉r♥s Y = 0.333

❙❡❝♦♥❞ ▼❛t❧❛❜ ♣r♦❣r❛♠ ✕ ❢♦❧❞❡r ✬❝♦❞❡❭✈✷✬ ❙♦❧✈❡ ❝❛❦❡ ❡❛t✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤♦✉t ✉♥❝❡rt❛✐♥t② ✉s✐♥❣ fzero t♦ ✜♥❞ t❤❡ r♦♦t ♦❢ t❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥

▼♦r❡ ♦♥ ❢✉♥❝t✐♦♥ ❛♣♣r♦①✐♠❛t✐♦♥

■♠♣r♦✈✐♥❣ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ ✇✐t❤ ♦t❤❡r ❦♥♦✇♥ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❢✉♥❝t✐♦♥✿ q✉❛s✐✲❧✐♥❡❛r✐s❛t✐♦♥

◮ ❘❡✈✐s✐t t❤❡ ♣r♦❜❧❡♠ ♦❢ ❛♣♣r♦①✐♠❛t✐♥❣ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥ f ◮ ▲✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ✐s ❢❛st ❜✉t ♥♦t ✈❡r② ❛❝❝✉r❛t❡ ◮ ❙✉♣♣♦s❡ ❤♦✇❡✈❡r t❤❛t ✇❡ ❦♥♦✇ ❛♥♦t❤❡r ❢✉♥❝t✐♦♥ g s✉❝❤ t❤❛t g(f (x)) = (g ◦ f )(x) ✐s ❝❧♦s❡ t♦ ❧✐♥❡❛r ✐♥ x ❛♥❞ t❤❛t g ✐s ✐♥✈❡rt✐❜❧❡ ◮ ❲❡ ❝❛♥ ✉s❡ g t♦ ❞♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿

✶✳ ❆t ❡❛❝❤ ♣♦✐♥t ❢♦r ✇❤✐❝❤ ✇❡ ❦♥♦✇ f (x)✱ ❝❛❧❝✉❧❛t❡ (g ◦ f )(x) ✷✳ ❆♣♣r♦①✐♠❛t❡ (g ◦ f )(x) ✉s✐♥❣ ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ❛♥❞ st♦r❡ t❤✐s

• (g ◦ f )(x)

✸✳ ❈❛❧❝✉❧❛t❡ g −✶

• (g ◦ f )(x)
• ✇❤✐❝❤ ✐s ♦✉r ❛♣♣r♦①✐♠❛t✐♦♥

f (x) t♦ f (x)

◮ ❆s ❧♦♥❣ ❛s (g ◦ f ) ✐s ❝❧♦s❡r t♦ ❧✐♥❡❛r t❤❛♥ ✐s f ✱ t❤❡♥ t❤✐s s❤♦✉❧❞ ❞❡❧✐✈❡r ❛ ❜❡tt❡r ❛♣♣r♦①✐♠❛t✐♦♥ t❤❛♥ ❞✐r❡❝t❧② ✐♥t❡r♣♦❧❛t✐♥❣ ♦♥ f

◗✉❛s✐✲❧✐♥❡❛r✐s❛t✐♦♥ ❡①❛♠♣❧❡

❙✉♣♣♦s❡ ✇❡ ✇❛♥t t♦ ❛♣♣r♦①✐♠❛t❡ ♠❛r❣✐♥❛❧ ✉t✐❧✐t② ✭✇✐t❤ ❛ ❈❘❘❆ ✉t✐❧✐t② ❢✉♥❝t✐♦♥✮ ✉s✐♥❣ ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ f (c) = c−γ ❚❤❡r❡ ❡①✐sts ❛♥ ✐♥✈❡rt✐❜❧❡ ❢✉♥❝t✐♦♥✿ g(x) = x− ✶

γ

s✉❝❤ t❤❛t (g ◦ f )(c) = c ✐s ❧✐♥❡❛r ❛♥❞ s♦ ❝❛♥ ❛❝❝✉r❛t❡❧② ❜❡ ✐♥t❡r♣♦❧❛t❡❞ ❧✐♥❡❛r❧② ❲❡ ❝❛♥ t❤❡♥ r❡❝♦✈❡r f (c) ❜② ❛♣♣❧②✐♥❣ t❤❡ ✐♥✈❡rs❡ ♦❢ g t♦ (g ◦ f )(c) ✇❤❡r❡ g −✶(y) = f (y) = y −γ

◗✉❛s✐✲❧✐♥❡❛r✐s❛t✐♦♥ ❡①❛♠♣❧❡

❈♦♥s✐❞❡r t❤r❡❡ ✈❡❝t♦rs ✕ c, f (c), g(f (c)) ✇❤❡r❡ (f , g) ❛r❡ ❢✉♥❝t✐♦♥s ❛s ❛❜♦✈❡ ✇✐t❤ γ = ✷       ✵.✶ ✵.✷ ✵.✺ ✶ ✷             ✶✵✵ ✷✺ ✹ ✶ ✵.✷✺             ✵.✶ ✵.✷ ✵.✺ ✶ ✷       ◮ ❚❤❡ tr✉❡ ♠❛r❣✐♥❛❧ ✉t✐❧✐t② ❛t c = ✶.✺ ✐s f (✶.✺) = ✶.✺−✷ = ✵.✹✹✹ ◮ ❆♣♣r♦①✐♠❛t✐♥❣ f ❛t c = ✶.✺ ❜② ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ②✐❡❧❞s

• f (✶.✺) = ✵.✻✷✺

◮ ❆♣♣r♦①✐♠❛t✐♥❣ (g ◦ f )(c) ❛t c = ✶.✺ ❜② ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥ ②✐❡❧❞s

• (g ◦ f )(✶.✺) = ✶.✺

◮ ■♥✈❡rt✐♥❣ (g ◦ f ) t♦ r❡❝♦✈❡r f ❛t c = ✶.✺ ②✐❡❧❞s

• f (✶.✺) = (✶.✺)−✷ = ✵.✹✹✹

❆❧t❡r♥❛t✐✈❡s t♦ ❧✐♥❡❛r ✐♥t❡r♣♦❧❛t✐♦♥

❆❧t❡r♥❛t✐✈❡ t♦ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ t❤❛t ▼❛t❧❛❜ ♣r♦✈✐❞❡s ✐♥❝❧✉❞❡✿ ◮ ❵♥❡❛r❡st✬✿ ♥❡❛r❡st ♥❡✐❣❤❜♦✉r ♠❛t❝❤✐♥❣ ◮ ❵s♣❧✐♥❡✬✿ ♣✐❡❝❡✇✐s❡ ❝✉❜✐❝ s♣❧✐♥❡ ◮ ❵♣❝❤✐♣✬✿ s❤❛♣❡✲♣r❡s❡r✈✐♥❣ ♣✐❡❝❡✇✐s❡ ❝✉❜✐❝ ✐♥t❡r♣♦❧❛t✐♦♥ ❘❡❝❛❧❧ ♦✉r ❛tt❡♠♣t t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ✈❛❧✉❡ ♦❢ ln(✶.✺)✿

approx1 = interp1(x, f, 1.5, 'linear'); (0.34) approx2 = interp1(x, f, 1.5, 'nearest'); (0.7) approx3 = interp1(x, f, 1.5, 'spline'); (0.467) approx4 = interp1(x, f, 1.5, 'pchip'); (0.442)

❚❤❡ tr✉t❤ ✐s ln(✶.✺) = ✵.✹✵✺✳

❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ■

❚♦ s❡❡ ❤♦✇ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❝❛♥ ❧❡❛❞ t♦ s❡r✐♦✉s ♠✐s❧❡❛❞✐♥❣ r❡s✉❧ts ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡✿ ◮ ❆ ❝♦♥s✉♠❡r ❧✐✈❡s ❢♦r ✷ ♣❡r✐♦❞s ❛♥❞ ❤❛s ❡♥❞♦✇♠❡♥t ♦❢ ✵✳✷ ✉♥✐ts ♦❢ ❝♦♥s✉♠♣t✐♦♥ t♦ s♣❧✐t ❜❡t✇❡❡♥ t❤♦s❡ t✇♦ ♣❡r✐♦❞s ◮ ❯t✐❧✐t② ✐s ❈❘❘❆ ✇✐t❤ γ = ✵.✺ ◮ ❚❤❡ ✐♥t❡r❡st r❛t❡ ❛♥❞ ❞✐s❝♦✉♥t r❛t❡ ❛r❡ ❡q✉❛❧ t♦ ③❡r♦ ◮ ❚❤❡ tr✉❡ ♦♣t✐♠❛❧ ✐s C✶ = C✷ = ✵.✶

❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ■■

P❧♦t ✉t✐❧✐t✐❡s ❛t t = ✶, ✷ ❛♥❞ ✈❛❧✉❡ ❛t t = ✶ ❢♦r ❝♦♥s✉♠♣t✐♦♥ ❛t t = ✶ ✈❛r②✐♥❣ ✐♥ t❤❡ ✐♥t❡r✈❛❧ (✵, ✵.✷)

• 6

❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ■■■

❚♦ s❡❡ ❤♦✇ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r ❝❛♥ ❧❡❛❞ t♦ s❡r✐♦✉s ♠✐s❧❡❛❞✐♥❣ r❡s✉❧ts ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❡①❛♠♣❧❡✿ ◮ ❆ ❝♦♥s✉♠❡r ❧✐✈❡s ❢♦r ✷ ♣❡r✐♦❞s ❛♥❞ ❤❛s ❡♥❞♦✇♠❡♥t ♦❢ ✵✳✷ ✉♥✐ts ♦❢ ❝♦♥s✉♠♣t✐♦♥ t♦ s♣❧✐t ❜❡t✇❡❡♥ t❤♦s❡ t✇♦ ♣❡r✐♦❞s ◮ ❯t✐❧✐t② ✐s ❈❘❘❆ ✇✐t❤ γ = ✵.✺ ◮ ❚❤❡ ✐♥t❡r❡st r❛t❡ ❛♥❞ ❞✐s❝♦✉♥t r❛t❡ ❛r❡ ❡q✉❛❧ t♦ ③❡r♦ ◮ ❚❤❡ tr✉❡ ♦♣t✐♠❛❧ ✐s C✶ = C✷ = ✵.✶ ◮ ◆♦✇ s✉♣♣♦s❡ ✇❡ ♦♥❧② ❦♥❡✇ t❤❡ ✈❛❧✉❡ ♦❢ s❛✈✐♥❣s ✭✉t✐❧✐t② ♦❢ s❛✈✐♥❣s t♦♠♦rr♦✇✮ ❛t C✶ = ✵.✵✵✶ ❛♥❞ C✶ = ✵.✵✶✾ ❛♥❞ ❤❛❞ t♦ ✐♥t❡r♣♦❧❛t❡ t❤❡ ✉t✐❧✐t✐❡s ❜❡t✇❡❡♥ t❤❡s❡ ♣♦✐♥ts

❆♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r

• ❆♣♣r♦①✐♠❛t❡❞ ♦♣t✐♠❛❧ ❝♦♥s✉♠♣t✐♦♥ ✇♦✉❧❞ ❜❡ C✶ = ✵.✵✻✺; C✷ = ✵.✶✸✺

❇❛❝❦ t♦ s❡❝♦♥❞ ▼❛t❧❛❜ ♣r♦❣r❛♠ ✕ ❢♦❧❞❡r ✬❝♦❞❡❭✈✷✬ ❙♦❧✈❡ ❝❛❦❡ ❡❛t✐♥❣ ♣r♦❜❧❡♠ ✇✐t❤♦✉t ✉♥❝❡rt❛✐♥t② ✉s✐♥❣ fzero t♦ ✜♥❞ t❤❡ r♦♦t ♦❢ t❤❡ ❊✉❧❡r ❡q✉❛t✐♦♥ ❛♥❞ ✐♥✈❡rs❡ ♠❛r❣✐♥❛❧ ✉t✐❧✐t② t♦ r❡❞✉❝❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❡rr♦r

❈♦♥❝❧✉s✐♦♥

❈♦♥❝❧✉s✐♦♥

◮ ❲❤❡♥ s♦❧✈✐♥❣ t❤✐♥❣s ♥✉♠❡r✐❝❛❧❧②✱ ♦♥❡ ❞♦❡s ♥♦t ❦♥♦✇ ✇❤❛t t❤❡ tr✉❡ ❛♥s✇❡r ✐s ◮ ■t ✐s ❞✐✣❝✉❧t t♦ ❛❜s♦❧✉t❡❧② ✈❡r✐❢② t❤❛t t❤❡r❡ ✐s ♥♦ ♠✐st❛❦❡ ◮ ❈❛❧❧s ❢♦r ❞✐s❝✐♣❧✐♥❡❞ ❝♦❞✐♥❣ ♣r❛❝t✐❝❡s✿

◮ ❲r✐t❡ ♠❛♥② ♠❛♥② ♠❛♥② ❝❤❡❝❦s ❛♥❞ ✇❛r♥✐♥❣s ✐♥t♦ ②♦✉r ❝♦❞❡ ◮ ❖♥❝❡ ②♦✉ t❤✐♥❦ ②♦✉r ♣r♦❣r❛♠ ✇♦r❦s ✲ tr② t♦ ❜r❡❛❦ ✐t ❜② str❡ss✲t❡st✐♥❣ ✲ s❡❧❡❝t ❡①tr❡♠❡ ♣❛r❛♠❡t❡rs ✲ ♠❛② ❤❡❧♣ ②♦✉ ✜♥❞ ❜✉❣s