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ҰҨҰү �қ ҩҝҘҰҠұ �ҡ ҝүҤҝҚ �қ �ҥҡҮҰҦ

ҲқҝүҨ y0ҝ f1 Ҥұ min ҲқҝүҨ x0 ҦۡҦҘ f1(x) + f2(y) Ҥұ min ҲқҝүҨ ҘҡҜ (x0, y0) ∈ Rnx

x × Rnyy �ҝҟҡңҝҜ Ҙ ��

�ҫңҝҘ ҲқҝүҨ ҰҝҙҪ ҜҦҝқ ҜҨҪҠ ҝҟҡңҝҜҝ ҝҟҩҨ �f2 Ҥұ min

ҥҡҤҝҤҩҦҙ ҰҞҪҡҜҤ ҧҲҡҨ ҲҝҨҝҝҨҦ ҲҝҡҠҡҰү ҲҝқҝүҨҙ �ҲҝҡҠҡҰүҜ ҲҝқҝүҨҜ Ҥң ҲҘ ҝҨҡҡҦҝ ҝҘҮҦ ҥҡҘҙҜ ҥҡҰүҦҙ ҙ�f(x, y, z) = x4 + y4 + z4 − 4xyz �JJ f(x, y) = y2 − sin3(x) �J

�f(x, y)=! x3y

x2+y2 , (x, y) ̸=(0, 0)0, (x, y) = (0, 0)

�JJ f(x, y)= xy1+sin(x)ecos(y)

�J �ҲҡұҘҰҙ ҲҡҠҡҰүҜ ҜқҝүҨҜ ҲҘ ҝҨҡҡҦ Қ

ҲҤҙүҦ σ ҡң ҝҟҡңҝҜ �σ(a, b) :="

(yi − axi − b)2 ҰҡқҚҨ (x1, y1), . . . , (xN , yN) ∈ R2 ҲҝҨҝұ ҲҝқҝүҨ ҧҲҨҡҜҙ қXJLJ� -FBTU TRVBSFT �

#"x2i

"xi"

xi N

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$=

#"xiyi"yi

$ҲҦҡҡүҦ ҥҝҦҡҨҡҦҜ ҲқҝүҨҝ R2 ҙ ҠҤҟҝҦ ҥҝҦҡҨҡҦ

�ҝңҡҰҬҜ�ҝҟҡңҝҜ ��

f(x, y)=x2+y2(1−x)3 �ҞҦҰ �ҪҰҤҦ ҜҦҝҩҟ f ҞҘ ҡҦҝүҦ ҥҝҦҡҨҡҦ ҘҡҜҝ Ҝқҡҟҡ ҲҡҠҡҰү ҜқҝүҨ ұҡ R2 f→R Ҥ ҥҘ Ҙ

�қҟҘ ҡҦҝүҦ ҥҝҦҡҨҡҦ ҲҝҟҬҤ ұҡ ҞҘ ҡҦҝүҦ ҥҝҦҡҩүҦ ҲҝқҝүҨ ҫҝҩҨҡҘ ұҡ R2 f→R Ҥ ҥҘ ҙ�f(x, y) = (1 + ey)cos(x)− yey �ҞҦҰ

�(0, 0) ҙ NJO ұҡ f Ҥ ҞҘ �0 ҙ NJO ұҡ f |L ҥҝҮҦҮҤ (0, 0) ҢҰқ L Ұұҡ Ҥң ҰҝҙҪұ ҟҡҨҨ �ҥҝҨҡҤҝҬ R2 f→ R ҡҜҡ Қ�ҥҡҡҙҝҡҟҝ ҥҡҡҤҡҤұ ҥҡҜҙҚҙ (0, 0) Ҳҙҡҙҩҙ ҜҙҝҚ ҡҝҝү ҝҠҠҰұ �f(x, y) = (y − x2)(y − 3x2) �ҞҦҰ

�ҜқҝүҨ ҫҘҙ ҲҩҬҘҲҦ ҘҤ ∂2xxf ұ ҟҡҨҨ �∂2

xxf + ∂2yyf = 0 -BQMBDF ҲҘҝҝұҦ ҲҦҡҡүҦ R2 ⊇ Df

f→ R ҡҜҲ Ҙ ���ҡҦҝүҦ NJO�NBY ҫҘ ҧҡҘ f Ҥ �ҝҟҡңҝҜ

ҲҰқҚҝҦ {∂2xixj

f |x0} ҜҮҡҰҠҦҜҝ gradx(f)|x0

= 0 �ҥҡҡүҲҦұ ҟҡҨҨ �ҲҝҬҡҮҰҙ ҥҡҡҦҪҬ ҜҰҡҞҚ Rn ⊇ Dff→ R ҡҜҲ ҙ

{∂2yiyjf |y0} ҜҮҡҰҠҦҜҝ grady(f)|y0 = 0 �ҝҟҡңҝҜ �ҲҝҬҡҮҰҙ ҥҡҡҦҪҬ ҜҰҡҞҚ ҥҡҨҲұҦ ҲҬҤҟҜ x ! y ҡҜҲ �Ҳҡҙҝҡҟ

ҲҝҟҝҨҜ ҲҝҠҨҡқҰҝҘҝүҙ ҲҝұҪҤ ҧҲҡҨ ҲҝҡҠҡҰү ҲҝқҝүҨ ҧҝҡҦ�ұҝҬҡҟ ҲҘ ҧңҤ �Ҳҡҙҝҡҟ ҲҰқҚҝҦ�f(x, y) = ln(arctan y

x) · sin(%

x2 + y2) ҜҡҮүҨҝҬҜ Ҥұ ҧҝҮҡүҜ ҲҝқҝүҨ ҲҘ ҝҨҡҡҦҝ ҝҘҮҦ Қ

ҠҤҟҝҦҜ ҥҝҦҡҨҡҦҜұ ҟҡҨҨ�f(x)=3"

i=1||x−x(i)|| ҜқҝүҨҤ қҪ ҥҡүҟҰҦҜ ҥҝңҩ ҰҡқҚҨ�R2 ҙ x(1),x(2),x(3) ҲҝқҝүҨ ҪҙүҨ қ

Ҥ Ҳҝҝұ {x0, x(i)} ҥҡҰұҡҜ ҧҡҙ ҲҝҡҝҞҜ Ҥң �ҝҟҡңҝҜ �{x0 ̸=x(i)} �ұ Ңң ҲҦҡҡү ҡң ҝҟҡңҝҜ x0 ҜқҝүҨҙ ҤҙүҲҦ f Ҥұ

�120◦

ҙ ҜҪҝҙүҝ Ballr(0) ҙ ҲҡҤҡҙҘҡҮҨҰҬҡқ ҜҬҡҮҰ ҜҡҮүҨҝҬ Rn ⊃ Ballr(0)f→ R1 ҡҜҲ 3PMMF ҠҬұҦ Ҥұ ҜҤҤңҜ Ҝ

�grad(f)|x0= 0 Ҝҙұ x0 ∈ Ballr(0) ҜқҝүҨ ҲҦҡҡү �ҝҟҡңҝҜ �∂Ballr(0)

�ҝҟҡңҝҜ �f(x) ̸= xn+1 ұ Ңң (x, xn+1) ∈ Rn+1 ҜқҝүҨ ҪҙүҨ �ҲҡҤҡҙҘҡҮҨҰҬҡқ ҜҡҮүҨҝҬ Rn f→ R ҡҜҲ ��

� ҜҤҘұ � Ҳҡҙ ҤҡҚҰҲ ҝҘҰ �dist&(x, xn+1),Γf

'=dist

&(x, xn+1),(x̃, f(x̃))

'ҜҰҝҙҪұ x̃ ∈ Rn ҜқҝүҨ ҲҦҡҡү Ҙ

�(x̃, f(x̃)) ҜқҝүҨҙ Γf ҫҰҚҤ үҡұҦҜ ҰҝұҡҦҤ ҢҨҝҘҦ (x, xn+1), (x̃, f(x̃)) ҰұҡҜ ҙ

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