Àkernerdm/teaching/2019.hedva... · 2020-01-02 · p077 pic in ken ir f tik l pg die hessfogler...
TRANSCRIPT
���������� ҍҏҚ҈ қҒ҄ґ҅ҍ � ҁ҈҄҆ۀ�� ҩҦ Ҳҡҙ ҤҡҚҰҲ ����� ҝҡҲҩ
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
$·#ab
$=
#"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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