고뇌의 고통이 없는 켤레 경사도 기법 개론(1과 1/4판)
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VisCom TR Series - Copyrightc©Jonathan Richard Shewchuk and VisCom Commune.
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�ª EPê(conjugate gradient)�ªú îê£ ,�°.Lî¯â� b�ü(Jacobi)�ª,/ �,�ª E
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1Ù9� - Jonathan Richard Shewchuk.Ù[C@ - An Introduction to the Conjugate Gradient Method without the Agonizing Pain.Ùf[��u�j#°�W�uÊç(Natrual Science and Engineering Research Council of Canada)� 1967[�W$¡PzYyY¡&³(National Science Foundation)��Ù(Grant ASC-9318163)ú�I�ØqP°.�[t�[>Y@Áù9��>©�6j#°�W�uÊç(NSERC),yY¡&³(NSF),ÓùyvAÙ�>©¿�I��Kt�Ký°.
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copyright c©Jonathan Richard Shewchuk and VisCom Commune 2
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1 ���qqq������
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��qEþ� ����\�&�M9°.����þ�ú #ê�© �G°°.�þ�ùÂÙÛ�
³ 3�ªEPêõ�Á°÷6,qC¢�[:z<ê{}LqN3�ªEPê�ª��ü�LKü}
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ê�ªù°üÍ×�èßð�IsîY:�°.
Ax = b (1)
�: x�?ò�¯â�L, b�NL��¯â�6 A�NL��A�(square),Âa(symmetric),j�A
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|ú�y ���q£¹°L£��ê#|�°èU_Ç,�¦%A£�Å�{°.�èßðù`yÛ
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±µY üp©Ã�. �±µ A� X�Å�(triangular factors)�� 0� I¨ Ù�� ��:÷¿ Ù�� AÃ
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2 vvv���ªªª
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copyright c©Jonathan Richard Shewchuk and VisCom Commune 3
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±µú vè � æ© Â[�õ PÌ£ ,�6,�[�� ¯âõ #Í4�Ú PÌý°.ße�� �ýß
[�¿v袰.A� n× n±µ�L x® b�¯â,�, n× 1±µ�°.½é 1ù°üY��vÇ¢°.
A11 A12 · · · A1n
A21 A22 A2n
......
...
An1 An1 · · · Ann
x1
x2
...
xn
=
b1
b2...
bn
.
� ¯â� 4:(inner product)ù xT y�L ße� ¦∑n
i=1 xiyiú #Í6°. xT y = yTx�°. x® y� �
p 8 xT y = 0�°.��:÷¿ xT y® yTAx®�� 1× 1±µ�ü�vÇéùße�÷¿D�ý°.
�c?� 0�I¨¯â x�©°üÙ#éú�T£:,±µ A�j�AÙÒ(positive-definite)�°.
xTAx > 0 (2)
��tIX¢�yõ���G¢°Lì� ��.��]ù�°��[: �]�I¦qtj�
AÙÒ ±µ��©�Mù,YqN3°ò3Ã����© �,�ÑÏ��I¨°.�õ�© �
æ©t�j�AÙÒ(positive-definite)��,� 2òÍé�qN3�³úyX��U_Ç�Å��°.
���÷¿ (AB)T = BTATY (AB)−1 = B−1A−1��|Å¢�ƨ#éú��L�r �.
3 2òòòÍÍÍééé
2òÍé(quadratic form)ù�³&©°üY�ùÍ×�ße�ú � 2ò¥½�°.
f(x) =12xTAx− bTx+ c (3)
A�±µ�L x® b�¯â�°.�ýL c�ße�\½�°.�����t A�Âa:�L(symmetric)
j�AÙÒ EÍ,Ax = b�©õ�Ì � f(x)õ/�Ü£½�üú�³&Ã�²¢°.
���;��t���°üY���³¢ìC�C[Cõ��L°j¢�]úz<£,�°.
A =
3 2
2 6
, b =
3
−8
, c = 0. (4)
copyright c©Jonathan Richard Shewchuk and VisCom Commune 4
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
èßð Ax = b��� 1�#Í#�°.��:÷¿© x� n��#e8(Ú��)�pò>�° (�:�
�� #e8ù �� n − 1 òÙú ��°).� [C� ©� x = [2,−2]T�°.Âÿü� �òé f(x)� ��
2�#Í&°. f(x)�#Lx�?jù�� 3Y�°. A�j�AÙÒ��:[� f(x)��©A�ü�v
8ùj]xÍ×�P�Y�ù?j�°. (��© è��Ì��&z< 9°)
[�� 1] 2òÙxÍèßð�. ©�xÛ��pò>�°
2òé��Ð�(gradient)�°üY��A�ý°.
f ′(x) =
∂∂x1
f(x)
∂∂x2
f(x)...
∂∂xn
f(x)
. (5)
�Ð��sq�> x�©, f(x)��$À���³ú�òÅ�¯â���°.�� 4ù½é 4Y�
ù\½õ½é 3�Â� �uq��Ð�¯âõz< L�°.P�?j�j]8�±�t��Ð��
0�°. f ′(x)� 0�üêÀ¥÷¿� f(x)ù/�Ü£½�°.
°��Ø¢�zú�©½é 5õ½é 3�Â� �°üúîꣽ�°.
f ′(x) =12ATx+
12Ax− b. (6)
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 2] 2òÍé f(x)����. �v8�/9>ù Ax = b�©�°.
[�� 3] 2òÍé�#Lx. �ÍÙMxùô�¢ f(x)ú �°.
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[�� 4] 2òÍé� �Ð� f ′(x). ?� x�  �, �Ð� ¯â� f(x)� �$ ¾3 \ç � �³ú �ýÅ6 #Lx�
�p¢°.
A�Âa�8��Aéù°üY����Üý°.
f ′(x) = Ax− b. (7)
�Ð�õ 0÷¿ 8Íý�uL� �xÍèßð½é 1úu�°.� t¿ Ax = b�©� f(x)��
G>(critical point)�°. A�Âa�<�I¦�j�AÙÒ(positive-definite)�8©� f(x)�/��°.
0�t Ax = b�©� f(x)ú/�Ü � xúüü÷¿�u£½�°. (A�Âa:��M÷8½é 6��
ªEPê�ª�èßð 12 (AT + A)x = b�©õü3ý°�,úÃ�sL�°.�: 1
2 (AT + A)�Âa
±µ�°.)
Âa j�AÙÒ±µù·�©3^ù��ú L�ú«?qE���> p�t� f®xÍèßð�
© x = A−1bP��[GõU_Ã�.½é 3ú�Ö÷¿ A�Âa EÍ (�±µ�j�AÙÒ(positivie-
definite) ��[G{�)°ü���¥úÃ�½�°. (ÙÀ C1).
f(p) = f(x) +12(p− x)TA(p− x). (8)
A�j�AÙÒ(positive-definite)�L 8,Ù#é 2��©ó��¨�?� p 6= x�©j½�°.�
� x� f�;�/��úy¢°.
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j�AÙÒ(positive-definite)±µ�[¢�$�[: �©��±µ� 2òÍé f(x)�j]8�ý°
�,�°.A��cj�AÙÒ�I¦�8°ô� ���°.A�ü�AÙÒ�þ½��Ú��j�
AÙÒ±µ�©ÙÒõ�ö@Y,��� 2�I�æõ��qnù?j�ý°. A�"�±µ EÍ
��î�¢©�{�EÍ�°;©��¦ù #��x,Óù�A¢ f ú���#e8(hyperplane)�
ý°. A� æ�t z<¢ qE EÍê I¦�8, x� K$>(saddle point)� ü6 / ��# �ª EPê
�ª#�ìC£,�°.�� 5�� ¢� ��úÃ�sL�°. b® c�ùj]Í�/�>�q%
�õ@A¢°.� #j]Í?j���³úyX�M�°.
xÍèßðú · Ì q²× Ã�� [C¿ ºÍ£«?/ �Y �ª EPê �ª #� �ª�ù ��
1® �� #e8� pòõ �©t� I¦� �� 2® �� ¥½� /�Üõ �© �[: �©� � �
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4 /// ������ªªª
/ ��ª�t�,qE��> x(0)�tè� �j]x���±÷¿y� K4²�°.© x�?Û
&�«3ü�S&ú�T£:«��³G x(0), x(1) · · ·úD¢°.
#�³G�t f��$�ò3�� ��³úxØ¢°.��³ù f ′(x(i))��Â�³�°.½é 7�
0ò8,��³ù −f ′(x(i)) = b−Ax(i)�°.
��të����r©b£=��A�õ��¢°.¡ò(error) e(i) = x(i) − x�©¿Ùâv�#%
ýFqK�#õ#Í4�¯â�°.#"�(residual) r(i) = b−Ax(i)ùAÝ¢ b÷¿Ùâv�#ò��
�#õ #Í6°. r(i) = −Ae(i)�ù Ö3 N ½ �L,� #"�� @v ¡òõ ±µ Aõ �Ì � b® �ù
W�÷¿ ºÞ¢ ,÷¿ Ç ½ �°.Ì |Å¢ ,ù r(i) = −f ′(x(i))��� ,�6,#"�(residula)õ /Â
��ª��±�³Y�ù,÷¿f� �b¢°.üxÍ[C�t�³�ó��A�ú:Ì � 14<
�t f�¢°.� t¿ “#"�(residual)”��� Ìqõ �3 ü8 “/ � �³”��� �yõ C¤ý
êÀ�r©��.
x(0) = [−2,−2]T�t ;�¢°L �A �.�ü ½± � �zù / �(steepest descent)�³ú 0
�t�� 6 (a)���ìx\�q%�Fq�,�°.�,°üY��^¿ÏqE>úD£,�°.
x(1) = x(0) + αr(0). (9)
[C� � �³ú 0� v�# �±¢ Vú xØ£ , ��°. (�,qE αõ xØ£ , �õ @A �
,�°.)
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[�� 5] (a)j� AÙÒ ±µú æ¢ 2òé. (b) ü� AÙÒ ±µú æ¢ 2òé. (c) "�±µ(�ýL j� ÙAÙÒ).
GM� �±ú �#� xÛ� ©� �¦�°. (d)ÙAÙÒ ±µ� :. ©� K$ >�� :[� / �Y �ª EPê
�ª�:Ìü�M�°. 3òÙ�#��\�t�"�±µ�èK$?jú��½�°.
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[�� 6]/ ��ª. (a)x(0) = [−2,−2]T�t;�©t f�/ ���³÷¿¢³GõD¢°. (b)�8�pòx
\� �� >� |�t fõ /�Ü � >ú ü�°. (c)� j]xù � 8(2òÍé j]8Y Ò_x �³� ½� e8)�
pò �x�°. (d)/9>�t��Ð���;³G��Ð�®�p¢°.
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Ò_x\�t�Ð��Ò_xY�p �V�t f�/�ú �°.
[�� 8]��,/ ���ªù [−2,−2]T �t;� L [2,−2]T�t½¶¢°.
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��©A�ýj]x�°.j]x�/9>�t αùv� �?
�Æ: y:Ûú �©, α� �³ ꥽(directional derivative) ddαf(x(1))� 0�: fõ /�Ü¢°. �
©ñ(chain rule)� �©, ddαf(x(1)) = f ′(x(1))T d
dαx(1) = f ′(x(1))T r(0) �°. � ½éú 0÷¿ L, r(0)®
f ′(x(1))���pü3 � α�xØüqb¢°(�� 6 (d)õÃ�).
� � ¯â� /�>�t t¿ �p� üqb � �[: �î� �°.�� 7� Ò_xú 0�t °
j¢ >��t �Ð� ¯âõ Ã�sL �°. j]x(�� 6 (c))æ� ��� >�t ��� j]x �Ð�
(slope)��>�t� 2òÍé�����Ð�(gradient)õÒ_�xæ��°ú: (�� 7,>xÜU
v)f��¯â�£�®�°.�ý�¯â�ùÒ_xú0��: f���íúvÇ¢°. f��ý
�Ð�¯â�£�� 0 V-�Ð��Ò_x��p �V-�t/�ú �°.
αú@A �æ©, f ′(x(1)) = −r(1)��î� 8°üúuú½�°.
rT(1)r(0) = 0
(b−Ax(1))T r(0) = 0
(b−A(x(0) + αr(0)))T r(0) = 0
(b−A(x(0))T r(0) − α(Ar(0))T r(0) = 0
(b−Ax(0))T r(0) = α(Ar(0))T r(0)
rT(0)r(0) = αrT
(0)(Ar(0))
α =rT(0)r(0)
rT(0)Ar(0)
�õ?�Â�©Ã8,/ �(Steepest Descent)�ªù°üY�°.
r(i) = b−Ax(i), (10)
α(i) =rT(i)r(i)
rT(i)Ar(i)
, (11)
x(i+1) = x(i) + α(i)r(i). (12)
copyright c©Jonathan Richard Shewchuk and VisCom Commune 12
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
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��;��Ð��©�p �:[�#Í#�Ç\�°.
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scent)�ª�GSüÌù±µ-¯âU��©_Íü�Ú,°±&ê��ÏÚ #�{\½��°.
½é 12�jº�X� −AúU L búÌ 8°üúu�°.
r(i+1) = r(i) − αAr(i). (13)
r(0)úGS �橽é 10ú¢¥GS �,ù��è�Å ��,��ó��Ä�t�½é 13õ
G�©t PÌ£ ½ �°.½é 11Y ½é 13?��t #Í#� ±µ-¯â U Arù ¢ ¥� GS 8 ý°.
�é�ú�Ì ��Äú£:f��[C>ù½é 13ú�©A�ü�½�� x(i) ÷¿ÙâqEü
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�����qE>�½¶£½�°�,�°.� ¢³>ù½é 10ú�Ì �s�:÷¿AÝ¢#"
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/ �(Steepest Descent)�ª�½¶�¢Ûu�;�,� Û��Lî¯â�¢Ýì¢�©õ
£½�êÀ ®�£¿��²L¢°.
5 LLLîîî¯âââ(eigenvector)®®®LLLîîî(eigenvalue)���¢¢¢LLLööö
���xͽ¡¢Y@ú½�¢ó�,Lî¯â(eigenvector)®Lî(eigenvalue)�X| �êX�N
½�{}°.� Û�ú�òZ�P���õ�òZ�P®üæ °8,Lî �g� �,ú��L
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°.� Û��cLî¯â¦Lî�¦ �,��©ÍL&& ú��L�°8�<ú&Clq
êý°.
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ªùNLý�ú½± �æ©Lî¯âõGS£�Å�{°.
5.1 LLLîîî���gggLLLîîî?
±µ B� Lî¯â v� Bõ :Ì � ºÞú �ê �Í� ºÜ� �q#� M� 0� I¨ ¯â�°(AÝ
3�Â�³÷¿ºÞü�,ùC½). v�£��º $#��³��Â�þ½����,�÷¿î�
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
�Mú,�°.°è©, Bv = λv qEße�\½ λ�U&¢°.�: λ� B�Lî�°.���
\½ α�©,¯â αvê�èô�¢Lî λõ �Lî¯â�°.�� B(αv) = αBv = λαv��:[
�°.°è©,Lî¯âõ¾�õ��$#v�ê�;&Lî¯â�°.
�(·êE�b£«?�Īù[[ BõqE¯â�G�©t�Ä:Ì �©õu¢°. B� #
� Lî¯â� �Ä:÷¿ :Ìþ :,°ü � �� �ÏÚ #� \á� �f£ ½ �°. |λ| < 1 EÍ�
8, i�X¢Â¿�: Biv = λiv� 0¯â¿½¶£,�°(�� 9). |λ| > 1�8,Biv�X¢Â¿v�,�
°(10).�¥ B�:Ìþ:�°,¯â� |λ|��0�v�$#�I�3ý°.
[�� 9] v� −0.5�Lî�Âÿü� B�Lî¯â. i��� 8, Biv� 0�½¶¢°.
[�� 10]���t v�Lî 2õ���Lî¯â�°. i��� 8, Biv�X¢Â¿�S¢°.
B� Âa±µ EÍ([[ �©� Mù EÍ�ê � °), B� xÍ: ë� n �� Lî¯âõ �
�°. �õ v1, v2, · · · , vn�� v� �. � �¦ù î� � M°. �� � Lî¯â� 0� I¨ :¾¢ \
½� ¾�� ºEþ ½ �� :[�°. ��� Lî¯â� �ê�3 Âÿü� Lîú ���Ú, �õ
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��½ê�Lt¿°õ½ê�°.�õ�q,¨#±µ I�Lîù?� 1�L, 0¯â�I¨���¯
â� I�Lî¯â�°.
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Á-�<�t�òXL� ���Á-ù #�¯âõ±ô�N²�°ô¯â�¦÷¿�s �,�°.
Lî¯â���¦ {vi}� Rn��9õ�Ø�\áúL²©Ã�(Âa±µ B�xÍë� n��Lî¯
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
â�ú��°).��� n-òÙ¯â���Lî¯â��xÍS¦÷¿vÇþ½�°.Z¢±µY¯â�
UùÛ�ªY�:Ìüt¿���Lî¯â�U©�� B�îY�ë�:÷¿SP£½�°.
�� 11�t¯â x��Lî¯â v1Y v2�¦÷¿#Í#�°. Bõ x�U �,ù BõLî¯â�
� �� U � ¦ú u � ,Y �°. Bõ �Ä©t U© #�8, Bix = Biv1 + Biv2 = λi2v2� ý°.?
�Lî�¾�� 1ð�÷8, Bix� 0�½¶ú£,�°. (�� xúu� L��Lî¯â�?��
Bõ�Ä:÷¿U£EÍ 0�½¶ �:[�°).Lî�| #�ê 1ðÀú �°8, x�X¢
Â� �S£ ,�°.�,� �¿ ½X©u ¡��� ±µ� ßX!¤ �E(spectral radius)� |Å�ú Ù�
��î�°.±µ�ßX!¤�Eù°üY�°.
ρ(B) = max |λi|, λi ù B�Lî¯â
x� 0��ò3½¶ �õÙ 8 ρ(B)ù 1ð�Ib L,� 8�ù,�^°.
[�� 11]¯â v(ìx ÜUv)� Lî¯â�(>x ÜUv)� xÍ S¦÷¿ vÇþ ½ �°. �® �[ý Lî� λ1 =
0.7Y λ2 = −2 �°. B� �Ä:÷¿ :Ìü8 ¢ Lî¯â� 0� ½¶ L °ô #� �S¢°. 0�t Bix �è
�S¢°.
xÍ ë� n �� Lî¯âõ ° ��� G � üÂa ±µw( )� U&¢°� Pìú s�£ �Å
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�,ÙT ±µ(defective matrix)� ±ô "�ù ��Üý Lî¯â(generalized eigenvectors)® ��Üý Lî
(generalized eigenvalue)õ�Ì �Ûu£½�°.Bix� 0�½¶ �æ¢�Å?ÛS&÷¿?���
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< �,ùÌÎq²×�°.
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�õ�Ì �°üY���<£½�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
Bv = λv
vTBv = λvT v.
j�AÙÒ(positive-definite)A��� �, vTBv�j�° (��I¨ v� �).0�t λ�èj�
½��{°.
5.2 bbb���üüü(Jacobi)���ÄÄĪªª
0��\½¶ �YAúI�,�Zu�P��t �õ��Ú�êÓ�ü�,ù]ÁI¨°.XÌ
îÌ¢ YAú U_Ã�: Ax = bú �� æ¢ b�ü(Jacobi)�ª�°.±µ A� � ÙÛ÷¿ Ûý(split)ý
°: A�Â��Ûú�¿ L°ô�Û�ù?� 0 D®Â�x�Ûù?� 0�L°ô�Û�ù A®
�ù E.0�t A = D + E�°.��ü�ªù°üY�°:
Ax = b
Dx = −Ex+ b
x = −D−1Ex+D−1b
x = Bx+ x, whereB = −D−1E, z = D−1b. (14)
D�Â�±µ�t¿Ö3�±µúu£½�°.�¨#éù°üY�ù>Üéú�©Ö3�Ä�ª
÷¿ºÞ£½�°.
x(i+1) = Bx(i) + z. (15)
sq� è� ¯â x0� ©,� ½éù �´� ¯â�ú f�¢°.Íý� � �Ä�ªú �© ��:÷
¿#Í#�¯â� # #��;�Yüp°ú:© x�Ì�«×��õ��°. x�½é 15�L
A>��LÙò�Ú,�� x(i) = x�: x(i+1)ê�è x®�ù�ü�:[�°.
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�E(spectral radius)õ �,�°.��t���`�\b��Ûý�ªúxØ �°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
���qE¯â x(0)�tè�¢°L �.��Ä�©,Bú�¯â�:Ì¢�� zú�@Y�Ì
¢°.��ijG�tqE��§q���?
#� ¯âõ Íý� �y � �© L �� °ô ¯â�� ¦÷¿ ��� ÙYú °è ¢ ¥ :Ì© Ã
�.�Äú�©#Í#� x(i)úAÝ¢© x®¡ò¨ e(i)�¦÷¿vÇ©Ã�.� 8½é 15�°üY�
�ý°.
x(i+1) = Bx(i) + z
= B(x+ e(i)) + z
= Bx+ z +Be(i)
= x+Be(i) ½é 14��©,
∴ e(i+1) = Be(i). (16)
��Äù x(i)� “AÝ¢ÙÛ”��³ús�M��(x�LA>��:[�°);�¥�Ä:�°¡ê
³ù�³ú��°.½é 16õ�©<�&N½��>ù ρ(B) < 1 EÍ� i�X¢Â¿?� 8¡ò
¨ e(i)� 0�½¶¢°�,�°.0�t#�¯â x(0)õqN3xØ �=�/[: ©��³úyX�
G¢°.
]Á x(0)õqN3xØ �=�0�sq�´Ì¡ò4�t x�½¶ �Ú��Å¢�Äì½��
�³úyX3ý°. ����³ùßX!¤�E ρ(B)��³�ü©ýCÑ|Å 6,ßX!¤�Eù
½¶��êõ@A¢°. vj� B�Lî¯â��ÏÚ�$ÀLîú��� (�, ρ(B) = λj )Lî¯â�
L �.Lî¯â��xÍS¦÷¿vÇý#�¡ò e(0)� vj��³÷¿�Ûú��°8,��Û��
$�ý3½¶£,�°.
B���:÷¿ÂaêI¦L(üÀ A�Âa���ê)Ý°;£�ê?ô°. ��b�ü(Jacobi)�
ª� ½¶�ê� ρ(B)� ¾3 _Íü6, � ù Z¢ Z¢ A� µ²�°. ݱ 3ê b�ü �ªù ?�
A�©½¶ �,ùI¦6,î�qj�AÙÒ(positivie-definite) A�©t꽶 �Mú½�
°.
5.3 uuu���:::���
�¢�]�úu�:÷¿Ã��æ©,���½é 4�#Í&�õ�qòL¢°.$9,LîYLî
¯âõü��ª��Å °.A���©Lî λõ������Lî¯â v�©°ü���¢°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 12] A�Lî¯â� 2òÍé f(x)��©A�ýj]8�9�³Y�X¢°. ���Lî¯â��[ýLî�
��:Å�°. ���Lîù©¾ü�EP��7ôAê�ü»¢°.
Av = λv = λIv
(λI −A)v = 0.
Lî¯â� 0¯â�I¦°.0�t λI −A�"�±µ�qb¢°.0�t°ü���¢°.
det(λI −A) = 0.
λI − A�±µéú"�°¨é(characteristic polynomial)��Ùô°.�,ù λ�¢ nò°¨é÷¿,
�°¨é��(Y)�±µ A�Lî�ý°.½é 4��� A�"�°¨éù°üY�°.
det
λ− 3 −2
−2 λ− 6
= λ2 − 9λ+ 14 = (λ− 7)(λ− 2),
°¨é���Lî�t¿±µ�Lîù 7Y 2�°. λ = 7�:Lî¯â�°üY��u£½�
°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
(λI −A)v =
4 −2
−2 1
v1v2
= 0
∴ 4v1 − 2v2 = 0.
��Aéú�T ����©��¿Lî¯â�°.�õ�q, v = [1, 2]T�Lî¯â�°.�ù�ª
÷¿Lî 2�©¾ �Lî¯â¿ [−2, 1]Tõu£½�°.�� 12�tǽ���®��,�Lî¯
â���¾¢ÍÙ�9Y�X¢°�,úǽ�°.Z¢Lî�¾8Ì�¢EPõ��°�,êÝ
£½�°. (ü½Lîù�� 5(b)® 5(d)�Ã��,Y��9ú0��ô£: f����¥ú�
y¢°.)
�Cb�ü(Jacobi)�ª��ô �?åúÃ�.½é 4���\½�ú�Ì �°üY�ù�Äé
úu�°.
x(i+1) = −
13 0
0 16
0 2
2 0
x(i) +
13 0
0 16
2
−8
=
0 − 23
− 13 0
x(i) +
23
− 43
B�Lî¯âõu 8Lî� −
√2/3�: [
√2, 1]T�L,Lî�
√2/3�: [−
√2, 1]T�°.��ù
�� 13(a)�t ��÷¿ #Í# �°.��ù A� Lî¯â® �X � M÷6,j]8� 9Yê \[� {
°.
�� 13(b)�b�ü(Jacobi)�ª�½¶�úÃ�sL�°.�NLý��½±ü8t0���ê�¢
E¿��Ä�ªú�©��:÷¿#Í#�¡ò¨������Lî¯â�Û��qN3ºÜ ��
U_È÷¿��©£½�°.(�� 13(c), (d), (e)).�� 13(f)�ÜU$vèõ�©Lî¯â�Û�ú�ý
L�°.���Ûù�� 11�tÃ��®�����Lî��©@Aý�꿽¶¢°.
����<ú�©� Û�� “Lî¯â”��,�¾ê��L�úÃ6���æ©p½����q
6h\¢L[êu�I¦�,�ÍîÌ¢êu��PìúÝì&NU÷8^9°.
6 /// ���(Steepest Descent)���ªªª���½½½¶¶¶ÛÛÛuuu
6.1 ¢¢¢¥¥¥������©©©üüü���
/ � �ª� ½¶ú �© � æ© Íx e(i)� Lî λeõ ��� Lî¯â�L �. � 8 #"
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 13] b�ü �ª� ½¶�. (a) B� Lî¯â�� ��� LîY ¥Í #Í#�°. A� Lî¯â® µý ��
Lî¯â� Ã]8� 9� I¦°. (b)b�ü �ª� [−2,−2]T�t è� � [2,−2]T�t ½¶¢°. (c,d,e)¡ò ¯â
e(0), e(1), e(2)(ìx ÜUv)® ��� Lî¯â �Û(>x ÜUv). (f) ÜU$ù �ü 4� ¡ò ¯â� Lî¯â �Ûú
vÇ¢°. ¡òõvÇ ����Lî¯â�Ûù���Lî�0��\£½���ê¿ 0�½¶¢°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
�(residual)r(i) = −Ae(i) = −λee(i)ê�èLî¯â�°.é 12õ�Ì �°üúuú½�°.
e(i+1) = e(i) +rT(i)r(i)
rT(i)Ar(i)
r(i)
= e(i) +rT(i)r(i)
λerT(i)Ar(i)
(−λee(i))
= 0.
�� 14� � �ª� · ¢ ¥�� ©� êµ ��õ Ã�u°.> x(i)� ÍÙ� 9 �ÏÚ #� n�
�°.0�t#"��ÍÙ�|îú�¿�ýÅ3ý°. α(i) = λ−1e õxØ¥÷¿���: ½¶úuú
½�°.
[�� 14] / �(Steepest Descent)�ªù ¡ò¨� #� Lî¯â¿ vÇþ EÍ ¢¥� �Ä�÷¿ AÝ¢ ©�
½¶¢°.
ÌÎ��: Ûuúæ©, e(i)õLî¯â�xÍS¦÷¿vÇ©b 6,Z¢�Lî¯â��A�
�p(orthonormal)�qb¢°.ÙÀ C2�t±µ A�Âa�EÍ n���pLî¯â�U&¥ú�<¢
°.Lî¯â����¾�¿ºE£½��:[����Lî¯âõA�Ü£½�°.�©3Lî¯â
õxØ 8°üY�ùîÌ¢"�ú �°.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 21
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
vTj vk =
1, j = k,
0, j 6= k.(17)
¡ò¨úLî¯â�xÍS¦÷¿vÇ �.
e(i) =n∑
j=1
ξjvj , (18)
�: ξj� e(i)��Û»£��°.é 17Yé 18¿Ùâ°ü�¨#éúu�°:
r(i) = −Ae(i) = −∑
j
ξjvj , (19)
||e(i)||2 = eT(i)e(i) =
∑j
ξ2j , (20)
eT(i)Ae(i) = (
∑j
ξjvTj )(∑
j
ξjλjvj)
=∑
j
ξ2jλj , (21)
||r(i)||2 = rT(i)r(i) =
∑j
ξ2jλ2j , (22)
rT(i)Ar(i) =
∑j
ξ2jλ3j , (23)
é 19� r(i) �èLî¯â�Û�¦÷¿vÇþ½�üúÃ�s6,���Û�£�� −ξjλj�°.é
20Y 22��¿�ÍL�ß(Pythagoras)�ªY�°.
�CÛuú©Ã�.é 12�t°üúuú½�°.
e(i+1) = e(i) +rT(i)r(i)
rT(i)Ar(i)
r(i)
= e(i) +
∑j ξ
2jλ
2j∑
j ξ2jλ
3j
r(i) (24)
�����tÍý� e(i)� #�Lî¯â�Û�ú��EÍ α(i) = λ−1e õxØ¥÷¿�¢¥��
©�½¶¥úÃU°.�C e(i)�������?�Lî¯â�ô�¢LîX λõ���EÍõU_
Ã�.é 24õ�Ì �°üúu�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 15]/ �ùLî�?��°8�¥«�Ä�tAÝ¢©�½¶¢°.
e(i+1) = e(i) +λ2∑
j ξ2j
λ3∑
j ξ2j
(−λe(i))
= 0
�� 15���ª�è��:÷¿©�½¶¥úÃ�L�°.?�Lî��ô� �:[�ÍÙù
uÍ(spherical)�üqqEæX�tè� �[G{�#"�(residual)�u�|îú�ýÅ3ý°. αõ
xØ �,ùX�t®�ô��¿ α(i) = λ−1¿xØ¢°.
� #,=��t¿°ôLî�U&£EÍqEé÷¿ α(i)õxØ Ì�êLî¯â�Û�ú?
�C$£½�{°.0�t�:xØ£½���ªù�[�ÍÊ�ý°.Pì,é 24�#Í#��Û½
� λ−1j �©�|Xe�÷¿Ç½�°.�|X ξ2j� e(i)��Û��ÏÚÀ�Û�Íx¿æõ�3
¢°.@Y:÷¿��Ä:�° e(i)�¥ù�Û��ÏÚ�Ù�ìC¿£���q#�ꢰ (üÀ�
Ù&�I¦��).�¢�î¿/ ��ªY�ªEPê�ªù N(rougher)�LÙô°.�¿��
ü�ªùßXÌ(smoother) Ú,��î�?�Lî¯â�Û����Ä:�°vq��:[�°./Â
�Y�ªEPê�½¡[¶��Ù�t[[ßXâ¿�G éüL���ìC¿�ßXÌ�I¦°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 16]����¯â�ô�¢�C�i(norm)ú �°.
6.2 ������½½½¶¶¶���
��: EÍ�©/ ��ª�½¶�ú�b� ²8�C�i(energy norm)||e||A = (eTAe)1/2ú
A�©b¢° (�� 16úÃ�).�i(norm)ùîÁý�i(Euclidean norm)ð°Ø��ÑÍ6qE8�
t�ÌÎ��ß Ïi(norm)�°;é 8úU_Ã8 ||e||Aõ/�Ü �,�@v f(x(i))õ/� �,�
úN½�°.�i(norm)ú�Ì �°üúuú½�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
||e(i+1)||2A = eT(i+1)Aei+1
= (eT(i) + α(i)r
T(i))A(e(i) + α(i)r(i)) (½é 12��©)
= eT(i)Ae(i) + 2α(i)r
T(i)Ae(i) + α2
(i)rT(i)Ar(i) (½é 14��©)
= ||e(i)||2A + 2rT(i)r(i)
rT(i)Ar(i)
(−rT
(i)r(i)
)+
(rT(i)r(i)
rT(i)Ar(i)
)2
rT(i)Ar(i)
= ||e(i)||2A −(rT
(i)r(i))2
rT(i)Ar(i)
= ||e(i)||2A
(1−
(rT(i)r(i))
2
(rT(i)Ar(i))(e
T(i)Ae(i))
)
= ||e(i)||2A
(1−
(∑
j ξ2jλ
2j )
2
(∑
j ξ2jλ
3j )(∑
j ξ2jλj)
)(¨#é 21, 22, 23��©)
= ||e(i)||2Aω2, ω2 = 1−(∑
j ξ2jλ
2j )
2
(∑
j ξ2jλ
3j )(∑
j ξ2jλj) (25)
Ûuù ω�\¢úü�,�µ²�°.�|X®Lî�qN3½¶��³úyX��õÃ��æ©
n = 2 EÍ�©@Yõîê£,�°. λ1 ≥ λ2�L �.±µ A�ßX!¤S&½(spectral condition
number)� κ = λ1/λ2 ≥ 1¿A�ý°. e(i)��Ð� (Lî¯â��©A�ü�_vG�©)�è�>�
�U:�6 µ = ξ2/ξ1�ý°.�C°üéúu�°.
ω2 = 1− (ξ21λ21 + ξ22λ
22)
2
(ξ21λ1 + ξ22λ2)(ξ21λ31 + ξ22λ
32)
= 1− (κ2 + µ2)2
(κ+ µ2)(κ3 + µ2) (26)
ω�ù/ ��ª�½¶�êõ@A 6 µ® κ�¢¥½¿�� 17Y���²�°.���
�¿Xtà ����õÝ £½�°.�c e(0)�Lî¯â�8�Ð� µ� 0�ü6 (ÓùX¢),�:
����t ω� 0�ÿúǽ�°.0�t½¶ù��:÷¿�Øq�°.Z¢Lî�?�ô� °8
S&½ κ� 1�ü6,�:�è ω� 0�ý°.
�� 18ù �� 17� P ��� ��� ��� ©¾ � �õ Ã�L �°.�� �ò Íé�ù Lî¯â
��©A�ü�_vG�t�²P°.�� 18(a)® 18(b)�ÀS&½õ����°./ �ùè�>�
Ï� ^3 xØü}°8 �ò3 ½¶¢° (�� 18(a)).� # ��:÷¿ κ� À ú ��8 �Í #D �
úà ° (�� 18(b)).�¥«��ùÀS&½�·#D�õ�$�[:÷¿Ã�L�°: f(x)�u
îÍ×õ��6/ ��ªù�uî�jÅúµ°�° 6$��± �G �\×�ý°.��
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
18(c)® 18(d)�t S& ½� �ù �t¿ �ò Íéù $� uÍ� ý°.�: ½¶ù è�>� [G {�
�ò3�Øq�°.
[�� 17] / �(Steepest Descent)�ª� ½¶ê ω. ½¶ê ω� µ (¡ò¨ e(i)� �Ð�)Y κ (A� S& ½)�
¥½�°. ½¶ù µ® κ��ù�:�ò°. LAý±µ�t µ = ±κ�:/J�½¶úà °.
κõ\½¿£:(A�LAüq�÷t¿),�³¢y:Ûú�©é 26� µ = ±κ�:/Â�ÿúN½
�°.�� 17�t�x��©��q�� y¢õ�xúǽ�°.�� 19�Íý�PÌ©µÏ±µ
A�©/J�è�>ú�ýL�°.�è�>�ù ξ2/ξ1 = ±κ¿A�ü�xæ�n °. ω�\¢(/
J�è�>�©¾)ù µ2 = κ2¿zA¥÷¿�üú½�°.
ω2 ≤ 1− 4κ4
κ5 + 2κ4 + κ3
=κ5 − 2κ4 + κ3
κ5 + 2κ4 + κ3
=(κ− 1)2
(κ+ 1)2
ω ≤ κ− 1κ+ 1
.(27)
é 27�Ù#é��� 20��²K�°.±µ�#D\×��ú½À(�,S&½ κ�Á½À),/Â
��½¶�ê�ÌÎ�²�°.< 9.2�tÂa�6j�AÙÒ ±µ�°üY�ùS&½õ��°8
é 27� n > 2 EÍ�ê�;&��¢°�,ú�<¢°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 18]� P �C� �� 17� P � ���� ©¾ � �> ��õ vÇ¢°. (a)� À κ,�ù µ. (b)#D ½¶� �.
κ® µ?�¾°. (c)�ù κ® µ. (d)�ù κ®À µ.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 27
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 19] ìx�ù / � �ª�t /J� ½¶ú Ã�� è�>ú vÇ L �°. >xù ½¶ YA�t $X�
³G�úÃ�L�°. �è��/J��>�t�Øq�EÍ,�°ü³G�è/J��y�t�Øq�°. ���
³G�tj]89(ç_ÜUv)õAÝ& 45ê¿pò 3ý°. ��t κ� 3.5�°.
κ = λmax/λmin,
��O/ÂLîY/�Lî�ü(ratio)�°./ ��ª�½¶@Y�°üY�°.
||e(i)||2 ≤(κ− 1κ+ 1
)i
||e(0)||A, �ýL (28)
f(x(i))− f(x)f(x(0))− f(x)
=12e
T(i)Ae(i)
12e
T(0)Ae(0)
(½é 8��©)
=(κ− 1κ+ 1
)2i
.
7 ���ªªª���³³³���ªªª
7.1 ���ªªª���
/ �(Steepest Descent)�ªù [[ �� 8Y �� �;� �y Ò_°Ï �³ú °è Ò_¢°. �¢
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[�� 20]/ ��½¶�ù±µ�S&½(condition number)���¥�0�JÜý°.
Ú �©3 �ù �³� Ò_ú #�qt � ¥ ½±£ ,� I¦� #� Ò_ �³� © �b£ $
ýõ ¢ ¥� �¿ � ½ �°8 Ì -� Mú«?I�%q� °üY �°: Íx t¿ �p � Ò_ �³
d(0), d(1), · · · , d(n−1)��¦úP�Ã�.�Ò_�³�©tÍý�¢¥�³G¿O$ x®�$�«Ï
³G«��êÀ �,�°.�:�Å¢�±$ýõN½�°8,?�Ò_�³�© 1¥�Ò_�½
± 8üL,@v n³Gó�©�êµ£,�°.
�� 21ù_v9úÒ_�³÷¿PÌ �z< L�°.� (½e)³G�t�AÝ¢ x1-_v®�X
£:«�Ò_ú�± 6,�¥« (½�)³G�t�Ù �©úu3ý°. e(1)� d(0)®½��ús@©
Ã�.�õ��Ü 8,�³G�t°üY��æXõxØ �,�°.
x(i+1) = x(i) + α(i)d(i) (29)
¢¥Ò_¢ d(i) �³÷¿�X÷¿Ì�\Ò_��Å{êÀ �æ©, e(e+1)� d(i)�t¿�p©b
¢°�Pìú�Ì � α(i) úu¢°.�¢S&ú�Ì �°üéúuú½�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 21]�p�³�ª. IÖ3ê��ªùÍý��y»úNL�úEÍ���Ì£½�°
dT(i)e(i+1) = 0
dT(i)(e(i) + α(i)d(i)) = 0 (½é 29�� �)
α(i) = −dT(i)e(i)
dT(i)d(i)
(30)
ݱ 3ê��ªùPì8?{��ª�°.��ªúPÌ � α(i)õGS ²8 e(i)õNIb�
�Ú, e(i)õK°�,ù�y©õNL�°�,�t¿[C����y�²��\× ,�°.
�¢[C>ú©@ ²8,�p �Ò_�³�I¦� A-�p Ò_�³úPÌ �b¢°.�¯â
d(i) ® d(j)�t¿ A-�pZ��ª(conjugate)�ü²8°üS&ú�T©b¢°.
dT(i)Ad(i) = 0
�� 22 (a)� A-�p¯â�qN3Ã���õÃ�sL�°.����xÈæ�;³üq�°L\
\©Ã�.�� � Û� �� 22 (a)� j�ú ä!It ÍÙ� Ù�¤ Ã�:«� !I¾D°L f�© Ã
�.� 8¯â���� 22 (b)�¤�p �,�¤Ã�,�°
^¿ÏÅuP¨ù e(i+1)® d(i)�t¿ A-�p(�� 23 (b))õ�Øqb¢°�,�°.��pS&ù/
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 22]�� (b)¯â�û���p t¿�� (a)¯â�ûù A-�p�°.
 ��ªY�ô��¿*_�³ d(i)ú0�t/�>úü�,Yô� °.�õÝ �æ©,�³
꥽õ 0÷¿�LI�®���qÃ�.
d
dαf(x(i+1)) = 0
f ′(x(i+1))T d
dαx(i+1) = 0
−rT(i+1)d(i) = 0
dT(i)Ae(i) = 0
é 30�îê�ªú�Ì �,*_�³��t¿ A-�p�üêÀ � α(i)�°üY��vÇý°.
α(i) = −dT(i)Ae(i)
dT(i)Ad(i)
(31)
=dT(i)r(i)
dT(i)Ad(i)
(32)
½é 30Y�µý�éùGS�� °.�é�t*_¯â�#"�(residual) â®ô� °8,�
éù@v/ ��ª�éY�ù,�ý°�>�s@ �(½é 11ú÷L è¡).
��ª� n��³G4� xõGS£½�°�,ú�< �æ©,¡ò¨(error term)ú*_�³�
�xÍS¦÷¿°üY��vÇ©Ã�.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 23]�ª �³ �ªù n�� �Ä 4� ½¶¢°. (a)� ¥« ³G�t� qE �³ d(0)õ 0� ½± ý°. e(1)�
��è d(0)� A-�p©b¢°�C¢S&ú�Ì �/�> x(i)�xØý°. (b)#�¡ò e(0)�ç_ÜUv¿vèý
A-�p �Û� ¦÷¿ vÇ£ ½ �°. �ª �³ �ª� � �Ä ³G� ¢ ¥ ½±þ :�° �� �Û |�t #�
�Û�ò»¿C$ý°.
e(0) =n−1∑j=0
δjd(j) (33)
� : δj ù �³¢ ½¡: �põ �© uú ½ �°.*_ �³�ù \Ò A-�p�t¿, dT(k)Aõ ½é
33�jº����X�U 8 #�Ò_�³úC½¢°ô?�Ò_�³�Âÿ � δj �úC$
£½�°:
dT(k)Ae(0) =
∑j
δjdT(k)Ad(j)
dT(k)Ae(0) = δ(k)d
T(k)Ad(k) (d¯â� A-�p��� �uq�)
δ(k) =dT(k)Ae(0)
dT(k)Ad(k)
=dT(k)A(e(0) +
∑k−1i=0 α(i)d(i))
dT(k)Ad(k)
(d¯â� A �p��� �uq�)
=dT(k)Ae(k)
dT(k)Ad(k)
(½é 29�� �îê ) (34)
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
½é 31Y 34�� �,Íý� α(i) = −δ(i)�úN½�°.�Pìù¡ò¨�¢^¿Ï[>úCW
¢°.°ü½é�tÃ��®��, xõu� ��Û # #õüI#��YAù@v¡ò¨ú�Û
»¿ #BC$©#��,÷¿Ç½�°.(�� 23(b)õÃ�)
e(i) = e(0) +i−1∑j=0
α(j)d(j)
=n−1∑j=0
δ(j)d(j) −i−1∑j=0
δ(j)d(j)
=n−1∑j=i
δ(j)d(j) (35)
n¥��Ä��#8,?��Û�C$üq e(n) = 0�ý°;�<�.
7.2 ������-ÙÙÙyyy!!!(Gram-Schmidt)���ªªª���
�C A-�p*_�³÷¿�Øq��¦ {d(j)}úu �� 8ý°.°±ß¥3ê�õ�³ 3f�
��ª���Ú��ª��¿�ª��-Ùy!YA(conjugate Gram-Schmidt process)�°.
n��xÍë�¯â u0, u1, · · · , un−1õ��L�°L �.üÀð� : xØ�� ��
�� _v 9ú � n �� xÍ ë� ¯â¿ 8 ½ �ú ,�°.Ò_ �³ d(i)õ ��q 4� æ© Íx uiõ
��L£�,��t�;�?�Ò_�³¯â(�� 24õÇ,)® A-�p �M�?��Ûú�°.°
�L#t+I��¯â��¿�;�?�Ò_�³Y A-�p Ò_�³ d(i)�ý°.°ô¿ �8
d(0) = u0¿�L, i > 0 ?�EÍ�©°üY��Ò_�³¯âõu£½�°.
d(i) = ui +i−1∑k=0
βikd(k) (36)
�: βik� i > k EÍ� �A�ý°.��úu �æ �, δjõu£:PÌ¢,Yô�¢�ª
úPÌ£½�°.
dT(i)Ad(j) = uT
i Ad(j) +i−1∑k=0
βikdT(k)Ad(j)
0 = uTi Ad(j) + βijd
T(j)Ad(j), i > j (d¯â�P�� A-�p��� �)
βij = −uT
i Ad(j)
dT(j)Ad(j)
(37)
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
�ª�³ (conjugate direction)�ª���-Ùy!�ªÜ (Gram-Schmidt conjugation)úPÌ£:�[
C>ù ^¿Ï Ò_ �³ú @A � æ© �;� ?� Ò_ �³ ¯â�ú -?ý� ��L �qb ¢°�
,�6,Ìu#;�Ò_�³ú@A �æ©t�?� O(n3)��S��Å °�,�°.Pì³æ£
��9¯âõ�Ö÷¿�ªÜ(conjugation)õ½± �Ò_�³�ú@A �°8,�ª�³�ªù@
v�Íß�$ª(�� 25)Yô�¢,�ý°.@Y:÷¿�ª�³�ªù�ªEPê�ª� -�ªEP
ê�ª�è�ª�³�ª�¢[ê�°-#Í#�¢�¢[Cõ©@£:«�$�PÌü�MU°.
�ª�³�ªú�© �Ú�ªî: Å�� (�ªEPê�ªê�ô���°)�� 25��� 21õ
� ,� ÝY °� ,ú NIòý� ,�°.�ª �³ �ªú ½±£ :(�ª EPê �ªú j¥ �),
��ª�Pì�q& (Óù¾��ºÍý)W��tÇ:�p�³ (orthogonal direction)�ª÷¿[Cõ
�L��,Y�°�,ú�r �.
[�� 24] � ¯â� ��-Ùy! �ªÜ(Gram-Schmidt conjugate).� �� xÍ ë� ¯â u0® u1 ¯â�t è�¢°.
d(0) = u(0)¿ zA¢°. ¯â u1ù d0� A-�p(Z� conjugate) u∗® d0� e± u+� �Û÷¿ Û©þ ½ �°.
�ªÜ�ó A-�pÙÛ��+3üq d1 = u∗�ý°
7.3 ¡¡¡òòò¨���///:::���
�ª�³�ªù°üY�ù&y��"���°:��ªù�¥�³G�tÒ_� Ìý©æ4�t
/x�©õu¢°.Ò_�´Ìý���q%�«? span{d(0), d(1), · · · , d(i−1)}�Ò_¯â d(0), d(1), · · · ,
d(i−1)��©f�ü� i-òÙÙÛW��6�õ Di�LvÇ �. e(i) ù e(0) + Di¿ÙâxØþ½�
°.���t � “/x�©”��3Xáy�«?�ª�³�ª�/x�©õü�°�,ù��ª
� e(0) +Di�ÙÛW�4�túxØ£: ||e(i)||A�/��ü�úxØ¢°�,�°(�� 26ú
Ã�).Pì qE ¡��ù ÙÛ W� e(0) + Di 4�t ||e(i)||A ú /�Ü¥÷¿� �ª EPê (conjugate
gradient)îê �ꢰ.
ô�¢ �ª÷¿ ¡ò¨ �è *_ �³� xÍ S¦÷¿ vÇþ ½ �÷6(½é 35),¡ò¨� �C�
i(norm)ù°üY�ù¦SvÇ÷¿#Í7½�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 25]9�³³æ¯âõPÌ¢�ª�³�ªù�Íß�$ª(Gauss Elimination)÷¿êN²K�°.
||e(i)||A =n−1∑j=i
n−1∑k=i
δ(j)δ(k)dT(j)Ad(k) (½é 35��©îê)
=n−1∑j=i
δ2(j)dT(j)Ad(j) (d¯â� A-�p��� �îê)
�¦S(summation)vÇé���?�¨ùI�PÌü�MùÒ_�³���[üq�°. e(0) +Di
ÙÛW��txØý���°ô¯â e��¦SvÇéúÝ$°ú:#Í#�,Yô�¢¨�ú�K
b 6,��@v e(i)���è/��C�i(norm)ú��°�,ú�<¢°.
½éú �© /:�ú �< �÷¦, �E �[:÷¿ �õ �©£ ½ �êÀ ©Ã�. �ª �³ �ª
� �z YAú è�:÷¿ vÇ£ ½ �� /x� �ªù �� 22® �� Íý� �z � W�Y “�q
&(stretched)”W�út¿üp �,�°.�� 27 (a)® 27 (c)��ª�³�ª� R2 W�Y R3W��t
qN3 ô� ��õ Ã�sL �°; � ���t t¿ ½�÷¿ Ã�� x�ù t¿ �p(orthogonal)¢°.
�8�,�� 27(b)® 27(d)�t�ô�¢��úLî¯â9ú0��²ÍÙÍ#Lx�uÍ(spherical)�
üêÀ¢,�°.����t½�÷¿Ã��x�ùt¿ A-�p�°.
�� 27(a)úÃ8�ª�³�ª�#�8AX x(0)�tè� 6,Ò_�³ d(0)õ�Ì �¢³G�
ô �°ü8AX x(1)>�t'8L�°.�:^¿Ï¡ò¯â e(1)�Ò_�³ d(0)Y A-�p�°.�
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 26]����tq�3\©���� e(0) + D2 = e(0) + span{d(0), d(1)}�°. ÍÙù�C�i(norm)��A¢
ú �#Lx�°. �³G�½±ý�,�ª�³�ªù e(0) + D2 \�t ||e(i)||A �/��ü�> e(2)õü3
ý°.
©3üù x(1)�qN3ÙÛW� x(0) +D1�t/�>���,úÃ$£½�ú«?»ù�� 27(b)�#
Í#�°: e(1)® d(0)�t¿ A-�p��:[���Ã���q&W�(stretched space)�t�¯â�½
�ú�ØL�°.� ¯â e(1)ùô�¢�C�i(norm)�¡òõ � (�,ô�¢ ||e||Aõ���)#L
x�úvÇ �ôîÙ����ö�°.0�tÙÛW� x(0) +D1��¿ x1�n���ôîÙ-�, x(1)Y
ô�¢ �C� i(norm)� ¡òõ � #Lx-� ? � ?e8� ü6 �: � ?e8Y #Lx� ? �
�>��¿ x(1)�°.
�,ù Pì �ý g��¢ �ù I¦°; �y 7.1<�t U_Æ �® ��, Ò_ �³Y ¡ò¨� t¿
A-�ª��Pì�@vÒ_�³ú0� fõ/�ÜèÅ�,(0�t ||e||Aõ/�ÜèÅ�,)Yô�
°.� #�ª�³�ª��¥«³Gõ½± ��¥«Ò_�³ d(1) �³÷¿ ||e||Aú½±¢��
ê�;& ||e||A��;�PÌ°ÏÒ_�³ d(0) �³�t/���î�ý°�,úqN3Ã$£½�
ú«? i¥��Äú½± L#8,qN3 f(x(i))�ÙÛW� x(0) +Di ;��t/�Üþ½��,�«?
d(0)® d(1)�t¿ A-�p��:[��� 27(b)�t½�÷¿�&°.��tÒ_�³ d(1)�© xõ³
¢°,ù<�¢Ú,��î� d(0) ¯â� xõ|î÷¿ L x(1)õ�#�Ù�© x(1)�t? �:[
�°.XòÙ�C��õÌÎ�Ã�u°.�� 27(c)® 27(d)����|î��ù���ÍÙý��²
K �°. x(1)� ½Ù ÍÙ� æ� n� �L, x2)� 4Ù ÍÙ�� n� �°.�� ��ú �î& U_ Ã�:
e8 x(0) +D2�ÀÍÙú�Y 6�òL�÷6,�e8ù4ÙÍÙY x(2)�t? �e8�ý°.©
x��e8I��ÍÙ��|î��°.
�� 27(c)õÃ8Xt��[ú°èL\½�°.��� ÛY4� x(1) æX�t��ÚÙÛW�
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 27] �ª �³ �ª� /:�. (a) �òÙ [C. ½�÷¿ �#� x�ù t¿ �p(orthogonal)¢°. (b)
“�q&(stretched)”W��t vÇý ô�¢ [C. ��t ½�÷¿ �#� x�ù t¿ A-�p�°. (c) XòÙ [C,
xõ |î÷¿ � � �� ôî ÍÙ�� Ã�L �°. �x x(0) + D1ù ½Ù ÍÙ�� © x(1)>�t ?¢°. e8
x(0) +D2�4ÙÍÙ�© x(2)>�t?¢°. (d)XòÙ[Cõ�q&W�(stretched space)÷¿¦Dú:?å.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 37
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
x(0) +D2\ú(q°¦8t¡ò�i(norm)||e||õ/�¿£½��V÷¿�ô ²L¢°L�A �;
�¢ÚÍý�°©½��VùÒ_�³ d(1)�t«q(½�{°.�c�Ò_�³ d(1)�/�>ú�
¿�ýÅL�°8Íý�@võµ�£½�°.�Ò_¯â d(1)�/�>úO$�ýÅL�°LÃ$
£½���î����?
�� 27(d)��»úÃ�u°. d(1)� d(0)� � A-�p�t¿,����tt¿½�÷¿�#L�
°.�C� Û�e8 x(0) + D2��e8�¢$�[� ,�¤f� LI�õ�°Ã�;�:Ã3
ü�$8ù�� 27(b)®ô�£,�°.> x(2)ù[��|î�þ,�L,> x�[��I�Å��3ü
�Ú,�æX� x(2)�t½�÷¿O$I�¿4²�æX�þ,�°. d(1)Y d(0)�½�÷¿�#L�÷
t¿,Ò_�³ d(1)�AÝ& x(2)õ³ 3ü6,� x(2)�e8 x(0) + D2 \���>��ÏÚ© x��
$�«Ï>�þ,�°.e8 x(0) +D2� x(2)�n���u�? �e8�°.��t�¥«³Gõ½
± 3ü8,D2® A-�p ��³÷¿Ó�� x(2)�t© x¿O$4²�3þ,�°.
�� 27(d)�t Xá �� §q���õ �© � Z °ô �ªù � Û �ê� © x� tt ÙÛ W�
x(0) +Di�t�Ó��êÀC¢üq��uâú�÷¿!I¾�L�°L\\ �,�°.�³G�°
Ý$ �ÙÛW�(expanding subspace)D�òÙ� #B�q#L,�:�°uâùS�BÌ� Û�
«�°�¤½�êÀ�î¿×�,�°.�CW�úé� r²�� 27(b)�¤Ã�3��8�,��
¿�ª�³�ª�ý°.
�ª �³ �ª� Z°ô |Å¢ "�� �� ��� #Í# �°. Íý� � ¥� ½± ³G�t #e
8(hyperplane)x(0) +Di� x(i)�n���ÍÙ�? 3ý°�,úU_ÃU°. 4<�z<¢�®��
���>�t���#"�(residual)���>ú�#�ÍÙ�v8Y�p¢°�Pìú�r �.�,
ù@v r(i)� Di®�p¢°�,ú�y¢°.�õ½¡:÷¿Ã��æ©é 35� −dT(i)Aõ���jº
X�U©Ã�.
−dT(i)Ae(i) = −
n−1∑j=i
δ(j)dT(i)Ad(j) (38)
dT(i)r(j) = 0, i < j (d- â� A-�p��� �.) (39)
�¨#éù°ô�ª÷¿îꣽê�°.Ò_�³�©¢¥�³Gõ½± L#8X÷¿Ì�
\ù��³÷¿Ò_£�Å�{°�Pìú\� �;¡ò¨ù�;�?�Ò_�³�©¨\ A-�
p�6, r(i) = −Ae(i)�t¿#"�(residual)��;�Ò_�³YsC#�p�ý°.
Ò_�³� u¯â�ú�Ì ���qP÷t¿, u0, · · · , ui−1�f� �ÙÛW���¿ Di�6,#
"�(residual)r(i)�� ¢�; u¯â®�p¢° (�� 28úÃ�).�,ù½é 36Y r(j)�4:úD �
�<£½�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
dT(i)r(j) = uT
i r(j) +i−1∑k=0
βijdT(k)r(j) (40)
0 = uTi r(j), i < j (½é 39�� �) (41)
X÷¿ PÌþ ¨#é� # Ì ��Ú.� ¨#éù ½é 40(�ýL �� 28)÷¿ Ùâ uú ½ �� °
üY�ùé�°.
dT(i)r(i) = uT
i r(i). (42)
[�� 28] Ò_ �³ d(0)® d(1)� ¯â u0, u1õ �Ì � ��qP÷t¿, ��ù ô�¢ ÙÛW� D2(ç_ ��)õ
f�¢°. ¡ò¨ e(2)� D2� A-�p�6, #"�(residual)r(2)� D2® �p�°. °ü �Äú æ« ^¿Ï Ò_ �³
d(2)ù��«��Ò_�³¯â�¿f�ü�ÙÛW�D2�A-�p êÀ (u2õ�Ì �)��q�°. d(2)õf�£
: u2õ��L��-Ùy!�ªÜõ:Ì �f� �:[�, d(2)® u2��>�ù D2�e±¢e8�n��°.
� <ú ��÷8t ¢ �� Ì s�£ ,ù / � �ªú PÌ£ :® �ô��¿ �ª �³ �ª
�è ±µ-¯â U � �Sú � �Ä è��° ¢ ¥� êÀ £ ½ �°� >�°. �õ æ©t� #"
�(residual)úGS£:°üY�ù>ÜéúPÌ �b¢°.
r(i+1) = −Ae(i+1)
= −A(e(i) + α(i)d(i)
= r(i) − α(i)Ad(i) (43)
8 ���ªªªEEEPPPêêê(conjugate gradient)���ªªª
�ª EPê �ªú °Ø� ��t �ª EPê �ª� ¢ 4Ì� ��«�ê s�ü� M� ,� �\£
,���,�Å¢?�$X��C @P°.Pì�ªEPê�ª���,ù�ª�³�ªYô�¢Ú,
³�Ò_�³ú��:#"�(residual)õ�ªÜ ���°�,�° (�, ui = r(i)¿zA¢°).
copyright c©Jonathan Richard Shewchuk and VisCom Commune 39
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
� ¢ xØ �ªù �² �� �î¿ ¦ý:�°.Íx / � �ª�t #"�(residual)ú PÌ �
,� � ô�°÷¦ �ª EPê �ª�t� PÌ � G£ �î� {°.�«,#"�(residual)ù �;� Ò
_ ¯â® �p¢°� ^ù "�ú ��L �°(½é 39). 0�t #"�(residual)¯â� 0¯â� I¦�8
¨\�;�Ò_�³�xÍë� ^Ò_�³úf�£½��°.#"�¯â� 0¯â EÍ���
�¿©�uq�EÍ�t¿[C�ü�M�°.#|�Ã3ü9��,#"�(residual)úPÌ �Ú��
ÌΦý: �î�U&¢°.
#"� ¯âõ PÌ � ,� qE �yõ ���� f�© Ã�.Ò_ ¯â�� #"�(residual)ú �Ì
���q��:[�,#"�¯â��f� �ÙÛW�,� span{r(0), r(1), · · · , r(i−1)}ù Di®ô�
°.Z¢ ��� #"�(residual)� �; *_ �³Y �p�t¿ �;� #"� ¯â ?�®ê �è �p¢
°((�� 29÷S).0�t½é 41ù°üY��°èvÇý°.
rT(i)r(j) = 0, i 6= j (44)
43�t�y¿ÏPìù���^#"�(residual)r(i)��;#"�(residual)® Ad(i−1)�xÍS¦�
��,�°. d(i−1) ∈ Di�ú\� 8,�Pìù@v���^¿ÏÙÛW� Di+1��;ÙÛW� Di®
ÙÛW� ADi�¦÷¿ÙâÍ�ý°�,�°.
D(i) = span{d(0),Ad(0),A2d(0), · · · ,Ai−1d(0)}
= span{r(0),Ar(0),A2r(0), · · · ,Ai−1r(0)}.
�ÙÛW�ú¾�¿�ÙÛW�(Krylov subspace)��L �Ú,�,ù #�¯â�qE±µú�
Ä:÷¿:Ì �f�£½�°.�W�ù�Í^ù"�ú��L�°: ADi� Di+1�j¥üq�L,
°ü#"� ri+1Y Di+1Y�p t¿(½é 39), ri+1� DiY A-�põ�Ú°�PìúN½�°. ri+1�
d(i)õC½¢�;�?�Ò_�³Y�y A-�põ�ØL��:[���-Ùy!@ªÜ�z��³©
��,�°!
½é 37�#Í#���-Ùy!\½� βij = −uTi Ad(j)/d
T(j)Ad(j)�ú\� �;�éú�� 3��
qÃ�.Íx½é 43Y riú4: 8°üY�ù@Yõu�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 29]�ªEPê�ª�t,�³G�t^¿u©��#"�(residual)��;�³G�t#Í.Ï#"�(residual)
X�� ��p�6,^Ä3xØü�Ò_�³ù�;�?�#"�®Ò_�³�©A-�p�üêÀ��q�°.
(Ò_ �³ù #"� ¯âõ �Ì � ��°). r(2)® d(2)� �>ù D(2)(q�3 �²� ÙÛW�)® e±¢ e8� n�
�°. �ªEPê�ª�t d(2)� r(2)® d(1)�xÍS¦�°.
rT(i)r(j+1) = rT
(i)r(j) − α(j)rT(i)Ad(j)
α(j)rT(i)Ad(j) = rT
(i)r(j) − rT(i)r(j+1)
rT(i)Ad(j) =
1
α(i)rT(i)r(i), i = j,
−1α(i−1)
rT(i)r(i), i = j + 1, (é 44�� �)
0, �½�EÍ.
∴ βij =
1
α(i−1)
rT(i)r(i)
dT(i−1)Ad(i−1)
, i = j + 1
0, i > j + 1, (é 37�� �.)
�Á�¤ÂÙÛ� βij ¨ùP�K£¹°.�C^¿Ï¯â� A-�p�üêÀ �æ©Ì�\�;
�PÌ°ÏÒ_¯âú?�õ9$£�Å�{°.� ¢|Å¢$>��ªEPê�ª�|Å¢>�
6,�ÄèW�Ä!ê®è�Ä!ê8�t O(n2)�t O(m)¿��ý°.�: mù A� 0�I¨�Û�
½�°.�CÙâ#� β(i) = βi,i−1�9cÍ÷¿v�£,�°.XÌ��&��8°üY�°.
β(i) =rT(i)r(i)
dT(i−1)r(i−1)
(½é 32�� �)
=rT(i)r(i)
rT(i−1)r(i−1)
(½é 42�� �)
�C��«�z<¢4Ì?�?I #¿Aý©Ã�.�ªEPê�ª��°üY�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 30]�ªEPê�ª
d(0) = r(0) = b−Ax(0), (45)
α(i) =rT(i)r(i)
dT(i)Ad(i)
(½é 32® 42�� �) (46)
x(i+1) = x(i) + α(i)d(i), (47)
r(i+1) = r(i) − α(i)d(i),
β(i+1) =rT(i+1)r(i+1)
rT(i)r(i)
(48)
d(i+1) = r(i+1) + β(i+1)d(i) (49)
�� 30ù �ª EPê �ªú Íý� PÌ L �� �C� :Ì°ú :� u3 ü� � ú Ã�s
L �°. Pì � �ª�t #Í#� �Ð�(gradient)� t¿ �ª� I¦� :[� “�ª EPê(conjugate
gradient)”��Ìq�°��Gý,��L£½�°.��ªù�Ð�õ�ªÜ �Ò_�³úu�:
[� “�ªÜý�Ð�(conjugated gradient)”�ª��L �,�ÌAÝ¢vÇ�°.
9 ���ªªªEEEPPPêêê���½½½¶¶¶ÛÛÛuuu
�ª EPê� n ¥� �Ä ó� °Òý°. �©°8 �ª EPê �ª� ½¶ Ûu� · [îú �qb £
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
«?ìC¿ � �ªú :Ì 8 Ùô�½> �¤� ¡ò :[� #"�(residual)� >> ÙAÝ©�6,£�
¡ò(cancellation error):[�Ò_¯â�P�� A-�p��>>\ìý°.�¤�¡ò��¢[C�/
 ��ª�t©@¢�®��°Û½���,£�¡ò��¢[C�Ö3©@£½��[C�I
¦°.��¤¡ò��©Ò_¯â�P���ª��·q��[C:[� 1960[Â��(:«�½¡
G���ªEPê�ª�[îú��MU÷6, 1970[Â��qt�Äú�©©õu ��ª÷¿�
�ªEPê�ª�îY:���@Y��vý�ó�bü¿t°è[îú�3ý°.
è��� �ªEPê�¢>©êº°°.¡�(�ªEPê�ìC¿:Ìü�[C��¾��
CX#vt n¥��Ľ±�Ý� ¢EÍ�ÂÙÛ��:[�,½¶Ûu�|Å¢�yõ��°.½
¶ÛuùÙô�½>¡òõ��æ¢,�I#�,AÝ¢NLý�÷¿©@£½{�[C��©�
ªEPê�ª�îÌ °�>ú�< �æ¢,�°.
�ªEPê�ª��¥«�ijG�/ ��ª��¥«�ijG®ô� °.0�t 6.1<
�4Ìú�¿�Ì ��ªEPê�ª�³¢¥��Ä��½¶ �S&úz<£½�°.
9.1 °°°¹¹¹¢¢¢°°°¨éééLLLòòò���
�ªEPê�ª��³G�t, e(i)� e(0) +Di¿ÙâxØüL,�: Di�°üY�ùÙÛW��ú�
yU_ÃU°.
Di = span{r(0), Ar(0), A2r(0), · · · , Ai−1r(0)}
= span{Ae(0), A2e(0), A3e(0), · · · , Aie(0)}.
�®�ù¾�¿�(Krylov)ÙÛW�ùZ¢���Û^ù"�ú��°.LAý i�©,¡ò¨ù°
üY��vÇý°.
e(0) =
I +i∑
j=1
ψjAj
e(0)
G½ ψj� α(i), β(i)® [Gü6,AÝ¢ [G�ù ��t |Å � M°.|Å¢ ,ù 7.3� #® ��
�<,�,�ªEPê�ª� ||e(i)||Aõ/�Ü �G½ ψjõxØ¢°�Pì�°.
æ é�t \ÒK� vÇù °¨é÷¿ vÇþ ½ �°. Pi(λ)õ ò½ i� °¨é�� �. Più �
¿ �# ±µ� ¤ ½ �÷6, ��ù ô� 3 GSý°. �õ �q, �� P2(λ) = 2λ2 + 1��L 8,
P2(A) = 2A2 + I�ý°.� ¢õ����v�ªù Pi(A)v = Pi(λ)v(Av = λv,A2v = λ2v, · · · .)®��
Lî¯â�tîÌ °.
�� Pi(0) = 1�üêÀ ²L 8,¡ò¨ù°üY��vÇþ½�°.
e(i) = Pi(A)e(0),
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
�ªEPê�ªù ψj G½õxØ£:,�°¨éúxØ¢°.�°¨éú e(0)�:Ì£:�îY�©
SP �./ ��ª�Ûu�t®�ô��¿, e(0)õ�p³æLî¯â��xÍS¦÷¿vÇ¢°.
e(0) =n∑
j=1
ξjvj ,
�ýL°ü�é�ú��q7½�°.
e(i) =∑
j
ξjPi(λj)vj
Ae(i) =∑
j
ξjPi(λj)λjvj
‖e(i)‖2A =∑
j
ξ2j [Pi(λj)]2λj .
�ª EPê �ªù � vÇéú /�Ü � °¨éú ü÷6, � �ª� ½¶ �ê� IXý ^Iê /J
� Lî¯â� ½¶ � Aêð ^ú ½� {°. Λ(A)� A� Lî�� �¦�� 8 ¡ò� �C�
i(norm)ù°üY�°.
‖e(i)‖2A ≤ minPi
maxλ∈Λ(A)
[Pi(λ)]2∑
j
ξ2jλj
= minPi
maxλ∈Λ(A)
[Pi(λ)]2‖e(0)‖2A (50)
��31ù Lî 2® 7ú ��� �C� © æ� vÇéú /�Ü � °¨éú = �� [ê� ò
½� © Ã�sL �°. P0(0) = 1ú �T � ò½ 0� °¨éù # �� {÷6, �� 31(a)�t Ã
� �® �� P0(λ) = 1�°. ò½ 1� /: °¨éù P1(λ) = 1 − 2x/9�L, ��31(b)�t Ç ½ �°.
P1(2) = 5/9, P1(7) = −5/9�L,�ª EPê� �¥« �Ä ó ¡ò¨� �C�i� #�X 5/9ð ¾
�MüúN½�°.��31(c)��¥��Äóé50�@Y� 0�ÿúÃ�u°.�,ù 2ò°¨é��
>(P2(0) = 1, P2(2) = 0, P2(7) = 0)ú�#êÀ£½��:[�°.��:÷¿ò½ n�°¨éù n+1�
>�ú�#¿À��½�÷t¿ n��t¿°ôLî�ú½Ì£½�°.
��«�� f�õ �© �ªEPê�ª� n¥� �Äó� AÝ¢ @Yõ u�°� ,ú Ì � N ½ �
°: ÌÍ� �ù Lî�� U&¢°8 �ªEPê�ª� Ì �ò3 ½¶¢°� ,� ¢ �<�°.X¢
½u�Ùô�½>A}êõ��°L£:,AÝ¢©õGS �æ©Åuü��Äì½��Ê©bt¿
°ôLî��½�ý°. (�ô[Ò�¢¢��°ô� ���°: x(0)� A��ÙLî¯â�Y�
y A−�põ�ØL��EÍ�°.�c x(0)õ;�£:#Í#�M�Lî¯â�ù��©¾ �Lî
��èé 50�#Í#�Mú,�°.� #XtEL¢�®��,� ¢Lî¯â��èÙô�½
>�¤�¡ò��©°è#Í#3þ,�°.)
copyright c©Jonathan Richard Shewchuk and VisCom Commune 44
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 31] i¥� �Äú ½±¢ �ó �ª EPê �ª� v�# ½¶°��� Pi(0) = 1�� sq� C¢ S&�t i ò
°¨é Pi���Lî�tv�# 0��«Ï��µ²�°.
Lî�� λminY λmax P�� Ý�Y 3 Ûjüq �� ,ð,��31(d)® �� ��� ?� ��
EÍ�ªEPê�ª�ÌÎý½·£½�üúN½�°.�,ù�ªEPê�ª�té50�ú�
3���°¨éúxØ �,�ÌÎÑq��:[�°.
��±µA�Lî��"�úNL�°8�ô½¶ú��q4�°¨éúCK£½ê���,�
�t��$��: EÍ,�Lî�� λminY λmaxP��Lò3Ûj L,|Äü�Lî�½�:
q�ù½�t¿°ôLî�#Í#6,Ùô�½>�¤�¡ò��f �EÍõ�A¢°.
9.2 ���üüüËËË���(Chebyshev)°°°¨ééé
¢ �� îÌ¢ ?��ªù î¢ �� >� ©t� I¦� ©æ[λmin, λmax]�t é50ú /�Ü � ,�
°.�õ½±£½��°¨é�ù�üË�(Chebyshev)°¨éú��÷¿¢°.
ò½ i��üË�(Chebyshev)°¨éù°üY�°.
Ti(ω) =12[(ω +
√ω2 − 1)i + (ω −
√ω2 − 1)i].
(�é�°¨é�¤Ã��M�°8, i� 1�# 2�EÍ�©�qÃ�.)=���üË�(Chebyshev)°
¨é�� ��32� �²K �°.�üË�(Chebyshev)°¨é�ù A�� ω ∈ [−1, 1]�t |Ti(ω) ≤ 1| "
�ú��6 (Pì -1Y 1�P��t�ô¢°),?�°¨é��qt |Ti(ω)|�ù ω /∈ [−1, 1] A���
copyright c©Jonathan Richard Shewchuk and VisCom Commune 45
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 32]ò½ 2, 5, 10, 49��üË�(Chebyshev)°¨é�
t/ÂXõ 3ý°.y} �MùvÇ÷¿�b� 8, |Ti(ω)|�����#Í#��\����
t� ¢¢�$�ò3��¢°.
½é 50�°üY�ù Pi(λ)õxØ¥��©/�Üü�,ùÙÀ C3�tz<ý°.
Pi(λ) =Ti(λmax+λmin−2λ
λmax−λmin)
Ti(λmax+λminλmax−λmin )
� °¨éù �� λmin ≤ λ ≤ λmax�t �üË�(Chebyshev)°¨é� �ô "�ú ��°(��33÷S).
Û?� Pi(0) = 1��� ÅuP¨ú ?T 3 6,Û�� λminY λmax P�� u��t / 1ú ��
°.0�té50÷¿Ùâ°ü�é���¢°.
‖e(i)‖A ≤ Ti
(λmax + λmin
λmax − λmin
)−1
‖e(0)‖A
= Ti
(κ+ 1κ− 1
)−1
‖e(0)‖A
= 2
[(√κ+ 1√κ− 1
)i
+(√
κ− 1√κ+ 1
)i]−1
‖e(0)‖A. (51)
P�Í\ÒK��¥«�½(�Õ, addend)� i���¥�0� 0÷¿½¶¢°.0�t�ªEPê�½¶
ùXÌc¢Ù#é÷¿°üY��vÇ �,���:�°.
‖e(i)‖A ≤ 2(√
κ− 1√κ+ 1
)i
‖e(0)‖A (52)
copyright c©Jonathan Richard Shewchuk and VisCom Commune 46
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 33] ��: EÍ�t �� λmin = 2® λmax = 7�t é 50ú /�Ü � °¨é P2(λ). � Mxù ò½ 2�
Chebyshev°¨é� ¾�õ ºÍ¢ ,�°. � ¥� �Ä �ó� ¡ò¨� �C� i(norm)ù #�� 0.183�õ J�
M�°. 2��Lî��U& �,÷¿�yN²K�Ï�� 31(c)®üp �.
�ªEPê��¥«³G�/ ���¥«³G®ô� °.é51�t i = 1¿�8,/ ��[
¢½¶@Y é28úu�°.�,ù�� 31(b)��²�xÍ°¨é�EÍ°.
�� 34��ªEPê��Ä�°�½¶�¢êv�°.ìC¿:Ì°ú:,�ªEPê�^ùLî
Ûj#^ùè�æX#��î:[�Â��EÍé 52�èP �½¶�êðÌ�ò3½¶¢°.
é 52®é 28úüp 8,�ªEPê(conjugate gradient)�ª�/ � (steepest descent)�ªÃ°Ì�
ò3½¶ �°�,�<Ý ° (�� 35÷S).� #�ªEPê�?��ijG�/ ��ªÃ°
Ì �ò3 ½¶ � ,ù I¦°; �õ �8,�ª EPê �ª� �¥« ³G� / � �ª� �¥« ³
G®ô� °.é52�tU©K��½ 2��©t�ªEPê�ª��Ù�ijG�tc���ê9
õÃ�½�3ý°.
10 ÄÄÄ!!!êêê
/ ��#�ªEPê�ª��³G�t�$�ùGSè�ú�Å¿ ��Sù±µ-¯âU��°.
��:÷¿,±µ-¯âU�ù±µ� 0�I¨Ù��½õm��£EÍ O(m)�GSè�ú�Å¿¢°.
�[t� 1<�t#�ý°j¢[C��tÃ� A� �±µ�Lm ∈ O(n)�°.
ε��½¿¡ò�¾�õv��æ©?Û¢�Äú½± �õÙ¢°L�A �:�, ‖e(i)‖ ≤ ε‖e(0)‖.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 47
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 34]\×½�¥½¿t��ªEPê(�Ä�°)�½¶. �� 20Yüp �.
é 28ù/ ��ªúPÌ ��¢Gõ�T �æ©Åuü�/Â��Äì½�°üY�üúÃ
�u°.
i ≤⌈
12κ ln
(1ε
)⌉,
¢`,é 52�� 8�ªEPê�ª�Åu �/Â��Äì½�°üY�°.
i ≤⌈
12√κ ln
(2ε
)⌉.
/ �ù O(mκ)�è�Ä!êõ��6,�ªEPê� O(m√κ)�è�Ä!êõ��°�@Á�°.�
NLý�?�W�Ä!ê� O(m)�°.
d−òÙ A���t �ò ÍÙ EG [C�ú î¢òÛY î¢Å�ª÷¿ �P£ :� Ä!ê� [[
κ ∈ O(n2/d)� ý°. 0�t, / � �ªù �òÙ [C�� © O(n2)� è�Ä!êõ ���� �
ªEPê� O(n3/2)�°.Z¢XòÙ[C��©/ �ù O(n5/3)�Ä!ê���ªEPê�ªù
O(n4/3)�è�Ä!êõ��°.
11 èèè���YYY[[[ÒÒÒ
Xtz<¢/ �Y�ªEPêNLý��t,=���Ù: ,��f�ü}°;"&,è�>úqN
3xØ LsC[Ò �=�[C�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 35]�ªEPê�¢¥��Ä��X �æ©Åuü�/ ���Äì½.
11.1 èèè���
è��[ �z<£,ù��M°.�� x��©Â�: 8AX��°8,�,úè� x(0)¿
PÌ �.�©�M°8, x(0) = 0÷¿�q�;xÍèßðú{�EÍ/ ��#�ªEPê�ªù@
v ½¶ 3 üq �°.üxÍ /�Ü(14<�t z<)� � �� ��/�� U&£ ½ �L è� æX�
xØ�0�q�/�>�½¶£��@Aü6,qEEÍ��½¶��Øq���©�Mú�õ@A
�:[�«°¿Ï[C�ý°.
11.2 [[[ÒÒÒ
/ ��#�ªEPê�/�>�êµ£:#"�(residual)� 0�üL,��é11�# 48��ó�¢
¥��ÄúÌ 3¢°8,Û?� 0�ý°.0�t#"�(residual)� 0�ü8�èGS�[Òüqb¢
°. #"�(residual)� >Üé Í×(47)GS�t �¤� ¡ò� �:ü� EÍ�� #"�� 0� I¨Úê
0÷¿GSþ½��°;�[C�é45õ�Ì �GSú^¿è�¥÷¿t©@þ½�°.
Ã��Eͽ¶�°;&�Øq��;�NLý��[Òü�õÙ¢°.¡ò¨ú�Ì£½{�:
[� #"�(residual)� ¾�� P;� <èý � ¿ 4²�8 [Ò � ,� Ã��°; [[ � ÷¿
#�#"�(residual)� 0ð�ù εúU©uù�ùúPÌ¢°. (‖r(i)‖ < ε‖r(0)‖). �C���
ÙÀ B�tǽ�°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
12 ;;;���ýýý
;�ý�±µ�S&½(condition number)õ�x �æ¢�ª�°.±µM�±µ Aõ�P �Âa�
8tj�AÙÒ ±µ�L,�±µúÖ3GS£½�°L �.�©°8°üéú�qt�?:÷¿
Ax = bõ�½�°.
M−1Ax = M−1b. (53)
�� κ(M−1A) � κ(A)�$#M−1A�Lî�� A�Lî�ðÌ�?��°8,Ù��[Cð
é 53�[C��Īú�©Ì�½¶¢°.[C�±µM�# A��Âa�Lj�AÙÒ±µ��
Ì�ê±µM−1Aù��:÷¿�©�M°�,�°.
Âa�6j�AÙÒ ?�±µM�© EET = M "�ú���±µ E� (î�¢,ùI¦�
�)U& �:[�,�q²Óù�£½�°. (�¢±µ E�&ªßÅ(Cholesky)Û©®�ù�ª÷¿u
£½�°.)±µM−1A® E−1AE−T�ô�¢Lî�ú��°.�,��� ��î� v�Lî λõ
���±µM−1A�Lî¯â�:, ET v�°üY��Lî λõ���±µ E−1AE−T�Lî¯â�
�:[�°.
(E−1AE−T )(ET v) = (ETE−T )E−1Av = ETM−1Av = λET v.
xÍèßð Ax = b�°ü�[C¿ºÍþ½�°.
E−1AE−T x = E−1b, x = ETx,
�[C�tÍý�Íx x�©�L#t x�©}°.±µ E−1AE−T�Âa�8tj�AÙÒ��
:[�, x� / ��# �ªEPê� �© u£ ½ �°.�ª EPê �ªú PÌ � � [Cõ ©@
�YAú “ºÞý;�ý�ªEPê�ª(Transformed preconditioned conjugate gradient method)”��Ùô
°.
d(0) = r(0) = E−1b− E−1AET x(0),
α(i) =rT(i)r(i)
dT(i)E
−1AE−T d(i),
x(i+1) = x(i) + α(i)d(i),
r(i+1) = r(i) − α(i)E−1AE−T d(i),
β(i+1) =rT(i+1)r(i+1)
rT(i)r(i)
,
d(i+1) = r(i+1) + β(i+1)d(i).
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
��ªù±µ EõGS©b¢°�[C��°.� #c��s�©tº½õXÞ 8±µ EõC
$£½�°. r(i) = E−1r(i)® d(i) = ET d(i)¿�L x(i) = ETx(i)® E−TE−1 = M−1��¨#éú:Ì
8,°üY �� “ºÞü� Mù ;�ý �ª EPê �ª(Untransformed preconditioned conjugate gradient
method)”õuú½�°.
r(0) = b−Ax(0),
d(0) = M−1r(0),
α(i) =rT(i)M
−1r(i)
dT(i)Ad(i)
,
x(i+1) = x(i) + α(i)d(i),
r(i+1) = r(i) − α(i)Ad(i),
β(i+1) =rT(i+1)M
−1r(i+1)
rT(i)M
−1r(i),
d(i+1) = M−1r(i+1) + β(i+1)d(i).
±µE���Aé�t�#Í#�M�°;��t�±µM−1��Å °.�ù�ª��©,±µEõ
PÌ �M�;�ý/ ��ªú��q4�,ê� °.
;�ý�M�îî�ù±µM−1A�S&½��©@AüL,EÍ�0�Lî��Á ß!�A
ê��©@Aý°.[C��¥�Ä��q(:�°¢¥BM−1r(i) U�ú½±©b �:[��f
�üÌú\©£½�úAê¿ Aõ��P �½¶�êõm�½��;CS&�õü�,�°. (<
è:÷¿M Z�M−1ú GS£ �Å� {°; �Å¢ ,ù ³� ±µM−1ú ¯â� :Ì°ú : �f �
îYõGS �,�°.)� ¢C¢S&��t,g�3ê�Ù¢� �����Ú��t�½�0¤
�®���Ù�°Û,�°.
�[:÷¿,;�ý� �ò Íéú ÌÎ uÍ� �²3 ��²� èê�°.0�t Lî�ù t¿�3
�?¢°.°;;�ý�(perfect preconditioner)�¢��M = A�°;�;�ý��©,M−1A�S&½
� 1� ü6,�ò Íéù °¹¢ uÍ� üq ³ ¢ ¥� �Ä�÷¿ê ©õ u£ ½ �°.� #,ݱ 3
ê,;�ý³G���Mx = bõ{�,�qt�;�ý��@�îÌ¢;�ý��I¦°.
�$�³¢;�ý��Â��Û��±µ AYô�¢Â�±µ�°.�;�ý�õ:Ì �YA
ùÂ�;�ýZ�b�ü;�ý���ö÷¿N²K�÷6,�YAù_v9�³÷¿�òÍé�¾
�õºE �,Y�°. (üpõ 8,°;;�ý�M = A�Lî¯â9ú0��òÍé�¾�õS<
¢°.)Â�±µ��±µúu �,ù�Í�³ ��,�±µùÃ�e©¢½u�;�ý��<�°.
Â�;�ý�ó��C[C��òÍé����#Lx����36�Ã��°.��3Yüp 8,�A
copyright c©Jonathan Richard Shewchuk and VisCom Commune 51
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 36]�C[CõÂ�;�ý¢�óu��òÍé�#Lx�
Aê�xü}°�,�<� °. 3.5�ÏS&½�Â� 2.8«�³\ý°.]Á,� ¢�xù n� 2 x
Íèßð�tÌîÌ °.
Ì Ap¢ ;�ý�� Ý°; &ªßÅ ;�ý(incomplete Cholesky preconditioning)�°.&ªßÅ Û©
�±µ Aõ LLT�Í׿۩ ��Á�°.��t LùI�X�±µ(lower triangular matrix)�°.Ý°
;¢&�ßÅÛ©�ýÓ�´Ìü�M�ºÍ�°; A� LLT®�ù±µUÍ׿�Pü�Ú,�: L�
A®ô�¢Í׿ 0�I¨Ù��#Í#b¢°; L�°ôÙ��ù?�£²�°. LLTõ;�ý�¿P
Ì �æ©, LLTw = z�©��XÞ�ª÷¿GSý°. (LLT��±µù@�<è:÷¿GS£�Å
�{°).ݱ 3êÝ°;¢&�ßÅ;�ý�¨\KA:���M°.
�ù ;�ý��� (� | �Ù� �Í Ä! °) ��ü}°.;�ý�õ qE ,ú xØ ��� ��,
Â�?[C�t�ªEPêõPÌ£:��$�sC#;�ý�õ�Ì �[Cõ�qb¢°�Pì
���:÷¿�I���L�°.
13 ���ªªªEEEPPPêêê
�ªEPê�Âa�6tj�AÙÒ�I¨,�ýLî�qA�±µ�I¨EÍ�êPÌþ½�°.°
üY�ù/��ç[CõÃ�.
minx‖Ax− b‖2 (54)
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
�[C�©�é54�yÛú 0÷¿�÷¿tu£½�°.
ATAx = AT b. (55)
�� A�A�Í�ü"�(non-singular)±µ�� 8,é 55�©� Ax = b�©®ô� °.�� A�
A� ±µ� I¦L Ax = b� º½Ã° Ì�ù xÍ ë� �Aéú ��� Y Cc(overconstrained) E
Í, Ax = b� ©� U&£ ½ê �L U& � Mú ½ê �°. �� � xÍ�Aé� ¡òCUú ¦¢ ½
é54ú/�Ü � xúü�,ù¨\� °.
ATA�Âa�Lj(��� x�©,xTATAx = ‖Ax‖2 ≥ 0)�°.��Ax = b�Y�Cc(underconstrained)�
I¦8,ATA�ü"�±µ�L,/� ��#�ªEPê�ªY�ù�ª�é55õ��æ©PÌþ½
�°.³�¢��[C� ATA�S&½� A����S&½�CU�L,0�t½¶��ý°�,�°.
��t�Á:÷¿|Å¢P¨ù±µ ATA� AÃ°Ñ �±µ��:[�<è:÷¿Í�ü�M
�°�,�°.Âê, Adõu¢�� ATAdõu¥÷¿�, ATA�� d�U¢\×�ý°.Z¢ Adõ��
�êY4: � dTATAd (é46)õGS 8½X: KA��³\ý°.
14 üüüxxxÍÍÍ���ªªªEEEPPPêêê���ªªª
�ªEPê�ªù�òÍé(quadratic form)�/�>úü�Ú�PÌþ<�I¦��Ð� f ′õGS£
½��qE��¥½ f(x)�©tê/�Üõæ©PÌþ½�°.��W¡zG,êE�¡å,�ýL
üxÍç�(regression)#Y�ù°j¢[ê�/:Ü[C�ÿÌþ½�°.
14.1 üüüxxxÍÍÍ���ªªªEEEPPPêêê���ªªª������ÅÅÅ
üxÍ�ªEPê�ªúîê£:xÍNLý�Yüp ����ºÜ��°:#"�(residual)�Â
¢&�:½é�PÌþ½{÷6,�< αõGS �,�ÌÎÄ! 6, β�©=��°ôxØ�
ª��°�,�°.
üxÍ�ªEPê�ª�t,#"��¨\�Ð��ÙÒõ�¿¢,÷¿zAý°; r(i) = −f ′(x(i)).
Ò_�³ùxÍ�ªEPê�ª�¤#"��¢��-Ùy!(Gram-Schmidt)�ªÜõ�Ì �GS
ý°.� Ò_ �³ú 0� �x Ò_ú � ,ù xÍ EÍ� ü© ÌÎ q²Í6,°j¢ <ò� PÌþ
½�°.xÍ�ªY��, f(x(i) + α(i)d(i))õ/�Ü � α(i)�ù�Ð��Ò_�³��p êÀ¥
÷¿�üú½�°.�õæ© [f(x(i) +α(i)d(i))]T d(i)� 0�ü�úü�qE[ê�NLý���êP
Ì£½�°.
xÍ �ª EPê �ª�t� β� � ¢ = �� ôX(Ý>) vÇ�� �°. üxÍ �ª EPê �
ª�t�t¿°ô�vÇ��Ì�\ôX�I¦°;�u��ù/x�xØ�X| �I�êSP
copyright c©Jonathan Richard Shewchuk and VisCom Commune 53
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
L�°.���xØ�ÏÚ #�GS�Ì��:[�xÍ�ª�PÌü}Ï�ª�-ýõß(Fletcher-
Reeves)é�6,°ô #�m�-ýü�ò(Polak-Ribiere)é�°:
βFR(i+1) =
rT(i+1)r(i+1)
rT(i)r(i)
, βPR(i+1) =
rT(i+1)(r(i+1) − r(i)
rT(i)r(i)
.
�ª�-ýõß�ªùè�>�Ù �/�>Y?Û&�«Ð:�½¶ 6,m�-ýü�ò�ªù�
]¡ ���Ù&½¶ �ML�Ä£½�°.� #m�-ýü�ò�ª�[[Ì�ò3½¶¢°.
°±&m�-ýü�ò�ª�½¶ù β = max{βPR, 0}÷¿xØ¥÷¿�Ã$þ½�°.�úPÌ
�,ù βPR < 0 EÍ��ªEPê�ªú^¿è� �,Y�°.�ªEPêõ^¿è� �,
ù�;�Ò_�³�ú?��L�$� 3 � �EP�³ú0�^Ä3�ªEPê�ªú½±
�,�°.
üxÍ�ªEP�ª��Å�°üY�°:
d(0) = r(0) = −f(x(0)),
Find α(i) that minimizes f(x(i) + α(i)d(i)),
x(i+1) = x(i) + α(i)d(i),
r(i+1) = −f ′(x(i+1)),
β(i+1) =rT(i+1)r(i+1)
rT(i)r(i)
or β(i+1) = max
{rT(i+1)(r(i+1) − r(i))
rT(i)r(i)
, 0
},
d(i+1) = r(i+1) + β(i+1)d(i).
üxÍ�ªEPê�ªùxÍ�ªEPê�ª�½¶Ã$���$�:Ìü�M�°.¥½ f��
ò¥½®îP �M÷8Mú½À,Ò_�³�ùÌý�ª�(conjugacy)ú��°. (üxÍ�ªEP
ê�ª�tê “�ª�”���Ìq��;&�yõ��°�,�O<�©�,�°.)Z°ô[C��
�: ¥½ f��ù��/�>ú��½�°�,�°.�ªEPê�ªù;�/�>�½¶£,�
�Ã$£½{÷6,î�q f� ¢(lower bound)�{�EÍ��/�Sòü�G£½ê�°.
�� 37ùüxÍ�ªEPê�ªú��÷¿Ã�L�°.�� 37(a)�°½���/�õ��L��
¥½�°.�� 37(b)��ª�-ýõß(Fletcher-Reeves)éúPÌ¢üxÍ�ªEPê�ª�½¶úÃ�
L�°.���txÍ�EÍ�¤îY:��M°;�¥½�/�Ü ���Íq·°.�� 37(c)���
37(b)�/#�xÒ_�³ú0�_8ú<³¢8úÃ�L�°.�<³8�°½�/�>�U&¥�
î� �;�xÒ_ù�«Ï/�>�©¾ � αúü�°.�� 37(d)�m�-ýü�ò(Polak-Ribiere)
�ª�Ì#ù½¶�úà °.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 54
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 37]üxÍ �ª EPê �ª� ½¶. (a)°½� �� /�® /Âõ �� Ä!¢ ¥½. (b)�ª�-ýõß(Fletcher-
Reeves)�ª�½¶E¿.xÍ�ªYµý�³G��½¶ �M�°. (c)/#��xÒ_ú0ôv8<³@Y. (d)
m�-ýü�ò(Polak-Ribiere)�ª�½¶E¿.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 55
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
14.2 ������:::���xxxÒÒÒ___
f ′��0�, f ′T d� 0�ü�©õü��ôNLý�úPÌ �,�� °.�õ�q, f ′� α�°
¨é�:,°¨é��úu �îñ: NLý�úPÌ£½�°.� #Íý�©ÌNLý��©
t�L²£,�°.
�[ê��Ä�ª÷¿�ä-�á(Newton-Raphson)�ªY£xª(}=� secant method)��°.��
ªù?� f��òyÛ�� �b¢°.�ä-�á�ªù f(x+αd)� α� � 2òyÛúGS£½
�qb¢°.
�ä-�á�ªùì� (Taylor)�½�Põ�Ì¢°.
f(x+ αd) ≈ f(x) + α
[d
dαf(x+ αd)
]α=0
+α2
2(56)
= f(x) + α[f ′(x)]T d+α2
2dT f ′′(x)d
d
dαf(x+ αd) ≈ [f ′(x)]T d+ αdT f ′′(x)d (57)
�: f ′′(x)�°üY�ù½èK(Hessian)±µ�°.
f ′′(x) =
∂2f∂x1∂x1
∂2f∂x1∂x2
· · · ∂2f∂x1∂xn
∂2f∂x2∂x1
∂2f∂x2∂x2
∂2f∂x2∂xn
......
...
∂2f∂xn∂x1
∂2f∂xn∂x2
· · · ∂2f∂xn∂xn
.
¥½ f(x+ αd)�é 57� 0�üêÀ¥÷¿��P:÷¿/�Ü£½�÷6�õ�©°üúu�°.
α = − f ′Td
dT f ′′d.
�²� ì� (Tayler)�½� f(x + αd)õ j]x÷¿ �P¢°; j]x� /�>«� 4²�° (��
39úÃ�).Pì f��òÍé��8,�j]x�P�AÝ¢,�ý°.��î� f ′′��¿Z�¢±µ
A��:[�°.��:÷¿,Ò_�³ù f ′′-�p (f ′′-orthogonal) EÍ��ª�ú �°L¢°.� #
“�ª”��y�G�©t��Ú,�� f ′′� x�0�º �:[�°. f ′′� x�0��ò3º 8º
£½À Ò_ �³� P�� �ª�� P��°.�8 x(i)� ©� �«Í8 �«Ð½À, f ′′� �Ä�° º
�Aê�vq�°.è�>�©��«Ð½ÀüxÍ�ªEPê�ª�½¶��xÍ�ªEPê�ª
Yüæ©�°.
�ò� I¨ ¥½� AÝ¢ �x Ò_ú ½± � æ©t� f ′T� 0� þ :«� �xú 0� Ò_ú �
Ä©b� ¢°; 0�t #� �ª EPê �ª �Ä 4� °½� �ä-�á �Ä� j¥üq �ú ½ �°.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 56
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 38]üxÍ�ªEPê�ªùs�:&è�úPÌ£:ÌîY:�°.
[�� 39]�òÙ¥½(ìx)õ/�Ü ��ä-�á�ª. > x�tè� � 1ò, 2ò꥽õGS L,�õ�Ì �
¥½�¢ 2ò�Põ¢°(>x). j]x�/�>�t^¿Ï> zõxØ¢°. ½¶£:«�� ¢³Gõ�Ä¢°.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 57
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 40] 1òÙ ¥½(ìx)ú /�Ü � £xª. > x�t è� � t¿ °ô � �>�t 1ò ꥽õ GS 6
(��t� α = 0® α = 2), �õ �Ì � ¥½� ¢ 2ò �P(>x)ú ½±¢°. α = 0 æX® α = 2 æX�t
� Mx� ?� ô�¢ �Ð�õ �°� ,� î� �. �ä-�á �ªY ��, ^¿Ï > z� j]x� /�>�t
xØü6,½¶þ:«��Ä¢°.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 58
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
f ′Td® dT f ′′d��ù�³G�°GSüqb¢°.� ¢GSù dT f ′′d��©u:÷¿��Üþ½
�°8üÌ�¾�M��, f ′′ ±µ;���Ä:÷¿GSüqb� t¿NLý��PÌ£½{úA
ê¿�ý°.�ÙÿÌ�t, f ′′�Â��Û�úPÌ ��P: �xÒ_ú½± �� ¢[Cõç
�£½�°.]Á f ′′õ;ÅGS£½{�¥½�ê�°.
f ′′õGS �MLAÝ¢�xÒ_ú½± �æ©£xª(secant method)� α = 0® α = σ ��
�t¿°ôæX�t f(x + αd)� 1ò꥽õGS L�õ�Ì � 2ò꥽õ�P¢°.�: σ�
0�I¨����ù½�°:
d2
dα2f(x+ αd) ≈
[ ddαf(x+ αd)]α=σ − [ d
dαf(x+ αd)]α=0
σσ 6= 0 (58)
=[f ′(x+ σd)]T − [f ′(x)]T d
σ,
�½� α® σ� 0��«Ð½ÀÌ^ù�P�ý°.é 58úì� �½(é ??)� 3¥«¨÷¿�ô8
°üúu�°:d
dαf(x+ αd) ≈ [f ′(x)]T d+
α
σ
{[f ′(x+ σd)]T d− [f ′(x)]T d
}.
꥽õ 0÷¿��q f(x+ αd)õ/�Ü � αõ°üY��u£½�°:
α = −σ [f ′(x)]T d[f ′(x+ σd)]T d− [f ′(x)]T d
(59)
�ä-�á �ªY ��, £xª(secant method)�è f(x + αd)õ j]x Í׿ �P ��, ¢ >�t
� 1ò, 2ò ꥽õ GS � ,� I¦� t¿ °ô � >�t 1ò ꥽õ GS � (�� 40)j]x
ú xØ¢°. £xª� �Ä �z� � ¥ « ³G�t� ��� σõ xØ¢°; �q�� �Ä ³G�t�
x + σd��;³G� x�üêÀxØ¢°.°è©, α[i]�£x¥ i¥«�ijG�tGSý α��
8 σ[i+1] = −α[i]�°.
�ä-�áY £xª ?� x� ©� ?Û& �«Í8 [Òüqb ¢°.�#X3 0ù A}êõ Åu 8
½¶�ìC£½ê���,�#X3mùA}êõÅu 8Ý�Å 3GS��²�L"»¢��úu
�G¢°.�� f ′′(x)� x�©��º£EÍ�ª���ò3��ü�:[�°.0�t,�ò��Ù
AÝ¢ �x Ò_� [[ Ì #ù Aþ�° (�õ �q,�ä-�á�# £xª�t C¢ý ½� �Ä�ú ½
± �,�°).ݱ 3êÙAÝ¢�xÒ_ù � ��³�I¨Ò_�³úf� 3þ½ê�°.
��: ©ªù� ¢P×õyý*P �,�6 (rT d�j½�I¨�?),�Å¢EÍ d = r¿zA
��ªEPê�ªú^¿è� �,�°.
��ª�ÌÀ[C����ª�/�>Y/Â>úuÛ �G¢°�,�°.üxÍ�ªEPê�
ª�@Y�è�>�� 3�U 6,�ä-�áÓù£xªúPÌ¢�ªEPê�ª���/�>�
��tè�ü}°8©¾æX¿½¶£� ��¾°.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 59
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
[�� 41]m�-ýü�ò(Polak-Ribiere)éú�Ì¢üxÍ�ªEPê�ªúÂ�;�ý±µ¿;�ý¢@Y. W��
“a�K” /�>sº#Lx�ÌÙÍ(Ìþ)��«×PüúÃ�L�°.
����ªù#ö�$>ú��°.�ä-�á�ªùÌ#ù½¶�êõ��6, dT f ′′d��ò3 (�,
O(n)è�K�)GSþ½��EÍ� (Óù�Pþ½��EÍ�)xÒý°.£xªù f� 1òyÛ�ú
Åu ��,� �ª� �Wù 7�yâ σõ v�# � xØ �=� _Íý°.�� �ª ½� °ô °j¢
�ªêÖ3îꣽ�°.�õ�q fõ���t¿°ô�>�tb��(sampling)¥÷¿� f� 1òê
¥½SòGS£�Å{� f(x+ αd)õ�P �j]xúf� �,ê� °.
14.3 ;;;���ýýý(Preconditioning)
üxÍ�ªEPê�ªù f ′′õ�P 8tM−1r�Ö3GSþ½��;�ý±µ(preconditioner matrix)
MúxØ �;�ýõ£½�°.xÍ�ªEPê�ª�t;�ý±µù�òÍéúºÞ �u®î
P 3 ��²L ¢°; üxÍ �ª EPê �ª� ;�ý ±µù � ¢ ºÞú x(i) ��� ��� © ½
±¢°.
;� ½èK ±µ f ′′õ GS � ,� �#X3 mù üÌú �:�ê � ±µ� Â� �Ûú GS �
�õ ;�ý ±µ¿ �Ì � ,� [[ îÌ °.� # x� �� /�>�t ?Û& FqK �ú :� ½
èK ±µ� Â� �Û |� j(�)� I¨ ,� �ú ½ê �°� ,� s� �. ;�ý ±µù j� AÙ
Ò(positive-definite)�qb t¿ j� I¨ Â� �Ûù ´Ìþ ½� {°.ý: ©@þù ½èK� j
copyright c©Jonathan Richard Shewchuk and VisCom Commune 60
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
�AÙÒ�úÃ$£½{ú:��;�ýõ �M�,�° (M = I¿zA).�� 41ùÂ��Û�©
;�ýõ ½±¢ m�-ýü�ò(Polak-Ribiere)üxÍ �ª EPê �ª� ½¶ú Ã�� ,÷¿ �� 37Y
ô�¢¥½�°.��t�³G�;�ýõæ©© x�t f ′′����Â��ÛúPÌ �½ªúPÌ
�°2.
ÙÙÙÀÀÀ
A.[[[´ ���uuu
�ª�³�ªù 1908[Ùy!(Schmidt)��©[14]/#¿�vü}ú,÷¿Ã�6, 1948[�kß(Fox),
´ßÅ(Huskey),�ýLéÇá(Wilkinson)��©[7] ë�:÷¿�>ü}°. 50[Â#���ªEPê�
ª�½ß1ß(Hestenes)®[10]ß/Z(Stiefel)�[15]�©ë�:÷¿�>ü}°;�ýLO�q,�P�ù
Wô÷¿�ªEPê�ª�ªî: ÷L[¶÷¿�;��4Ìú;: �°[11].ËüË�(Chebyshev)
°¨é÷¿��ªEP꽶¢G�b¦�(Kaniel)��©��ü}°[12].�ªEPê�ªù 1971[�
��(Reid)��©Â�? �±µ�:Ìü��Ä:�ª÷¿Â|Üü}°[13].
�ªEPê�ªùÚ�ü�(Davidon)��u[4],�ýL�ª�(Fletcher)®7Ú(Powell)��u[5]õ�
�÷¿ 1964[�ª�(Fletcher)®ýõß(Reeves)��©üxÍ[C¿��Üü}°[6]. ÙAÝ¢�xÒ
_úPÌ¢üxÍ�ªEPê�ª�½¶ù°¦�(Daniel)��©Ûuü}°[3]. üxÍ�ªEPê�
ª�t βõxØ �[C�££!(Gilbert)®dèÑ(Nocedal)��©f�ü}°[8].
P¥(Golub)Y¤ýqý(O’Leary)� 70[Â|���ªEPê�ªY [´ý�P®�¢[¶�Â
¢suúCW �°.�óÂÙÛ��u�üÂaèßð�#>ú�:°.xÍèßð���õæ¢�
Ä:�ª�¢SP�u��ª!(Barrett)#��©�ØqP°[1].
B.NNNLLLýýý ���SSS���
�<�����(code)��fL�tf�ý� NLý �îñ: uÇúÃ�L�°.
B1./// ���(Steepest Descent)
sq��³ A, b�©,è�>ù x�6,/Â�Äç½õ imax,¡ò´Ì©æõ ε < 1�°.
2��z:@v�© xõü�,�t¿,��©õü�æ©©õ�Ì¢ÙA±æ�©¾¢°.��t�;�ýõ�¢½¶�úÃ
��æ©�yNL��©õ�Ì �^ù;�ý±µúu¢,�°.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 61
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
i⇐= 0r ⇐= b−Axδ ⇐= rT rδ0 ⇐= δWhile i < imax andδ > ε2δ0 do
q ⇐= Arα⇐= δ
rT q
x⇐= x+ αrIf i is divisible by 50
r ⇐= b−Axelse
r ⇐= r − αqδ ⇐= rT ri⇐= i+ 1
�NLý ù�Ä� imax ç�êµ $# ||r(i)|| ≤ ε||r(0)||�ü8|³¢°.
#"�� �ô >Üé÷¿ GSü�� 50¥ �Äþ :�° ¢ ¥Bù AÝ¢ ÷¿ &GSüq �:ý
Ùô�½> ¡êõ C$¢°.]Á �� PÌý 50��� ½� ��� ½�°; À n� ©√n� :¾ °.
¡ò´ÌX�¾°8,#"�(residual)õ½A£�Å�;Å{° (ìC¿½Aù$�PÌü�M�°).¡
ò´ÌX��G�Ùô�½>A}ê�¢G��?£EÍ, δ�GSý�� δ < ε2δ0 �Ý �*P
�8�üqb 6,�*P@Y�÷��8AÝ¢#"�(residual)�°èGSüqb 6 δê°èe�
üqb¢°.�,ùÙô�½>�¤�¡ò��©NLý ��ê[Òü�,ú��°.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
B2.���ªªªEEEPPPêêê(Conjugate Gradient)
sq��³ A, b�©,è�>ù x�6,/Â�Äç½õ imax,¡ò´Ì©æõ ε < 1�°.
i⇐= 0r ⇐= b−Axd⇐= rδnew ⇐= rT rδ0 ⇐= δnew
While i < imax andδnew > ε2δ0 doq ⇐= Adα⇐= δnew
dT q
x⇐= x+ αdIf i is divisible by 50
r ⇐= b−Axelse
r ⇐= r − αqδold ⇐= δnew
β ⇐= δnew
δold
d⇐= r + βdi⇐= i+ 1
�;< B1�������4Ìú÷S �.
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
B3.;;;���ýýýýýý���ªªªEEEPPPêêê(Preconditioned Conjugate Gradients)
sq� �³ A, b� ©,è�>ù x�6,;�ý ±µùM (Rè:÷¿ A�þ ½ê �ü),/ �Ä ç½
õ imax,¡ò´Ì©æõ ε < 1�°.
i⇐= 0r ⇐= b−Axd⇐= M−1rδnew ⇐= rT dδ0 ⇐= δnew
While i < imax andδnew > ε2δ0 doq ⇐= Adα⇐= δnew
dT q
x⇐= x+ αdIf i is divisible by 50
r ⇐= b−Axelse
r ⇐= r − αqs⇐= M−1rδold ⇐= δnew
δnew ⇐= rT sβ ⇐= δnew
δold
d⇐= s+ βdi⇐= i+ 1
“s ⇐= M−1r”ù;�ý�õ:Ì©b¥úRè L�÷6,�:;�ý��±µ�Í×õ���M
ú½ê�°.
�è�;< B1�������4Ìú÷S �.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 64
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
B4.���äää-���ááá(Newton-Raphson)���ªªªúúú���ÌÌÌ¢¢¢;;;���ýýý���ªªª���-ýýýõõõßßß(Fletcher-Reeves)
üüüxxxÍÍÍ���ªªªEEEPPPêêê
sq�¥½ f�©,è�>� x�6,�ªEPê�ª�/Â�Ä� imax,�ªEPê�ª�¡ò´Ì
X� ε < 1�L,�ä-�á�ª�/Â�Äç½� jmax,�ä-�᪡ò´ÌX� ε < 1�°.
i⇐= 0k ⇐= 0r ⇐= −f ′(x)d⇐= rδnew ⇐= rT rδ0 ⇐= δnew
While i < imax andδnew > ε2δ0 doj ⇐= 0δd ⇐= dT dDo
α⇐= − [f ′(x)]T ddT f ′′(x)d
x⇐= x+ αdj ⇐= j + 1
While j < jmax andα2δd > ε2
r ⇐= −f ′(x)δold ⇐= δnew
δnew ⇐= rT rβ ⇐= δnew
δold
d⇐= r + βdk ⇐= k + 1If k = n or rT d ≤ 0
d⇐= rk ⇐= 0
i⇐= i+ 1
�NLý ù/Â�Äç½ imaxõ#Y $#, ||r(i)|| ≤ ε||r(0)|| EÍ�[Ò¢°.
����ä-�á�Ä�° αdõ x�Ì¢°;��Äù αd�sq�´ÌX� ¿Fq�: (||αd|| ≤ ε)
Óù�Äç½� jmaxõ#Y£:[Ò¢°.�ò��ÙAÝ¢�xÒ_ù jmaxõ�ù÷¿PÌ $
#½èK(Hessian)±µ f ′′(x)õÂ��Û�÷¿�P �£½�°.
üxÍ�ªEPê�ªùÒ_�³� � ��³�I¨,÷¿GSþ:�°^¿è�ý° (d⇐=
r¿zA).�ù½ n�©½¶�êõm��æ©,� n¥��Ä�°�袥B&è�ý°.
α�GSù ‘0÷¿#��(divide-by-zero)’¡êõ�÷Ƚ�°.�,ùè�> x(0)�Ù �/�
�?Û&�²�M$# f� 2òyÛ�� �MúEÍ��f£½�°.;��EÍÌ#ùè�>
úxØ $#ÌÄ!¢�xÒ_úPÌ �©@£½�°.ó��EÍ���ªEPê�ª�/�Ü
�:¦¢NLý �I©½�°.
�è�;< B1�������4Ìú÷S �.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 65
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
B5.£££xxxªªªúúú���ÌÌÌ¢¢¢;;;���ýýýmmm���-ýýýüüü���òòò(Polak-Ribiere)üüüxxxÍÍÍ���ªªªEEEPPPêêê
sq�¥½ f�©,è�>� x�6,�ªEPê�ª�/Â�Ä� imax,�ªEPê�ª�¡ò´Ì
X� ε < 1�L,£xª�<7�yâ� σ0,£xª�/Â�Äç½� jmax,£xª¡ò´ÌX� ε < 1�
°.
i⇐= 0k ⇐= 0r ⇐= −f ′(x)Calculate a preconditionerM ≈ f ′′(x)s⇐= M−1rd⇐= sδnew ⇐= rT dδ0 ⇐= δnew
While i < imax andδnew > ε2δ0 doj ⇐= 0δd ⇐= dT dα⇐= −σ0
νprev ⇐= [f ′(x+ σ0d)]T dDo
ν ⇐= [f ′(x)]T dα⇐= α ν
νprev−ν
x⇐= x+ αdνprev ⇐= νj ⇐= j + 1
While j < jmax andα2δd > ε2
r ⇐= −f ′(x)δold ⇐= δnew
δmid ⇐= rT sCalculate a preconditionerM ≈ f ′′(x)s⇐= M−1rδnew ⇐= rT sβ ⇐= δnew−δmid
δold
k ⇐= k + 1If k = n or β ≤ 0
d⇐= sk ⇐= 0
elsed⇐= s+ βd
i⇐= i+ 1
�NLý ù/Â�Äç½ imaxõ#Y $#, ||r(i)|| ≤ ε||r(0)|| EÍ�[Ò¢°.
���£xª�Ä�° αdõ x�Ì¢°;��Äù αd�sq�´ÌX� ¿Fq�: (||αd|| ≤ ε)Ó
ù�Äç½� jmaxõ#Y£:[Ò¢°.�ò��ÙAÝ¢�xÒ_ù jmaxõ�ù÷¿PÌ �½
±£½�°.7�yâ σ0ùé 59� σú@A 6£xªú�Ì¢/�Ü����³G�PÌý°.Ý
± 3ê,�7�yâ�½¶úæ©SAüqb£½ê�°.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 66
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
m�-ýü�ò(Polak-Ribiere)β 7�yâ� δnew−δmid
δold¿ ��
rT(i+1)s(i+1)−rT
(i+1)s(i)
rT(i)s(i)
® �L, � ù Z¢rT(i+1)M
−1(r(i1)−r(i))
rT(i)M
−1r(i)�°.;�ý ±µ M� ¨\ j� AÙÒ(positive-definite)�êÀ s�õ �Ð�b ¢°.
;�ý��Û±µ�Í×õ���,ùI¦°.
� üxÍ �ª EPê �ªù m�-ýü�ò(Polak-Ribiere)β 7�yâ� ü½� :�° ^¿ è�ý°
(d⇐= r¿zA).�ù½ n�©½¶�êõm��æ©,� n¥��Ä�°�袥B&è�ý°.
üxÍ�ªEPê�ªù=��xØúC袰:;�ýõ£,��®�©�Mù�,�ä-�á(Newton-
Raphson)�ª, £xª(Secant method)Óù �Í� �ª, �ª�-ýõß(Fletcher-Reeves)Óù m�-ýü�
ò(Polak-Ribiere)#�xØ��Å.æ�tCèýxØú�©°j¢ºÍ�� °. (m�-ýü�ò(Polak-
Ribiere)õxØ �,ùsC#��� °.)
copyright c©Jonathan Richard Shewchuk and VisCom Commune 67
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
C.888¢¢¢���<<<
C1. Ax = b���©©©������òòòÍÍÍéééúúú///���ÜÜÜèèèÇÇÇ°°°
A�Âa��L�A �. x� Ax = bõ�T �>�6,�òÍé (é 3)õ/�Ü �>��L L, e�
¡ê¨��L �.� 8°ü��T¢°.
f(x+ e) =12(x+ e)TA(x+ e)− bT (x+ e) + c (by Equation 3)
=12xTAx+ eTAx+
12eTAe− bTx− bT e+ c (by symmetry of A)
=12xTAx− bTx+ c+ eT b+
12eTAe− bT e
= f(x) +12eTAe.
A�j�AÙÒ(positive-definite)�L 8,���¨ù 0�I¨?� e� �¨\j(positive)�°;
0�t x� fÂ/�Ü¢°.
C2.ÂÂÂaaa±±±µµµùùù n���������pppLLLîîî¯âââõõõ������°°°
qE±µ��/�¢ #�Lî¯â(eigenvector)õ��°.�õÝ �æ© det(A− λI)� λ�°¨é
���,�s@ 8,/�¢ #(|�)�\�©õ��°;�õ λA� �.±µ A−λAI�±µé� 0�
6,0�t"�±µ(singular)�t¿ A−λAI)v = 0 0�I¨¯â v���èU&©b¢°.Av = λAv�
t¿,�¯â v��¿Lî¯â�°.
?�Âa±µù n���pLî¯âõ��°.�õ�< �æ©, 4×4±µ�EÍ�©�,��
�¥úÃ�,�6,���¾�õ��±µ¿�<�¢��Ü�ë���3�£,�°.±µ A�/�
¢ #�Lî λA®Lî¯â võ��°�,ù�y�<ü}°. x1 = v/||v||�L 8��³æ£�õ
��°.t¿�p 6³æ£�õ������¯â x2, x3, x4õxØ � (� ¢¯â�ù��-Ùy!
�pÜõ�©üú½�°). X = [x1, x2, x3, x4]�L �. xi�A��p(orthonormal)�t¿ XTX = I�
6,XT = X−1�°.Z¢ i 6= 1�©, xTi Axi = xT
i Ax1 = xTi λAx1 = 0�t¿°üY�°.
XTAX =
xT
1
xT2
xT3
xT4
A[x1x2x3x4] =
xT1
xT2
xT3
xT4
[λAx1Ax2Ax3Ax4]
=
λA 0 0 000 B0
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
�:,B� 3× 3Âa±µ�°.B�W λBõ �Lî¯â wõ�Kb�¢°.�: wõ�¥«Ù�
� 0�L#"�Ù�� w®ô�¢ 4�Ù�¯â�L �.� 8°ü�<� °.
X−1AXw = XTAXw =
λA 0 0 000 B0
w = λBw
°è©, AXw = λBXw�6,0�t Xw� A�Lî¯â�ý°.Z¢, xT1 Xw = [1000]w = 0�t¿,
x1Y Xw��p�°.0�t/�¢����pLî¯â�U&¢°!
Âa ±µ� n �� �p Lî¯âõ ��°� 4Ì� ÌÎ ��Üý �Áù �,ª÷¿ �<ý°.æ�
��t,��� 3×3±µ (B®�ù)� 3���pLî¯âõ��°L�A �;��ú�� w1, w2, w3�
�L �.� 8Xw1,Xw2,Xw3� A�Lî¯â��6,X����A��p�t¿�p¯â�ú�p
¯â¿��(mapping) t¿�Lî¯â��è�p�°.0�t A� 4���pLî¯âõ��°.
C3.���üüüËËË���(Chebyshev)°°°¨ééé���///:::���
�üË�(Chebyshev)°¨éùé50Y�ùvÇ�/�Ü�/:�°.���°¨é�©æ [−1, 1]�4Ù
�t¾�� 1úJ�MêÀC¢ý°¨é�|�t©æ [−1, 1]��t�$�ò3¾���� �°¨
é��:[�°.
ò½ i��üË�°¨éù°üY�L,
Ti(ω) =12[(ω +
√ω2 + 1)i + (ω −
√ω2 − 1)i],
��©æ [−1, 1]�©°üY��vÇþ½�°.
Ti(ω) = cos(i cos−1 ω), − 1 ≤ ω ≤ 1.
�vÇ�t (�ýL�� 32�t),�üË�°¨é�°üY�ù"�ú �°�,úN½�°.
|Ti(ω) ≤ 1, − 1 ≤ ω ≤ 1
�ýL -1Y 1P��t�ò3�ô¢°:
Ti
(cos(κπi
))= (−1)k, k = 0, 1, · · · , i.
Ti� i��©�©æ [−1, 1]4��� Ti� i+ 1���(extrema)P��n °.�õ�q T5(ω)� 5
�©��� 32®�°.
copyright c©Jonathan Richard Shewchuk and VisCom Commune 69
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VCTR-2006-F-002 Ls�L��{��ªEPê�ª�Á (1 14:)-��{†,�ô�‡,î�|††,�êõ‡‡ b
îP¢EÍ¿°ü¥½õÃ�.
Pi(λ) =Ti(λmax+λmin−2λ
λmax−λmin)
Ti(λmax+λmin
λmax−λmin)
æ� ¥½� A�� [λmin, λmax]� © ±Ti(λmax+λmin
λmax−λmin)−1� ©æ 4�t �ô¢°. Pi(λ)� Pi(0) =
1���ÅuS&ú�T¢°.
�<ù�³¢�põPÌ¢°.ò½ i °¨é÷¿t Qi(0) = 1�8t©æ [λmin, λmax]�© PiÃ
° #ù °¨é Qi(λ)� U& � M�°.�õ �< � æ©,� ¢ °¨é� U&¢°L �A �; �
8 qjadnl [λmin, λmax]4�t Qi(λ) < Ti(λmax+lambdamin
λmax−λmin)−1�°.�,ùO Pi − Qi� λ = 0�t©õ
�°�,�°.0�t Pi −Qi� iò°¨é�8t/�¢ i+ 1��©õ��3ü�Ú,��Ý� ¢�
�°.� ¢?¿��© iò�üË�°¨é�é 50ú/:Ü¥ú�<£½�°.
D.���åååYYYCCC
°ü �[� ©, (°ô � {� ¢) � Û� AÝ¢ SÁú PÌ � Ùô�½> �¤� ¡ò� {°L
�A¢°.
1. (#�);�ýý�ªEPê�ªù;�ý±µ�ße�úU©ê�³ú��Müú�< �.
°è©,��ù 0�I¨\½ γ�©,;�ý±µ γMúPÌ �uù x(0), x(1), · · ·�;�ý±
µMúPÌ �uù³G�Yô�¥úÃ��.
2. (L�,� #�y¿Ï[C)Âa�6j�AÙÒ n× n±µ A�j¥ý Ax = b�©õu ²L
¢°L�A °.ݱ 3ê,xͽ½z�Aê:?<÷¿�ªEPê�ª�¢�r�#�
M�°.�:¾ê�L{úÆó¢Lî¯âÅA�#Í#t�¾ê�3±µ A���� d��t
¿°ôLî(eigenvalue)úN²s}° (Lî¯â�N²s�M�°).� #���Lî�v�
#|Äüq#Í#���N�G¢°.
¾êù[[¢P���t,¡�I^ ��t°üNLý�ú|v$ý3ü}°.
Choose an arbitrary starting pointx(0)
For i⇐= 0 to d− 1r(i) ⇐= b−Ax(i)
Remove an arbitrary eigenvalue from the list and call itλi
x(i+1) ⇐= x(i) + λ−1i r(i)
Lî��¥PÌü�EÍ�{÷6;[Òè�Lî@À(list)ùü3ý°.
(a) �NLý��[Ò£:, x(d)� Ax = b�©�ÿúÃ��. ¡(!¿??<ús�ª3�qÃ�.
2òÙ��½¶ú�²Ã�;Lî¯â9ú�ÿ�����Lîú$9xØ ²L©Ã�.
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(NLý��X|ú ��õ�[:÷¿z< �,��$|Å °;�3� 8����
<��Å¢½éúCè �.)
(b) �NLý�� dç��Äú�©AÝ¢©õü�� ��,� Ûù�¥u3ü� x(i)��
¢¢ ©� �«Ï ú �õ Ù£ ,�°.� �Ä :�° Lî @À�t qE Lîú x
Ø � ,� ^ù�õ @A � Â�� ªYú Cè© Ã�. (°è©,qE ¿½¿ Lîú P
̩b ��?)
(c) Ùô�½>�¤�¡ê��q(EÍ�NLý�ùqN3~��þ½���?
(d) Ùô�½>�¤�¡ê�ÝÂü�,ú��æ©��Äè��°@À�tqELîúx
Ø©b ��Â���ªúCè �. ¡(!¿�[C�»ù[C (b)�»Yô� °.
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÷÷÷LLL[[[¶¶¶
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