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10

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1000 2000 3000 4000 5000 6000 7000 8000 9000

Run

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Tim

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Order of Matrix

Quadtreedgemm()

�3HKJ;L?M N�� n � � � *{L?U?U?HYU?J [\HK^wN c T L?U?HYb?MOc R NCiOi c;M{^`L?WY[OHKb?WYH R]P [\HYc;Ufc�U � c l NCM $ SqP WYWKNCU?J;N

0

50

100

150

200

250

300

1000 2000 3000 4000 5000 6000 7000 8000 9000

MF

lop/

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Order of Matrix

Quadtreedgemm()

Maximum

�IHYJ;L?MON � n � �<� � c;b � i{c T L"U?HKb?M c R N]i iOc;M{^`L?Wa[\HYb?WKH R�P [OHKc;Uwc;U � c l NCM $ S?P WKWYN]U?J;N

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Order of Matrix

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�IHKJ�L?MON � n � � � *|L?U"U?HKU?J [OHK^wN c T L?U?HKb"MOc R N]i iOc;M{^`L"WY[\HYb?WKH R�P [OHKc;Uwc;U � R [ P U?N

0

50

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350

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1000 1500 2000 2500 3000 3500 4000 4500

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Order of Matrix

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�IHKJ;L"MON�� n ��� � � c;b � i{c T L?U?HYb?MOc R NCiOiOc�M|^>L?WY[OHKb?WYH R�P [\HKc�Ufc;U � R [ P U?N

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Tim

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�IHYJ;L?M N � n � � � *{L?U?U?HYU?J [\HK^wN c T L?U?HYb?MOc R NCiOiOc�M|^>L?WY[OHKb?WYH R�P [\HKc�Ufc;U � U5[\NCMOb?M HKiON � ���

0

50

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Order of Matrix

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Order of Matrix

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� <�" � - "�" � $ 9�-!; "=< 7=2�=>- " 2 <6? 2 � 4 $'&8" - � -!; ;'5�7=- "%$ - � &�< � � 2 # � $ � -:�%� � - &8"%$ 7 9 ^ *P� � 2 � & 2�� < " 2 ? 3-� � < - � < 2 � � � 2�� 2 � 2 <6? # "�= $ 7 <�*6 ? VXW 9�-!; "%< 7 2='- " 2 <6?�% " � $�< 7 " � < 3 <6? -�42 " " < 9 V 362�� $ � 2 " 3!7 $ - " � " - # 2�42 " " * � � $�$ 7=7 < 7 - ? -� �

" � 2 � < 9 VHW 9�- ; "%< 7=2='- " 2 <6? 5 � 2 ? 3 � 2�� $'? � 7 < " - " 2 <6? � � 2 " � < 5 " & 2!� <!" 2 ? 3�2 � � $ 7 "9�- � < 7=- # � $ � -:�%( $ ; " 2 <6? 6*� &7�*]-`% ( $ ; " 2 <6? b *\[F^ * � 2�� <!" 2 ? 3 2 � 5 � $&� � 2 " � VXW 9�-!; "=< 7=2�=>- " 2 <6? � <6? � 2 ? 3!5�� -!7. - " 7 2 ; $ � 9 < 7 7 $ - � <6? � # $ " <6?�� " � $ � ; <6&�$ < 9 " � 2 �2� < 7 4 * � -�7�% ( $ ; " 2 <6? [8* 7 *� &^�� 2!� <!" 2 ? 3 2 �5 ?�? $ ; $ � � -!7 " <6? ?�<6? � 2 ? 3!5�� -!7 . - " 7=24; $ � *

� � 2 � <6? �!" . -4 $ � j�� �� g ong & 2!� <!" 2 ? 3 5 ? ?�$ ; $ � � -!7 " *�6 9 � � 2 � ­ %�" � $'? " � $ ? - " 5�7%-�� 2�� $ ? � 7 <!" - " 2 <!? 3 $ ?�$ 7=- "%$�� # " ­ � 2�� % 2 ? 9�-!; " % 24. & � 2 ; 2 " ��" & 2�� <!" " � $ . - " 7=24; $ �;� < " � - "�� � *I��DI ��� �"!���� ��4 *�( * /2,.4 (q4�� �A,�� �"/ !2��-��P F PL���

( $'$ -� � < " � $ ; < . & 2� $ 7 2 � � 5 $ � < 9 ( $ ; " 2 <6?�-8*Z7 - ?���0 *Z7�*576�576�: �ZH���@�J�D O N�� J"# < L\G�C���>AR

� � $ 9@5 ? ; " 2 <!? �0T - ?�� ­ - � & 7 $ � $'? "%$�� 2 ? 102 365�7 $ � [ *]- - ?�� [ *\0 � 7 $ � &�$ ; " 2�� $ ��" � -!7 $�& 5�7 $ ��"9@5 ? ; " 2 <6? -� - ?�� ��< ? <!" 7 5 ? 2 ? � & � -!; $!* 6 . & � $ . $ ? " 2 ? 3 " � $ . "=< 7=5 ? 2 ?`� & � -!; $ ;>- ? # $X��<!?�$9 2

# " 367=- � 5 -����" 7 $ & � - ;'2 ? 3 � # " W ��� 2�� $ - � � $ . # � 2 ? 3 V 2 ? - ?�<!" � $ 7�. - " 7 2 C�*�� - " 7 2 C . 5�� �" 2 & � 24;>- " 2 <6? � < 9 " � $ 9 < 7=. ��� ��� - ?�� � ��� -!7 $ � $ 7 " 5 � $ 9@5��09 < 7�-!;'; 5�. 5�� - " 2 ? 3 V- ?�� 5 & � - " 2 ? 3 � % # 5 "�" � $ " 7 $ �#5�2 7 $ $9C " 7%- . $ . < 7 " "=< -!;'; 5�. 5�� - "%$�& -!7 " 2 -� � 5�. � * � � 5 � %- "%$ . & < 7%- 7 " . - " 7=2 C 2 � & - � � $&� "=< $ -!; � -� 3 < 7=2 " � . 9 < 7 � "=< 7=2 ? 3 2 ? "%$ 7=. $&� 2 - "%$ 7 $ � 5�� " � *576�576�� � �IN���LMD0HIRS@W< RSN <��

� � $ -� 3 < 7=2 " � . � 9 < 7 " � $ 9@5 ? ; " 2 <6? �)T - ?�� ­ -!; " 5 -����" 7 $+" 5�7 ? V � % 7%- " � $ 7 " � - ? V %# $ ;>- 5 � $ 2 " 2 � $ - � 2 $ 7 "=< 5 & � - "%$ � 5 � 2 ? 3 V � *�6 " 2 � $ - � " "=< " 7%- ? � &�< � $ - . - " 7 2 C - 9 "%$ 7 " � $9�-!; "%< 7 2=>- " 2 <!? 2 � ; < . & � $ "=$!*576�576�� � D RSN ;=D R�N�� H���< J ! G�H$#� 24 $ . - " 7=2 C . 5�� " 2 & � 24;>- " 2 <6?@%6" � $ # - � $ ;>- � $�< 9 VXW 9�-!; "%< 7 2=>- " 2 <!? 2 � 5 ? 9 < � � $&� 2 ? & 7%-!; �" 24; $�% � $ 7 $�"%< - ? b b # � < ;54 � � $ $ ( $ ; " 2 <6?�0 *Z7�*\- � *P� � $ b b # � < ;54 2 � 9�- ; "%< 7 $&� 5 � 2 ? 3 " � $

� < <6& � 2 ? 1 24365�7 $2[8*! 6*H� � $ � < <6& � < 9 " � - " ; <`��$ � $ 7 $" 7=- ? � � - "=$&� 9 < 7�5 � $ � 2 " � � $ � $ � � < 7 ��$ 72 ?�� 2 ; $ � - ��� 5 363 $ � "%$�� # " � 2 � $ � 7 ® ^ *576�576 8 � NPO < LMDFCMGs< H�� L\G �?N H��� �ID OQCS<?L\G�� DFCEG�<?H

��< � &�$ ;'2 -�� ;>- � $ � 9 < 7 � - ?�� � 2 ? 3 2 ��$ ? " 2 " " . - " 7 2 ; $ � -!7 $ ?�$ ; $ � � -!7 " 2 ? ­ < 7 T # $ ;>-!5 � $" � $ " 7%- 7 $ �!" -!7 2 � $ 2 ? � % < 7=24362 ? -��!" < 7 � 5�7=2 ? 3 " � $ 9�-!; "%< 7 2=>- " 2 <!? * 6 9 " � $ 7 $ 2 � - ? 2 ��$ ?#" 2 " ". - " 7 2 C 2 ? � % 2 " � 2��� # $ 9�-!; "%< 7 $&� ; < 7=7 $ ; " ��" � 2 * $!*!% V W �� � � 2 " � 2�� a 5 � " # $�" 7 $ - "=$&� - �- ��$ ? � $ . - " 7 2 C�* � � $ 2 ��$'? " 2 " " . - " 7=2 ; $ � 2 ? V - 7 $ ?�$ � $ 7�9�- ; "%< 7 $&�@% a 5 � " . 5�� " 2 & �42 $&�@% - ?��" � $ . 5�� " 2 & �42 ;>- " 2 <!? -�43 < 7 2 " � . � <8$ � " 7 $ - " " � $ . - ��� & $ ; 2 -� ;'- � $ � � � $ $ ( $ ; " 2 <6?80 *Z7�* 0 � *9

� $ 7 < . - " 7=24; $ � � $ - � "%< � < . $ � 24. & � 2�� $&� � < 7 4 2 ?y­ - ?�� T 7 $ � 5�� " 2 ? 3�. < � " �!" 2 ? 2 ��$ ?#" 2 " "" 7=- ? � 9 < 7=. - " 2 <6? � ¬ T ¬�� �� � � � � � - ?�� ­�¬ � � � � ���� � � � � � * 6 ? # <!" � 9@5 ? ; " 2 <6? � % " � $. - " 7 2 C "=< # $ # $ 9�- ; "%< 7 $&� < 7 $ � 24. 2 ? - "=$&� 2 � -�47 $ - � " $ �42 . 2 ? - "%$&�@% � < ?�< ; � - ? 3 $ � . 5 � " # $. - ��$ "%< � < 7 � % - ?�� V 2 � " 7 2�� 2 -����" " � $�" � $ 2 ��$'? " 2 " " . - " 7 2 C�*� 5 $ "%< " � $ � - "�� 2 � � 2 � 2 "%$�� # " . - " 7 2 C . 5�� " 2 & �42 ;>- " 2 <!?�% � & $ ; 2 -��� 7 � " � - ?�� � - � " �

� 2 � 2 " 9@5 ? ; " 2 <6? � -!7 $ ?�$ $&��$�� "=< 2 ? 2 " 2 -� 2�= $ - ?�� ��$ ; < 7=- "%$ � & 7 <6&�$ 7 �!" - ?�� $�� ; 2 $'? " ��" � � $ $( $ ; " 2 <6? � 0 *Z7�*Z7 - ?��80 *Z7�*][ � *P� � $ 7 $ 2 � ? < � &�$ ;'2 -� . - " 7 2 C "%< 2 ? 2 " 2 -�42= $ 2 ? T < 7 ­ % � < � & $ ; 2 -�� 7 � " � � 2 � 2 " 9@5 ? ; " 2 <6? � -!7 $ 5 ? ?�$ ; $ � � -!7 " * � - � " � �82 � 2 " 9@5 ? ; " 2 <6? � -!7 $ -� � < 5 ?�?�$ ; $ � � -!7 " � 2 ? ; $V % W % - ?�� �� . 5 � " -� � - " � # $ ��$ ; < 7=- "%$�� - " " � $�$'?�� < 9 # < " � 9@5 ? ; " 2 <!? � *576�576�5 � O OPB�> B�J�DFCEG�H$# � - ? " -� 3 < 7=2 " � . � 9 < 7 VHW 9�-!; "=< 7=2�=>- " 2 <6? ��< ?�<!" -!; ;'5�. 5�� - "=$ V $9C8& �42 ; 2 " �!" *�� � 5 -��!" %$ - ; � V� 2 � � "=< 7 $�� 2 ? " � $ . $ . < 7 " 9 < 7 " � $ $ � 24. 2 ? - "=$&� & < 7 " 2 <6? < 9 � * Y < � $ � $ 7 %�"%< " -4 $- � � - ? " -!3 $�< 9 " � $ # � < ;54 $�� $n� ;'2 $ ? ;�" < 9�. - " 7=2 C . 5�� " 2 & �42 ;>- " 2 <!?�% V . 5 � " # $ -!;'; 5�. 5�� - "%$��$9C8& �42 ; 2 " �!" 9 < 7 " � $ �#5 - � " 7 $ $ . - " 7 2 C -� 3 < 7=2 " � . * ( < V 2 � -!;'; 5�. 5�� - "%$�� $DC8& � 24;'2 " ��" *

I��� �� ��� ���

� G � G��� � � $'$ ( $ ; " 2 <6?* 6* b � & 7 < �82 ��$ � " � $ 9@5 ? ; " 2 <6?���� ���������! 9 < 7 VHW 9�-!; "=< 7=2�=>- " 2 <6?� � 2 " � < 5 " & 2�� <!" 2 ? 3 � *�� � 2 � 9@5 ? ; " 2 <6?���< $ � ?�<!" ; 7 $ - "%$ - . - " 7=2 C V % # 5 " � "%< 7 $ � " � $ Y < 5 � $D�� < � ��$ 77 $ A $ ; " 2 <6? � 2 ? " � $ = $ 7 < $&� &�< 7 " 2 <6? � < 9 � - � 2 " # $ ; < . $ � W * � ( <�% " � 2 � 9@5 ? ; " 2 <6?& $ 7 9 < 7=. � 9 $ � $ 7 A <6& � " � - ? " � $ �#5 - � " 7 $'$ . - " 7 2 C -�43 < 7 2 " � . ��� 2 ; � -!;'; 5�. 5�� - "%$ � V $DC �& �42 ;'2 " �!" *� �xw*=Z-/.#" %%$ N/'�7�%0#$ '�N/7 :�#&<J@�'(%C&'(%�@E%.U:�-;:�<QNH.{<&$�.+%=?%�@(@�-�'('/X

9 -

0

500

1000

1500

2000

2500

1000 2000 3000 4000 5000 6000 7000 8000 9000

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in b

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Order of Matrix

dgeqrf()quadtree

1 24365�7 $H[ * 7¥¬ 1 � <6& ; < 5 ? " < 9 VHW 9�- ; "%< 7=2='- " 2 <6? -�43 < 7=2 " � . �� � $ � G � G��� � 2 # 7%- 7 " � - � ; < . & 2�� $�� 9@7 < . � ;'7%- " ; � <!? " � $ ( � 6 . -!; � 2 ?�$ � % # 5 " � 2 ? 4 $��"%< " � $ � - ?�� � " 5 ?�$���� � G ( �42 # 7=-!7 " * � � $ ( 5 ? �0$ 7 9 < 7=. - ? ; $ � 2 # 7%-!7 " & 7 < �82 ��$ � - � - ?��`�" 5 ?�$&� � G � G���2% - ?�� " � 2 � � - � 5 � $�� <6? " � $ ( 5 ? . -!; � 2 ?�$ � *

I���� ��� �K�I- / � ��4 *�(q!�� �� PL"!n*

576�76�: �)J�<?@ ;K< B�H C

� � $ A <6& ; < 5 ? " � 9 < 7 ��� ����� ���! - ?�� " � $ � 5�- � " 7 $'$ -�� 3 < 7 2 " � . - 7 $ 367%- & � $&� 2 ? 102 3 �5�7 $ [8*Z7�*>� � $ A <6& ; < 5 ? " 9 < 7 " � $ �#5 - �8" 7 $ $ . - " 7 2 C 2 �;� 2 3 � $ 7 # $ ;>-!5 � $ 2 " -!;'; 5�. 5�� - "%$ �V $9C8& �42 ;'2 " �!" % ��� 2� $ ��� ���������! � "%< 7 $ � a 5 � " " � $ Y < 5 � $ � < � � � $ ; "%< 7 �0� � $'$ ( $ ; " 2 <6? � [8*\[8*\[- ?��.[8* : � *

9 0

� � $ A <6& ; < 5 ? " � 9 < 7 ��� ���������! � $ 7 $ - &�& 7 < C 2 . - "=$&� - "�7 � ?�� 0 A <6& ��� -�7�% ( $ ; " 2 <6?B[8*\-8*! &^ �" � $ -!; " 5 -� ? 5�. # $ 7 < 9�A <6& � . - " # $ � � 243 � " ��" � 2 3 � $ 7 � 2 ? ; $ " � $ # � < ;54 Y < 5 � $ � < � � $ 7�-�43 <��7=2 " � .:2 ? ;'5�7 ��� < . $ $9C " 7=- A <6& ����� $'? ; < . & 5 " 2 ? 3)2 ? "=$ 7=. $&� 2 - "=$ 7 $ � 5�� " � � -�7�% ( $ ; " 2 <6?B[8*\-8*\-F^ * <6? � $ �#5 $ ?#" ��" %�" � $ A <6& � 9 < 7 ��� ���������! -!7 $ ; <6? � $ 7 �!- " 2�� $% # 5 " # " - 9�-!; "=< 7 . 5 ; � � $ � �" � - ? !*

� � $ A <6& ; < 5 ? " � 9 < 7 " � $ �#5 - � " 7 $ $ -�43 < 7=2 " � . � $ 7 $ " -��� 2 $&� # " $9C " 7=- ; < ��$ 2 ? " � $-�43 < 7=2 " � . *� <6? � $ �#5 $ ?#" ��" % 2 " � A <6& ; < 5 ? " � -!7 $ & 7 $ ;'2 � $ * 1 2 ?�� 2 ? 3 - ? $ �#5 - " 2 <!? "=< ; < 5 ? "" � $�? 5�. # $ 7 < 9 A <6& � 9 < 7 " � $ �#5 - � " 7 $'$ -� 3 < 7=2 " � . 2 � � 2 � ;'5�� " 2 ? & - 7 " # $ ;>- 5 � $ 2 " 2 � # - � $&�<6? � < . $�$ � - # < 7%- "%$ 7 $ ; 5�7=7 $'? ; $ 7 $ � - " 2 <6? � * < . & �42 ;'- " 2 ? 3 " � $ A <6& $ � 5�- " 2 <6? 9@5�7 " � $ 7 2 � " � $V 7 $+" 5�7 ?�$&� # " " � $ 9@5 ? ; " 2 <6? ­ � " � 2 � V � - � - & -!7 " 24;'5�� -!7 " 7 2 - ? 3!5�� -!7 & - "="=$ 7 ? "%< 2 "�" � - "-��� < ��� " � $�� XZ[�Y ��$ ; < 7=- " 2 <6? "%< - � < 2 � . - ? " . 5�� " 2 & �42 ;>- " 2 <!? � *576�76�� �$BID !�CELMNPN3� J #?< L\G C ��> � N LMRSBIR��5� %.� ;� � J #?< L\G C ��>

� � $ . -!; � 2 ?�$ � 5 � $�� "%< 7 5 ? " � $ � $�"=$ � " � -!7 $ ��$ � ; 7=2 # $&� 2 ? ( $ ; " 2 <6?�-8* 0 *� � $ 7=5 ? ? 2 ? 3 " 2 . $ � - ?�� .BA <6&� � < 9 ��� ���������! - ?�� " � $ �#5 - � " 7 $ $ -� 3 < 7=2 " � . <6? " � $9 < 5�7�. -!; � 2 ?�$ � -!7 $ 362�� $'? 2 ? 1 24365�7 $ � [8*\[ " � 7 < 5�3 � [8*� 3-8*( 2 ? ; $ 2 " ��< $ � ?�< " -!;'; 5�. 5�� - "%$ V $9C8& �42 ; 2 " �!" %���� ����� ���! . $ - � 5�7 $ � 9�- � "=$ 7 " � - ? " � $

�#5 - � " 7 $ $ -�� 3 < 7 2 " � . - � $9C8& $ ; "%$&� * � 5 " ��� $'? " � $& � <!" � 9 < 7 .BA <6&� � -!7 $ ; < . & -!7 $���% " � $�#5 - � " 7 $ $ -� 3 < 7=2 " � . $ ?�� � 5 & < 5 "%& $ 7 9 < 7=. 2 ? 3 � � ������� �! * 6 ? 9�-!; "&% ��� ���������! - & &�$ -!7 � "=<& $ 7 9 < 7=. �#5�2 "%$ & <8< 7 ��" 7 $ � - " 2�� $ "%< " � $ � < 7 4 2 " ��< $ � <6? -���09 < 5�7�. -!; � 2 ?�$ � % $ � $ ? <6? " � $( 5 ? . -!; � 2 ? $ � ��� $ 7 $ . - ? 5 9�- ; " 5 7 $ 7 � � � 2 # 7%- 7=2 $ � � $ 7 $ 5 � $&� *

� ? " � $��0< � $ 7 � -�� $'? 3 $ � 1 24365�7 $ � [8*\[ - ?��<[8* : � %6" � $ �#5 - � " 7 $'$ VHW 9�- ; "%< 7=2='- " 2 <6? & $ 7 �9 < 7 . � - " -BA <6& 7=- "%$ ?�$ - 7 ��" - � 3 < < � - � . - " 7 2 C . 5�� " 2 & � 24;>- " 2 <6? � 1 24365�7 $ � 08*� _- - ?�� 0 *! 30 �9 7

��� 24; � 2 � $ - � 2�!" $9C8& � -!2 ? $&� # " " � $ -!; ;'5�. 5�� - " 2 <6? < 9 V * V 2 � -!; ;'5�. 5�� - "=$&� " � 7 < 5�3 � . 5�� �" 2 & � 24;>- " 2 <6?@% - ?�� ; < . & -!7 2 ? 3 " � $ A <!& ; < 5 ? " � 2 ? 1 2 365 7 $�[8*Z7�%�" � $ � $ . 5�� " 2 & � 24;>- " 2 <6? � -!7 $. < � "�< 9 " � $ � < 7 4 " � - ")" � $ �#5 - �8" 7 $ $ -� 3 < 7=2 " � . ��< $ � * ( < 2 " 2 � $DC8&�$ ; "%$�� " � - "�" � 2 � VHW9�-!; "%< 7 2=>- " 2 <!? -� 3 < 7=2 " � . � < 5�� � & $ 7 9 < 7=. - " -2A <6& 7=- "%$�? $ -!7 ��" - � 3 <8<`� - � . - " 7=2 C . 5�� " 2 �& �42 ;>- " 2 <!? *� 5 " <6? " � $ ��< � $ 7 � -��� $ ? 3 $%#��� ��������� ��< $ � ?�< " $ � $ ? 7 $ -!; � -*�#5 -!7 "%$ 7 < 9 " � $.BA <6&� � & $ 7 9 < 7=. - ? ; $�< 9 ��� ����� �! *�6 " . - " # $ � �42 3 � " �!" 5 ? 9�-!2 7 ; < . & - 7=2 ? 3 " � $ A <6& 7=- "%$ �< 9 " � $ �#5 - � " 7 $'$ -�43 < 7 2 " � . - ?�� < 9 ��� ����� ���! � 2 ? ; $�" � $ �#5 - �8" 7 $ $ # $'?�$ � " � 9@7 < . " � $ A <6&`�7%- "%$ # < < � " 2 ? 3 . - " 7=2 C`� . 5�� " 2 & � 2 ;'- " 2 <!? -�� 3 < 7 2 " � . * � 5 ")" � $ � 2�� $ 7 $'? ; $ 2 ? &�$ 7 9 < 7 . - ? ; $�< 9

��� ����� �! - ?�� < 9 ��� ���������! � � < 5�� � ?�<!" # $ - 9�-!; "%< 7 < 909 < 5 7 *G � ? <!"%$�� 2 ? ( $ ; " 2 <6? [8* :�% ��� ���������! � - � ; < . & 2�� $�� 9@7 < . � < 5�7 ; $ ; <`��$ - ?�� � 2 ? 4 $��"%< " � $ . - ? 5 9�- ; " 5 7 $ 7 � � � � G ( <6? " � $���< � $ 7 � -�� $'? 3 $ � " � $ & $ 7 9 < 7=. - ? ; $ < 9 ��� ���������! . 2 3 � " # $ 2 . & 7 < � $&� <6? " � $ ��< � $ 7 � -��� $'? 3 $ � 2 " � . < 7 $ " 2�42 ? 3 < 7 . < 7 $ 2 ? 9 < 7=. - " 2 <6?- # < 5 " " � $ ��< � $ 7 � -��� $ ? 3 $!* 6 ? ; <6? " 7=- � "&% � 2 " � < 5 " - ? " " 5 ? 2 ? 3 " � $ �#5 - �8" 7 $ $ -�� 3 < 7 2 " � .& $ 7 9 < 7=. � � $ �� % � $ . <6? � " 7%- " 2 ? 3 2 " �2� " 7 $ ? 3 " � * 6 " � & $ 7 9 < 7=. - ? ; $ ��$ ; 7 $ - � $ � ; < . & -!7 $�� "=<. - " 7 2 C . 5�� " 2 & �42 ;'- " 2 <6? � ;>- ? # $�$DC & � -!2 ?�$&� # " " � $�? - " 5�7 $�< 9 " � $ VHW -�43 < 7=2 " � . ��� 24; � 2 �

� $ � � . $ . < 7 "�9@7 2 $ ?�� ��" " � - ? . - " 7 2 C . 5�� " 2 & � 24;>- " 2 <6? � " � 2 �P� -!. $ $DC8& � - ? - " 2 <6? ��< $ � ? <!" � < 7 49 < 7 " � $ . -!3 ? 2 " 5 � $ < 9 " � $+��$ ;'7 $ - � $ 2 ? &�$ 7 9 < 7 . - ? ; $ < 9 ��� ���������! ; < . & -!7 $�� "=< ��� ����� �! *G 3 - 2 ? " � $ � ; " - ?�$ � 102 365�7 $ � [8* ® - ?�� [8* b � � - � 2 ?#"=$ 7 $ � " 2 ? 3 7 $ � 5�� " � *�� 5 � " � 2�4 $ " � $

�0< � $ 7 � -��� $'? 3 $% ��� ���������! � - � ; < . & 2� $�� 9@7 < . � < 5 7=; $ ; < ��$ - ?�� � < � 5� $ 7 � 9@7 < . " � $� - . $ ��$ ;'7 $ - � $ 2 ? & $ 7 9 < 7=. - ? ; $!*/G � � <�% ; < . & -!7=2 ? 3 1 24365�7 $ � 0 *! _[ - ?���[8* b % 2 " 2 � ;�� $ -!7 " � - "� " 7=2 � 2 ? 32 � - - ? 2 � � 5 $ - " < 7 � $ 7 - 2 7 b ��$ � & 2 "%$�< 5�7�; < . &�$ ? � - " 2 ? 39 < 7�2 " � ( $ ; " 2 <6?�-8*Z7�*\- � *

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- ?�� " � $ " 2 ? ;'7 $ - � $ � � -!7 & ��" - 9 "%$ 7 < 7 � $ 7 0'[ 2 2 * � � $ � 2 7 " 5 -�� � . $ . < 7 " & 7 < # � $ . . - ? 249 $ � " �2 " � $ �49 - " -�� - 7=3 $ 7 < 7 ��$ 7�9 < 7 ��� ���������! " � - ? ��� ����� �! � ��� 2 ; � � " - 7 "%$�� � - � 2 ? 3 & 7 < # � $ . �- " < 7 ��$ 7 - [ 2 2 � # $ ;'-!5 � $ " � $ 7 $ 2 � <6? ��" <6?�$ . - " 7=2 C 9 < 7 ��� ���������! & �45 � % & $ 7 � - & � % � < . $� ; 7%- " ; �.� & -!; $ � � � ����� � ? $'$&� �+� & -!; $ 9 < 7 - " � $ - � "�" � 7 $'$ 9@5���� . - " 7=24; $ � *

� � $ & � <!" < 9 " � $ &�$ 7 9 < 7 . - ? ; $ < 9 " � $ �#5 - �8" 7 $ $ -�� 3 < 7 2 " � . <6? " � $ � ; " - ?�$ - &�& $ -!7 �"%< # $ - � "%$'& 9@5 ? ; " 2 <6?�% � 2 " � $ -!; � &�$ -4 - " - ? < 7 � $ 7 " � - " 2 � - &�< � $ 7 < 9 " � <�* Y - ?�� �42 ? 3" � $�& - ��� 2 ? 3 - &�& $ -!7 � "%< � - � $ -367 $ - "=$ 7�; < � " <6? " � $ � ; " - ?�$ � 2 " � " � $ VXW 9�-!; "=< 7=2�=>- " 2 <6?-�43 < 7=2 " � . * � 5 " " � $ < � $ 7%-��� &�$ 7 9 < 7 . - ? ; $�<6? " � $ � ; " - ?�$ 2 � 7 $ � &�$ ; " - # � $ * � < � " ?�< " - # �!" %" � $ &�$ 7 9 < 7 . - ? ; $ < 9 " � $ �#5 - � " 7 $'$ -�43 < 7=2 " � . ; <6? " 2 ? 5 $ � "=< 2 . & 7 < � $ 2 ? � & 2 "=$ < 9 " � $ 9�-!; "" � - " - ?�<!" � $ 7 � $ � $ � < 9 " � $ � 2 $ 7=-!7=; � " � & -!3!2 ? 3 � 2 � # $ 2 ? 3 5 � $&� * ( <�% a 5 � " - � � 2 " � . - " 7 2 C. 5�� " 2 & � 24;>- " 2 <6?�%�" � $ 7 $ � 5�� " � <6? " � $ � ; " - ?�$���$ . <6? � " 7=- "%$ " � $ � $ � � <6? < 9 " � $ �#5 - �8" 7 $ $. - " 7 2 Cj¬ fMj�� �E� � � jlj�m �� � g�� f ��� � � jlh�j � olf � j���j � ksr fMj�m jnm k ��� f_g j ����� olf �Wt g fMk �� j�� ����olk2��g � �)j��Ik �� ��� � t g fMk �_ ���M���]� k t+� j��v�_j�ksr m � olf_g � j�h � jlong �7olh2��0$ 7 9 < 7 . - ? ; $ <!? " � $ ( 5 ? � � 102 365�7 $ � [8* 9 - ?��E[8*! 2 9 < 7 " � $ � ? "%$ 7 & 7 2 � $<7 [ 2 - ?�� 102 3 �5�7 $ � [8*! - ?��8[8*� 3- � 2 � -� � < "%$ ��42 ? 3 % 2 ? � 2 � $ 7 $'? " � -3" � * 6 " 2 � ;�� $ - 7 " � - " " � $ 7 $ - 7 $ � " 7=2 � 2 ? 3& 7 < # � $ . � 9 < 7 ��� ����� �! - ?�� ��� ����� ���! - " < 7 ��$ 7 � " � - " -!7 $ - &�< � $ 7 < 9 " � < � � $ $ 102 3 �5�7 $ � 0 *! ® - ?��8[8*! 2 2 ? & -!7 " 2 ; 5�� -!7 � * � � $�&�$ 7 9 < 7 . - ? ; $ ��$ ;'7 $ - � $ - " " � $ � $�< 7 � $ 7 � 2 � ?�<!" - �

# - � 9 < 7 ��� ���������! # $ ;'-!5 � $ 2 " � < � $ 7=-�� & $ 7 9 < 7=. - ? ; $�<6? " � $ ( 5 ? � � a 5 � " - � 2 " � - � <!? " � $( � 6 � � 2 � m � olf � < � $ 7 " � - ? 2 " � - � 9 < 7 ��� ����� �! * Y < � $ � $ 7 % 5 ? � 2�4 $ " � $ ( � 6 � % � G � G���/<6?" � $ ( 5 ? � 2 � & - 7 "�< 9 " � $ ( 5 ? ��$ 7 9 < 7=. - ? ; $ � 2 # 7%-!7 " * � ?�$ � < 5�� � $9C8& $ ; " . - ? 5 9�-!; " 5�7 $ 7 �� 5 &�& � 2 $&� ; <`��$ "=< # $ - � $�� ; 2 $'? " - � & < � � 2 # � $% # 5 " " � $ � $ 7 $ � 5�� " � � < ?�<!" ��$ . <6? � " 7=- "%$ " � 2 � *� $ " " � $ �#5 - � " 7 $ $ -�43 < 7=2 " � . ��< $ � ?�<!" � $'$ . "=< " -4 $ - ? " &�$ 7 9 < 7 . - ? ; $ � 2 " - " -�� <!? " � $( 5 ? � % $ � $'? " � < 5�3 � " � $ VXW 9�-!; "=< 7=2�=>- " 2 <6? 2 � ?�<!" - � $n� ; 2 $ ?#" - � . - " 7 2 C . 5�� " 2 & � 2 ;'- " 2 <6? *

� � $ 7=-�� % " � $ � 5�- � " 7 $'$ -�43 < 7 2 " � . ��$ . <6? � " 7%- # ��" � - ?�� � $ � -�� � $ � $ � � < 9 " � $ . $ . < 7 "9 :

� 2 $ 7%- 7=; � " � 2 " � < 5 " $DC " 7%- < 7 � &�$ ;'2 -� 4 ?�< � � $&� 3 $�% - ?�� " � $�&�$ 7 9 < 7 . - ? ; $ < 9 " � $ VHW 9�-!; "=< 7 �2='- " 2 <6? 7 $ . - 2 ? � ;�� < � $�"%<" � $�& $ 7 9 < 7=. - ? ; $�< 9 " � $ . - " 7=2 C`� . 5�� " 2 & � 2 ;'- " 2 <6? -�43 < 7 2 " � . *

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Order of Matrix

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1 24365�7 $ [8*][�¬�� 5 ?�? 2 ? 3 " 24. $)< 9 5 ? 2 & 7 < ; $ � � < 7 VHW 9�-!; "%< 7 2=>- " 2 <!? <6? �0< � $ 7 � -�� $'? 3 $

0

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1000 2000 3000 4000 5000 6000 7000 8000 9000

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Order of Matrix

Quadtreedgeqrf()

Maximum

1 2 3!5�7 $H[8* :�¬ � A <6&� � < 905 ? 2 & 7 < ; $ � � < 7 VHW 9�- ; "%< 7=2='- " 2 <6? <6? �0< � $ 7 � -��� $'? 3 $9 b

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Order of Matrix

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1 2 365 7 $H[8* ® ¬ � 5 ?�? 2 ? 3 " 2 . $ < 905 ? 2 & 7 < ; $ � � < 7 VXW 9�-!; "=< 7=2�=>- " 2 <6? <6? � ; " - ?�$

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1000 1500 2000 2500 3000 3500 4000 4500

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Order of Matrix

Quadtreedgeqrf()

Maximum

1 24365�7 $ [8* b ¬� A <6&� � < 9 5 ? 2 & 7 < ; $ � � < 7 VXW 9�-!; "=< 7=2�=>- " 2 <6? <6? � ; " - ?�$9:9

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Order of Matrix

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1 2 3!5�7 $ [ * 9 ¬ � 5 ?�? 2 ? 3 " 24. $�< 9 5 ? 2 & 7 < ; $ � � < 7 VXW 9�-!; "%< 7 2=>- " 2 <!? <6? � ? "=$ 7 & 7=2 � $X7'[ 2

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1000 2000 3000 4000 5000 6000 7000 8000 9000

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Order of Matrix

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Maximum

102 365�7 $ [8*! 2 ¬ � A <6&� � < 9 5 ? 2 & 7 < ; $ � � < 7 VHW 9�- ; "%< 7=2='- " 2 <6? <6? � ? "%$ 7 & 7=2 � $H7 [ 2 2:2

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Order of Matrix

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1 24365�7 $ [8*! O¬ � 5 ?�? 2 ? 3 " 2 . $ < 905 ? 2 & 7 < ; $ � � < 7 VXW 9�-!; "=< 7=2�=>- " 2 <6? <!? � � " 7=- [ 2

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Order of Matrix

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Maximum

1 24365�7 $H[ *� _-�¬ � A <6& � < 9 5 ? 2 & 7 < ; $ � � < 7 VXW 9�-!; "=< 7=2�=>- " 2 <6? <6?�� � " 7%- [ 2 2

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