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Lesson 18Linear partial differential equations
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2
� ;I�LEZI�WIIR�XLEX�3()W�GER�FI�VIHYGIH�XS EPQSWX�FERHIH MR½RMXI�HMQIRWMSREPIUYEXMSRW�YWMRK�'LIF]WLIZ�ERH�YPXVEWTLIVMGEP�TSP]RSQMEPW
� 8LIWI�MR½RMXI�HMQIRWMSREP�IUYEXMSRW�GER�FI WSPZIH�EHETXMZIP] YWMRK�+MZIRW�VSXE�XMSRW�
8LI�I\EGX IVVSV�MR�VIWMHYEP MW�GEPGYPEXIH� [LMGL�GER�FI�YWIH�XS�HIGMHI�GSR�ZIVKIRGI
� 8LMW�PIGXYVI�[I�WII�LS[�XLIWI�MHIEW�GER�FI�YWIH�JSV�PMRIEV�4()W
� 8LI�VIWYPXMRK�QIXLSH�[SVOW�JSV�EVFMXVEV]�PMRIEV�4()W�SR�E�VIGXERKPI
� 8LI�½VWX�WXIT�MW�XS�HS JYRGXMSR�ETTVS\MQEXMSR�SR�E�WUYEVI
3
2D Function Approximation
4
� -R��(��[I�ETTVS\MQEXIH�JYRGXMSRW�YWMRK 'LIF]WLIZ�WIVMIW�
f(x) �n�1�
k=0
fkTk(x) = (T0(x), . . . , Tn�1(x))
�
��f0...
fn�1
�
��
*YRGXMSRW�EVI�MHIRXM½IH�[MXL ZIGXSVW
� -R��(��[I�YWI�E XIRWSV�TVSHYGX�SJ�'LIF]WLIZ�WIVMIW�
f(x, y) �n�1�
k=0
m�1�
j=0
fkjTk(x)Tj(y)
= (T0(x), . . . , Tn�1(x))
�
��f00 · · · f0(m�1)...
. . ....
f(n�1)0 · · · f(n�1)(m�1)
�
��
�
��T0(y)
...Tm�1(y)
�
��
*YRGXMSRW�EVI�MHIRXM½IH�[MXL QEXVMGIW
5
� -R��(��[I�ETTVS\MQEXIH�JYRGXMSRW�YWMRK 'LIF]WLIZ�WIVMIW�
f(x) �n�1�
k=0
fkTk(x) = (T0(x), . . . , Tn�1(x))
�
��f0...
fn�1
�
��
*YRGXMSRW�EVI�MHIRXM½IH�[MXL ZIGXSVW
� -R��(��[I�YWI�E XIRWSV�TVSHYGX�SJ�'LIF]WLIZ�WIVMIW�
f(x, y) �n�1�
k=0
m�1�
j=0
fkjTk(x)Tj(y)
= (T0(x), . . . , Tn�1(x))
�
��f00 · · · f0(m�1)...
. . ....
f(n�1)0 · · · f(n�1)(m�1)
�
��
�
��T0(y)
...Tm�1(y)
�
��
*YRGXMSRW�EVI�MHIRXM½IH�[MXL QEXVMGIW
6
–1 1
� -R��(��[I�GEPGYPEXIH�XLI�'LIF]WLIZ�WIVMIW�F]�IZEPYEXMRK f(x) EX�XLI 'LIF]WLIZTSMRXW xn�
6
–1 1
� -R��(��[I�GEPGYPEXIH�XLI�'LIF]WLIZ�WIVMIW�F]�IZEPYEXMRK f(x) EX�XLI 'LIF]WLIZTSMRXW xn�
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
y
x
� -R��(��[I�IZEPYEXI f(x, y) EX�E TVSHYGX�SJ�'LIF]WLIZ�TSMRXW xn,m := xn � x�m�
7
� -R��(��[I�LEZI
Cnf(xn) �
�
��f0...
fn�1
�
��
[LIVI Cn GER�MW�ZMI[IH�EW�ER n � n QEXVM\
� ;I�[MPP�WLS[�XLEX�MR��( [I�LEZI
Cnf(xn,m)C�m �
�
��f00 · · · f0(m�1)...
. . ....
f(n�1)0 · · · f(n�1)(m�1)
�
��
8
� -R��(��[I�LEZI
Cnf(xn) �
�
��f0...
fn�1
�
��
[LIVI Cn GER�MW�ZMI[IH�EW�ER n � n QEXVM\
� ;I�[MPP�WLS[�XLEX�MR��( [I�LEZI
Cnf(xn,m)C�m �
�
��f00 · · · f0(m�1)...
. . ....
f(n�1)0 · · · f(n�1)(m�1)
�
��
9
� *SV�ER]�½\IH y� [I�GER�I\TERH
f(x, y) =��
k=0
fk(y)Tk(x)
[LIVIfk(y) =
�2
�
� 1
�1
f(x, y)Tk(x)�1 � x2
x
� -X�JSPPS[W�XLEX�XLI�WQSSXLRIWW�SJ fk(y) MW�MRLIVMXIH�JVSQ f
� ;I�GER�XLYW�I\TERH
fk(y) =��
j=0
fkjTj(y)
10
� *SV�ER]�½\IH y� [I�GER�I\TERH
f(x, y) =��
k=0
fk(y)Tk(x)
[LIVIfk(y) =
�2
�
� 1
�1
f(x, y)Tk(x)�1 � x2
x
� -X�JSPPS[W�XLEX�XLI�WQSSXLRIWW�SJ fk(y) MW�MRLIVMXIH�JVSQ f
� ;I�GER�XLYW�I\TERH
fk(y) =��
j=0
fkjTj(y)
11
� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]
Cnf(xn, y) =:
�
��fn0 (y)...
fnn�1(y)
�
��
� ;I�GER�XLYW�I\TERH
Cmfnk (xm) =:
�
��fnm
k0...
fnmk(m�1)
�
��
� 8LMW�KMZIW�YW
Cnf(xn, xm)�C�m =
�
��fn0 (x�
m)...
fnn�1(x
�m)
�
��C�m
= Cm
�fn0 (xm), . . . , fn
n�1(xm)�
=
�
��
fnm00 · · · fnm
0(m�1)...
. . ....
fnm(n�1)0 · · · fnm
(n�1)(m�1)
�
��
12
� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]
Cnf(xn, y) =:
�
��fn0 (y)...
fnn�1(y)
�
��
� ;I�GER�XLYW�I\TERH
Cmfnk (xm) =:
�
��fnm
k0...
fnmk(m�1)
�
��
� 8LMW�KMZIW�YW
Cnf(xn, xm)�C�m =
�
��fn0 (x�
m)...
fnn�1(x
�m)
�
��C�m
= Cm
�fn0 (xm), . . . , fn
n�1(xm)�
=
�
��
fnm00 · · · fnm
0(m�1)...
. . ....
fnm(n�1)0 · · · fnm
(n�1)(m�1)
�
��
13
� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]
Cnf(xn, y) =:
�
��fn0 (y)...
fnn�1(y)
�
��
� ;I�GER�XLYW�I\TERH
Cmfnk (xm) =:
�
��fnm
k0...
fnmk(m�1)
�
��
� 8LMW�KMZIW�YW
Cnf(xn, xm)�C�m =
�
��fn0 (x�
m)...
fnn�1(x
�m)
�
��C�m
= Cm
�fn0 (xm), . . . , fn
n�1(xm)�
=
�
��
fnm00 · · · fnm
0(m�1)...
. . ....
fnm(n�1)0 · · · fnm
(n�1)(m�1)
�
��
14
� *SV�ER]�½\IH y� [I�I\TERH�RYQIVMGEPP]
Cnf(xn, y) =:
�
��fn0 (y)...
fnn�1(y)
�
��
� ;I�GER�XLYW�I\TERH
Cmfnk (xm) =:
�
��fnm
k0...
fnmk(m�1)
�
��
� 8LMW�KMZIW�YW
Cnf(xn, xm)�C�m =
�
��fn0 (x�
m)...
fnn�1(x
�m)
�
��C�m
= Cm
�fn0 (xm), . . . , fn
n�1(xm)�
=
�
��
fnm00 · · · fnm
0(m�1)...
. . ....
fnm(n�1)0 · · · fnm
(n�1)(m�1)
�
��
15
Partial derivatives
16
17
18
19
20
21
22
23
24
25
26
27
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.F.
1 0 0 0 …
0 1
40 0 …
- 23
0 1
60 …
0 - 38
0 1
8…
1
60 - 4
150 …
0 1
80 - 5
24…
Å Å Å Å Å
+
1 0 - 23
0 1
60 0 0 …
0 1
40 - 3
80 1
80 0 …
0 0 1
60 - 4
150 1
100 …
0 0 0 1
80 - 5
240 1
12…
Å Å Å Å Å Å Å Å Å
.F.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
=
28
Boundary conditions
29
� ;I�[ERX�XS�WSPZI�4()W�[MXL�FSYRHEV]�GSRHMXMSRW
� *SV�I\EQTPI� 4SMWWSR�[MXL�(MVMGLPIX XEOIW�XLI�JSVQ
u(�1, y) = g1(y), u(1, y) = g2(y)
u(x, �1) = h1(x), u(x, 1) = h2(x)
�u(x, y) = f(x, y)
� ;I�GER�VITLVEWI�XLMW�MR�XIVQW�SJ�XLI�GSIJ½GMIRXW�SJ u
30
� )\TERH
u(x, y) = (T0(x), T1(x), . . .)U
�
��T0(y)T1(y)
...
�
��
� 8LIR�[I�LEZI
�u(�1, y)u(1, y)
�=
�T0(�1) T1(�1) . . .T0(1) T1(1) . . .
�U
�
��T0(y)T1(y)
...
�
��
=
�1 �1 . . .1 1 . . .
�U
�
��T0(y)T1(y)
...
�
��
=: BU
�
��T0(y)T1(y)
...
�
��
31
� )\TERH
u(x, y) = (T0(x), T1(x), . . .)U
�
��T0(y)T1(y)
...
�
��
� 8LIR�[I�LEZI
�u(�1, y)u(1, y)
�=
�T0(�1) T1(�1) . . .T0(1) T1(1) . . .
�U
�
��T0(y)T1(y)
...
�
��
=
�1 �1 . . .1 1 . . .
�U
�
��T0(y)T1(y)
...
�
��
=: BU
�
��T0(y)T1(y)
...
�
��
32
� 8LYW�XLI�GSRHMXMSR �u(�1, y)u(1, y)
�=
�g1(y)g2(y)
�
FIGSQIWBU = G�
[LIVI
G = (g1, g2)
=
�
��g10 g20
g11 g21...
...
�
��
33
� 7MQMPEVP]� XLI�GSRHMXMSR �u(x, �1)u(x, 1)
�=
�h1(x)h2(x)
�
FIGSQIWUB� = H
[LIVI
H =�h1, h2
�
=
�
��h10 h20
h11 h21...
...
�
��
34
� -R�SXLIV�[SVHW� 4SMWWSR�[MXL�(MVMGLPIX
u(�1, y) = g1(y), u(1, y) = g2(y)
u(x, �1) = h1(x), u(x, 1) = h2(x)
�2u
�x2+
�2u
�y2= f(x, y)
FIGSQIW
BU = G�,
UB� = H,
D2US�0�2 + S0�2UD�
2 = F
35
� %RSXLIV�I\EQTPI� ,IPQLSPX^�[MXL�(MVMGLPIX
u(�1, y) = g1(y), u(1, y) = g2(y)
u(x, �1) = h1(x), u(x, 1) = h2(x)
�2u
�x2+
�2u
�y2+ k2u = f(x, y)
FIGSQIW
BU = G�,
UB� = H,�D2 + k2S�
0�2
�US�
0�2 + S0�2UD�2 = F
Sylvester matrix equations
36
37
� 'SRWMHIV�E 7]PZIWXIV�W�QEXVM\�IUYEXMSR
AX + XB = F
[LIVI A, B, F EVI n � n QEXVMGIW
� %R]�IUYEXMSRAXC + DXB = F
GER�FI�VIHYGIH�XS�XLMW�JSVQ�EWWYQMRK C ERH D EVI�MRZIVXMFPI�F]�MRZIVXMRK�SR�IEGLWMHI
D�1AX + XBC�1 = D�1FC�1
-J D ERH C EVI EPQSWX�FERHIH �EW�MW�XLI�GEWI�JSV�YW � XLIR�ETTP]MRK�XLI�MRZIVWIXS�E�ZIGXSV�MW�ER O(n) EPKSVMXLQ� WS�ETTP]MRK�XS�IEGL�GSPYQR�SJ A� B ERH FMW O
�n2
�STIVEXMSRW
38
� 'SRWMHIV�E 7]PZIWXIV�W�QEXVM\�IUYEXMSR
AX + XB = F
[LIVI A, B, F EVI n � n QEXVMGIW
� %R]�IUYEXMSRAXC + DXB = F
GER�FI�VIHYGIH�XS�XLMW�JSVQ�EWWYQMRK C ERH D EVI�MRZIVXMFPI�F]�MRZIVXMRK�SR�IEGLWMHI
D�1AX + XBC�1 = D�1FC�1
-J D ERH C EVI EPQSWX�FERHIH �EW�MW�XLI�GEWI�JSV�YW � XLIR�ETTP]MRK�XLI�MRZIVWIXS�E�ZIGXSV�MW�ER O(n) EPKSVMXLQ� WS�ETTP]MRK�XS�IEGL�GSPYQR�SJ A� B ERH FMW O
�n2
�STIVEXMSRW
39
� 'SRWMHIV�E 7]PZIWXIV�W�QEXVM\�IUYEXMSR
AX + XB = F
[LIVI A, B, F EVI n � n QEXVMGIW
� %R]�IUYEXMSRAXC + DXB = F
GER�FI�VIHYGIH�XS�XLMW�JSVQ�EWWYQMRK C ERH D EVI�MRZIVXMFPI�F]�MRZIVXMRK�SR�IEGLWMHI
D�1AX + XBC�1 = D�1FC�1
-J D ERH C EVI EPQSWX�FERHIH �EW�MW�XLI�GEWI�JSV�YW � XLIR�ETTP]MRK�XLI�MRZIVWIXS�E�ZIGXSV�MW�ER O(n) EPKSVMXLQ� WS�ETTP]MRK�XS�IEGL�GSPYQR�SJ A� B ERH FMW O
�n2
�STIVEXMSRW
40
� *SV�XLI 7]PZIWXIV�W�QEXVM\�IUYEXMSR
AX + XB = F
½VWX HMEKSREPM^I A = V �V �1 ERH B = W�W�1
� 2S[�PIXXMRK Y = V �1XW [I�LEZI
V �V �1X + XW�W�1 = V �Y W�1 + V Y �W�1 = F
� -R�SXLIV�[SVHW��Y + Y � = V �1FW
41
� *SV�XLI 7]PZIWXIV�W�QEXVM\�IUYEXMSR
AX + XB = F
½VWX HMEKSREPM^I A = V �V �1 ERH B = W�W�1
� 2S[�PIXXMRK Y = V �1XW [I�LEZI
V �V �1X + XW�W�1 = V �Y W�1 + V Y �W�1 = F
� -R�SXLIV�[SVHW��Y + Y � = V �1FW
42
� *SV�XLI 7]PZIWXIV�W�QEXVM\�IUYEXMSR
AX + XB = F
½VWX HMEKSREPM^I A = V �V �1 ERH B = W�W�1
� 2S[�PIXXMRK Y = V �1XW [I�LEZI
V �V �1X + XW�W�1 = V �Y W�1 + V Y �W�1 = F
� -R�SXLIV�[SVHW��Y + Y � = V �1FW
43
� &YX�[I�LEZI
�Y + Y � =
�
���1y00 · · · �1y0(n�1)
.... . .
...�ny(n�1)0 · · · �ny(n�1)(n�1)
�
�� +
�
���1y00 · · · �ny0(n�1)
.... . .
...�1y(n�1)0 · · · �ny(n�1)(n�1)
�
��
=
�
��(�1 + �1)y00 · · · (�1 + �n)y0(n�1)
.... . .
...(�n + �1)y(n�1)0 · · · (�n + �n)y(n�1)(n�1)
�
��
� 8LYW �Y + Y � = P MW�WSPZIH�F]
yij =pij
�i + �j
� 3RGI�[I�WSPZI �Y + Y � = V �1FW � [I�VIGSZIV X = V Y W�1
44
� &YX�[I�LEZI
�Y + Y � =
�
���1y00 · · · �1y0(n�1)
.... . .
...�ny(n�1)0 · · · �ny(n�1)(n�1)
�
�� +
�
���1y00 · · · �ny0(n�1)
.... . .
...�1y(n�1)0 · · · �ny(n�1)(n�1)
�
��
=
�
��(�1 + �1)y00 · · · (�1 + �n)y0(n�1)
.... . .
...(�n + �1)y(n�1)0 · · · (�n + �n)y(n�1)(n�1)
�
��
� 8LYW �Y + Y � = P MW�WSPZIH�F]
yij =pij
�i + �j
� 3RGI�[I�WSPZI �Y + Y � = V �1FW � [I�VIGSZIV X = V Y W�1
45
� &YX�[I�LEZI
�Y + Y � =
�
���1y00 · · · �1y0(n�1)
.... . .
...�ny(n�1)0 · · · �ny(n�1)(n�1)
�
�� +
�
���1y00 · · · �ny0(n�1)
.... . .
...�1y(n�1)0 · · · �ny(n�1)(n�1)
�
��
=
�
��(�1 + �1)y00 · · · (�1 + �n)y0(n�1)
.... . .
...(�n + �1)y(n�1)0 · · · (�n + �n)y(n�1)(n�1)
�
��
� 8LYW �Y + Y � = P MW�WSPZIH�F]
yij =pij
�i + �j
� 3RGI�[I�WSPZI �Y + Y � = V �1FW � [I�VIGSZIV X = V Y W�1
46
� &YX�[I�LEZI
�Y + Y � =
�
���1y00 · · · �1y0(n�1)
.... . .
...�ny(n�1)0 · · · �ny(n�1)(n�1)
�
�� +
�
���1y00 · · · �ny0(n�1)
.... . .
...�1y(n�1)0 · · · �ny(n�1)(n�1)
�
��
=
�
��(�1 + �1)y00 · · · (�1 + �n)y0(n�1)
.... . .
...(�n + �1)y(n�1)0 · · · (�n + �n)y(n�1)(n�1)
�
��
� 8LYW �Y + Y � = P MW�WSPZIH�F]
yij =pij
�i + �j
� 3RGI�[I�WSPZI �Y + Y � = V �1FW � [I�VIGSZIV X = V Y W�1
47
Complexity
� A = V �V �1 ERH B = W�W�1
� V �1FW
� 7SPZMRK �Y + Y � = V �1FW
� X = V Y W�1
O�n3
�O
�n3
�
O�n3
�
O�n3
�
O�n3
�
Reducing PDEs to Sylvester equations
48
1 0 - 23
0 1
6…
0 1
40 - 3
80 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
1 0 0 0 Å
0 1
40 0 Å
- 23
0 1
60 Å
0 - 38
0 1
8Å
… … … … Å
= F
K 1 -1 1 -1 …1 1 1 1 …
O.U =
U.
1 1-1 11 1
-1 1Å Å
=
GT
H
1 0 - 23
0 1
6…
0 1
40 - 3
80 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
1 0 0 0 Å
0 1
40 0 Å
- 23
0 1
60 Å
0 - 38
0 1
8Å
… … … … Å
= F
K 1 -1 1 -1 …1 1 1 1 …
O.U =
U.
1 1-1 11 1
-1 1Å Å
=
GT
H
Row reduction
1
2
�1 1
�1 1
�1
2
�1 1
�1 1
�
1
2
�1 �11 1
�1
2
�1 �11 1
�
1 0 - 23
0 1
6…
0 1
40 - 3
80 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
1 0 0 0 Å
0 1
40 0 Å
- 23
0 1
60 Å
0 - 38
0 1
8Å
… … … … Å
= F
=
=
PT
R
K 1 0 1 0 …0 1 0 1 …
O.U
U.
1 00 11 00 1Å Å
1 0 - 23
0 1
6…
0 1
40 - 3
80 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
1 0 0 0 Å
0 1
40 0 Å
- 23
0 1
60 Å
0 - 38
0 1
8Å
… … … … Å
= F
= PTK 1 0 1 0 …0 1 0 1 …
O.U
= RU.
1 00 11 00 1Å Å
�1
14
� �1
14
�D�
2 D�2
1 0 - 23
0 1
6…
0 1
40 - 3
80 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
1 0 0 0 Å
0 1
40 0 Å
- 23
0 1
60 Å
0 - 38
0 1
8Å
… … … … Å
= F
PTK 1 0 1 0 …0 1 0 1 …
O.U
= RU.
1 00 11 00 1Å Å
�1
14
� �1
14
�D�
2 D�2– –
=
= RU.
1 00 11 00 1Å Å
F
0 0 - 53
0 - 56
…
0 0 0 - 58
0 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
1 0 0 0 Å
0 1
40 0 Å
- 23
0 1
60 Å
0 - 38
0 1
8Å
… … … … Å
=
= RU.
1 00 11 00 1Å Å
F
0 0 - 53
0 - 56
…
0 0 0 - 58
0 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
1 0 0 0 Å
0 1
40 0 Å
- 23
0 1
60 Å
0 - 38
0 1
8Å
… … … … Å
= F
0 0 - 53
0 - 56
…
0 0 0 - 58
0 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å
- 53
0 1
60 Å
0 - 58
0 1
8Å
… … … … Å
= F
0 0 - 53
0 - 56
…
0 0 0 - 58
0 …
0 0 1
60 - 4
15…
0 0 0 1
80 …
Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
0 0 4 0 0 0 …0 0 0 6 0 0 …0 0 0 0 8 0 …0 0 0 0 0 10 …Å Å Å Å Å Å Å
.U.
0 0 0 0 Å0 0 0 0 Å
- 53
0 1
60 Å
0 - 58
0 1
8Å
… … … … Å
U =
�U11 U12
U21 U22
�2 � 2 2 � �
� � 2 � � �
= F
U =
�U11 U12
U21 U22
�2 � 2 2 � �
� � 2 � � �
- 53
0 - 56
…
0 - 58
0 …1
60 - 4
15…
0 1
80 …
Å Å Å Å
.U22.
4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
4 0 0 0 …0 6 0 0 …0 0 8 0 …0 0 0 10 …Å Å Å Å Å
.U22.
- 53
0 1
60 Å
0 - 58
0 1
8Å
… … … … Å
= F
U =
�U11 U12
U21 U22
�2 � 2 2 � �
� � 2 � � �
- 53
0 - 56
…
0 - 58
0 …1
60 - 4
15…
0 1
80 …
Å Å Å Å
.U22.
4 0 0 0 Å0 6 0 0 Å0 0 8 0 Å0 0 0 10 Å… … … … Å
+
4 0 0 0 …0 6 0 0 …0 0 8 0 …0 0 0 10 …Å Å Å Å Å
.U22.
- 53
0 1
60 Å
0 - 58
0 1
8Å
… … … … Å
Truncate to get Sylvester’s equation!
60
Recover solution
61
� ;I�LEZI�XLYW�GEPGYPEXIH U22� ERH�[I�NYWX�RIIH�XS�VIGSZIV U11, U12 ERH U21
� ;I�YWI�XLI FSYRHEV]�GSRHMXMSRW�
P� =
�1 0 1 0 · · ·0 1 0 1 · · ·
��U11 U12
U21 U22
�
=
�
���U11 +
�1 0 1 0 · · ·0 1 0 1 · · ·
�U21
U12 +
�1 0 1 0 · · ·0 1 0 1 · · ·
�U22
�
���
[VMXMRK P� = (P1, P2) [LIVI P1 MW 2 � 2 ERH P2 MW 2 � �� [I�KIX
U12 = P2 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U22
62
� ;I�LEZI�XLYW�GEPGYPEXIH U22� ERH�[I�NYWX�RIIH�XS�VIGSZIV U11, U12 ERH U21
� ;I�YWI�XLI FSYRHEV]�GSRHMXMSRW�
P� =
�1 0 1 0 · · ·0 1 0 1 · · ·
��U11 U12
U21 U22
�
=
�
���U11 +
�1 0 1 0 · · ·0 1 0 1 · · ·
�U21
U12 +
�1 0 1 0 · · ·0 1 0 1 · · ·
�U22
�
���
[VMXMRK P� = (P1, P2) [LIVI P1 MW 2 � 2 ERH P2 MW 2 � �� [I�KIX
U12 = P2 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U22
63
� ;I�LEZI�XLYW�GEPGYPEXIH U22� ERH�[I�NYWX�RIIH�XS�VIGSZIV U11, U12 ERH U21
� ;I�YWI�XLI FSYRHEV]�GSRHMXMSRW�
P� =
�1 0 1 0 · · ·0 1 0 1 · · ·
��U11 U12
U21 U22
�
=
�
���U11 +
�1 0 1 0 · · ·0 1 0 1 · · ·
�U21
U12 +
�1 0 1 0 · · ·0 1 0 1 · · ·
�U22
�
���
[VMXMRK P� = (P1, P2) [LIVI P1 MW 2 � 2 ERH P2 MW 2 � �� [I�KIX
U12 = P2 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U22
64
� 7MQMPEVP]� [I�KIX
R =
�U11 U12
U21 U22
��
����
1 00 11 0...
...
�
����
=
�
�������������
U11 + U12
�
����
1 00 11 0...
...
�
����
U21 + U22
�
����
1 00 11 0...
...
�
����
�
�������������
� 8LYW�[I�KIX�JSV R =
�R1
R2
�
U21 = R2 � U22
�
����
1 00 11 0...
...
�
����
65
� 7MQMPEVP]� [I�KIX
R =
�U11 U12
U21 U22
��
����
1 00 11 0...
...
�
����
=
�
�������������
U11 + U12
�
����
1 00 11 0...
...
�
����
U21 + U22
�
����
1 00 11 0...
...
�
����
�
�������������
� 8LYW�[I�KIX�JSV R =
�R1
R2
�
U21 = R2 � U22
�
����
1 00 11 0...
...
�
����
66
� 7MQMPEVP]� [I�KIX
R =
�U11 U12
U21 U22
��
����
1 00 11 0...
...
�
����
=
�
�������������
U11 + U12
�
����
1 00 11 0...
...
�
����
U21 + U22
�
����
1 00 11 0...
...
�
����
�
�������������
� 8LYW�[I�KIX�JSV R =
�R1
R2
�
U21 = R2 � U22
�
����
1 00 11 0...
...
�
����
67
� *MREPP]� [I�LEZI
U11 = P1 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U21
= R1 � U12
�
����
1 00 11 0...
...
�
����
� *SV�XLIWI�XS�QEXGL� [I�RIIH
P1 � BU21 = P1 � B�R2 � U22B�
�
= R1 ��P2 � BU22
�B� = R1 � U12B�
� 8LYW�[I�[ERXBR = P�B�
� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW
BH = G�B�
68
� *MREPP]� [I�LEZI
U11 = P1 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U21
= R1 � U12
�
����
1 00 11 0...
...
�
����
� *SV�XLIWI�XS�QEXGL� [I�RIIH
P1 � BU21 = P1 � B�R2 � U22B�
�
= R1 ��P2 � BU22
�B� = R1 � U12B�
� 8LYW�[I�[ERXBR = P�B�
� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW
BH = G�B�
69
� *MREPP]� [I�LEZI
U11 = P1 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U21
= R1 � U12
�
����
1 00 11 0...
...
�
����
� *SV�XLIWI�XS�QEXGL� [I�RIIH
P1 � BU21 = P1 � B�R2 � U22B�
�
= R1 ��P2 � BU22
�B� = R1 � U12B�
� 8LYW�[I�[ERXBR = P�B�
� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW
BH = G�B�
70
� *MREPP]� [I�LEZI
U11 = P1 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U21
= R1 � U12
�
����
1 00 11 0...
...
�
����
� *SV�XLIWI�XS�QEXGL� [I�RIIH
P1 � BU21 = P1 � B�R2 � U22B�
�
= R1 ��P2 � BU22
�B� = R1 � U12B�
� 8LYW�[I�[ERXBR = P�B�
� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW
BH = G�B�
71
� *MREPP]� [I�LEZI
U11 = P1 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U21
= R1 � U12
�
����
1 00 11 0...
...
�
����
� *SV�XLIWI�XS�QEXGL� [I�RIIH
P1 � BU21 = P1 � B�R2 � U22B�
�
= R1 ��P2 � BU22
�B� = R1 � U12B�
� 8LYW�[I�[ERXBR = P�B�
� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW
BH = G�B�
72
� *MREPP]� [I�LEZI
U11 = P1 ��
1 0 1 0 · · ·0 1 0 1 · · ·
�U21
= R1 � U12
�
����
1 00 11 0...
...
�
����
� *SV�XLIWI�XS�QEXGL� [I�RIIH
P1 � BU21 = P1 � B�R2 � U22B�
�
= R1 ��P2 � BU22
�B� = R1 � U12B�
� 8LYW�[I�[ERXBR = P�B�
� -R�XIVQW�SJ�XLI�SVMKMREP�ZEVMEFPIW�XLMW�FIGSQIW
BH = G�B�
73
� )\EQTPI� (MVMGLPIX
� ;I�LEZIB(h1, h2) =
�h1(�1) h2(�1)h1(1) h2(1)
�
ERH �g1
g2
�B� =
�g1(�1) g1(1)g2(�1) g2(1)
�
� 8LYW�GSQTEXMFMPMX]�MW�IUYMZEPIRX�XS GSRXMRYMX]�
h1(�1) =x��1
u(x, �1) =y��1
u(�1, y) = g1(�1)
h2(�1) =x��1
u(x, 1) =y�1
u(�1, y) = g1(1)
h1(1) =x�1
u(x, �1) =y��1
u(1, y) = g2(�1)
h2(1) =x�1
u(x, 1) =y�1
u(1, y) = g2(1)
74
� )\EQTPI� (MVMGLPIX
� ;I�LEZIB(h1, h2) =
�h1(�1) h2(�1)h1(1) h2(1)
�
ERH �g1
g2
�B� =
�g1(�1) g1(1)g2(�1) g2(1)
�
� 8LYW�GSQTEXMFMPMX]�MW�IUYMZEPIRX�XS GSRXMRYMX]�
h1(�1) =x��1
u(x, �1) =y��1
u(�1, y) = g1(�1)
h2(�1) =x��1
u(x, 1) =y�1
u(�1, y) = g1(1)
h1(1) =x�1
u(x, �1) =y��1
u(1, y) = g2(�1)
h2(1) =x�1
u(x, 1) =y�1
u(1, y) = g2(1)
75
� )\EQTPI� (MVMGLPIX
� ;I�LEZIB(h1, h2) =
�h1(�1) h2(�1)h1(1) h2(1)
�
ERH �g1
g2
�B� =
�g1(�1) g1(1)g2(�1) g2(1)
�
� 8LYW�GSQTEXMFMPMX]�MW�IUYMZEPIRX�XS GSRXMRYMX]�
h1(�1) =x��1
u(x, �1) =y��1
u(�1, y) = g1(�1)
h2(�1) =x��1
u(x, 1) =y�1
u(�1, y) = g1(1)
h1(1) =x�1
u(x, �1) =y��1
u(1, y) = g2(�1)
h2(1) =x�1
u(x, 1) =y�1
u(1, y) = g2(1)
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