Post on 12-Mar-2020
Chapter 11 polar coordinate
Speaker: Lung-Sheng Chien
Book: Lloyd N. Trefethen, Spectral Methods in MATLAB
Discretization on unit diskConsider eigen-value problem on unit disk
2u uλ∆ = − with boundary condition ( )1, 0u r θ= =
We adopt polar coordinate cossin
x ry r
θθ
=⎧⎨ =⎩
, then
22
1 1rr ru u u u u
r r θθ λ∆ = + + = − ( ]0,1r∈ [ ]0,2θ π∈, on and
, and non-periodic Chebyshev grid in rUsually we take periodic Fourier grid in θ
[ ]0,1r∈1
Chebyshev grid in [ ]1,1x∈ −
12
xr +=
Chebyshev grid in [ ]0,1r∈
Observation: nodes are clustered near origin 0r = , for time evolution problem,
we need smaller time-step to maintain numerical stability.
[ ]1,1r∈ −2
( )( )
cos cos
sin sin
x r r
y r r
θ θ π
θ θ π
⎧ = = − +⎪⎨
= = − +⎪⎩( ) ( ), 0,0x y ≠
1 to 2 mapping ( ),r θ
Asymptotic behavior of spectrum of Chebyshev diff. matrix
2NDIn chapter 10, we have showed that spectrum of Chebyshev differential matrix
(second order) approximates
with eigenmode2
2
4k kπλ = −( ), 1 1, B.C. 1 0xxu u x uλ= − < < ± =
2ND1 4
max 0.048Nλ ≈ −Eigenvalue of is negative (real number) and2
2
4k kπλ = −Large eigenmode of does not approximate to2ND2
Since ppw is too small such that
resolution is not enough
Mode N is spurious and localized near boundaries 1x = ±
Grid distribution
[ ]0,1r∈ [ ]1,1r∈ −1 2
Preliminary: Chebyshev node and diff. matrix [1]
Consider 1N + Chebyshev node on [ ]1,1− , cosjjxNπ⎛ ⎞= ⎜ ⎟
⎝ ⎠for 0,1,2, ,j N=
x0x1x2x3x/ 2NxNx
1− 1
6N =
0 1x =1x2x3 0x =4x5x61 x− =x
Even case:
Uniform division in arc
Odd case:
0 1x =1x2x03x4x51 x− =x
5N =
Preliminary: Chebyshev node and diff. matrix [2]
Given 1N + Chebyshev nodes , cosjjxNπ⎛ ⎞= ⎜ ⎟
⎝ ⎠and corresponding function value jv
We can construct a unique polynomial of degree N , called ( ) ( )0
N
j jj
p x v S x=
=∑
( ) 1 0 j k
k jS x
k j=⎧
= ⎨ ≠⎩is a basis.
( ) ( ) ( )1
0 0
N NN
i j j i ij jj j
p x v S x D v= =
′ = =∑ ∑ where differential matrix ( )NN ijD D is expressed as
2
002 1,
6N ND +=
22 1,6
NNN
ND += − ( )2
,2 1
jNjj
j
xD
x−
=−
for 1,2, , 1j N= −
( )1,
i jN iij
j i j
cDc x x
+−=
− for , , 0,1,2, ,i j i j N≠ = where2 0,1 otherwisei
i Nc
=⎧= ⎨⎩
with identity 0,
NN Nii ij
j j iD D
= ≠
= − ∑
Second derivative matrix is 2N N ND D D= ⋅
Preliminary: Chebyshev node and diff. matrix [3]
Let ( )p x be the unique polynomial of degree with N≤ ( )1 0p ± = and ( )j jp x v=
define 0 j N≤ ≤( )j jw p x′′= and ( )j jz p x′= for
, then impose B.C. We abbreviate 2Nw D v= ⋅ ( )1 0p ± = ,that is, 0 0Nv v= =
In order to keep solvability, we neglect 0 Nw w= ( )2 2 1: 1,1: 1N ND D N N= − −,that is,
0w
1w
2w
1Nw −
Nw
0v
1v
2v
1Nv −
Nv
zero
2ND
neglect
neglect
=
zero
( )1: 1,1: 1N ND D N N= − −Similarly, we also modify differential matrix as
Preliminary: DFT [1]
Given a set of data point { }1 2, , , NNv v v R∈ with 2N m= is even,
2hNπ
=
Then DFT formula for { }jv
( )1
2ˆ expN
k j jj
v v ikxNπ
=
= −∑ for , 1, , 1,k m m m m= − − + −
( )1
1 ˆ exp2
m
j k jk m
v v ikxπ =− +
= ∑ for 1,2, ,j N=
Definition: band-limit interpolant of functionδ − , is periodic sinc function ( )NS x
( ) ( )( ) ( )
( )( )
sin / sin12 2 / tan / 2 2 tan / 2
mikx
Nk m
x h mxhS x P eh x m xπ
π π=−
= =∑
If we write 1
N
j k j kk
v v vδ δ−=
= = ∗∑ , then ( ) ( )1
1 ˆ2
m Nikx
k k N kk m k
p x P e v v S x xπ =− =
= = −∑ ∑
Also derivative is according to ( ) ( ) ( )1
1
N
j j k N j kk
w p x v S x x=
′= = −∑
Preliminary: DFT [2]
Direct computation of derivative of ( ) ( )( ) ( )
sin /2 / tan / 2N
x hS x
h xπ
π=
( ) ( )
, we have
( )
( ) ( )1
0 0 mod
1 1 cot , 0 mod2 2
jN j
j NS x jh j N
⎧ =⎪= ⎨ ⎛ ⎞− ≠⎜ ⎟⎪
⎝ ⎠⎩
( ) ( )( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
1 1 1 15 5 5 5
1 1 1 15 5 5 5
1 1 1 1 15 5 5 5 5 5
1 1 1 15 5 5 5
1 1 1 15 5 5 5
0 2 3 4
0 2 3
2 0 2
3 2 0
4 3 2 0
j k
S h S h S h S h
S h S h S h S h
D S x x S h S h S h S h
S h S h S h S h
S h S h S h S h
⎛ ⎞− − − −⎜ ⎟
− − −⎜ ⎟⎜ ⎟
= − = − −⎜ ⎟⎜ ⎟−⎜ ⎟⎜ ⎟⎝ ⎠
Example:
is a Toeplitz matrix.
Second derivative is ( ) ( )( )
( )( ) ( )
2
22
2
1 0 mod 6 3
1, 0 mod
2sin / 2
jN j
j Nh
S xj N
jh
π⎧− − =⎪⎪= ⎨ −⎪− ≠⎪⎩
Preliminary: DFT [3]
( ) ( )( )
( )( ) ( )
2
22
2
1 0 mod 6 3
1, 0 mod
2sin / 2
jN j
j Nh
S xj N
jh
π⎧− − =⎪⎪= ⎨ −⎪− ≠⎪⎩
For second derivative operation ( ) ( ) ( ) ( )2 2
1 1,
N N
j j k N j k N kk k
w p x v S x x D j k v= =
′′= = − ≡∑ ∑
second diff. matrix is explicitly defined by using Toeplitz matrix (command in MATLAB)
( ) ( )( )
( )( )
( )( )
( )( )( )
2
22:2
22 22 2
2
2
2
csc 2 / 2 / 2csc / 2 / 2
11 ,1 6 3 2sin 1: 1 / 26 3
csc / 2 / 2csc 2 / 2 / 2
N
N N j k
hh
D S x x toeplitzh N h
hh
h
ππ
⎛ ⎞⎜ ⎟−⎜ ⎟⎜ ⎟ ⎛ ⎞−⎜ ⎟ ⎜ ⎟= − = = − −⎜ ⎟ ⎜ ⎟− − −⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠
Fornberg’s idea : extend radius to negative image [1]
[ ]1,1r∈ − [ ]0, 2θ π∈( ), x y unit disk∈ and
5rN = (odd): to avoid singularity of coordinate transformation 0r =1
( ) ( )( )( ), , mod 2u r u rθ θ π π= − +6Nθ = (even): to keep symmetry condition2
θ
r
01 r=
1r
2r
3r
4r
51rNr r− = =
0θ 1θ 2θ 3θ 4θ 5θ 6θ0 π 2Nθ
θ π=
0r =
coordinater θ− x y− coordinate
y1θ2θ
3θ
4θ 5θ
6θ x
In general
θ
r
01 r=
1r
kr
1kr +
1rNr −
1rNr− =
0θ 1θ 2θ mθ 1mθ +
0 π 2Nθθ π=
0r =
r θ− coordinate
2r
1Nθθ −
x
y
1θ
2θ
mθ
1mθ +
x y− coordinate
Nθθ
1Nθθ −
2 1rN k= + 2N mθis odd, and is even, then=
Fornberg’s idea : extend radius to negative image [2]
: non-unifr∆ 2Nθ
πθ∆ =
1 : 1 1i rr i r i N
and
= − ∆ ≤ ≤ −
: 1j j j Nθθ θ= ∆ ≤ ≤( )1rN Nθ−Active variable , total number is
Redundancy in coordinate transformation [1]
( ) ( )( )( ) ( ), , , mod 2 ,r r x yθ θ π π− + ↔2 to 1 mapping
θ
r
01 r=
1r
2r
3r
4r
51rNr r− = =
0θ 1θ 2θ 3θ 4θ 5θ 6θ0 π 2Nθ
θ π=
x
y1θ2θ
3θ
4θ 5θ
6θ
x y− coordinatecoordinater θ−
redundant
0r =
y1θ2θ
3θ
4θ 5θ
6θ x
Redundancy in coordinate transformation [2]
( ) ( )( )( ) ( ), , , mod 2 ,r r x yθ θ π π− + ↔2 to 1 mapping
x y− coordinatecoordinater θ−y
1θ2θ
3θ
4θ 5θ
6θ
θ
r
01 r=
1r
2r
3r
4r
51rNr r− = =
0θ 1θ 2θ 3θ 4θ 5θ 6θ0 π 2Nθ
θ π=
x
0r =
y1θ2θ
3θ
4θ 5θ
6θ
redundant
x
8.5
2.6180
-0.7236
0.3820
-0.2764
0.5
-10.4721
-1.1708
2
-0.8944
0.6180
-1.1056
2.8944
-2
-0.1708
1.618
-0.8944
1.5279
-1.5279
0.8944
-1.6180
0.1708
2
-2.8944
1.1056
-0.618
0.8944
-2
1.1708
10.4721
-0.5
0.2764
-0.382
0.7236
-2.618
-8.5
rND =
0r
0r
1r
1r
2r
2r
3r
3r
4r
4r
5r
5r
Redundancy in coordinate transformation [3]2 1 5rN k= + = is odd, then Chebyshev differential matrix
rND is expressed as
rND =
-1.1708
2
-0.8944
0.6180
-2
-0.1708
1.618
-0.8944
0.8944
-1.6180
0.1708
2
-0.618
0.8944
-2
1.1708
neglect
Redundancy in coordinate transformation [4]
( ) ( ), ,N Ni j N i N jD D
− −= −Symmetry property of Chebyshev differential matrix :
-1.1708
2
-0.8944
0.6180
-2
-0.1708
1.618
-0.8944
0.8944
-1.6180
0.1708
2
-0.618
0.8944
-2
1.1708
rND =
-1.1708
2
-0.8944
0.6180
-2
-0.1708
1.618
-0.8944
0.8944
-1.6180
0.1708
2
-0.618
0.8944
-2
1.1708
-1.1708
2
-2
-0.1708
Permute column by
r
TND P =
( )1, 2, 4,3P =
is symmetric
0.8944
-1.6180
-0.618
0.8944
1E =
2E =
( )1
21 2
3
4
uu
E Euu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
is NOT symmetric
is faster ?
41.6
21.2859
-1.8472
0.7141
-0.9528
-8
-68.3607
-31.5331
7.3167
-1.9056
2.2111
17.5724
40.8276
12.6833
-10.0669
5.7889
-3.6944
-23.6393
-23.6393
-3.6944
5.7889
-10.0669
12.6833
40.8276
17.5724
2.2111
-1.9056
7.3167
-31.5331
-68.3607
-8
-0.9528
0.7141
-1.8472
21.2859
41.6
2r r rN N ND D D= ⋅ =
0r
0r
1r
1r
2r
2r
3r
3r
4r
4r
5r
5r
Redundancy in coordinate transformation [5]2 1 5rN k= + = is odd, then Chebyshev differential matrix 2
rND is expressed as
2rND =
-31.5331
7.3167
-1.9056
2.2111
12.6833
-10.0669
5.7889
-3.6944
-3.6944
5.7889
-10.0669
12.6833
2.2111
-1.9056
7.3167
-31.5331
neglect
Redundancy in coordinate transformation [6]
( ) ( )2 2
, ,N Ni j N i N jD D
− −=Symmetry property of Chebyshev differential matrix :
2rND =
-31.5331
7.3167
-1.9056
2.2111
12.6833
-10.0669
5.7889
-3.6944
-3.6944
5.7889
-10.0669
12.6833
2.2111
-1.9056
7.3167
-31.5331-31.5331
7.3167
12.6833
-10.06691D = is NOT sym.
Permute column by ( )1, 2, 4,3P =-3.6944
5.7889
2.2111
-1.9056is NOT sym.
2D =
2r
TND P =
-31.5331
7.3167
-1.9056
2.2111
12.6833
-10.0669
5.7889
-3.6944
-3.6944
5.7889
-10.0669
12.6833
2.2111
-1.9056
7.3167
-31.5331
Row-major indexing: remove redundancy [1]
( ),ij i ju u r θ
( ) ( )( ),1 ,2 , , 1 , 2 ,, , , , , , ,T
i i i i m i m i m i Nu u u u u u uθ+ +
1 1ri N j Nθ≤ ≤≤ ≤ − 1Define active variable for and
total number of active variables is 2 6 12k Nθ⋅ = ⋅ =11
12
13
uuu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
14
15
16
uuu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1
NOT ( )1 4 6 24rN Nθ− ⋅ = ⋅ = 2
31u =
θ
r
01 r=
1r
2r
3r
4r
51rNr r− = =
0θ 1θ 2θ 3θ 4θ 5θ 6θ0 π 2Nθ
θ π=
0r =
4
51 2 3 4 5 67 8 9 10 11 12
6
Index order21
22
23
uuu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
24
25
26
uuu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
78redundant 9
2u =10
1112
Index order
2 1rN k= + is odd, and
Row-major indexing: remove redundancy [2]
0 j k≤ ≤ , then 1 0k j k jr r− + ++ =
cos : 0i rr
ir i NNπ⎛ ⎞
= ≤ ≤⎜ ⎟⎝ ⎠
suppose since
( ) ( ) ( )2 11
1cos cos cosrN k
k j k jr r r
k j k j k jr r
N N Nπ π π
π = +− + +
⎛ ⎞ ⎛ ⎞− − + += = − − ⎯⎯⎯⎯→− = −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
2 ,N mθ
π πθ∆ = = : 1j j j Nθθ θ= ∆ ≤ ≤2N mθ = is even, and
( )( )mod 2m j jθ π π θ+ + =Hence for andj m jθ π θ ++ =1 ,j m≤ ≤
0 1r =1rkr01kr +1rNr −1rNr− =
rk jr −1k jr + +
1 0k kr r ++ =
1 0k j k jr r− + ++ =
1 1 0rNr r− + =
Row-major indexing: remove redundancy [3]
( ) ( )( )( )1 1, , mod 2k i j k i ju r u rθ θ π π+ + + += − +From symmetry condition, we have
( ) ( )1 , ,k i j k i m ju r u rθ θ+ + − +=
0 i k≤ ≤for 0 j m≤ ≤ , symmetry condition impliesand( ) ( )1 , ,k i m j k i ju r u rθ θ+ + + −=
Therefore, we have two important relationships
1, , 1,
2, 1, 2,
1, 1, ,
11
1r
k j k m j m j
k j k m j m j
N j m j k m j
u u uu u u
u u u
+ + +
+ − + +
− + +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
1
1, , 1,
2, 1, 2,
1, 1, ,
11
1r
k m j k j j
k m j k j j
N m j j k j
u u uu u u
u u u
+ +
+ + −
− +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠
2
θ
r
01 r=
1r
2r
31 r− =
00 θ= 1θ 2θ 3θ π=
Kronecker product [1]
1 2 3
4 5 6
11
12
13
uuu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
21
22
23
uuu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1u =
2u =
1
2( ),ij i ju u r θ
1 2i≤ ≤ 1 3
Define active variable 3
4j≤ ≤andfor
56
Separation of variable: assume matrix A acts on r-dir and matrix B acts on dirθ −
( )( )( )
1 1,11 12
2,21 222
,
,
j j
jj
r ua aAu
ua ar
θ
θ
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ = ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠is independent of j
( )( )( )( )
1 11 12 13 ,1
2 21 22 23 ,
3 31 32 33 ,3
,,,
i i
i i
i i
r b b b uBu r b b b u
r b b b u
θθθ
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
i
Let 1 6
2
uX R
u⎛ ⎞
= ∈⎜ ⎟⎝ ⎠
be row-major
index of active variable 2 is independent of
Kronecker product [2]
11 12 6 6
21 22
a B a BA B R
a B a B×⎛ ⎞
⊕ ≡ ∈⎜ ⎟⎝ ⎠
Kronecker product is defined by
( ) 11 12 1 11 1 12 2
21 22 2 21 1 22 2
a B a B u a Bu a BuA B X
a B a B u a Bu a Bu+⎛ ⎞⎛ ⎞ ⎛ ⎞
⊕ = =⎜ ⎟⎜ ⎟ ⎜ ⎟+⎝ ⎠⎝ ⎠ ⎝ ⎠
Case 1: A I=
( )( )( )( )
1 11 12 13 ,1
2 21 22 23 ,2
3 31 32 33 ,3
,,,
i i
i i
i i
r b b b uBu r b b b u
r b b b u
θθθ
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
( ) 1
2
BuI B X
Bu⎛ ⎞
⊕ = ⎜ ⎟⎝ ⎠
Case 2: B I=
( )
11 21
11 12 12 22
13 2311 1 12 2
21 1 22 2 11 21
21 12 22 22
13 23
u ua u a u
u ua u a uA I X
a u a u u ua u a u
u u
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟+⎜ ⎟⎜ ⎟ ⎜ ⎟
⎜ ⎟ ⎜ ⎟⎜ ⎟+⎛ ⎞ ⎝ ⎠ ⎝ ⎠⎜ ⎟⊕ = =⎜ ⎟+ ⎜ ⎟⎛ ⎞ ⎛ ⎞⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟+⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠
( )( )( )
1 1,11 12
2,21 221
,
,
j j
jj
r ua aAu
ua ar
θ
θ
⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ = ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠
Kronecker product [3]
Case 3: permutation, if permute 2 3θ θ↔
θ
r
01 r=
1r
2r
31 r− =
00 θ= 1θ 2θ 3θ π=
1 2 3
4 5 6θ
r
01 r=
1r
2r
31 r− =
00 θ= 1θ 2θ π=3θ
1 2 3
4 5 6
11
12
13
uuu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
21
22
23
uuu
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
11
13 1
12
uu P uu
θ
⎛ ⎞⎜ ⎟ = ⋅⎜ ⎟⎜ ⎟⎝ ⎠
21
23 2
22
uu P uu
θ
⎛ ⎞⎜ ⎟ = ⋅⎜ ⎟⎜ ⎟⎝ ⎠
( )1,3, 2Pθ = 1
2
uX
u⎛ ⎞
= =⎜ ⎟⎝ ⎠
1
2
uX
u⎛ ⎞
= =⎜ ⎟⎝ ⎠
( ) 1
2
P uI P X X
P uθ
θθ
⎛ ⎞⊕ = =⎜ ⎟
⎝ ⎠
Kronecker product [4]
Case 4: ( )1 2,A diag a a=
( ) 1 1
2 2
a BuA B X
a Bu⎛ ⎞
⊕ = ⎜ ⎟⎝ ⎠
( )( )( )( )
1 1 11 12 13 1,1
1 1 2 1 21 22 23 1,2
1 3 31 32 33 1,3
,,,
r b b b ua Bu r a b b b u
r b b b u
θθθ
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
( )( )( )( )
2 1 11 12 13 2,1
2 2 2 2 21 22 23 2,2
2 3 31 32 33 2,3
,,,
r b b b ua Bu r a b b b u
r b b b u
θθθ
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟
⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠2
2
1 1 rr ru u u ur r θθ λ+ + = −
2,2 2 2
1 2
1 1 1, , , Nk
diag Dr r r θθ
⎛ ⎞⊕⎜ ⎟
⎝ ⎠
1i =
2i =
22
1 1rr ru u u u
r r θθ λ+ + = − , on [ ]1,1r∈ − [and ]0,2θ π∈
2 1rN k= + 2N mθ = is even, thenis odd, and
cos : 1ir
ir i kNπ⎛ ⎞
= ≤ ≤⎜ ⎟⎝ ⎠
: 1j j j Nθθ θ= ∆ ≤ ≤Active variable is
Total number is ,kNθ NOT ( )1rN Nθ− ⋅
Non-active variable active variable [1]
0, 0 2i jr, that is θ π> < ≤
2 22
1 1r rN N r ND u D u D u u
r r θλ+ + = −
x
y1θ2θ
3θ
4θ 5θ
6θ( )1 0u r = ± =
Note that differential matrix rND ( )1rN +is of dimension
0,
1,
1,
,
r
r
j
j
N j
N j
uu
u
u−
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
Neglect due to B.C.,acts on
Non-active variable active variable [2]
2rND
rND
1,
2,
,
j
j
k j
uu
u
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1,
2,
1,r
k j
k j
N j
uu
u
+
+
−
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
However and act on
NOT active variable, how to deal with?
From previous discussion, we have following relationships which can solve this problem
( )
1, , 1,
2, 1, 2,
1, 1, ,
: 1:1
r
k j k m j m j
k j k m j m j
N j m j k m j
u u uu u u
k
u u u
+ + +
+ − + +
− + +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = −⎜ ⎟ ⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
( )
1, , 1,
2, 1, 2,
1, 1, ,
: 1:1
r
k m j k j j
k m j k j j
N m j j k j
u u uu u u
k
u u u
+ +
+ + −
− +
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟= = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠⎝ ⎠
⎟ and
( ): 1:1k −where is a permutation matrix
Non-active variable active variable [3]
( ),ij i ju u r θ 1 1ri N j Nθ≤ ≤≤ ≤ − 1Recall for and
and evaluate at ( ),i jr θConsider Chebyshev differential matrix , rr ND acts on u
( )( ) ( ) ( ), ,, row of ,r rr N i j r N jD u r i D uθ θ= − ⋅ i
We write in matrix notation
( )( ) ( ), ,1 ,2 , , 1 , 2 , 1,r rr N i j i i i k i k i k i ND u r d d d d d dθ + + −=
0r > 0r <
1,
2,
,
j
j
k j
uu
u
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1,
2,
1,r
k j
k j
N j
uu
u
+
+
−
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
0r >
0r <
Question: How about if we arrange equations on ( ), i jr iθ ∀ when fixed j
Non-active variable active variable [4]
( )
( )( )
( )( )( )
( )
111 12 1, 1, 1 1, 2 1, 1
221 22 2, 2, 1 2, 2 2, 1
,1 ,2 , , 1 , 2 , 1
1,1 1,2 1,1
2
1
,
,
,
,
,
,
r
r
r
r
r
jk k k N
jk k k N
k j k k k k k k k k k NN
k k k kk j
k j
N j
rd d d d d d
r d d d d d d
r d d d d d dD u
d d d dr
r
r
θ
θ
θ
θ
θ
θ
+ + −
+ + −
+ + −
+ + ++
+
−
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
=⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1, 1 1, 2 1, 1
2,1 2,2
1,1 1,2 1, 1, 1 1, 1
r
r r r r r r
k k k k k N
k k
N N N k N k N N
d d
d d
d d d d d
+ + + + + −
+ +
− − − − + − −
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1,
2,
,
j
j
k j
uu
u
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1,
2,
1,r
k j
k j
N j
uu
u
+
+
−
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
,
1,
1,
k m j
k m j
m j
uu
u
+
− +
+
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
0r >
0r >
0r <
0r <
We only keep operations on active variable ( )0,u r θ>
( ), ju r θ
That is, only consider equation ( )( ),rN i jD u r θ for 0ir >
1 2
3 4rN
E ED
E E⎛ ⎞
= ⎜ ⎟⎝ ⎠
abbreviate
( )1 2rND E E=Later on, we use the same symbol
Non-active variable active variable [5]
Define permutation matrix ( ) ( )( ) 11: , 1: 1: 1
1
r
I
P k N k
⎛ ⎞⎜ ⎟⎜ ⎟= − − + =⎜ ⎟⎜ ⎟⎝ ⎠
k
k
1,
2,
,
j
j
k j
uu
u
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1,
2,
1,r
k j
k j
N j
uu
u
+
+
−
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
( )( ), jP u r θ
1,
2,
,
j
j
k j
uu
u
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
1,
2,
,
m j
m j
k m j
uu
u
+
+
+
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
,
1,
1,
k m j
k m j
m j
uu
u
+
− +
+
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
( ), ju r θ ≡Let
Active variable with the same indexing in r-direction
Non-active variable active variable [6]
Moreover, we modify differential matrix according to permutation P by
( )( ) ( ) ( )( ), ,r r
TN j N jD u r D P Pu rθ θ=
( )11 12 1, 1, 1 1, 2 1, 1
21 22 2, 2, 1 2, 2 2, 11 2
,1 ,2 , , 1 , 2 , 1
r
r
r
r
k N k k
k N k kTN
k k k k k N k k k k
d d d d d d
d d d d d dD P E E
d d d d d d
− + +
− + +
− + +
⎛ ⎞⎜ ⎟⎜ ⎟= =⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
0r > 0r <
where
such that for 1 j m≤ ≤
( )( )1, 1,
2, 2,, 1 2
, ,
,r
j m j
j m jr N j
k j k m j
u uu u
D u r E E
u u
θ
+
+
+
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
( )( )1, 1,
2, 2,, 1 2
, ,
,r
m j j
m j jr N m j
k m j k j
u uu u
D u r E E
u u
θ
+
++
+
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
and
( ) ( ) ( )( )1 2, , , , , ,j j k jr r rθ θ θEvaluated at
Non-active variable active variable [7]
( ) ( ) ( ) ( )( )1,1 1,2 1, 1, 1 1, 2 1,2
2,1 2,2 2, 2, 1 2, 2 2,2, 1 2 1 2
,1 ,2 , , 1 , 2 ,2
, , , , , ,r
m m m m
m m m mr N m
k k k m k m k m k m
u u u u u uu u u u u u
D u r r r E E
u u u u u u
θ θ θ
+ +
+ +
+ +
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
( ) ( ) ( ) ( )( )1, 1 1, 2 1,2 1,1 1,2 1,
2, 1 2, 2 2,2 2,1 2,2 2,, 1 2 2 1 2
, 1 , 2 ,2 ,1 ,2 ,
, , , , , ,r
m m m m
m m m mr N m m m
k m k m k m k k k m
u u u u u uu u u u u u
D u r r r E E
u u u u u u
θ θ θ
+ +
+ ++ +
+ +
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
,1
,2
,
,
i
ii
i m
uu
U
u
⎛ ⎞⎜ ⎟⎜ ⎟≡ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
, 1
, 2
,2
i m
i mi
i m
uu
V
u
+
+
⎛ ⎞⎜ ⎟⎜ ⎟≡ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠
( )1 1
2 21 2 1 2, ,
T T
T T
T Tk k
U VU V
X X X X X
U V
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= = =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
define and active variable
( ) ( ) ( )( ) ( ) ( ), 1 2 1 1 2 2 2 1, , , , | |rr N mD u r r E X X E X Xθ θ = +To sum up
Non-active variable active variable [8]
( )1 1
2 21 2
T T
T T
T Tk k
U VU V
X X X
U V
⎛ ⎞⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠( ) ( )1 1 2 2| | | | | | | | |
TT T T T T T T Tj j k kmem X U V U V U V U V=
Note that under row-major indexing, memory storage of
is
( ) ( )2 1 1 1 2 2| | | | | | | | | |TT T T T T T T T
j j k kmem X X V U V U V U V U=but
( ) ( )2 1| mk
m
Imem X X I mem X
I⎧ ⎫⎛ ⎞⎪ ⎪= ⊕ ⋅⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
If we adopt Kronecker products, then
( ) ( ) ( )( ) ( ), 1 2 1 2, , , ,r
m mr N m
m m
I ID u r r E E mem X
I Iθ θ
⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= ⊕ + ⊕⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭
11 12 1
12 22 2
1 2
n
n
m m mn
a B a B a Ba B a B a B
A B
a B a B a B
⎛ ⎞⎜ ⎟⎜ ⎟⊕ =⎜ ⎟⎜ ⎟⎝ ⎠
Definition: Kronecker product is defined by
Non-active variable active variable [9]
The same reason holds for second derivative operator
1 22,
3 4rr N
D DD
D D⎛ ⎞
= ⎜ ⎟⎝ ⎠
( ) ( ) ( )( ) ( )2, 1 2 1 2, , , ,
r
m mr N m
m m
I ID u r r D D mem X
I Iθ θ
⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= ⊕ + ⊕⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭
( )2, 1 2rr ND D D=
0r
neglect equation on
<( )2
, 1 2r
Tr ND P D D=
then
( ) ( ) ( ), ,1 1, row of matrix
r r
Tr N i j r N
i
D u r i D Pr r
θ⎡ ⎤ = × −⎢ ⎥⎣ ⎦collect equations on each :1ir i k≤ ≤
1 21 m m
m m
I IR E E
I Ir r⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ⎪ ⎪→ ⊕ + ⊕⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭
we have
1
2
1 2
1/1/ 1 1 1, , ,
1/k
k
rr
R diagr r r
r
⎛ ⎞⎜ ⎟ ⎛ ⎞⎜ ⎟= = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎝ ⎠
where
Non-active variable active variable [10]
directionθ −We write second derivative operator on as
( ) ( )( )
( )( ) ( )
2
22
2
1 0 mod 6 3
1, 0 mod
2sin / 2
jN j
j Nh
S xj N
jh
π⎧− − =⎪⎪= ⎨ −⎪− ≠⎪⎩
( ) ( )( )22,N N k jD S x x
θθ ≡ − where
( )1 , 1 2 0ii k j m r≤ ≤ ≤ ≤ >for
( )( ),1
,22 2 2, , ,
,2
, row of row of
i
i iN i j N N
i
i m
uu U
D u r j D j DV
u
θ θ θθ θ θθ
⎧ ⎫⎛ ⎞⎪ ⎪⎜ ⎟ ⎧ ⎫⎛ ⎞⎪ ⎪ ⎪ ⎪⎜ ⎟= − = −⎨ ⎬ ⎨ ⎬⎜ ⎟⎜ ⎟ ⎪ ⎪⎝ ⎠⎩ ⎭⎪ ⎪⎜ ⎟⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭
( )( )
,1
,22 2 2, , ,
,2
,
i
i iN i N N
i
i m
uu U
D u r D DV
u
θ θθ θ θθ
⎛ ⎞⎜ ⎟
⎛ ⎞⎜ ⎟= = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠
( )2 2, ,2 2
1 1, iN i N
ii
UD u r D
Vr rθ θθ θθ⎛ ⎞⎛ ⎞ = ⎜ ⎟⎜ ⎟
⎝ ⎠ ⎝ ⎠so
θimplies
Non-active variable active variable [11]
( )( )
( )
1 12 2,2 ,2
1 11 11
22 22,,2 22 2
2, 2 222
22 ,2,2
1 1
,11,1
,11
N N
NNN
kk
NNkkk
U UD DV Vr r
rU U
DDrrD u V Vr
rr
U DD rVr
θ θ
θθ
θ
θθ
θ θ
θθθ
θθ
θθ
θ
⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎛⎜ ⎟ ⎜ ⎟⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟ = =⎜ ⎟⎝ ⎠ ⎝⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎜ ⎟⎛ ⎞ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠
k
k
UV
⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎞⎜ ⎟⎟⎜ ⎟⎠⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
If we adopt Kronecker products, then
( )( )
( )
( ) ( )
1
22 2 2, ,2
,,1
,
N N
k
rr
D u R D mem Xr
r
θ θθ θ
θθ
θ
⎛ ⎞⎜ ⎟
⎛ ⎞⎜ ⎟ = ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠
Non-active variable active variable [12]
summary
( ) ( ) ( )( ) ( )2, 1 2 1 2, , , ,
r
m mr N m
m m
I ID u r r D D mem X
I Iθ θ
⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= ⊕ + ⊕⎨ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭
1
( ) ( )( ) ( ), 1 2 1 21 1 , , , ,
r
m mr N m
m m
I ID u r r R E E mem X
I Ir r rθ θ
⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ⎪ ⎪⎛ ⎞→ = ⊕ + ⊕⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ⎝ ⎠ ⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭
( )( )
( )
( ) ( )
1
22 2 2, ,2
,,1
,
N N
k
rr
D u R D mem Xr
r
θ θθ θ
θθ
θ
⎛ ⎞⎜ ⎟
⎛ ⎞⎜ ⎟ = ⊕⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎜ ⎟⎝ ⎠
2
3
Note that
( ) ( ) ( )( )( )( )
( )
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
1 1 1 1 2 1 2
2 2 1 2 2 2 21 2 2
1 2 2
, , , ,, , , ,
, , ,
, , , ,
m
mm
k k k k m
r r r rr r r r
r r r
r r r r
θ θ θ θθ θ θ θ
θ θ θ
θ θ θ θ
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
so that all three system of equations are of the same order.
Non-active variable active variable [13]
22
1 1rr ru u u
r r θθ[ ]1,1r∈ −u , on andλ+ + = − [ ]0,2θ π∈
Discretization on ( )mem X
( ) ( )2L mem X mem Xλ⋅ = −
( )
( ) ( ) ( )
1 2 1 2
2 2,
2 21 1 2 2 ,
m m m m
m m m m
N
m mN
m m
I I I IL D D R E E
I I I I
R D
I ID RE D RE R D
I I
θ
θ
θ
θ
⎧ ⎫ ⎧ ⎫⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪ ⎪ ⎪ ⎪= ⊕ + ⊕ + ⊕ + ⊕⎨ ⎬ ⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎪ ⎪ ⎪ ⎪⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩ ⎭ ⎩ ⎭
+ ⊕
⎛ ⎞ ⎛ ⎞= + ⊕ + + ⊕ + ⊕⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
then
1
2
1 2
1/1/ 1 1 1, , ,
1/k
k
rr
R diagr r r
r
⎛ ⎞⎜ ⎟ ⎛ ⎞⎜ ⎟= = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎜ ⎟⎝ ⎠
where
Example: program 28
2u uλ∆ = − with boundary condition ( )1, 0u r θ= =
25rN = ( ),k kVλis even, let eigen-pair beis odd and 20Nθ =
( ) ( ) ( )2 21 1 2 2 ,
m mN
m m
I IL D RE D RE R D
I I θθ⎛ ⎞ ⎛ ⎞
= + ⊕ + + ⊕ + ⊕⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
sparse structure of
Dimension : 1 12 20 240
2rN Nθ−
= × =
Example: program 28 (mash plot of eigenvector)
1 Eigenvalue is sorted, monotone increasing and normalized to first eigenvalue
1 2 nλ λ λ≤ ≤ ≤ ≤
Eigenvector is normalized by supremum norm, kk
k
VVV
∞
←2
1 5.7832λ = 2 3 14.682λ λ= = 4 5 26.3746λ λ= = 6 30.4713λ = 7 8 40.7065λ λ= = 9 10 42.2185λ λ= =
Example: program 28 (nodal set)
Exercise 1
2u uλ∆ = − with boundary condition ( )1,2 0u r = =
25rN = is odd and 20Nθ = is even, let eigen-pair be ( ),k kVλ
{ }1 2r< <on annulus
{ }1 2r< <[ ]1,1x∈ −Chebyshev node on Chebyshev node on
112
xr += +
( ) ( )2 2 2, , ,r rr N r N N NL D RD I R D
θ θθ= + ⊕ + ⊕
Exercise 1 (mash plot of eigenvector)1 Eigenvalue is sorted, monotone increasing and normalized to first eigenvalue
1 2 nλ λ λ≤ ≤ ≤ ≤
Eigenvector is normalized by supremum norm, kk
k
VVV
∞
←2