def: characteristic polynomial (of degree n) qr - algorithm note: 1) qr – algorithm is different...
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DEF: nnCALet )det()( xAIxp Characteristic Polynomial (of degree n)
QR - Algorithm
Note: 1) QR – Algorithm is different from QR-Decomposition
2) a procedure to calculate the eigenvalues of a matrix
The eigenvalues of A are the n roots of its characteristic polynomial
123 The set of these roots is called the spectrum of A and
denoted by },,{)( 1 nA
for i=1:50 [Q,R]=qr(A); A=R*Q;end
QR Algorithm:(Basic version)
Find QR factorization
Multiply in the reverse order
DEF: nnCALet
)det()( xAIxp Characteristic Polynomial (of degree n)
Eigenvalues & Eigenvectors
The eigenvalues of A are the n roots of its characteristic polynomial
123 The set of these roots is called the spectrum of A and
denoted by },,{)( 1 nA
4 nA 21)det(
5 nAtrace 21)(
6 If A is symmetric then all eigenvalues are reals
DEF: nnCALet the nonzero vectors x that satisfy)( If A
Eigenvalues & Eigenvectors
xAx are called eigenvectors.
DEF:
similar
,BA Xmatrix r nonsingula AXXB 1
ation transformsimilarity called is X
DEF:
A similar to
y orthogonal B Qmatrix orthogonal AQQB T
THM: similar ,BA )()( BA same eigenvalues
Special Matrices
nna
a
a
22
11
Diagonal
nn
n
ac
b
ac
ba
1
22
11
Tridiagonal
nn
n
a
a
aa
22
111
Upper triangular
nnnn
n
aa
aa
aa
1,
2221
111
Upper Hessenberg
n
n
a
b
a
ba
1
2
11
Upper bidiagonal
Schur Factorization
Schur Factorization:
TAQQ *
nnCA
x
xx
)(unitary * IQQQ
Schur Factorization:
TAQQT
x
x
orthogonal Qsymmetric If nnRA
1) A and T are similar2) They have same eigenvalues3) Eigenvalues of T are the diagonal entries
Example:Compute all eigenvalues of A:
Schur Factorization
TAQQ *
x
xx
IDEA: We need to construct the Schur factorization
for i=1:50 [Q,R]=qr(A); A=R*Q;end
QR Algorithm:(Basic version)
>> eig(A)‘-100.0111 -10.0988 -0.8901
110
1101
01100
A
A5 =
-100.0111 0.0000 0.0000 0.0000 -10.0988 0.0000 0 0.0000 -0.8901
A3 =
-100.0111 0.0010 0.0000 0.0010 -10.0988 0.0007 0 0.0007 -0.8901
All A’s are similarCompute all eigenvalues of A:
Schur Factorization
TAQQ *
x
xx
IDEA: We need to construct the Schur factorization
for i=1:50 [Q,R]=qr(A); A=R*Q;end
QR Algorithm:(Basic version)
AA 1
111 RQA
112 QRA
222 RQA
112 QRA
11*12 )( QAQA
iiii QAQA *
1
All A’s have same eigenvalues
ngularupper triakA
Compute all eigenvalues of A:
Schur Factorization
TAQQ *
x
xx
IDEA: We need to construct the Schur factorization
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0
0
0
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0
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000
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0
AHHH 123AHH 12AH1A
QR
-fac
tori
zati
on
QAAHHH 123
Idea
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0
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0
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0
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AHHH 123AHH 12AH1A
QR
-fac
tori
zati
on
QAAHHH 123
TAQQT
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0
0
0
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00
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0
*2
*112 HAHHH*
11AHHAH1A
Bad
Idea
All zeros are destroyed
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0
0
0
Idea TAQQT
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0
0
0
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0
*2
*112 HAHHH*
11AHHAH1AB
ad Id
ea
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0
0
0
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0
0
*11AQQAQ1A
Go
od
Idea
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0
0
zeros are not destroyed
):2,:2(
):2,1()1,1(
11
11
nnAe
nAA
111
11
ˆ0
01
):2,:2(
):2,1()1,1(
QnnAe
nAA
AQA 11 *11AQQ
111
111
ˆ):2,:2(
ˆ):2,1()1,1(
QnnAe
QnAA
Idea TAQQT
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0
*11AQQAQ1A
Go
od
Idea
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0
*112 AQQQ
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0
*2
*112 QAQQQ
AQQ*
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0
It is not upper triangular
HessenbergGivens
Idea TAQQT
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A
Go
od
Idea
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0
H
x
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000
00
0
T
PHASE-I PHASE-II
QR-Algorithm
for i=1:50 [Q,R]=qr(A); A=R*Q;end
QR Algorithm:
for i=1:50 [Q,R]=qr(H); H=R*Q;end
QR Algorithm:
General matrix A Hessenberg H
O(n^3)
All H’s are Hessenberg (why?)We can upper triangularize H with a sequence of Givens
HGGQH TTn 11
),1,( ii iiGG
O(n^2)
6n^2
for k=1:n-1 v=house(A(k+1:n,k); A(k+1:n,k:n)=(I-2*v*v’/v’*v)*A(k+1:n,k:n); A(1:n,k+1:n)=A(1:n,k+1:n)(I-2*v*v’/v*v);end
Reduce H
Idea
A H
T
PHASE-I
TAQQT
PHASE-II
for i=1:50 [Q,R]=qr(H); H=R*Q;end
QR Algorithm:
H
If A is symmetric matrix, then H is Hessenberg and symmetric al tridiagonA
A symmetric
The work reduces to 50% or less
Example
Slow Convergence
A H
T
PHASE-I
PHASE-IIHA = 1 2 3 4 4 5 6 7 2 1 5 0 4 2 1 0
H = 1.0000 -5.0000 -1.5395 -1.2767 -6.0000 8.5556 0.1085 3.6986 0 -5.9182 -2.1689 -1.1428 0 0 -0.1428 3.6133
>> eig(A)' 11.1055 -3.8556 3.5736 0.1765
H20 = 11.1055 -4.7403 3.9060 -4.0296 -0.0000 -3.8526 -0.6899 1.2559 0 -0.0322 3.5706 0.1571 0 0 -0.0000 0.1765
H5 = 11.1328 4.5792 3.9941 -4.0372 -0.0905 -3.8908 0.5327 -1.2342 0 -0.1006 3.5815 0.1342 0 0 -0.0000 0.1765
Any solution for low convergence
Remark:
Remark:Deflation
Remark:We assume that H is unreduced. If not then we have
Decouple and Deflation
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H0
The problem decouples into two smaller problem
x
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H
0
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H
0
A of spectrum },,{)( If 1 nA sI-A of spectrum },,{)(then 1 sssIA n
QR Algorithm with shift
TPHASE-II
QR Algorithm with shift:
H A20 = 11.1055 -4.7403 3.9060 -4.0296 -0.0000 -3.8526 -0.6899 1.2559 0 -0.0322 3.5706 0.1571 0 0 -0.0000 0.1765
>> eig(A)' 11.1055 -3.8556 3.5736 0.1765
1-i
i1, rate with 0
iia
The theorem says that if we shift an exact eigenvalue, then in exact arithmetic deflation occurs in one step. )(
,k choice good a knnh
nnnn hh
h
h
1,
21
11
nnnn hh
h
h
1,
21
11
H H
end
I;Q*RH
I);-qr(HR][Q,
shift determine
50:1kfor
k
k
k
Example:
QR Algorithm with shift
TPHASE-II
QR Algorithm with shift:
end
I;Q*RH
I);-qr(HR][Q,
shift determine
50:1kfor
k
k
k
H
>> eig(A)' 11.1055 -3.8556 3.5736 0.1765
1-i
i1, rate with 0
iia
H = 1.0000 -5.0000 -1.5395 -1.2767 -6.0000 8.5556 0.1085 3.6986 0 -5.9182 -2.1689 -1.1428 0 0 -0.1428 3.6133
H1 = 3.3236 -5.5508 4.4353 2.6929 -8.0984 2.9128 -6.1528 -2.4688 0 -2.2250 1.1900 1.0296 0 0 -0.0017 3.5736
H2 = 2.4924 -9.7341 -2.5648 2.3834 -5.9560 5.0131 3.0854 -2.9507 0 -1.4702 -0.0791 0.0767 0 0 -0.0000 3.5736
nnk h
deflation
)(,k choice good a knnh
Example:
QR Algorithm with shift
H = 1.0000 -5.0000 -1.5395 -1.2767 -6.0000 8.5556 0.1085 3.6986 0 -5.9182 -2.1689 -1.1428 0 0 -0.1428 3.6133
H1 = 3.3236 -5.5508 4.4353 2.6929 -8.0984 2.9128 -6.1528 -2.4688 0 -2.2250 1.1900 1.0296 0 0 -0.0017 3.5736
H2 = 2.4924 -9.7341 -2.5648 2.3834 -5.9560 5.0131 3.0854 -2.9507 0 -1.4702 -0.0791 0.0767 0 0 -0.0000 3.5736
nnk h
Theory: quadratic convSymmetric case : cubic
H5 = 11.0798 5.2490 -3.2292 -0.0099 0.1708 1.2672 0 -0.0000 -3.8640
H4 11.0926 5.2199 -3.2299 -0.0368 0.1589 1.2723 0 -0.0029 -3.8649
H3 = 11.1418 5.2542 2.9806 -0.1409 0.0453 -1.4768 0 -0.1735 -3.8005
deflation
Eigenvalues of Symmetric Matrices
A T
D
PHASE-I
PHASE-II
for i=1:50 [Q,R]=qr(H); H=R*Q;end
QR Algorithm:
T
If A is symmetric matrix, then H is Hessenberg and symmetric al tridiagonA
A symmetricT is tridiagonal
O(n)
Schur Factorization:
DAQQT
n
1
orthogonal Qsymmetric If nnRA
kkk qAq Moreover,
Wilkinson shift:
QR Algorithm with wilkinson shift
TPHASE-II
QR Algorithm with shift:
end
I;Q*RH
I);-qr(HR][Q,
shift determine
50:1kfor
k
k
k
H
nnakfor choice reasonable a
nn
nn
ab
ba
b
ba
T
1
11
1
11
A more effective choice is to shift by the eigenvalue of
nn
nn
ab
baT
1
11
That is closer to na2
12)( nn bddsignda
2/)( 1 nn aad
Wilkinson has shown that: cubically convergent.
Can we use this shift to non-symmetric matrix