Math 504 Fall 2016 Notes Week 6, Lecture 2 Emre Mengi Department of Mathematics Koç University ˙ Istanbul, Turkey Emre Mengi Week 6, Lecture 2
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Math 504 Fall 2016 NotesWeek 6, Lecture 2
Emre Mengi
Department of MathematicsKoç University
Istanbul, Turkey
Emre Mengi Week 6, Lecture 2
Outline
QR Factorization
QR Factorization by Householder Triangularization
Emre Mengi Week 6, Lecture 2
QR Factorization
Emre Mengi Week 6, Lecture 2
Reduced and Full QR Factorizations
Let A ∈ Cn×p with n ≥ p.
The reduced QR factorization of A
A = QR
Q ∈ Cn×p has orthonormal columns, R ∈ Cp×p is uppertriangular.
The full QR factorization of A
A = QR
Q ∈ Cn×n is unitary, R ∈ Cn×p is upper triangular.
Emre Mengi Week 6, Lecture 2
Reduced and Full QR Factorizations
Let A ∈ Cn×p with n ≥ p.
The reduced QR factorization of A
A = QR
Q ∈ Cn×p has orthonormal columns, R ∈ Cp×p is uppertriangular.
The full QR factorization of A
A = QR
Q ∈ Cn×n is unitary, R ∈ Cn×p is upper triangular.
Emre Mengi Week 6, Lecture 2
Reduced and Full QR Factorizations
Let A ∈ Cn×p with n ≥ p.
The reduced QR factorization of A
A = QR
Q ∈ Cn×p has orthonormal columns, R ∈ Cp×p is uppertriangular.
The full QR factorization of A
A = QR
Q ∈ Cn×n is unitary, R ∈ Cn×p is upper triangular.
Emre Mengi Week 6, Lecture 2
Reduced and Full QR Factorizations
Reduced QR factorization can be obtained from the full QRfactorization.
Let
A =
Q1︸︷︷︸n×p
Q2︸︷︷︸n×(n−p)
R1︸︷︷︸p×p0︸︷︷︸
(n−p)×p
be the full QR factorization.
ThenA = Q1R1
is the reduced QR factorization.
Emre Mengi Week 6, Lecture 2
Reduced and Full QR Factorizations
Reduced QR factorization can be obtained from the full QRfactorization.
Let
A =
Q1︸︷︷︸n×p
Q2︸︷︷︸n×(n−p)
R1︸︷︷︸p×p0︸︷︷︸
(n−p)×p
be the full QR factorization.
ThenA = Q1R1
is the reduced QR factorization.
Emre Mengi Week 6, Lecture 2
Reduced and Full QR Factorizations
Reduced QR factorization can be obtained from the full QRfactorization.
Let
A =
Q1︸︷︷︸n×p
Q2︸︷︷︸n×(n−p)
R1︸︷︷︸p×p0︸︷︷︸
(n−p)×p
be the full QR factorization.
ThenA = Q1R1
is the reduced QR factorization.
Emre Mengi Week 6, Lecture 2
QR Factorization and the Gram-Schmidt Procedure
The Gram-Schmidt procedure applied to the columns of
A =[
a(1) a(2) . . . a(p)]
can be expressed as
a(1) = r11q(1),
a(2) = r12q(1) + r22q(2),
...
a(p) = r1pq(1) + r2pq(2) + · · ·+ rppq(p).
Emre Mengi Week 6, Lecture 2
QR Factorization and the Gram-Schmidt Procedure
Hence,
[a(1) a(2) . . . a(p)
]︸ ︷︷ ︸A
=[
q(1) q(2) . . . q(p)]︸ ︷︷ ︸
Q
r11 r12 . . . r1p0 r22 . . . r2p...
. . ....
0 0 rpp
︸ ︷︷ ︸
R
.
Gram-Schmidt procedure yields a reduced QR factorization.
Emre Mengi Week 6, Lecture 2
QR Factorization and the Gram-Schmidt Procedure
Hence,
[a(1) a(2) . . . a(p)
]︸ ︷︷ ︸A
=[
q(1) q(2) . . . q(p)]︸ ︷︷ ︸
Q
r11 r12 . . . r1p0 r22 . . . r2p...
. . ....
0 0 rpp
︸ ︷︷ ︸
R
.
Gram-Schmidt procedure yields a reduced QR factorization.
Emre Mengi Week 6, Lecture 2
QR Factorization by HouseholderTriangularization
Emre Mengi Week 6, Lecture 2
Householder Reflectors
For a given vector v ∈ Cn, would like a unitary matrix H ∈ Cn×n
such that
Hv = ‖v‖2e(1) =
‖v‖2
0...0
.
One possibility is the Householder reflector
H = In − 2uu∗
whereu = (v − ‖v‖2e(1))/‖v − ‖v‖2e(1)‖2.
H reflects about the subspace S⊥ where S = span{u}.
Emre Mengi Week 6, Lecture 2
Householder Reflectors
For a given vector v ∈ Cn, would like a unitary matrix H ∈ Cn×n
such that
Hv = ‖v‖2e(1) =
‖v‖2
0...0
.
One possibility is the Householder reflector
H = In − 2uu∗
whereu = (v − ‖v‖2e(1))/‖v − ‖v‖2e(1)‖2.
H reflects about the subspace S⊥ where S = span{u}.
Emre Mengi Week 6, Lecture 2
Householder Reflectors
For a given vector v ∈ Cn, would like a unitary matrix H ∈ Cn×n
such that
Hv = ‖v‖2e(1) =
‖v‖2
0...0
.
One possibility is the Householder reflector
H = In − 2uu∗
whereu = (v − ‖v‖2e(1))/‖v − ‖v‖2e(1)‖2.
H reflects about the subspace S⊥ where S = span{u}.
Emre Mengi Week 6, Lecture 2
Householder Reflectors
Example: Letting v =
[43
],
u =
([43
]−[
50
])/
∥∥∥∥[ 43
]−[
50
]∥∥∥∥ =1√10
[−1
3
],
H =
[1 00 1
]− 2
10
[−1
3
] [−1 3
]=
[4/5 3/53/5 −4/5
]
Emre Mengi Week 6, Lecture 2
Householder Reflectors
Example: Letting v =
[43
],
u =
([43
]−[
50
])/
∥∥∥∥[ 43
]−[
50
]∥∥∥∥ =1√10
[−1
3
],
H =
[1 00 1
]− 2
10
[−1
3
] [−1 3
]=
[4/5 3/53/5 −4/5
]
Emre Mengi Week 6, Lecture 2
Householder Reflectors
Example: Letting v =
[43
],
u =
([43
]−[
50
])/
∥∥∥∥[ 43
]−[
50
]∥∥∥∥ =1√10
[−1
3
],
H =
[1 00 1
]− 2
10
[−1
3
] [−1 3
]=
[4/5 3/53/5 −4/5
]
Emre Mengi Week 6, Lecture 2
The Algorithm
1st Column
A =
x x . . . xx x xx x x...
......
x x x
7→
x x . . . x0 x x0 x x...
......
0 x x
= Q(1)A
Q(1) = I − 2u(1)[u(1)]∗
u(1) = (a(1) − ‖a(1)‖2e(1))/‖a(1) − ‖a(1)‖2e(1)‖2
Emre Mengi Week 6, Lecture 2
The Algorithm
1st Column
A =
x x . . . xx x xx x x...
......
x x x
7→
x x . . . x0 x x0 x x...
......
0 x x
= Q(1)A
Q(1) = I − 2u(1)[u(1)]∗
u(1) = (a(1) − ‖a(1)‖2e(1))/‖a(1) − ‖a(1)‖2e(1)‖2
Emre Mengi Week 6, Lecture 2
The Algorithm
2nd Column
A(2) := Q(1)A =
x x . . . x0 x x0 x x...
......
0 x x
7→
x x . . . x0 x x0 0 x...
......
0 0 x
= Q(2)Q(1)A
Q(2) =
[1 00 I − 2u(2)[u(2)]∗
]u(2) = (h(2) − ‖h(2)‖2e(1))/‖h(2) − ‖h(2)‖2e(1)‖2h(2) = A(2)(2 : n,2)
Emre Mengi Week 6, Lecture 2
The Algorithm
2nd Column
A(2) := Q(1)A =
x x . . . x0 x x0 x x...
......
0 x x
7→
x x . . . x0 x x0 0 x...
......
0 0 x
= Q(2)Q(1)A
Q(2) =
[1 00 I − 2u(2)[u(2)]∗
]u(2) = (h(2) − ‖h(2)‖2e(1))/‖h(2) − ‖h(2)‖2e(1)‖2h(2) = A(2)(2 : n,2)
Emre Mengi Week 6, Lecture 2
The Algorithm
kth column
A(k) =
[R(k) N(k)
0 M(k)
]7→[
R(k) N(k)
0 H(k)M(k)
]= Q(k)A(k)
where R(k) ∈ C(k−1)×(k−1) is upper triangular,M(k) ∈ C(n−k+1)×(n−k+1) is the block to be modified.
Q(k) =
[Ik−1 0
0 H(k) := I − 2u(k)[u(k)]∗
]u(k) = (h(k) − ‖h(k)‖2e(1))/‖h(k) − ‖h(k)‖2e(1)‖2h(k) = A(k)(k : n, k)
Emre Mengi Week 6, Lecture 2
The Algorithm
kth column
A(k) =
[R(k) N(k)
0 M(k)
]7→[
R(k) N(k)
0 H(k)M(k)
]= Q(k)A(k)
where R(k) ∈ C(k−1)×(k−1) is upper triangular,M(k) ∈ C(n−k+1)×(n−k+1) is the block to be modified.
Q(k) =
[Ik−1 0
0 H(k) := I − 2u(k)[u(k)]∗
]u(k) = (h(k) − ‖h(k)‖2e(1))/‖h(k) − ‖h(k)‖2e(1)‖2h(k) = A(k)(k : n, k)
Emre Mengi Week 6, Lecture 2
The Algorithm
After p steps, we have
Q(p) . . .Q(2)Q(1)A = R
where R ∈ Cn×p is upper triangular.
Emre Mengi Week 6, Lecture 2
The Algorithm
After p steps, we have
Q(p) . . .Q(2)Q(1)A = R
where R ∈ Cn×p is upper triangular.
This yields a full QR factorization
A =[Q(1)
]∗ [Q(2)
]∗. . .[Q(p)
]∗︸ ︷︷ ︸
Q
R.
Emre Mengi Week 6, Lecture 2
The Algorithm
After p steps, we have
Q(p) . . .Q(2)Q(1)A = R
where R ∈ Cn×p is upper triangular.
This yields a full QR factorization
A = Q(1)Q(2) . . .Q(p)︸ ︷︷ ︸Q
R.
Emre Mengi Week 6, Lecture 2
The Algorithm
Input: A ∈ Cn×p with n ≥ pOutput: Upper triangular R ∈ Cn×p and the vectors
u(1), . . . ,u(p).1: for k = 1,2, . . . ,p do2: v ← A(k : n, k)3: u(k) ←
(v − ‖v‖e(1)) /‖v − ‖v‖e(1)‖2
4: A(k : n, k : p)← A(k : n, k : p)− 2u(k)([u(k)
]∗A(k : n, k : p))
5: end for6: R ← A
Emre Mengi Week 6, Lecture 2
The Algorithm
Input: A ∈ Cn×p with n ≥ pOutput: Upper triangular R ∈ Cn×p and the vectors
u(1), . . . ,u(p).1: for k = 1,2, . . . ,p do2: v ← A(k : n, k)3: u(k) ←
(v − ‖v‖e(1)) /‖v − ‖v‖e(1)‖2
4: A(k : n, k : p)← A(k : n, k : p)− 2u(k)([u(k)
]∗A(k : n, k : p))
5: end for6: R ← A
Emre Mengi Week 6, Lecture 2
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