中国科学院李保滨矩阵分析课件八
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Suppose that floating-point arithmetic produces an orthogonal matrix Q + E and upper triangular matrix R + F ˜ = (Q + E)(R + F) = QR + QF + ER + EF = A + QF + ER + EF. A If E and F account for the roundoff errors, and if their entries are small relative to those in A, then the entries in EF are negligible, and ˜ ≈ A + QF + ER. A Since Q is orthogonal, QF F = F F , and A F = QR F = R F . This means that neither QF nor ER can contain entries that are large relative to those in A. ˜ ≈ A, which says that the algorithm is stable. Hence A
Elementary reflectors are not the only type of orthogonal matrices that can be used to reduce a matrix to an upper-trapezoidal form. Plane rotation matrices are also orthogonal and can be used to selectively annihilate any component in a given column.
Li Bao bin | UCAS
7 / 65
Norms and Inner Products | Orthogonal Reduction
Orthogonal Reduction and Least Squares
Orthogonal reduction can be used to solve the least squares problem with an inconsistent system Ax = b in which A ∈ Rm×n and m ≥ n. If ε denotes the difference ε = Ax − b, then, the general least square problem is to find a vector x that minimizes the quantity
Li Bao bin | UCAS
4 / 65
Norms and Inner Products | Orthogonal Reduction
Problem: Use Householder reduction to find an orthogonal matrix P such that PA = T is upper triangular with positive diagonal entries, where 0 −20 −14 27 −4 . A= 3 4 11 −2 Solution: To annihilate the entries below the (1,1)-position and to guarantee that t11 is positive, we set u1 = A∗1 − A∗1 e1 = A∗1 − 5e1 = (−5 3 4)T We obtain and R1 = I − 2 u1 uT . uT 1 u1
Norms and Inner Products
ÚSÈII
Norms and Inner Products
Baobin Li Email:libb@ucas.ac.cn
School of Computer and Control Engineering, UCAS
Li Bao bin | UCAS
1 / 65
−1 −1
.
2 u2 ˆ 2 = I − 2 uT If R and R2 = u u 2
ˆ 2 A2 = R and
25 10 0 10
5 25 −4 and R2 R1 A = 0 25 10 0 0 10
0 15 20 1 −20 12 −9 . P = R2 R1 = 25 −15 −16 12
Tk−1 Tk−1 . 0 Ak Eventually, all of the rows or all of the columns will be exhausted, so the final result is one of the two following upper-trapezoidal forms: The result after k − 1 steps is Rk−1 · · · R2 R1 A =
Li Bao bin | UCAS
ห้องสมุดไป่ตู้
8 / 65
Norms and Inner Products | Orthogonal Reduction
If A has linearly independent columns, then the least squares solution for Ax = b is obtained by solving the nonsingular triangular system Rx = c. We now have four different ways to reduce a matrix to an upper-triangular form
So that R1 A∗1 = ∓µ AA∗1 e1 = (t11 , 0, · · · , 0)T . Applying R1 to A yields R1 A = [R1 A∗1 |R1 A∗2 | · · · |R1 A∗n ] = where A2 is m − 1 × n − 1.
Li Bao bin | UCAS 2 / 65
Li Bao bin | UCAS
5 / 65
Norms and Inner Products | Orthogonal Reduction
To annihilate the entry below the (2,2)-position, set A2 = 0 −10 −25 −10
T 2
and
u2 = [A2 ]∗1 − [A2 ]∗1 e1 = 25 1 0 ˆ2 0 R then
(1) (2) (3) (4) Gaussian elimination Gram-Schmidt procedure Householder reduction Givens reduction
It’s natural to try to compare them and to sort out the advantages and disadvantages of each. First consider numerical stability. Strictly speaking, an algorithm is considered to be numerically stable if, under floating-point arithmetic, it always returns an answer that is the exact solution of a nearby problem.
Li Bao bin | UCAS
9 / 65
Norms and Inner Products | Orthogonal Reduction
The Householder or Givens reduction is a stable algorithm for producing the QR factorization of An×n .
Li Bao bin | UCAS
3 / 65
Norms and Inner Products | Orthogonal Reduction
If m = n, then the final form is an upper-triangular matrix. The elementary reflectors Ri described above are unitary matrices, so every product Rk · · · R1 is a unitary matrix.
Norms and Inner Products | Orthogonal Reduction
Orthogonal Reduction
A matrix A can be reduced to row echelon form by elementary row operation by Gaussian elimination. Gaussian elimination is not the only way to reduce a matrix. Elementary reflector Rk can accomplish the same purpose, which called Householder reduction. It proceeds as follows. For Am×n = [A∗1 |A∗2 | · · · |A∗n ], use x = A∗1 to construct the elementary reflector R1 = I − 2 uu∗ u∗ u where u = A∗1 ± µ A∗1 e1 ,
Li Bao bin | UCAS 6 / 65
Norms and Inner Products | Orthogonal Reduction
A sequence of plane rotations can be used to annihilate all elements below a particular pivot. This means that a matrix A can be reduced to an upper-trapezoidal form by using plane rotations. Such a process is usually called a Givens reduction Householder and Givens reductions are closely related to the results produced by applying the GramõSchmidt process to the columns of A. When A is nonsingular, Householder, Givens, and GramõSchmidt each produce an orthogonal matrix Q and an upper-triangular matrix R such that A = QR .
t11 tT 1 0 A2
,
Norms and Inner Products | Orthogonal Reduction
Thus all entries below the (1,1)-position are annihilated. Now apply the same procedure to A2 to construct an elementary ˆ 2 that annihilates all entries below the (1,1)-position in A2 . reflector R 1 0 Set R2 = ˆ 2 , then R2 R1 is an orthogonal matrix such that 0 R R2 R1 A = t11 tT 1 ˆ 2 A2 0 R
Elementary reflectors are not the only type of orthogonal matrices that can be used to reduce a matrix to an upper-trapezoidal form. Plane rotation matrices are also orthogonal and can be used to selectively annihilate any component in a given column.
Li Bao bin | UCAS
7 / 65
Norms and Inner Products | Orthogonal Reduction
Orthogonal Reduction and Least Squares
Orthogonal reduction can be used to solve the least squares problem with an inconsistent system Ax = b in which A ∈ Rm×n and m ≥ n. If ε denotes the difference ε = Ax − b, then, the general least square problem is to find a vector x that minimizes the quantity
Li Bao bin | UCAS
4 / 65
Norms and Inner Products | Orthogonal Reduction
Problem: Use Householder reduction to find an orthogonal matrix P such that PA = T is upper triangular with positive diagonal entries, where 0 −20 −14 27 −4 . A= 3 4 11 −2 Solution: To annihilate the entries below the (1,1)-position and to guarantee that t11 is positive, we set u1 = A∗1 − A∗1 e1 = A∗1 − 5e1 = (−5 3 4)T We obtain and R1 = I − 2 u1 uT . uT 1 u1
Norms and Inner Products
ÚSÈII
Norms and Inner Products
Baobin Li Email:libb@ucas.ac.cn
School of Computer and Control Engineering, UCAS
Li Bao bin | UCAS
1 / 65
−1 −1
.
2 u2 ˆ 2 = I − 2 uT If R and R2 = u u 2
ˆ 2 A2 = R and
25 10 0 10
5 25 −4 and R2 R1 A = 0 25 10 0 0 10
0 15 20 1 −20 12 −9 . P = R2 R1 = 25 −15 −16 12
Tk−1 Tk−1 . 0 Ak Eventually, all of the rows or all of the columns will be exhausted, so the final result is one of the two following upper-trapezoidal forms: The result after k − 1 steps is Rk−1 · · · R2 R1 A =
Li Bao bin | UCAS
ห้องสมุดไป่ตู้
8 / 65
Norms and Inner Products | Orthogonal Reduction
If A has linearly independent columns, then the least squares solution for Ax = b is obtained by solving the nonsingular triangular system Rx = c. We now have four different ways to reduce a matrix to an upper-triangular form
So that R1 A∗1 = ∓µ AA∗1 e1 = (t11 , 0, · · · , 0)T . Applying R1 to A yields R1 A = [R1 A∗1 |R1 A∗2 | · · · |R1 A∗n ] = where A2 is m − 1 × n − 1.
Li Bao bin | UCAS 2 / 65
Li Bao bin | UCAS
5 / 65
Norms and Inner Products | Orthogonal Reduction
To annihilate the entry below the (2,2)-position, set A2 = 0 −10 −25 −10
T 2
and
u2 = [A2 ]∗1 − [A2 ]∗1 e1 = 25 1 0 ˆ2 0 R then
(1) (2) (3) (4) Gaussian elimination Gram-Schmidt procedure Householder reduction Givens reduction
It’s natural to try to compare them and to sort out the advantages and disadvantages of each. First consider numerical stability. Strictly speaking, an algorithm is considered to be numerically stable if, under floating-point arithmetic, it always returns an answer that is the exact solution of a nearby problem.
Li Bao bin | UCAS
9 / 65
Norms and Inner Products | Orthogonal Reduction
The Householder or Givens reduction is a stable algorithm for producing the QR factorization of An×n .
Li Bao bin | UCAS
3 / 65
Norms and Inner Products | Orthogonal Reduction
If m = n, then the final form is an upper-triangular matrix. The elementary reflectors Ri described above are unitary matrices, so every product Rk · · · R1 is a unitary matrix.
Norms and Inner Products | Orthogonal Reduction
Orthogonal Reduction
A matrix A can be reduced to row echelon form by elementary row operation by Gaussian elimination. Gaussian elimination is not the only way to reduce a matrix. Elementary reflector Rk can accomplish the same purpose, which called Householder reduction. It proceeds as follows. For Am×n = [A∗1 |A∗2 | · · · |A∗n ], use x = A∗1 to construct the elementary reflector R1 = I − 2 uu∗ u∗ u where u = A∗1 ± µ A∗1 e1 ,
Li Bao bin | UCAS 6 / 65
Norms and Inner Products | Orthogonal Reduction
A sequence of plane rotations can be used to annihilate all elements below a particular pivot. This means that a matrix A can be reduced to an upper-trapezoidal form by using plane rotations. Such a process is usually called a Givens reduction Householder and Givens reductions are closely related to the results produced by applying the GramõSchmidt process to the columns of A. When A is nonsingular, Householder, Givens, and GramõSchmidt each produce an orthogonal matrix Q and an upper-triangular matrix R such that A = QR .
t11 tT 1 0 A2
,
Norms and Inner Products | Orthogonal Reduction
Thus all entries below the (1,1)-position are annihilated. Now apply the same procedure to A2 to construct an elementary ˆ 2 that annihilates all entries below the (1,1)-position in A2 . reflector R 1 0 Set R2 = ˆ 2 , then R2 R1 is an orthogonal matrix such that 0 R R2 R1 A = t11 tT 1 ˆ 2 A2 0 R