QR-decomposition. The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which A = QR
|
|
- Joella Perkins
- 5 years ago
- Views:
Transcription
1 QR-decomposition The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which In Matlab A = QR [Q,R]=qr(A); Note. The QR-decomposition is unique up to a change of signs of the columns of Q: A = (QD)( DR) with D = I
2 QR-decomposition The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which A = QR If A is n k with column rank l and l k n, then the ecomical QR-decomposition is an n l orthonormal matrix Q and an l k upper triangular matrix R for which In Matlab A = QR [Q,R]=qr(A, 0 );
3 QR-decomposition The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which A = QR If A is n k with column rank l and l k n, then the ecomical QR-decomposition is an n l orthonormal matrix Q and an l k upper triangular matrix R for which A = QR Note. The columns of Q form an orthonormal basis of the space spanned by the columns of A: the QR-decomp. represents the results of the Gram-Schmidt process.
4 QR-decomposition The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which A = QR Theorem. The QR-decomposition can be stably computed with Householder reflections.
5 QR-decomposition The QR-decomposition of an n k matrix A, k n, is an n n unitary matrix Q and an n k upper triangular matrix R for which A = QR Theorem. The QR-decomposition can be stably computed with Householder reflections: Let R be the computed R and Q = (H vk... H v ) with H vj the Householder reflection as actually used in step j. Then A + A = Q R for some A with A F nku A F. Note. The claim is not that Q is close to the Q that we would have been obtained in exact arithmetic, but that Q is unitary (product of Householder reflections).
6 Schur decomposition The Schur decomposition or of an n n matrix A is an n n unitary matrix U and an n u upper triangular matrix S such that In Matlab A = USU or, equivalently, AU = US [U,S]=schur(A);
7 Schur decomposition The Schur decomposition or of an n n matrix A is an n n unitary matrix U and an n u upper triangular matrix S such that In Matlab A = USU or, equivalently, AU = US [U,S]=schur(A); Theorem. If ST = TΛ is the eigenvalue decomposition of S, i.e., T is non-singular and Λ is diagonal, then A(UT) = (UT)Λ is the eigenvalue decomposition of A. In particular, Λ(A) = Λ(S) = diag(s) and C 2 (T) = C 2 (UT).
8 Schur decomposition The Schur decomposition or of an n n matrix A is an n n unitary matrix U and an n u upper triangular matrix S such that In Matlab A = USU or, equivalently, AU = US [U,S]=schur(A); The columns u j Ue j of U are called Schur vectors.
9 Schur decomposition The Schur decomposition or of an n n matrix A is an n n unitary matrix U and an n u upper triangular matrix S such that In Matlab A = USU or, equivalently, AU = US [U,S]=schur(A); Observation. Many techniques, where the eigenvalue decomposition of A is exploited, can be based on the Schur decomposition as well. For practical computations, the Schur decomposition is preferable, since it is stable: C 2 (U) =, while C 2 (UT) can be huge.
10 Schur decomposition The Schur decomposition or of an n n matrix A is an n n unitary matrix U and an n u upper triangular matrix S such that In Matlab A = USU or, equivalently, AU = US [U,S]=schur(A); Note. The first column u Ue of U is an eigenvector of A with eigenvalue λ e Se. Proof. S is upper triangular: Se = λ e.
11 Schur decomposition The Schur decomposition or of an n n matrix A is an n n unitary matrix U and an n u upper triangular matrix S such that In Matlab A = USU or, equivalently, AU = US [U,S]=schur(A); Note. The last column u n Ue n of U is an eigenvector of A with eigenvalue λ n, where λ n e n Se n. Proof. S is lower triangular: S e n = λ n e n.
12 Schur decomposition The Schur decomposition or of an n n matrix A is an n n unitary matrix U and an n u upper triangular matrix S such that In Matlab A = USU or, equivalently, AU = US [U,S]=schur(A); Note. The second column u 2 of U is an eigenvector of A (I u u )A(I u u ) with eigenvalue λ 2 e 2 Se 2. Proof. S is upper triangular: Se 2 = αe + λ e 2 for α = S,2. Hence, USU u 2 = αu + λ u 2.
13 Schur decomposition The Schur decomposition or of an n n matrix A is an n n unitary matrix U and an n u upper triangular matrix S such that In Matlab A = USU or, equivalently, AU = US [U,S]=schur(A); Note. The second column u 2 of U is an eigenvector of A (I u u )A(I u u ) with eigenvalue λ 2 e 2 Se 2. In A, the eigenvector u is deflated from A.
14 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Theorem. With a proper shift strategy: U k U, U is unitary S k S, S is upper triangular, AU = US: The QR-algorithm converges to the Schur decomposition.
15 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. Proof. S 0 Q = (S 0 σ I + σ I)Q = (Q R + σ I)Q = Q (R Q + σ I) = Q S
16 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. Proof. S k Q k = Q k S k AU k = S 0 Q Q 2... Q k = Q S Q 2... Q k = U k S k
17 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, Proof. (A σ k I)U k = U k (S k σ k I) = U k Q k R k
18 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, d) (A σ k I)U k = U k R k. Proof. U k (A σ k I) = R k U k
19 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, d) (A σ k I)U k = U k R k. Corollary. With x k U k e and τ k e R ke, we have (A σ k I)x k = τ k x k (the shifted power method). Proof. R k upper triangular R k e = τ k e.
20 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, d) (A σ k I)U k = U k R k. Corollary. With x k U k e and τ k e R ke, we have (A σ k I)x k = τ k x k (the shifted power method). With p(λ) (λ σ k )... (λ σ ), x k = τp(a)x 0 for some τ C.
21 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, d) (A σ k I)U k = U k R k. Corollary. With x k U k e and τ k e R ke, we have (A σ k I)x k = τ k x k (the shifted power method). Note that λ (k) x k Ax k = e U k AU ke = e S k e.
22 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, d) (A σ k I)U k = U k R k. Suppose v is the dominant eigenvector for A σi. With σ k = σ and x k U k e, for k, we have that (x k, v) 0, λ (k) e S ke λ, S k e λ (k) e 0, where λ is the eigenvalue of A associated v.
23 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, d) (A σ k I)U k = U k R k. Corollary. With x k U k e n and τ k e n R ke n, we have (A σ k I)x k = τ k x k (Shift & Invert). Proof. R k lower triangular R k e n = τ k e n.
24 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, d) (A σ k I)U k = U k R k. Corollary. With x k U k e n and τ k e n R ke n, we have (A σ k I)x k = τ k x k (Shift & Invert). Note that λ (k) x k Ax k = e n U k AU ke n = e n S k e n.
25 QR-algorithm Select U 0 unitary. Compute S 0 = U 0 AU 0 for k =, 2,,... do ) Select a shift σ k 2) Compute Q k unitary and R k upper triangular such that S k σ k I = Q k R k 3) Compute S k = R k Q k + σ k I 4) U k = U k Q k end for Lemma. a) U k unitary, b) AU k = U k S k. c) (A σ k I)U k = U k R k, d) (A σ k I)U k = U k R k. Suppose v is the dominant eigenvector for (A σi). With σ k = σ and x k U k e n, for k, we have that (x k, v) 0, λ (k) e n S ke n λ, S k e n λ (k) e n 0, where λ is the eigenvalue of A associated v.
26 Selecting shifts The QR-agorithm incorporates the Shift and Invert power method (for A ). Rayleigh Quotient Iteration is Shift and Invert with shifts equal to the the Rayleigh quotients, σ k = x k A x k. Theorem. The asymptotic convergence of RQI is quadratic. In this case, with x k = U k e n, σ k = e n S k e n. With The asymptotic convergence of this method is quadratic, we mean: the method produces sequences (x k ) that converge provided x 0 is close enough to some (limit) eigenvector, and for k large, the error reduces quadratically.
27 Selecting shifts The QR-agorithm incorporates the Shift and Invert power method (for A ). Rayleigh Quotient Iteration is Shift and Invert with shifts equal to the the Rayleigh quotients, σ k = x k A x k. Theorem. The asymptotic convergence of RQI is quadratic. In this case, with x k = U k e n, σ k = e n S k e n. RQI need converge, as the following example shows [ ] 0 Example. With A = and x 0 0 = e. RQI produces the sequence (x k ) = (e, e 2, e, e 2,...). Note that x k Ax k = 0. Observation. The shifts σ k = e n S k e n may lead to stagnation.
28 Selecting shifts The QR-agorithm incorporates the Shift and Invert power method (for A ). The Wilkinson shift is the absolute smallest eigenvalue of the 2 2 right lower block of S k.
29 Selecting shifts The QR-agorithm incorporates the Shift and Invert power method (for A ). The Wilkinson shift is the absolute smallest eigenvalue of the 2 2 right lower block of S k. Theorem. For σ k take the Wilkinson shift and take x k U k e n. Then, for some eigenpair (v, λ) of A, we have that (x k, v) 0, λ (k) e ns k e n λ, S k e n λ (k) e n 0. The convergence is quadratic (and cubic if A is Hermitian).
30 Selecting shifts The QR-agorithm incorporates the Shift and Invert power method (for A ). The Wilkinson shift is the absolute smallest eigenvalue of the 2 2 right lower block of S k. Theorem. For σ k take the Wilkinson shift and take x k U k e n. Then, for some eigenpair (v, λ) of A, we have that (x k, v) 0, λ (k) e ns k e n λ, S k e n λ (k) e n 0. The convergence is quadratic (and cubic if A is Hermitian). The QR-algorithm: if e n S k λ (k) e n 2 ɛ, then accept U k e n as an eigenvector of A deflate: delete the last row and column of S k and continu (the search for an eigenpair of the lower dimensional matrix).
31 Selecting shifts The QR-agorithm incorporates the Shift and Invert power method (for A ). The Wilkinson shift is the absolute smallest eigenvalue of the 2 2 right lower block of S k. Theorem. For σ k take the Wilkinson shift and take x k U k e n. Then, for some eigenpair (v, λ) of A, we have that (x k, v) 0, λ (k) e ns k e n λ, S k e n λ (k) e n 0. The convergence is quadratic (and cubic if A is Hermitian). The QR-algorithm: if e n S k λ (k) e n 2 ɛ, then accept U k e n as the nth Schur vector of A deflate: delete the last row and column of S k and continu (the search for an eigenpair of the lower dimensional matrix).
32 Deflation Consider the kth step of the QR-algorithm. Put u n U k e n. Note that Hence (I u n u n)u k = U k (I e n e n) (I u n u n )A(I u n u n )U k = U k (I e n e n )S k(i e n e n ). Deflating the nth Schur vector from A can easily be performed in the QR-algorithm: simply delete the last row and last column of the active matrix S k (assumming that U k e n is the nth Schur vector to required accuracy).
33 QR-algorithm Select U unitary. S = U AU, m = size(a, ), N = [ : m], I = I m. repeat until m = ) Select the Wilskinson shift σ 2) [Q, R] = qr(s σ I) 3) S RQ + σi 4) U(:, N) U(:, N)Q 5) if S(m, m ) ɛ S(m, m) %% Deflate m m, N [ : m], I I m S S(N, N) end if end repeat Theorem. U k U, U is unitary S k S, S is upper triangular, AU = US.
34 Observations. The QR-algorithm quickly converges towards to the eigenvalue as targeted by the Wilkinson shift (on average 8 steps of the QR algorithm seems to be required for accurate detection of the first eigenvalue). While converging to a target eigenvalue, other eigenvalues are also approximated. Therefore, the next eigenvalues are detected more quickly (from the 5th eigenvalue on, 2 steps appear to be sufficient). All eigenvalues are being computed (according to multiplicity). Computation of all eigenvalues (actually of the Schur decomposition of A) requires approximately 2n steps of the QR-algorithm. The order in which the eigenvalues are being computed can not be controled.
35 Initiation of the QR-algorithm Theorem. There is a unitary matrix U 0 (product of Householder reflections) such that S 0 U 0 AU 0 is upper Hessenberg. Start QR-alg. Bring A to upper Hessenberg form (i.e., S = S 0, U = U 0 ). Computation requires 4 3 n3 flop. Theorem. If S k is upper Hessenberg, then S k is upper Hessenberg. Moreover, if S k is m m, then Q k can be obtained as a product of m Givens rotations, i.e., rotations in the (j, j + ) plane. The steps 2 and 3 in the QR algorithm can be performed in (together) 3m 2 flop.
36 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) R = Here, = c s s c = G R 0 [ ] [ ] c s cos(φ ) sin(φ ) =. Empty matrix entries are 0. s c sin(φ ) cos(φ )
37 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) R 2 = Here, = c s s c [ ] [ ] c s cos(φ2 ) sin(φ 2 ) = s c sin(φ 2 ) cos(φ 2 ) = G 2R
38 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) R 3 = Here, = c s s c [ ] [ ] c s cos(φ3 ) sin(φ 3 ) = s c sin(φ 3 ) cos(φ 3 ) = G 3R 2
39 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) R 4 = Here, = [ ] [ ] c s cos(φ4 ) sin(φ 4 ) = s c sin(φ 4 ) cos(φ 4 ) c s s c = G 4R 3
40 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) R 4 = Here, = [ ] [ ] c s cos(φ4 ) sin(φ 4 ) = s c sin(φ 4 ) cos(φ 4 ) c s s c = G 4R 3
41 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) Then R = R n and Q = G n... G. Note. Q need not be formed explicitly: it suffices to store the sequences of cosines and sines.
42 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) Then R = R n and Q = G n... G, and S = RG... G n.
43 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) (G 3 (R 2 )G ) = c s s c c s s c
44 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) (G 3 (R 2 )G ) = c s s c c s s c
45 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) (G 3 (R 2 )G ) = c s s c Chasing the bulge.
46 Upper Hessenberg matrices Theorem. If S is an upper Hessenberg matrix and S = QR is the QR-decomposition, then Q is Hessenberg S RQ and S + σi are upper Hessenberg. Q can be obtained as the product of n Givens rotations: with R 0 S, R j = G j R j (j =,..., n ), where G j rotates in the (j, j+) plane (i.e., in span(e j, e j+ )) Then R = R n and Q = G n... G and S = RG... G n. Property. S =... G 4 (G 3 (G 2 G S)G )G 2... Only two sines and two cosines have to be stored.
47 Initiation of the QR-algorithm Theorem. There is a unitary matrix U 0 (product of Householder reflections) such that S 0 U 0 AU 0 is upper Hessenberg. Start QR-alg. Bring A to upper Hessenberg form (i.e., S = S 0, U = U 0 ). Computation requires 4 3 n3 flop. Theorem. If S k is upper Hessenberg, then S k is upper Hessenberg. Moreover, if S k is m m, then Q k can be obtained as a product of m Givens rotations, i.e., rotations in the (j, j + ) plane. The steps 2 and 3 in the QR algorithm can be performed in (together) 3m 2 flop.
48 Initiation of the QR-algorithm Theorem. There is a unitary matrix U 0 (product of Householder reflections) such that S 0 U 0 AU 0 is upper Hessenberg. Start QR-alg. Bring A to upper Hessenberg form (i.e., S = S 0, U = U 0 ). Computation requires 4 3 n3 flop. Theorem. If S k is upper Hessenberg, then S k is upper Hessenberg. Moreover, if S k is m m, then Q k can be obtained as a product of m Givens rotations, i.e., rotations in the (j, j + ) plane. The steps 2 and 3 in the QR algorithm can be performed in (together) 3m 2 flop. Observation. For computing the eigenvalues only, step 4 can be skipped.
49 Initiation of the QR-algorithm Theorem. There is a unitary matrix U 0 (product of Householder reflections) such that S 0 U 0 AU 0 is upper Hessenberg. Start QR-alg. Bring A to upper Hessenberg form (i.e., S = S 0, U = U 0 ). Computation requires 4 3 n3 flop. Theorem. If S k is upper Hessenberg, then S k is upper Hessenberg. Moreover, if S k is m m, then Q k can be obtained as a product of m Givens rotations, i.e., rotations in the (j, j + ) plane. The steps 2 and 3 in the QR algorithm can be performed in (together) 3m 2 flop. If the eigenvalue λ j is available, then the associated eigenvector can also be computed with Shift & Invert: solve (A λ j I)v j = e for v j. Note that the LU-decomposition can cheaply be computed if A is upper Hessenberg.
50 Initiation of the QR-algorithm Theorem. There is a unitary matrix U 0 (product of Householder reflections) such that S 0 U 0 AU 0 is upper Hessenberg. Start QR-alg. Bring A to upper Hessenberg form (i.e., S = S 0, U = U 0 ). Computation requires 4 3 n3 flop. Theorem. If S k is upper Hessenberg, then S k is upper Hessenberg. Moreover, if S k is m m, then Q k can be obtained as a product of m Givens rotations, i.e., rotations in the (j, j + ) plane. The steps 2 and 3 in the QR algorithm can be performed in (together) 3m 2 flop. Observation. The QR-algorithm requires approximately 8n 3 flop to compute the Schur decomposition to full accuracy.
51 Initiation of the QR-algorithm Theorem. There is a unitary matrix U 0 (product of Householder reflections) such that S 0 U 0 AU 0 is upper Hessenberg. Start QR-alg. Bring A to upper Hessenberg form (i.e., S = S 0, U = U 0 ). Computation requires 4 3 n3 flop. Theorem. If S k is upper Hessenberg, then S k is upper Hessenberg. Moreover, if S k is m m, then Q k can be obtained as a product of m Givens rotations, i.e., rotations in the (j, j + ) plane. The steps 2 and 3 in the QR algorithm can be performed in (together) 3m 2 flop. Observation. The QR-algorithm can not exploit any sparsity structure of A.
52 Benefits of the QR-RQ steps. The Shift & Invert power method is implicitly incorporated for one eigenvalue. The power method is implicitly incorporated for all other eigenvalues. Easy deflation is allowed. The computations are stable (when a stable qr-decomposition is used). When combined with an upper Hessenberg start: Upper Hessenberg structure is preserved, leading to realtively low computational costs per step. Simple error controle: the norm of the residual equals S k (n, n ). Effective shifts can easily be computed: with an eigenvalue of the 2 2 right lower block of S k, quadratic convergence is achieved and stagnation avoided.
53 Excellent performance of the QR algorithm relies on QR-RQ steps. (see previous transparant) A good shift strategy leading to fast convergence (quadratic and, if A is Hermitian, cubic) to one eigenvalue. While quickly converging to one eigenvalue, other eigenvalues are also approximated, yielding good starts for quick eigenvalue computation. Deflation allows a fast search for the next eigenvalue. Deflation is performed simply by deleting the last row and the last column of the active matrix. The upper Hessenberg structure is preserved, allowing relatively cheap QR steps.
54 Theorem. The QR-algorithm is stable: for the matrix U and the upper triangular matrix S we have that (A + A )U = US, U U I 2 us where A is an n n matrix such that A 2 u A 2
Orthogonal iteration to QR
Notes for 2016-03-09 Orthogonal iteration to QR The QR iteration is the workhorse for solving the nonsymmetric eigenvalue problem. Unfortunately, while the iteration itself is simple to write, the derivation
More informationComputation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated.
Math 504, Homework 5 Computation of eigenvalues and singular values Recall that your solutions to these questions will not be collected or evaluated 1 Find the eigenvalues and the associated eigenspaces
More informationComputing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm
Computing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo November 19, 2010 Today
More informationEIGENVALUE PROBLEMS. EIGENVALUE PROBLEMS p. 1/4
EIGENVALUE PROBLEMS EIGENVALUE PROBLEMS p. 1/4 EIGENVALUE PROBLEMS p. 2/4 Eigenvalues and eigenvectors Let A C n n. Suppose Ax = λx, x 0, then x is a (right) eigenvector of A, corresponding to the eigenvalue
More information13-2 Text: 28-30; AB: 1.3.3, 3.2.3, 3.4.2, 3.5, 3.6.2; GvL Eigen2
The QR algorithm The most common method for solving small (dense) eigenvalue problems. The basic algorithm: QR without shifts 1. Until Convergence Do: 2. Compute the QR factorization A = QR 3. Set A :=
More informationSolving large scale eigenvalue problems
arge scale eigenvalue problems, Lecture 5, March 23, 2016 1/30 Lecture 5, March 23, 2016: The QR algorithm II http://people.inf.ethz.ch/arbenz/ewp/ Peter Arbenz Computer Science Department, ETH Zürich
More informationComputing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm
Computing Eigenvalues and/or Eigenvectors;Part 2, The Power method and QR-algorithm Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo November 13, 2009 Today
More informationLecture 4 Basic Iterative Methods I
March 26, 2018 Lecture 4 Basic Iterative Methods I A The Power Method Let A be an n n with eigenvalues λ 1,...,λ n counted according to multiplicity. We assume the eigenvalues to be ordered in absolute
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationECS130 Scientific Computing Handout E February 13, 2017
ECS130 Scientific Computing Handout E February 13, 2017 1. The Power Method (a) Pseudocode: Power Iteration Given an initial vector u 0, t i+1 = Au i u i+1 = t i+1 / t i+1 2 (approximate eigenvector) θ
More informationEIGENVALUE PROBLEMS. Background on eigenvalues/ eigenvectors / decompositions. Perturbation analysis, condition numbers..
EIGENVALUE PROBLEMS Background on eigenvalues/ eigenvectors / decompositions Perturbation analysis, condition numbers.. Power method The QR algorithm Practical QR algorithms: use of Hessenberg form and
More informationIndex. for generalized eigenvalue problem, butterfly form, 211
Index ad hoc shifts, 165 aggressive early deflation, 205 207 algebraic multiplicity, 35 algebraic Riccati equation, 100 Arnoldi process, 372 block, 418 Hamiltonian skew symmetric, 420 implicitly restarted,
More informationSolving large scale eigenvalue problems
arge scale eigenvalue problems, Lecture 4, March 14, 2018 1/41 Lecture 4, March 14, 2018: The QR algorithm http://people.inf.ethz.ch/arbenz/ewp/ Peter Arbenz Computer Science Department, ETH Zürich E-mail:
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Reduction to Hessenberg and Tridiagonal Forms; Rayleigh Quotient Iteration Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical
More informationLecture 3: QR-Factorization
Lecture 3: QR-Factorization This lecture introduces the Gram Schmidt orthonormalization process and the associated QR-factorization of matrices It also outlines some applications of this factorization
More informationEECS 275 Matrix Computation
EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 http://faculty.ucmerced.edu/mhyang Lecture 17 1 / 26 Overview
More informationEigenvalues and eigenvectors
Chapter 6 Eigenvalues and eigenvectors An eigenvalue of a square matrix represents the linear operator as a scaling of the associated eigenvector, and the action of certain matrices on general vectors
More informationDiagonalizing Matrices
Diagonalizing Matrices Massoud Malek A A Let A = A k be an n n non-singular matrix and let B = A = [B, B,, B k,, B n ] Then A n A B = A A 0 0 A k [B, B,, B k,, B n ] = 0 0 = I n 0 A n Notice that A i B
More informationOrthogonal iteration to QR
Week 10: Wednesday and Friday, Oct 24 and 26 Orthogonal iteration to QR On Monday, we went through a somewhat roundabout algbraic path from orthogonal subspace iteration to the QR iteration. Let me start
More informationI. Multiple Choice Questions (Answer any eight)
Name of the student : Roll No : CS65: Linear Algebra and Random Processes Exam - Course Instructor : Prashanth L.A. Date : Sep-24, 27 Duration : 5 minutes INSTRUCTIONS: The test will be evaluated ONLY
More informationLecture 11. Linear systems: Cholesky method. Eigensystems: Terminology. Jacobi transformations QR transformation
Lecture Cholesky method QR decomposition Terminology Linear systems: Eigensystems: Jacobi transformations QR transformation Cholesky method: For a symmetric positive definite matrix, one can do an LU decomposition
More informationNumerical Methods I Eigenvalue Problems
Numerical Methods I Eigenvalue Problems Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 2nd, 2014 A. Donev (Courant Institute) Lecture
More informationNumerical Methods I: Eigenvalues and eigenvectors
1/25 Numerical Methods I: Eigenvalues and eigenvectors Georg Stadler Courant Institute, NYU stadler@cims.nyu.edu November 2, 2017 Overview 2/25 Conditioning Eigenvalues and eigenvectors How hard are they
More informationComputational Methods. Eigenvalues and Singular Values
Computational Methods Eigenvalues and Singular Values Manfred Huber 2010 1 Eigenvalues and Singular Values Eigenvalues and singular values describe important aspects of transformations and of data relations
More informationChasing the Bulge. Sebastian Gant 5/19/ The Reduction to Hessenberg Form 3
Chasing the Bulge Sebastian Gant 5/9/207 Contents Precursers and Motivation 2 The Reduction to Hessenberg Form 3 3 The Algorithm 5 4 Concluding Remarks 8 5 References 0 ntroduction n the early days of
More information5.3 The Power Method Approximation of the Eigenvalue of Largest Module
192 5 Approximation of Eigenvalues and Eigenvectors 5.3 The Power Method The power method is very good at approximating the extremal eigenvalues of the matrix, that is, the eigenvalues having largest and
More informationLast Time. Social Network Graphs Betweenness. Graph Laplacian. Girvan-Newman Algorithm. Spectral Bisection
Eigenvalue Problems Last Time Social Network Graphs Betweenness Girvan-Newman Algorithm Graph Laplacian Spectral Bisection λ 2, w 2 Today Small deviation into eigenvalue problems Formulation Standard eigenvalue
More informationThe QR Algorithm. Chapter The basic QR algorithm
Chapter 3 The QR Algorithm The QR algorithm computes a Schur decomposition of a matrix. It is certainly one of the most important algorithm in eigenvalue computations. However, it is applied to dense (or:
More informationSection 4.5 Eigenvalues of Symmetric Tridiagonal Matrices
Section 4.5 Eigenvalues of Symmetric Tridiagonal Matrices Key Terms Symmetric matrix Tridiagonal matrix Orthogonal matrix QR-factorization Rotation matrices (plane rotations) Eigenvalues We will now complete
More informationMatrix Algorithms. Volume II: Eigensystems. G. W. Stewart H1HJ1L. University of Maryland College Park, Maryland
Matrix Algorithms Volume II: Eigensystems G. W. Stewart University of Maryland College Park, Maryland H1HJ1L Society for Industrial and Applied Mathematics Philadelphia CONTENTS Algorithms Preface xv xvii
More information4.8 Arnoldi Iteration, Krylov Subspaces and GMRES
48 Arnoldi Iteration, Krylov Subspaces and GMRES We start with the problem of using a similarity transformation to convert an n n matrix A to upper Hessenberg form H, ie, A = QHQ, (30) with an appropriate
More informationNumerical Methods - Numerical Linear Algebra
Numerical Methods - Numerical Linear Algebra Y. K. Goh Universiti Tunku Abdul Rahman 2013 Y. K. Goh (UTAR) Numerical Methods - Numerical Linear Algebra I 2013 1 / 62 Outline 1 Motivation 2 Solving Linear
More informationMath 504 (Fall 2011) 1. (*) Consider the matrices
Math 504 (Fall 2011) Instructor: Emre Mengi Study Guide for Weeks 11-14 This homework concerns the following topics. Basic definitions and facts about eigenvalues and eigenvectors (Trefethen&Bau, Lecture
More informationLinear Algebra, part 3. Going back to least squares. Mathematical Models, Analysis and Simulation = 0. a T 1 e. a T n e. Anna-Karin Tornberg
Linear Algebra, part 3 Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2010 Going back to least squares (Sections 1.7 and 2.3 from Strang). We know from before: The vector
More informationEIGENVALUE PROBLEMS (EVP)
EIGENVALUE PROBLEMS (EVP) (Golub & Van Loan: Chaps 7-8; Watkins: Chaps 5-7) X.-W Chang and C. C. Paige PART I. EVP THEORY EIGENVALUES AND EIGENVECTORS Let A C n n. Suppose Ax = λx with x 0, then x is a
More informationCourse Notes: Week 1
Course Notes: Week 1 Math 270C: Applied Numerical Linear Algebra 1 Lecture 1: Introduction (3/28/11) We will focus on iterative methods for solving linear systems of equations (and some discussion of eigenvalues
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More informationLinear Algebra, part 3 QR and SVD
Linear Algebra, part 3 QR and SVD Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2012 Going back to least squares (Section 1.4 from Strang, now also see section 5.2). We
More informationImplementing the QR algorithm for efficiently computing matrix eigenvalues and eigenvectors
Implementing the QR algorithm for efficiently computing matrix eigenvalues and eigenvectors Final Degree Dissertation Degree in Mathematics Gorka Eraña Robles Supervisor: Ion Zaballa Tejada Leioa, June
More informationPreliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012
Instructions Preliminary/Qualifying Exam in Numerical Analysis (Math 502a) Spring 2012 The exam consists of four problems, each having multiple parts. You should attempt to solve all four problems. 1.
More informationThe QR Algorithm. Marco Latini. February 26, 2004
The QR Algorithm Marco Latini February 26, 2004 Contents 1 Introduction 2 2 The Power and Inverse Power method 2 2.1 The Power Method............................... 2 2.2 The Inverse power method...........................
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationMatrix decompositions
Matrix decompositions Zdeněk Dvořák May 19, 2015 Lemma 1 (Schur decomposition). If A is a symmetric real matrix, then there exists an orthogonal matrix Q and a diagonal matrix D such that A = QDQ T. The
More informationMath 411 Preliminaries
Math 411 Preliminaries Provide a list of preliminary vocabulary and concepts Preliminary Basic Netwon s method, Taylor series expansion (for single and multiple variables), Eigenvalue, Eigenvector, Vector
More informationMatrices, Moments and Quadrature, cont d
Jim Lambers CME 335 Spring Quarter 2010-11 Lecture 4 Notes Matrices, Moments and Quadrature, cont d Estimation of the Regularization Parameter Consider the least squares problem of finding x such that
More informationOrthogonal Transformations
Orthogonal Transformations Tom Lyche University of Oslo Norway Orthogonal Transformations p. 1/3 Applications of Qx with Q T Q = I 1. solving least squares problems (today) 2. solving linear equations
More informationNotes on Eigenvalues, Singular Values and QR
Notes on Eigenvalues, Singular Values and QR Michael Overton, Numerical Computing, Spring 2017 March 30, 2017 1 Eigenvalues Everyone who has studied linear algebra knows the definition: given a square
More informationECS231 Handout Subspace projection methods for Solving Large-Scale Eigenvalue Problems. Part I: Review of basic theory of eigenvalue problems
ECS231 Handout Subspace projection methods for Solving Large-Scale Eigenvalue Problems Part I: Review of basic theory of eigenvalue problems 1. Let A C n n. (a) A scalar λ is an eigenvalue of an n n A
More informationEigenvalues and Eigenvectors
Chapter 1 Eigenvalues and Eigenvectors Among problems in numerical linear algebra, the determination of the eigenvalues and eigenvectors of matrices is second in importance only to the solution of linear
More informationLecture 5 Basic Iterative Methods II
March 26, 2018 Lecture 5 Basic Iterative Methods II The following two exercises show that eigenvalue problems and linear systems of equations are closely related. From a certain point of view, they are
More informationApplied Mathematics 205. Unit V: Eigenvalue Problems. Lecturer: Dr. David Knezevic
Applied Mathematics 205 Unit V: Eigenvalue Problems Lecturer: Dr. David Knezevic Unit V: Eigenvalue Problems Chapter V.4: Krylov Subspace Methods 2 / 51 Krylov Subspace Methods In this chapter we give
More informationThe QR Decomposition
The QR Decomposition We have seen one major decomposition of a matrix which is A = LU (and its variants) or more generally PA = LU for a permutation matrix P. This was valid for a square matrix and aided
More informationOrthonormal Transformations and Least Squares
Orthonormal Transformations and Least Squares Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo October 30, 2009 Applications of Qx with Q T Q = I 1. solving
More informationLecture # 11 The Power Method for Eigenvalues Part II. The power method find the largest (in magnitude) eigenvalue of. A R n n.
Lecture # 11 The Power Method for Eigenvalues Part II The power method find the largest (in magnitude) eigenvalue of It makes two assumptions. 1. A is diagonalizable. That is, A R n n. A = XΛX 1 for some
More information5 Selected Topics in Numerical Linear Algebra
5 Selected Topics in Numerical Linear Algebra In this chapter we will be concerned mostly with orthogonal factorizations of rectangular m n matrices A The section numbers in the notes do not align with
More informationAlternative correction equations in the Jacobi-Davidson method
Chapter 2 Alternative correction equations in the Jacobi-Davidson method Menno Genseberger and Gerard Sleijpen Abstract The correction equation in the Jacobi-Davidson method is effective in a subspace
More informationLecture 2: Numerical linear algebra
Lecture 2: Numerical linear algebra QR factorization Eigenvalue decomposition Singular value decomposition Conditioning of a problem Floating point arithmetic and stability of an algorithm Linear algebra
More informationKrylov Subspace Methods to Calculate PageRank
Krylov Subspace Methods to Calculate PageRank B. Vadala-Roth REU Final Presentation August 1st, 2013 How does Google Rank Web Pages? The Web The Graph (A) Ranks of Web pages v = v 1... Dominant Eigenvector
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra)
AMS526: Numerical Analysis I (Numerical Linear Algebra) Lecture 16: Rayleigh Quotient Iteration Xiangmin Jiao SUNY Stony Brook Xiangmin Jiao Numerical Analysis I 1 / 10 Solving Eigenvalue Problems All
More informationOrthonormal Transformations
Orthonormal Transformations Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo October 25, 2010 Applications of transformation Q : R m R m, with Q T Q = I 1.
More informationOrthogonal iteration revisited
Week 10 11: Friday, Oct 30 and Monday, Nov 2 Orthogonal iteration revisited Last time, we described a generalization of the power methods to compute invariant subspaces. That is, starting from some initial
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 9
STAT 309: MATHEMATICAL COMPUTATIONS I FALL 2018 LECTURE 9 1. qr and complete orthogonal factorization poor man s svd can solve many problems on the svd list using either of these factorizations but they
More informationPolynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems
Polynomial Jacobi Davidson Method for Large/Sparse Eigenvalue Problems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan April 28, 2011 T.M. Huang (Taiwan Normal Univ.)
More informationKrylov Subspace Methods for Large/Sparse Eigenvalue Problems
Krylov Subspace Methods for Large/Sparse Eigenvalue Problems Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan April 17, 2012 T.-M. Huang (Taiwan Normal University) Krylov
More information4.2. ORTHOGONALITY 161
4.2. ORTHOGONALITY 161 Definition 4.2.9 An affine space (E, E ) is a Euclidean affine space iff its underlying vector space E is a Euclidean vector space. Given any two points a, b E, we define the distance
More informationMath 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm
Math 405: Numerical Methods for Differential Equations 2016 W1 Topics 10: Matrix Eigenvalues and the Symmetric QR Algorithm References: Trefethen & Bau textbook Eigenvalue problem: given a matrix A, find
More informationLecture 8 : Eigenvalues and Eigenvectors
CPS290: Algorithmic Foundations of Data Science February 24, 2017 Lecture 8 : Eigenvalues and Eigenvectors Lecturer: Kamesh Munagala Scribe: Kamesh Munagala Hermitian Matrices It is simpler to begin with
More informationBindel, Fall 2016 Matrix Computations (CS 6210) Notes for
1 Power iteration Notes for 2016-10-17 In most introductory linear algebra classes, one computes eigenvalues as roots of a characteristic polynomial. For most problems, this is a bad idea: the roots of
More informationThe German word eigen is cognate with the Old English word āgen, which became owen in Middle English and own in modern English.
Chapter 4 EIGENVALUE PROBLEM The German word eigen is cognate with the Old English word āgen, which became owen in Middle English and own in modern English. 4.1 Mathematics 4.2 Reduction to Upper Hessenberg
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationLinear Algebra in Actuarial Science: Slides to the lecture
Linear Algebra in Actuarial Science: Slides to the lecture Fall Semester 2010/2011 Linear Algebra is a Tool-Box Linear Equation Systems Discretization of differential equations: solving linear equations
More informationAMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences)
AMS526: Numerical Analysis I (Numerical Linear Algebra for Computational and Data Sciences) Lecture 19: Computing the SVD; Sparse Linear Systems Xiangmin Jiao Stony Brook University Xiangmin Jiao Numerical
More informationDirect methods for symmetric eigenvalue problems
Direct methods for symmetric eigenvalue problems, PhD McMaster University School of Computational Engineering and Science February 4, 2008 1 Theoretical background Posing the question Perturbation theory
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationHOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION)
HOMEWORK PROBLEMS FROM STRANG S LINEAR ALGEBRA AND ITS APPLICATIONS (4TH EDITION) PROFESSOR STEVEN MILLER: BROWN UNIVERSITY: SPRING 2007 1. CHAPTER 1: MATRICES AND GAUSSIAN ELIMINATION Page 9, # 3: Describe
More informationIntroduction to Applied Linear Algebra with MATLAB
Sigam Series in Applied Mathematics Volume 7 Rizwan Butt Introduction to Applied Linear Algebra with MATLAB Heldermann Verlag Contents Number Systems and Errors 1 1.1 Introduction 1 1.2 Number Representation
More informationLinear Analysis Lecture 16
Linear Analysis Lecture 16 The QR Factorization Recall the Gram-Schmidt orthogonalization process. Let V be an inner product space, and suppose a 1,..., a n V are linearly independent. Define q 1,...,
More information1 Last time: least-squares problems
MATH Linear algebra (Fall 07) Lecture Last time: least-squares problems Definition. If A is an m n matrix and b R m, then a least-squares solution to the linear system Ax = b is a vector x R n such that
More informationEigenvalue Problems. Eigenvalue problems occur in many areas of science and engineering, such as structural analysis
Eigenvalue Problems Eigenvalue problems occur in many areas of science and engineering, such as structural analysis Eigenvalues also important in analyzing numerical methods Theory and algorithms apply
More informationNumerical Linear Algebra
Numerical Linear Algebra Decompositions, numerical aspects Gerard Sleijpen and Martin van Gijzen September 27, 2017 1 Delft University of Technology Program Lecture 2 LU-decomposition Basic algorithm Cost
More informationProgram Lecture 2. Numerical Linear Algebra. Gaussian elimination (2) Gaussian elimination. Decompositions, numerical aspects
Numerical Linear Algebra Decompositions, numerical aspects Program Lecture 2 LU-decomposition Basic algorithm Cost Stability Pivoting Cholesky decomposition Sparse matrices and reorderings Gerard Sleijpen
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 4 Eigenvalue Problems Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction
More informationMATRICES ARE SIMILAR TO TRIANGULAR MATRICES
MATRICES ARE SIMILAR TO TRIANGULAR MATRICES 1 Complex matrices Recall that the complex numbers are given by a + ib where a and b are real and i is the imaginary unity, ie, i 2 = 1 In what we describe below,
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More information6.4 Krylov Subspaces and Conjugate Gradients
6.4 Krylov Subspaces and Conjugate Gradients Our original equation is Ax = b. The preconditioned equation is P Ax = P b. When we write P, we never intend that an inverse will be explicitly computed. P
More informationThis can be accomplished by left matrix multiplication as follows: I
1 Numerical Linear Algebra 11 The LU Factorization Recall from linear algebra that Gaussian elimination is a method for solving linear systems of the form Ax = b, where A R m n and bran(a) In this method
More informationThe Singular Value Decomposition and Least Squares Problems
The Singular Value Decomposition and Least Squares Problems Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 27, 2009 Applications of SVD solving
More informationNumerical Methods for Solving Large Scale Eigenvalue Problems
Peter Arbenz Computer Science Department, ETH Zürich E-mail: arbenz@inf.ethz.ch arge scale eigenvalue problems, Lecture 2, February 28, 2018 1/46 Numerical Methods for Solving Large Scale Eigenvalue Problems
More informationMatrix Theory, Math6304 Lecture Notes from September 27, 2012 taken by Tasadduk Chowdhury
Matrix Theory, Math634 Lecture Notes from September 27, 212 taken by Tasadduk Chowdhury Last Time (9/25/12): QR factorization: any matrix A M n has a QR factorization: A = QR, whereq is unitary and R is
More informationEigenvalue and Eigenvector Problems
Eigenvalue and Eigenvector Problems An attempt to introduce eigenproblems Radu Trîmbiţaş Babeş-Bolyai University April 8, 2009 Radu Trîmbiţaş ( Babeş-Bolyai University) Eigenvalue and Eigenvector Problems
More informationNumerical methods for eigenvalue problems
Numerical methods for eigenvalue problems D. Löchel Supervisors: M. Hochbruck und M. Tokar Mathematisches Institut Heinrich-Heine-Universität Düsseldorf GRK 1203 seminar february 2008 Outline Introduction
More informationarxiv: v1 [math.na] 5 May 2011
ITERATIVE METHODS FOR COMPUTING EIGENVALUES AND EIGENVECTORS MAYSUM PANJU arxiv:1105.1185v1 [math.na] 5 May 2011 Abstract. We examine some numerical iterative methods for computing the eigenvalues and
More informationThe University of Texas at Austin Department of Electrical and Computer Engineering. EE381V: Large Scale Learning Spring 2013.
The University of Texas at Austin Department of Electrical and Computer Engineering EE381V: Large Scale Learning Spring 2013 Assignment Two Caramanis/Sanghavi Due: Tuesday, Feb. 19, 2013. Computational
More informationMATRIX AND LINEAR ALGEBR A Aided with MATLAB
Second Edition (Revised) MATRIX AND LINEAR ALGEBR A Aided with MATLAB Kanti Bhushan Datta Matrix and Linear Algebra Aided with MATLAB Second Edition KANTI BHUSHAN DATTA Former Professor Department of Electrical
More informationG1110 & 852G1 Numerical Linear Algebra
The University of Sussex Department of Mathematics G & 85G Numerical Linear Algebra Lecture Notes Autumn Term Kerstin Hesse (w aw S w a w w (w aw H(wa = (w aw + w Figure : Geometric explanation of the
More informationNumerical Methods in Matrix Computations
Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices
More informationMAA507, Power method, QR-method and sparse matrix representation.
,, and representation. February 11, 2014 Lecture 7: Overview, Today we will look at:.. If time: A look at representation and fill in. Why do we need numerical s? I think everyone have seen how time consuming
More informationIassume you share the view that the rapid
the Top T HEME ARTICLE THE QR ALGORITHM After a brief sketch of the early days of eigenvalue hunting, the author describes the QR Algorithm and its major virtues. The symmetric case brings with it guaranteed
More information1 Number Systems and Errors 1
Contents 1 Number Systems and Errors 1 1.1 Introduction................................ 1 1.2 Number Representation and Base of Numbers............. 1 1.2.1 Normalized Floating-point Representation...........
More informationACM106a - Homework 4 Solutions
ACM106a - Homework 4 Solutions prepared by Svitlana Vyetrenko November 17, 2006 1. Problem 1: (a) Let A be a normal triangular matrix. Without the loss of generality assume that A is upper triangular,
More information