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

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

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 );

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.

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.

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).

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);

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).

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.

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.

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.

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.

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.

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.

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.

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

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

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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).

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).

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).

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.

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.

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.

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 )

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

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

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

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

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.

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.

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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