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QR-method lecture 2 SF2524 - Matrix Computations for Large-scale Systems QR-method lecture 2 1 / 23 Agenda QR-method 1 Decompositions (previous lecture) I I I 2 3 Jordan form Schur decomposition QR-factorization Basic QR-method Improvement 1: Two-phase approach I I Hessenberg reduction (previous lecture) Hessenberg QR-method 4 Improvement 2: Acceleration with shifts 5 Convergence theory QR-method lecture 2 2 / 23 Agenda QR-method 1 Decompositions (previous lecture) I I I 2 3 Jordan form Schur decomposition QR-factorization Basic QR-method Improvement 1: Two-phase approach I I Hessenberg reduction (previous lecture) Hessenberg QR-method 4 Improvement 2: Acceleration with shifts 5 Convergence theory Reading instructions Point 1: TB Lecture 24 Points 2-4: Lecture notes “QR-method” on course web page Point 5: TB Chapter 28 (Extra reading: TB Chapter 25-26, 28-29) QR-method lecture 2 2 / 23 Basic QR-method (previous lecture) Basic QR-method = basic QR-algorithm Simple basic idea: Let A0 = A and iterate: Compute QR-factorization of Ak = QR Set Ak+1 = RQ. QR-method lecture 2 3 / 23 Basic QR-method (previous lecture) Basic QR-method = basic QR-algorithm Simple basic idea: Let A0 = A and iterate: Compute QR-factorization of Ak = QR Set Ak+1 = RQ. Disadvantages Computing a QR-factorization is quite expensive. One iteration of the basic QR-method O(m3 ). QR-method lecture 2 3 / 23 Basic QR-method (previous lecture) Basic QR-method = basic QR-algorithm Simple basic idea: Let A0 = A and iterate: Compute QR-factorization of Ak = QR Set Ak+1 = RQ. Disadvantages Computing a QR-factorization is quite expensive. One iteration of the basic QR-method O(m3 ). The method often requires many iterations. QR-method lecture 2 3 / 23 Improvement 1: Two-phase approach We will separate the computation into two phases: × × × × × × × × × × × × × × × × × × × × × × × × × × × → Phase 1 × × × × × × × × × × × × QR-method lecture 2 × × × × × → Phase 2 × × × × × × × × × × × × × × × 4 / 23 Improvement 1: Two-phase approach We will separate the computation into two phases: × × × × × × × × × × × × × × × × × × × × × × × × × × × → Phase 1 × × × × × × × × × × × × × × × × × → Phase 2 × × × × × × × × × × × × × × × Phases Phase 1: Reduce the matrix to a Hessenberg with similarity transformations (Section 2.2.1 in lecture notes) QR-method lecture 2 4 / 23 Improvement 1: Two-phase approach We will separate the computation into two phases: × × × × × × × × × × × × × × × × × × × × × × × × × × × → Phase 1 × × × × × × × × × × × × × × × × × → Phase 2 × × × × × × × × × × × × × × × Phases Phase 1: Reduce the matrix to a Hessenberg with similarity transformations (Section 2.2.1 in lecture notes) Phase 2: Specialize the QR-method to Hessenberg matrices (Section 2.2.2 in lecture notes) QR-method lecture 2 4 / 23 Phase 1: Hessenberg reduction (previous lecture) Phase 1: Hessenberg reduction Compute unitary P and Hessenberg matrix H such that A = PHP ∗ QR-method lecture 2 5 / 23 Phase 1: Hessenberg reduction (previous lecture) Phase 1: Hessenberg reduction Compute unitary P and Hessenberg matrix H such that A = PHP ∗ Unlike the Schur factorization, this can be computed with a finite number of operations. QR-method lecture 2 5 / 23 Phase 1: Hessenberg reduction (previous lecture) Phase 1: Hessenberg reduction Compute unitary P and Hessenberg matrix H such that A = PHP ∗ Unlike the Schur factorization, this can be computed with a finite number of operations. Idea Householder reflector: P = I − 2uu ∗ where u ∈ Cm and kuk = 1, Apply one Householder reflector at a time to eliminate the column by column. QR-method lecture 2 5 / 23 Phase 2: Hessenberg QR-method A QR-step on a Hessenberg matrix is a Hessenberg matrix: * Matlab demo showing QR-step: hessenberg is hessenberg.m * QR-method lecture 2 6 / 23 Phase 2: Hessenberg QR-method A QR-step on a Hessenberg matrix is a Hessenberg matrix: * Matlab demo showing QR-step: hessenberg is hessenberg.m * Theorem (Theorem 2.2.4) If the basic QR-method is applied to a Hessenberg matrix, then all iterates Ak are Hessenberg matrices. QR-method lecture 2 6 / 23 Phase 2: Hessenberg QR-method A QR-step on a Hessenberg matrix is a Hessenberg matrix: * Matlab demo showing QR-step: hessenberg is hessenberg.m * Theorem (Theorem 2.2.4) If the basic QR-method is applied to a Hessenberg matrix, then all iterates Ak are Hessenberg matrices. Recall: basic QR-step is O(m3 ). QR-method lecture 2 6 / 23 Phase 2: Hessenberg QR-method A QR-step on a Hessenberg matrix is a Hessenberg matrix: * Matlab demo showing QR-step: hessenberg is hessenberg.m * Theorem (Theorem 2.2.4) If the basic QR-method is applied to a Hessenberg matrix, then all iterates Ak are Hessenberg matrices. Recall: basic QR-step is O(m3 ). Hessenberg structure can be exploited such that we can carry out a QR-step with less operations. QR-method lecture 2 6 / 23 Definition (Givens rotation) The matrix G (i, j, c, s) ∈ Rn×n corresponding to a Givens rotation is defined by I G (i, j, c, s) := −s c , I s c I which deviates from identity at row and column i and j. QR-method lecture 2 7 / 23 Definition (Givens rotation) The matrix G (i, j, c, s) ∈ Rn×n corresponding to a Givens rotation is defined by I G (i, j, c, s) := −s c , I s c I which deviates from identity at row and column i and j. ej Gx θ x ei QR-method lecture 2 7 / 23 Definition (Givens rotation) The matrix G (i, j, c, s) ∈ Rn×n corresponding to a Givens rotation is defined by I G (i, j, c, s) := −s c , I s c I which deviates from identity at row and column i and j. ej Properties G ∗ = G −1 Gx θ x ei QR-method lecture 2 7 / 23 Definition (Givens rotation) The matrix G (i, j, c, s) ∈ Rn×n corresponding to a Givens rotation is defined by I G (i, j, c, s) := −s c , I s c I which deviates from identity at row and column i and j. ej Properties G ∗ = G −1 Gz can be computed with O(1) operations Gx θ x ei QR-method lecture 2 7 / 23 Definition (Givens rotation) The matrix G (i, j, c, s) ∈ Rn×n corresponding to a Givens rotation is defined by I G (i, j, c, s) := −s c , I s c I which deviates from identity at row and column i and j. ej Properties G ∗ = G −1 Gz can be computed with O(1) operations Gx θ x ··· ei QR-method lecture 2 7 / 23 The Q-matrix in the QR-factorization of a Hessenberg matrix can be factorized as a product of m − 1 Givens rotators. QR-method lecture 2 8 / 23 The Q-matrix in the QR-factorization of a Hessenberg matrix can be factorized as a product of m − 1 Givens rotators. Theorem (Theorem 2.2.6) Suppose A ∈ Cm×m is a Hessenberg matrix. Let Hi be generated as follows H1 = A Hi+1 = GiT Hi , i = 1, . . . , m − 1 q where Gi = G (i, i + 1, (Hi )i,i /ri , (Hi )i+1,i /ri ) and ri = (Hi )2i,i + (Hi )2i+1,i and we assume ri 6= 0. Then, Hn is upper triangular and A = (G1 G2 · · · Gm−1 )Hn = QR is a QR-factorization of A. QR-method lecture 2 8 / 23 The Q-matrix in the QR-factorization of a Hessenberg matrix can be factorized as a product of m − 1 Givens rotators. Theorem (Theorem 2.2.6) Suppose A ∈ Cm×m is a Hessenberg matrix. Let Hi be generated as follows H1 = A Hi+1 = GiT Hi , i = 1, . . . , m − 1 q where Gi = G (i, i + 1, (Hi )i,i /ri , (Hi )i+1,i /ri ) and ri = (Hi )2i,i + (Hi )2i+1,i and we assume ri 6= 0. Then, Hn is upper triangular and A = (G1 G2 · · · Gm−1 )Hn = QR is a QR-factorization of A. Proof idea: Only one rotator required to bring one column of a Hessenberg matrix to a triangular. * Matlab: Explicit QR-factorization of Hessenberg qrg ivens.m ∗ QR-method lecture 2 8 / 23 Idea of Hessenberg QR-method: Do not explicitly compute the Q-matrix but only implicitly apply the Givens rotators QR-method lecture 2 9 / 23 Idea of Hessenberg QR-method: Do not explicitly compute the Q-matrix but only implicitly apply the Givens rotators: Let Ak−1 = (G1 G2 · · · Gm−1 )Rm and Ak = Rm (G1 G2 · · · Gm−1 ) = QR-method lecture 2 9 / 23 Idea of Hessenberg QR-method: Do not explicitly compute the Q-matrix but only implicitly apply the Givens rotators: Let Ak−1 = (G1 G2 · · · Gm−1 )Rm and Ak = Rm (G1 G2 · · · Gm−1 ) = (· · · ((Rm G1 )G2 ) · · · )Gm QR-method lecture 2 9 / 23 Idea of Hessenberg QR-method: Do not explicitly compute the Q-matrix but only implicitly apply the Givens rotators: Let Ak−1 = (G1 G2 · · · Gm−1 )Rm and Ak = Rm (G1 G2 · · · Gm−1 ) = (· · · ((Rm G1 )G2 ) · · · )Gm Complexity of one QR-step for a Hessenberg matrix We need to apply 2(m − 1) givens rotators to compute one QR-step. QR-method lecture 2 9 / 23 Idea of Hessenberg QR-method: Do not explicitly compute the Q-matrix but only implicitly apply the Givens rotators: Let Ak−1 = (G1 G2 · · · Gm−1 )Rm and Ak = Rm (G1 G2 · · · Gm−1 ) = (· · · ((Rm G1 )G2 ) · · · )Gm Complexity of one QR-step for a Hessenberg matrix We need to apply 2(m − 1) givens rotators to compute one QR-step. One givens rotator applied to a vector can be computed in O(1) operations. QR-method lecture 2 9 / 23 Idea of Hessenberg QR-method: Do not explicitly compute the Q-matrix but only implicitly apply the Givens rotators: Let Ak−1 = (G1 G2 · · · Gm−1 )Rm and Ak = Rm (G1 G2 · · · Gm−1 ) = (· · · ((Rm G1 )G2 ) · · · )Gm Complexity of one QR-step for a Hessenberg matrix We need to apply 2(m − 1) givens rotators to compute one QR-step. One givens rotator applied to a vector can be computed in O(1) operations. One givens rotator applied to matrix can be computed in O(m) operations. QR-method lecture 2 9 / 23 Idea of Hessenberg QR-method: Do not explicitly compute the Q-matrix but only implicitly apply the Givens rotators: Let Ak−1 = (G1 G2 · · · Gm−1 )Rm and Ak = Rm (G1 G2 · · · Gm−1 ) = (· · · ((Rm G1 )G2 ) · · · )Gm Complexity of one QR-step for a Hessenberg matrix We need to apply 2(m − 1) givens rotators to compute one QR-step. One givens rotator applied to a vector can be computed in O(1) operations. One givens rotator applied to matrix can be computed in O(m) operations. ⇒ the complexity of one Hessenberg QR step = O(m2 ) QR-method lecture 2 9 / 23 Givens rotators only modify very few elements. Several optimizations possible. ⇒ QR-method lecture 2 10 / 23 Show animation again: http://www.youtube.com/watch?v=qmgxzsWWsNc QR-method lecture 2 11 / 23 Show animation again: http://www.youtube.com/watch?v=qmgxzsWWsNc Acceleration still remains QR-method lecture 2 11 / 23 Outline: Basic QR-method Improvement 1: Two-phase approach I I Hessenberg reduction Hessenberg QR-method Improvement 2: Acceleration with shifts Convergence theory QR-method lecture 2 12 / 23 Improvement 2: Acceleration with shifts (Section 2.3) Shifted QR-method One step of shifted QR-method: Let Hk = H QR-method lecture 2 13 / 23 Improvement 2: Acceleration with shifts (Section 2.3) Shifted QR-method One step of shifted QR-method: Let Hk = H H − µI = QR H̄ = RQ + µI and Hk+1 := H̄. QR-method lecture 2 13 / 23 Improvement 2: Acceleration with shifts (Section 2.3) Shifted QR-method One step of shifted QR-method: Let Hk = H H − µI = QR H̄ = RQ + µI and Hk+1 := H̄. Note: Hk+1 = H̄ = RQ + µI = QR-method lecture 2 13 / 23 Improvement 2: Acceleration with shifts (Section 2.3) Shifted QR-method One step of shifted QR-method: Let Hk = H H − µI = QR H̄ = RQ + µI and Hk+1 := H̄. Note: Hk+1 = H̄ = RQ + µI = Q T (H − µI ))Q + µI QR-method lecture 2 13 / 23 Improvement 2: Acceleration with shifts (Section 2.3) Shifted QR-method One step of shifted QR-method: Let Hk = H H − µI = QR H̄ = RQ + µI and Hk+1 := H̄. Note: Hk+1 = H̄ = RQ + µI = Q T (H − µI ))Q + µI = Q T Hk Q QR-method lecture 2 13 / 23 Improvement 2: Acceleration with shifts (Section 2.3) Shifted QR-method One step of shifted QR-method: Let Hk = H H − µI = QR H̄ = RQ + µI and Hk+1 := H̄. Note: Hk+1 = H̄ = RQ + µI = Q T (H − µI ))Q + µI = Q T Hk Q ⇒ One step of shifted QR-method is a similarity transformation, with a different Q matrix. QR-method lecture 2 13 / 23 Idealized situation: Let µ = λ(H) Suppose µ is an eigenvalue: ⇒ H − µI is a singular Hessenberg matrix. QR-method lecture 2 14 / 23 Idealized situation: Let µ = λ(H) Suppose µ is an eigenvalue: ⇒ H − µI is a singular Hessenberg matrix. QR-factorization of singular Hessenberg matrices (Lemma 2.3.1) The R-matrix in the QR-decomposition of a singular unreduced Hessenberg matrix has the structure × × × × × × × × × × × × R= . × × 0 QR-method lecture 2 14 / 23 Idealized situation: Let µ = λ(H) Suppose µ is an eigenvalue: ⇒ H − µI is a singular Hessenberg matrix. QR-factorization of singular Hessenberg matrices (Lemma 2.3.1) The R-matrix in the QR-decomposition of a singular unreduced Hessenberg matrix has the structure × × × × × × × × × × × × R= . × × 0 * Matlab demo: Show QR-factorization of singular Hessenberg matrix in matlab * QR-method lecture 2 14 / 23 Shifted QR for exact shift: µ = λ If µ = λ is an eigenvalue of H, then H − µI is singular. Suppose Q, R a QR-factorization of a Hessenberg matrix and × × × × × R= × × × × × × Then, × × × 0 . * Prove on blackboard * QR-method lecture 2 15 / 23 Shifted QR for exact shift: µ = λ If µ = λ is an eigenvalue of H, then H − µI is singular. Suppose Q, R a QR-factorization of a Hessenberg matrix and × × × × × × R= × × × × × × × × 0 . Then, * Prove on blackboard * × × RQ = × × × × × × × × × × × × × × × 0 and QR-method lecture 2 15 / 23 Shifted QR for exact shift: µ = λ If µ = λ is an eigenvalue of H, then H − µI is singular. Suppose Q, R a QR-factorization of a Hessenberg matrix and × × × × × × R= × × × × × × × × 0 . Then, * Prove on blackboard * × × RQ = and × × × × × H̄ = RQ + λI = × × × × × × × × × × × QR-method lecture 2 × × × × × × × × 0 × × × × × × × × λ . 15 / 23 Shifted QR for exact shift: µ = λ If µ = λ is an eigenvalue of H, then H − µI is singular. Suppose Q, R a QR-factorization of a Hessenberg matrix and × × × × × × R= × × × × × × × × 0 . Then, * Prove on blackboard * × × RQ = and × × × × × H̄ = RQ + λI = × × × × × × × × × × × × × × × × × × × 0 × × × × × × × × λ . ⇒ λ is an eigenvalue of H̄. QR-method lecture 2 15 / 23 More precisely: Lemma (Lemma 2.3.2) Suppose λ is an eigenvalue of the Hessenberg matrix H. Let H̄ be the result of one shifted QR-step. Then, h̄n,n−1 = 0 h̄n,n = λ. QR-method lecture 2 16 / 23 Select the shift How to select the shifts? Shifted QR-method with µ = λ computes an eigenvalue in one step. QR-method lecture 2 17 / 23 Select the shift How to select the shifts? Shifted QR-method with µ = λ computes an eigenvalue in one step. The exact eigenvalue not available. How to select the shift? QR-method lecture 2 17 / 23 Select the shift How to select the shifts? Shifted QR-method with µ = λ computes an eigenvalue in one step. The exact eigenvalue not available. How to select the shift? Rayleigh shifts If we are close to convergence the diagonal element will be an approximate eigenvalue. QR-method lecture 2 17 / 23 Select the shift How to select the shifts? Shifted QR-method with µ = λ computes an eigenvalue in one step. The exact eigenvalue not available. How to select the shift? Rayleigh shifts If we are close to convergence the diagonal element will be an approximate eigenvalue. Rayleigh shifts: µ := rm,m . QR-method lecture 2 17 / 23 Select the shift How to select the shifts? Shifted QR-method with µ = λ computes an eigenvalue in one step. The exact eigenvalue not available. How to select the shift? Rayleigh shifts If we are close to convergence the diagonal element will be an approximate eigenvalue. Rayleigh shifts: µ := rm,m . Explanation The QR-method can be interpreted as equivalent to variant of Power Method applied to A. (Will be shown later) QR-method lecture 2 17 / 23 Select the shift How to select the shifts? Shifted QR-method with µ = λ computes an eigenvalue in one step. The exact eigenvalue not available. How to select the shift? Rayleigh shifts If we are close to convergence the diagonal element will be an approximate eigenvalue. Rayleigh shifts: µ := rm,m . Explanation The QR-method can be interpreted as equivalent to variant of Power Method applied to A. (Will be shown later) The QR-method can be interpreted as equivalent to variant of Power Method applied to A−1 . (Proof sketched in TB Chapter 29) ⇒ Rayleigh shifts can be interpreted as Rayleigh quotient iteration. QR-method lecture 2 17 / 23 Deflation QR-step on reduced Hessenberg matrix Suppose H0 H 1 H= , 0 H3 where H3 is upper triangular and let H̄0 H̄1 H̄ = ¯ , H2 H̄3 be the result of one (shifted) QR-step. QR-method lecture 2 18 / 23 Deflation QR-step on reduced Hessenberg matrix Suppose H0 H 1 H= , 0 H3 where H3 is upper triangular and let H̄0 H̄1 H̄ = ¯ , H2 H̄3 be the result of one (shifted) QR-step. Then, H¯2 = 0, H¯3 = H3 and H̄0 is the result of one (shifted) QR-step applied to H0 . QR-method lecture 2 18 / 23 Deflation QR-step on reduced Hessenberg matrix Suppose H0 H 1 H= , 0 H3 where H3 is upper triangular and let H̄0 H̄1 H̄ = ¯ , H2 H̄3 be the result of one (shifted) QR-step. Then, H¯2 = 0, H¯3 = H3 and H̄0 is the result of one (shifted) QR-step applied to H0 . * show proof * QR-method lecture 2 18 / 23 Deflation QR-step on reduced Hessenberg matrix Suppose H0 H 1 H= , 0 H3 where H3 is upper triangular and let H̄0 H̄1 H̄ = ¯ , H2 H̄3 be the result of one (shifted) QR-step. Then, H¯2 = 0, H¯3 = H3 and H̄0 is the result of one (shifted) QR-step applied to H0 . * show proof * ⇒ We can reduce the active matrix when an eigenvalue is converged. QR-method lecture 2 18 / 23 Deflation QR-step on reduced Hessenberg matrix Suppose H0 H 1 H= , 0 H3 where H3 is upper triangular and let H̄0 H̄1 H̄ = ¯ , H2 H̄3 be the result of one (shifted) QR-step. Then, H¯2 = 0, H¯3 = H3 and H̄0 is the result of one (shifted) QR-step applied to H0 . * show proof * ⇒ We can reduce the active matrix when an eigenvalue is converged. This is called deflation. QR-method lecture 2 18 / 23 Rayleigh shifts can be combined with deflation ⇒ * show Hessenberg qr with shifts in matlab * * http://www.youtube.com/watch?v=qmgxzsWWsNc * QR-method lecture 2 19 / 23 Outline: Basic QR-method Improvement 1: Two-phase approach I I Hessenberg reduction Hessenberg QR-method Improvement 2: Acceleration with shifts Convergence theory QR-method lecture 2 20 / 23 Convergence theory - TB Chapter 28 Didactic simplification for convergence of QR-method: Assume A = AT . QR-method lecture 2 21 / 23 Convergence theory - TB Chapter 28 Didactic simplification for convergence of QR-method: Assume A = AT . Convergence characterization (1) Artificial algorithm: USI - Unnormalized Simultaneous Iteration QR-method lecture 2 21 / 23 Convergence theory - TB Chapter 28 Didactic simplification for convergence of QR-method: Assume A = AT . Convergence characterization (1) Artificial algorithm: USI - Unnormalized Simultaneous Iteration (2) Show convergence properties of USI QR-method lecture 2 21 / 23 Convergence theory - TB Chapter 28 Didactic simplification for convergence of QR-method: Assume A = AT . Convergence characterization (1) Artificial algorithm: USI - Unnormalized Simultaneous Iteration (2) Show convergence properties of USI (3) Artificial algorithm: NSI - Normalized Simultaneous Iteration QR-method lecture 2 21 / 23 Convergence theory - TB Chapter 28 Didactic simplification for convergence of QR-method: Assume A = AT . Convergence characterization (1) Artificial algorithm: USI - Unnormalized Simultaneous Iteration (2) Show convergence properties of USI (3) Artificial algorithm: NSI - Normalized Simultaneous Iteration (4) Show: USI ⇔ NSI ⇔ QR-method QR-method lecture 2 21 / 23 Definition: Unnormalized simultaneous iteration (USI) A generalization of power method with n vectors “simultaneously” QR-method lecture 2 22 / 23 Definition: Unnormalized simultaneous iteration (USI) A generalization of power method with n vectors “simultaneously” (0) (0) V (0) = [v1 , . . . , vn ] ∈ Rm×n . Define V (k) := Ak V (0) . QR-method lecture 2 22 / 23 Definition: Unnormalized simultaneous iteration (USI) A generalization of power method with n vectors “simultaneously” (0) (0) V (0) = [v1 , . . . , vn ] ∈ Rm×n . Define V (k) := Ak V (0) . A QR-factorization generalizes the normalization step: Q̂ (k) R̂ (k) = V (k) . QR-method lecture 2 22 / 23 Definition: Unnormalized simultaneous iteration (USI) A generalization of power method with n vectors “simultaneously” (0) (0) V (0) = [v1 , . . . , vn ] ∈ Rm×n . Define V (k) := Ak V (0) . A QR-factorization generalizes the normalization step: Q̂ (k) R̂ (k) = V (k) . QR-method lecture 2 22 / 23 Convergence of USI QR-method lecture 2 23 / 23 Convergence of USI Assumptions: Let eigenvalues ordered and assume: |λ1 | > |λ2 | > · · · > |λn+1 | ≥ |λn+2 | ≥ · · · ≥ |λm |. QR-method lecture 2 23 / 23 Convergence of USI Assumptions: Let eigenvalues ordered and assume: |λ1 | > |λ2 | > · · · > |λn+1 | ≥ |λn+2 | ≥ · · · ≥ |λm |. Assume leading principal submatrices of Q̂ T V (0) are nonsingular, where Q̂ = (q1 , . . . , qn ) are the eigenvectors. QR-method lecture 2 23 / 23 Convergence of USI Assumptions: Let eigenvalues ordered and assume: |λ1 | > |λ2 | > · · · > |λn+1 | ≥ |λn+2 | ≥ · · · ≥ |λm |. Assume leading principal submatrices of Q̂ T V (0) are nonsingular, where Q̂ = (q1 , . . . , qn ) are the eigenvectors. Theorem (TB Theorem 28.1) Suppose simultaneous iteration is started with V (0) and assumptions above are satisfied. Let qj , j = 1, . . . , n be the first n eigenvectors of A. Then, as k → ∞, the columns of the matrices Q̂ (k) convergence linearly to qj (k) kqj − ±qj k = O(C k ), j = 1, . . . , n, QR-method lecture 2 23 / 23 Convergence of USI Assumptions: Let eigenvalues ordered and assume: |λ1 | > |λ2 | > · · · > |λn+1 | ≥ |λn+2 | ≥ · · · ≥ |λm |. Assume leading principal submatrices of Q̂ T V (0) are nonsingular, where Q̂ = (q1 , . . . , qn ) are the eigenvectors. Theorem (TB Theorem 28.1) Suppose simultaneous iteration is started with V (0) and assumptions above are satisfied. Let qj , j = 1, . . . , n be the first n eigenvectors of A. Then, as k → ∞, the columns of the matrices Q̂ (k) convergence linearly to qj (k) kqj − ±qj k = O(C k ), j = 1, . . . , n, where C = max1≤k≤n |λk+1 |/|λk |. QR-method lecture 2 23 / 23 Convergence of USI Assumptions: Let eigenvalues ordered and assume: |λ1 | > |λ2 | > · · · > |λn+1 | ≥ |λn+2 | ≥ · · · ≥ |λm |. Assume leading principal submatrices of Q̂ T V (0) are nonsingular, where Q̂ = (q1 , . . . , qn ) are the eigenvectors. Theorem (TB Theorem 28.1) Suppose simultaneous iteration is started with V (0) and assumptions above are satisfied. Let qj , j = 1, . . . , n be the first n eigenvectors of A. Then, as k → ∞, the columns of the matrices Q̂ (k) convergence linearly to qj (k) kqj − ±qj k = O(C k ), j = 1, . . . , n, where C = max1≤k≤n |λk+1 |/|λk |. * Show matlab demo on USI (video) * QR-method lecture 2 23 / 23
