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Bounds on the solution of Sylvester equation and something else Ninoslav Truhar Department of Mathematics, University of Osijek ntruhar@mathos.hr 24.04.2009 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 1 / 22 Coauthors:

  1. Estimating the size of the solution AX − XB = GF ∗ the main tool; Consider: factorized version of ADI iterations: 1 solve ( A − β i I ) X i +1 / 2 = X i ( B − β i I ) + GF ∗ for X i +1 / 2 ; 2 solve X i +1 ( B − α i I ) = ( A − α i I ) X i +1 / 2 − GF ∗ for X i +1 , where { α i } � = { β i } are given two sets of parameters. It holods X i +1 − X = ( A − α i I )( A − β i I ) − 1 ( X i − X )( B − β i I )( B − α i I ) − 1 ,     i i � ( A − α j I )( A − β j I ) − 1  ( X 0 − X ) � ( B − β j I )( B − α j I ) − 1  , =   j =0 j =0 In addition, if { α i } n j =1 = σ ( A ) or { β j } m j =1 = σ ( B ), N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 6 / 22

  2. Estimating the size of the solution AX − XB = GF ∗ the main tool; Consider: factorized version of ADI iterations: 1 solve ( A − β i I ) X i +1 / 2 = X i ( B − β i I ) + GF ∗ for X i +1 / 2 ; 2 solve X i +1 ( B − α i I ) = ( A − α i I ) X i +1 / 2 − GF ∗ for X i +1 , where { α i } � = { β i } are given two sets of parameters. It holods X i +1 − X = ( A − α i I )( A − β i I ) − 1 ( X i − X )( B − β i I )( B − α i I ) − 1 ,     i i � ( A − α j I )( A − β j I ) − 1  ( X 0 − X ) � ( B − β j I )( B − α j I ) − 1  , =   j =0 j =0 In addition, if { α i } n j =1 = σ ( A ) or { β j } m j =1 = σ ( B ), then X min { m , n } = X , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 6 / 22

  3. Estimating the size of the solution AX − XB = GF ∗ the main tool; Consider: factorized version of ADI iterations: 1 solve ( A − β i I ) X i +1 / 2 = X i ( B − β i I ) + GF ∗ for X i +1 / 2 ; 2 solve X i +1 ( B − α i I ) = ( A − α i I ) X i +1 / 2 − GF ∗ for X i +1 , where { α i } � = { β i } are given two sets of parameters. It holods X i +1 − X = ( A − α i I )( A − β i I ) − 1 ( X i − X )( B − β i I )( B − α i I ) − 1 ,     i i � ( A − α j I )( A − β j I ) − 1  ( X 0 − X ) � ( B − β j I )( B − α j I ) − 1  , =   j =0 j =0 In addition, if { α i } n j =1 = σ ( A ) or { β j } m j =1 = σ ( B ), then X min { m , n } = X , the Hamilton-Cayley theorem, p ( A ) ≡ 0 for p ( λ ) def = det( λ I − A ) and q ( B ) ≡ 0 for q ( λ ) def = det( λ I − B ). N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 6 / 22

  4. Low Rank ADI (LRADI) Method for Sylvester Equations N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 7 / 22

  5. Low Rank ADI (LRADI) Method for Sylvester Equations For A ∈ R n × n , B ∈ R m × m G ∈ R m × r and F ∈ R n × r , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 7 / 22

  6. Low Rank ADI (LRADI) Method for Sylvester Equations For A ∈ R n × n , B ∈ R m × m G ∈ R m × r and F ∈ R n × r , AX − XB = GF ∗ . consider Sylvester equation: N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 7 / 22

  7. Low Rank ADI (LRADI) Method for Sylvester Equations For A ∈ R n × n , B ∈ R m × m G ∈ R m × r and F ∈ R n × r , AX − XB = GF ∗ . consider Sylvester equation: Input : (a) A , B , G , and F ; (b) ADI shifts { β 1 , β 2 , . . . } , { α 1 , α 2 , . . . } ; (c) k , the number of ADI steps; N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 7 / 22

  8. Low Rank ADI (LRADI) Method for Sylvester Equations For A ∈ R n × n , B ∈ R m × m G ∈ R m × r and F ∈ R n × r , AX − XB = GF ∗ . consider Sylvester equation: Input : (a) A , B , G , and F ; (b) ADI shifts { β 1 , β 2 , . . . } , { α 1 , α 2 , . . . } ; (c) k , the number of ADI steps; Output: Z ( m × kr ) , D ( kr × kr ) , and Y ( n × kr ) such that ZDY ∗ ≈ X Z 1 = ( A − β 1 I ) − 1 G ; ( Y ∗ ) 1 = F ∗ ( B − α 1 I ) − 1 ; 1. 2. for i = 1 , 2 , . . . , k do Z i = Z i − 1 + ( β i +1 − α i )( A − β i +1 I ) − 1 Z i − 1 ; 3. ( Y ∗ ) i = ( Y ∗ ) i − 1 + ( α i +1 − β i ) ( Y ∗ ) i − 1 ( B − α i +1 I ) − 1 ; 4. 5. end for; 6. D = diag (( β 1 − α 1 ) I r , . . . , ( β k − α k ) I r ). N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 7 / 22

  9. Low Rank ADI (LRADI) Method for Sylvester Equations For A ∈ R n × n , B ∈ R m × m G ∈ R m × r and F ∈ R n × r , AX − XB = GF ∗ . consider Sylvester equation: Input : (a) A , B , G , and F ; (b) ADI shifts { β 1 , β 2 , . . . } , { α 1 , α 2 , . . . } ; (c) k , the number of ADI steps; Output: Z ( m × kr ) , D ( kr × kr ) , and Y ( n × kr ) such that ZDY ∗ ≈ X Z 1 = ( A − β 1 I ) − 1 G ; ( Y ∗ ) 1 = F ∗ ( B − α 1 I ) − 1 ; 1. 2. for i = 1 , 2 , . . . , k do Z i = Z i − 1 + ( β i +1 − α i )( A − β i +1 I ) − 1 Z i − 1 ; 3. ( Y ∗ ) i = ( Y ∗ ) i − 1 + ( α i +1 − β i ) ( Y ∗ ) i − 1 ( B − α i +1 I ) − 1 ; 4. 5. end for; 6. D = diag (( β 1 − α 1 ) I r , . . . , ( β k − α k ) I r ). Recall: if { α i } n j =1 = σ ( A ) or { β j } m j =1 = σ ( B ), N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 7 / 22

  10. Low Rank ADI (LRADI) Method for Sylvester Equations For A ∈ R n × n , B ∈ R m × m G ∈ R m × r and F ∈ R n × r , AX − XB = GF ∗ . consider Sylvester equation: Input : (a) A , B , G , and F ; (b) ADI shifts { β 1 , β 2 , . . . } , { α 1 , α 2 , . . . } ; (c) k , the number of ADI steps; Output: Z ( m × kr ) , D ( kr × kr ) , and Y ( n × kr ) such that ZDY ∗ ≈ X Z 1 = ( A − β 1 I ) − 1 G ; ( Y ∗ ) 1 = F ∗ ( B − α 1 I ) − 1 ; 1. 2. for i = 1 , 2 , . . . , k do Z i = Z i − 1 + ( β i +1 − α i )( A − β i +1 I ) − 1 Z i − 1 ; 3. ( Y ∗ ) i = ( Y ∗ ) i − 1 + ( α i +1 − β i ) ( Y ∗ ) i − 1 ( B − α i +1 I ) − 1 ; 4. 5. end for; 6. D = diag (( β 1 − α 1 ) I r , . . . , ( β k − α k ) I r ). Recall: if { α i } n j =1 = σ ( A ) or { β j } m j =1 = σ ( B ), min { m , n } ( β j − α j ) Z ( j ) Y ( j ) ∗ � then X = X min { m , n } = j =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 7 / 22

  11. Estimating the size of the solution N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 8 / 22

  12. Estimating the size of the solution Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 8 / 22

  13. Estimating the size of the solution Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , min { m , n } m n | τ ( ℓ, j − 1) | � ˆ | σ ( i , j − 1) | · � ˆ g i � f ℓ � � � � � X � ≤ � S �� T − 1 � | β j − α j | , | λ i − β j | | µ ℓ − α j | j =1 i =1 ℓ =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 8 / 22

  14. Estimating the size of the solution Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , min { m , n } m n | τ ( ℓ, j − 1) | � ˆ | σ ( i , j − 1) | · � ˆ g i � f ℓ � � � � � X � ≤ � S �� T − 1 � | β j − α j | , | λ i − β j | | µ ℓ − α j | j =1 i =1 ℓ =1 j − 1 λ i − α s � σ ( i , j − 1) = , σ ( i , 0) = 1 , i = 1 , . . . , m , λ i − β s s =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 8 / 22

  15. Estimating the size of the solution Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , min { m , n } m n | τ ( ℓ, j − 1) | � ˆ | σ ( i , j − 1) | · � ˆ g i � f ℓ � � � � � X � ≤ � S �� T − 1 � | β j − α j | , | λ i − β j | | µ ℓ − α j | j =1 i =1 ℓ =1 j − 1 λ i − α s � σ ( i , j − 1) = , σ ( i , 0) = 1 , i = 1 , . . . , m , λ i − β s s =1 j − 1 µ i − β s � τ ( i , j − 1) = , τ ( i , 0) = 1 , i = 1 , . . . , n . µ i − α s s =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 8 / 22

  16. Estimating the size of the solution Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , min { m , n } m n | τ ( ℓ, j − 1) | � ˆ | σ ( i , j − 1) | · � ˆ g i � f ℓ � � � � � X � ≤ � S �� T − 1 � | β j − α j | , | λ i − β j | | µ ℓ − α j | j =1 i =1 ℓ =1 j − 1 λ i − α s � σ ( i , j − 1) = , σ ( i , 0) = 1 , i = 1 , . . . , m , λ i − β s s =1 j − 1 µ i − β s � τ ( i , j − 1) = , τ ( i , 0) = 1 , i = 1 , . . . , n . µ i − α s s =1 F ∗ = F ∗ T g i is the i th row of ˆ ˆ f i is the i th column of ˆ G = S − 1 G ; ˆ N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 8 / 22

  17. Example N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 9 / 22

  18. Example Let A = S diag (1 , 2 , 3 , 4 , 5) S − 1 where   5 . 04 − 4 . 2 0 . 0013 − 0 . 003 0 . 002 − 3 . 91 − 3 . 67 0 . 0028 0 . 01 0 . 004     S = 0 . 001 0 . 0035 − 1 . 55 − 0 . 11 − 0 . 21     − 0 . 002 0 . 0011 0 . 32 − 1 . 82 0 . 42   − 0 . 004 0 . 0025 0 . 79 1 . 29 − 0 . 89 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 9 / 22

  19. Example Let A = S diag (1 , 2 , 3 , 4 , 5) S − 1 where   5 . 04 − 4 . 2 0 . 0013 − 0 . 003 0 . 002 − 3 . 91 − 3 . 67 0 . 0028 0 . 01 0 . 004     S = 0 . 001 0 . 0035 − 1 . 55 − 0 . 11 − 0 . 21     − 0 . 002 0 . 0011 0 . 32 − 1 . 82 0 . 42   − 0 . 004 0 . 0025 0 . 79 1 . 29 − 0 . 89 Let B = T diag (11 , 12 , 13 , 14 , 5 . 0001) T − 1 where  5 . 0443 − 4 . 1948 0 . 0022 0 . 0062 0 . 0023  − 3 . 9041 − 3 . 6609 0 . 0032 0 . 0105 0 . 0045   T = 0 . 0019 0 . 0101 − 1 . 5425 − 0 . 1093 − 0 . 2076  .     0 . 0005 0 . 0061 0 . 3232 − 1 . 8143 0 . 4213  0 . 0019 0 . 0088 0 . 7907 1 . 2944 − 0 . 8837 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 9 / 22

  20. Example N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 10 / 22

  21. Example � T � T � 2 � 2 . 2 5 0 0 0 4 . 5 0 0 0 G = , F = . 5 1 0 0 0 5 . 1 1 . 1 0 0 0 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 10 / 22

  22. Example � T � T � 2 � 2 . 2 5 0 0 0 4 . 5 0 0 0 G = , F = . 5 1 0 0 0 5 . 1 1 . 1 0 0 0 Consider AX − XB = GF ∗ , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 10 / 22

  23. Example � T � T � 2 � 2 . 2 5 0 0 0 4 . 5 0 0 0 G = , F = . 5 1 0 0 0 5 . 1 1 . 1 0 0 0 Consider AX − XB = GF ∗ , � X � = 4 . 4884 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 10 / 22

  24. Example � T � T � 2 � 2 . 2 5 0 0 0 4 . 5 0 0 0 G = , F = . 5 1 0 0 0 5 . 1 1 . 1 0 0 0 Consider AX − XB = GF ∗ , � X � = 4 . 4884 � X � ≤ 176 . 5 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 10 / 22

  25. Example � T � T � 2 � 2 . 2 5 0 0 0 4 . 5 0 0 0 G = , F = . 5 1 0 0 0 5 . 1 1 . 1 0 0 0 Consider AX − XB = GF ∗ , � X � = 4 . 4884 � X � ≤ 176 . 5 � − 1 �� vec ( W ) � ≤ 34469 ( I ⊗ A ) − ( B T ⊗ I ) � � X � ≤ � N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 10 / 22

  26. Error Bound for Sylvester equation N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 11 / 22

  27. Error Bound for Sylvester equation Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 11 / 22

  28. Error Bound for Sylvester equation Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , Let X k obtained by LRADI, s.t. X k ≈ X , δ X = X − X k N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 11 / 22

  29. Error Bound for Sylvester equation Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , Let X k obtained by LRADI, s.t. X k ≈ X , δ X = X − X k min { m , n } m n | τ ( ℓ, j − 1) | � ˆ | σ ( i , j − 1) | · � ˆ g i � f ℓ � � � � � δ X � ≤ � S �� T − 1 � | β j − α j | , | λ i − β j | | µ ℓ − α j | j = k +1 i =1 ℓ =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 11 / 22

  30. Error Bound for Sylvester equation Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , Let X k obtained by LRADI, s.t. X k ≈ X , δ X = X − X k min { m , n } m n | τ ( ℓ, j − 1) | � ˆ | σ ( i , j − 1) | · � ˆ g i � f ℓ � � � � � δ X � ≤ � S �� T − 1 � | β j − α j | , | λ i − β j | | µ ℓ − α j | j = k +1 i =1 ℓ =1 j − 1 λ i − α s � σ ( i , j − 1) = , σ ( i , 0) = 1 , i = 1 , . . . , m , λ i − β s s =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 11 / 22

  31. Error Bound for Sylvester equation Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , Let X k obtained by LRADI, s.t. X k ≈ X , δ X = X − X k min { m , n } m n | τ ( ℓ, j − 1) | � ˆ | σ ( i , j − 1) | · � ˆ g i � f ℓ � � � � � δ X � ≤ � S �� T − 1 � | β j − α j | , | λ i − β j | | µ ℓ − α j | j = k +1 i =1 ℓ =1 j − 1 λ i − α s � σ ( i , j − 1) = , σ ( i , 0) = 1 , i = 1 , . . . , m , λ i − β s s =1 j − 1 µ i − β s � τ ( i , j − 1) = , τ ( i , 0) = 1 , i = 1 , . . . , n . µ i − α s s =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 11 / 22

  32. Error Bound for Sylvester equation Consider AX − XB = GF ∗ , where A and B are diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 , Let X k obtained by LRADI, s.t. X k ≈ X , δ X = X − X k min { m , n } m n | τ ( ℓ, j − 1) | � ˆ | σ ( i , j − 1) | · � ˆ g i � f ℓ � � � � � δ X � ≤ � S �� T − 1 � | β j − α j | , | λ i − β j | | µ ℓ − α j | j = k +1 i =1 ℓ =1 j − 1 λ i − α s � σ ( i , j − 1) = , σ ( i , 0) = 1 , i = 1 , . . . , m , λ i − β s s =1 j − 1 µ i − β s � τ ( i , j − 1) = , τ ( i , 0) = 1 , i = 1 , . . . , n . µ i − α s s =1 F ∗ = F ∗ T g i is the i th row of ˆ ˆ f i is the i th column of ˆ G = S − 1 G ; ˆ N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 11 / 22

  33. Error Bound for Lyapunov equation N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 12 / 22

  34. Error Bound for Lyapunov equation Let X k be obtained by LRADI, s.t. X k ≈ X N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 12 / 22

  35. Error Bound for Lyapunov equation Let X k be obtained by LRADI, s.t. X k ≈ X How close is X k to X ? N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 12 / 22

  36. Error Bound for Lyapunov equation Let X k be obtained by LRADI, s.t. X k ≈ X How close is X k to X ? For the case of Lyapunov equation: AX − XA ∗ = GG ∗ , A = S Λ S − 1 diagonalizable N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 12 / 22

  37. Error Bound for Lyapunov equation Let X k be obtained by LRADI, s.t. X k ≈ X How close is X k to X ? For the case of Lyapunov equation: AX − XA ∗ = GG ∗ , A = S Λ S − 1 diagonalizable ( T . and Veseli´ c 2007): m m σ ( j , k ) 2 · � ˆ � X − X k � ≤ � S � 2 � � g k � 2 ( − 2 Re ( α j )) j = k +1 k =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 12 / 22

  38. Error Bound for Lyapunov equation Let X k be obtained by LRADI, s.t. X k ≈ X How close is X k to X ? For the case of Lyapunov equation: AX − XA ∗ = GG ∗ , A = S Λ S − 1 diagonalizable ( T . and Veseli´ c 2007): m m σ ( j , k ) 2 · � ˆ � X − X k � ≤ � S � 2 � � g k � 2 ( − 2 Re ( α j )) j = k +1 k =1 j − 1 1 1 λ k − ¯ α t � σ (1 , k ) = , and σ ( j , k ) = for j > 1 . λ k + α 1 λ k + α j λ k + α t t =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 12 / 22

  39. Error Bound for Lyapunov equation Let X k be obtained by LRADI, s.t. X k ≈ X How close is X k to X ? For the case of Lyapunov equation: AX − XA ∗ = GG ∗ , A = S Λ S − 1 diagonalizable ( T . and Veseli´ c 2007): m m σ ( j , k ) 2 · � ˆ � X − X k � ≤ � S � 2 � � g k � 2 ( − 2 Re ( α j )) j = k +1 k =1 j − 1 1 1 λ k − ¯ α t � σ (1 , k ) = , and σ ( j , k ) = for j > 1 . λ k + α 1 λ k + α j λ k + α t t =1 g k is the k th row of ˆ G = S − 1 G ˆ N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 12 / 22

  40. Perturbation Bound for Sylvester equation N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 13 / 22

  41. Perturbation Bound for Sylvester equation Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( G + δ G )( F + δ F ) ∗ , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 13 / 22

  42. Perturbation Bound for Sylvester equation Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( G + δ G )( F + δ F ) ∗ , Using N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 13 / 22

  43. Perturbation Bound for Sylvester equation Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( G + δ G )( F + δ F ) ∗ , Using A δ X − δ X B ≈ G δ F ∗ + δ G F ∗ − δ A X + X δ B , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 13 / 22

  44. Perturbation Bound for Sylvester equation Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( G + δ G )( F + δ F ) ∗ , Using A δ X − δ X B ≈ G δ F ∗ + δ G F ∗ − δ A X + X δ B , one gets: δ X ≈ δ X 1 + δ X 2 + δ X 3 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 13 / 22

  45. Perturbation Bound for Sylvester equation Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( G + δ G )( F + δ F ) ∗ , Using A δ X − δ X B ≈ G δ F ∗ + δ G F ∗ − δ A X + X δ B , one gets: δ X ≈ δ X 1 + δ X 2 + δ X 3 A δ X 1 − δ X 1 B = − δ A X , A δ X 2 − δ X 2 B = − X δ B , ( G δ F ∗ + δ G F ∗ ) . A δ X 3 − δ X 3 B = N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 13 / 22

  46. Perturbation Bound for Sylvester equation N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 14 / 22

  47. Perturbation Bound for Sylvester equation The following perturbation bound holds: N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 14 / 22

  48. Perturbation Bound for Sylvester equation The following perturbation bound holds: for A and B diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 . N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 14 / 22

  49. Perturbation Bound for Sylvester equation The following perturbation bound holds: for A and B diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 . and sufficiently small η = max {� δ A � , � δ B � , � δ G � , � δ F �} , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 14 / 22

  50. Perturbation Bound for Sylvester equation The following perturbation bound holds: for A and B diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 . and sufficiently small η = max {� δ A � , � δ B � , � δ G � , � δ F �} , � � δ X � κ ( T ) � S �� S − 1 δ A � + κ ( S ) � T − 1 �� δ BT � ≤ � X � + � S �� T − 1 �� S − 1 ( G δ F ∗ + δ GF ∗ ) T � � γ + O ( η 2 ) , � X � N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 14 / 22

  51. Perturbation Bound for Sylvester equation The following perturbation bound holds: for A and B diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 . and sufficiently small η = max {� δ A � , � δ B � , � δ G � , � δ F �} , � � δ X � κ ( T ) � S �� S − 1 δ A � + κ ( S ) � T − 1 �� δ BT � ≤ � X � + � S �� T − 1 �� S − 1 ( G δ F ∗ + δ GF ∗ ) T � � γ + O ( η 2 ) , � X � n 0 | µ j − λ j | τ ( j ) max σ ( j ) � γ = max j =1 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 14 / 22

  52. Perturbation Bound for Sylvester equation The following perturbation bound holds: for A and B diagonalizable; A = S diag ( λ i ) S − 1 , B = T diag ( µ i ) T − 1 . and sufficiently small η = max {� δ A � , � δ B � , � δ G � , � δ F �} , � � δ X � κ ( T ) � S �� S − 1 δ A � + κ ( S ) � T − 1 �� δ BT � ≤ � X � + � S �� T − 1 �� S − 1 ( G δ F ∗ + δ GF ∗ ) T � � γ + O ( η 2 ) , � X � n 0 | µ j − λ j | τ ( j ) max σ ( j ) � γ = max where: j =1 | σ ( i , j − 1) | | τ ( i , j − 1) | σ ( j ) | λ i − µ j | , τ ( j ) max = � ∆ Φ ( j ) � = max max = � ∆ Ψ ( j ) � = max | µ i − λ j | . i i N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 14 / 22

  53. Illustration of new perturbation bound N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 15 / 22

  54. Illustration of new perturbation bound Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( C + δ C ) , N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 15 / 22

  55. Illustration of new perturbation bound Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( C + δ C ) , The standard bound (N. J. Higham, 1996): √ � δ X � F 3Ψ ǫ + O ( ǫ 2 ) , ≤ � X � F N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 15 / 22

  56. Illustration of new perturbation bound Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( C + δ C ) , The standard bound (N. J. Higham, 1996): √ � δ X � F 3Ψ ǫ + O ( ǫ 2 ) , ≤ � X � F where Ψ = � P − 1 [ α ( X T ⊗ I m ) − β ( I n ⊗ X ) − δ I mn ] � / � X � F , P = I n ⊗ A − B T ⊗ I m N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 15 / 22

  57. Illustration of new perturbation bound Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( C + δ C ) , The standard bound (N. J. Higham, 1996): √ � δ X � F 3Ψ ǫ + O ( ǫ 2 ) , ≤ � X � F where Ψ = � P − 1 [ α ( X T ⊗ I m ) − β ( I n ⊗ X ) − δ I mn ] � / � X � F , P = I n ⊗ A − B T ⊗ I m � � � δ A � F , � δ B � F , � δ C � F ǫ = max , α β δ N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 15 / 22

  58. Illustration of new perturbation bound Consider perturbed Sylvester equation ( A + δ A )( X + δ X ) − ( X + δ X )( B + δ B ) = ( C + δ C ) , The standard bound (N. J. Higham, 1996): √ � δ X � F 3Ψ ǫ + O ( ǫ 2 ) , ≤ � X � F where Ψ = � P − 1 [ α ( X T ⊗ I m ) − β ( I n ⊗ X ) − δ I mn ] � / � X � F , P = I n ⊗ A − B T ⊗ I m � � � δ A � F , � δ B � F , � δ C � F ǫ = max , α = � A � F , β = � B � F and δ = � C � F α β δ N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 15 / 22

  59. Illustration of new perturbation bound N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 16 / 22

  60. Illustration of new perturbation bound Let A = S Λ S − 1 with Λ = diag (30 , 40 , 60 , 70 , 90) N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 16 / 22

  61. Illustration of new perturbation bound Let A = S Λ S − 1 with Λ = diag (30 , 40 , 60 , 70 , 90)  0 . 052 0 . 494 0 . 935 0 . 592 0 . 827  0 . 264 0 . 094 0 . 220 0 . 388 0 . 387     S = 0 . 757 0 . 971 0 . 189 0 . 935 0 . 445 .     0 . 435 0 . 424 0 . 280 0 . 826 0 . 009   0 . 506 0 . 270 0 . 674 0 . 051 0 . 280 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 16 / 22

  62. Illustration of new perturbation bound Let A = S Λ S − 1 with Λ = diag (30 , 40 , 60 , 70 , 90)  0 . 052 0 . 494 0 . 935 0 . 592 0 . 827  0 . 264 0 . 094 0 . 220 0 . 388 0 . 387     S = 0 . 757 0 . 971 0 . 189 0 . 935 0 . 445 .     0 . 435 0 . 424 0 . 280 0 . 826 0 . 009   0 . 506 0 . 270 0 . 674 0 . 051 0 . 280 B = T Ω T − 1 with Ω = diag (100 , 200 , 300 , 400 , 450) N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 16 / 22

  63. Illustration of new perturbation bound Let A = S Λ S − 1 with Λ = diag (30 , 40 , 60 , 70 , 90)  0 . 052 0 . 494 0 . 935 0 . 592 0 . 827  0 . 264 0 . 094 0 . 220 0 . 388 0 . 387     S = 0 . 757 0 . 971 0 . 189 0 . 935 0 . 445 .     0 . 435 0 . 424 0 . 280 0 . 826 0 . 009   0 . 506 0 . 270 0 . 674 0 . 051 0 . 280 B = T Ω T − 1 with Ω = diag (100 , 200 , 300 , 400 , 450)  85610 89420 42300 31670 97380  9709 64180 89230 36420 28030     T = 0 . 3910 0 . 2030 0 . 9910 0 . 6150 0 . 1740 .     0 . 3760 0 . 0680 0 . 4000 0 . 5970 0 . 9210   0 . 5150 0 . 8000 0 . 3400 0 . 6850 0 . 2600 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 16 / 22

  64. Illustration of new perturbation bound N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 17 / 22

  65. Illustration of new perturbation bound Further, � − 10 � T − 60 0 0 0 0 G = , F = G . − 60 400 0 0 0 0 N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 17 / 22

  66. Illustration of new perturbation bound Further, � − 10 � T − 60 0 0 0 0 G = , F = G . − 60 400 0 0 0 0 Perturbations are δ B = 10 − 9 · diag 10 − 5 , 10 − 5 , 1 , 1 , 1 � � , δ A = 0 , δ F = δ G = 0 . N. Truhar (Osijek, 2009) Sylvester equation 24.04.2009 17 / 22

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