Bounds on the solution of Sylvester equation and something else - - PowerPoint PPT Presentation

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: Ren-Cang Li Department of Mathematics, University of Texas at Arlington Zoran Tomljanovi´ c Department of Mathematics, University of Osijek Luka Grubiˇ si´ c Department of Mathematics, University of Zagreb Kreˇ simir Veseli´ c Fernuniversit¨ at, Hagen, currently Department of Mathematics, University of Osijek

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 2 / 22

Introduction

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W with given A ∈ Rn×n, B ∈ Rm×m, W ∈ Rn×m and unknown X ∈ Rn×m.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W with given A ∈ Rn×n, B ∈ Rm×m, W ∈ Rn×m and unknown X ∈ Rn×m. Lyapunov equation AX + XAT = W

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W with given A ∈ Rn×n, B ∈ Rm×m, W ∈ Rn×m and unknown X ∈ Rn×m. Lyapunov equation AX + XAT = W with given A ∈ Rn×n, W = W T ∈ Rn×n and unknown X ∈ Rn×n.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W with given A ∈ Rn×n, B ∈ Rm×m, W ∈ Rn×m and unknown X ∈ Rn×m. Lyapunov equation AX + XAT = W with given A ∈ Rn×n, W = W T ∈ Rn×n and unknown X ∈ Rn×n. Overview of problems

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W with given A ∈ Rn×n, B ∈ Rm×m, W ∈ Rn×m and unknown X ∈ Rn×m. Lyapunov equation AX + XAT = W with given A ∈ Rn×n, W = W T ∈ Rn×n and unknown X ∈ Rn×n. Overview of problems ♦ Estimating the size of the solution

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W with given A ∈ Rn×n, B ∈ Rm×m, W ∈ Rn×m and unknown X ∈ Rn×m. Lyapunov equation AX + XAT = W with given A ∈ Rn×n, W = W T ∈ Rn×n and unknown X ∈ Rn×n. Overview of problems ♦ Estimating the size of the solution ♦ Perturbation bound

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W with given A ∈ Rn×n, B ∈ Rm×m, W ∈ Rn×m and unknown X ∈ Rn×m. Lyapunov equation AX + XAT = W with given A ∈ Rn×n, W = W T ∈ Rn×n and unknown X ∈ Rn×n. Overview of problems ♦ Estimating the size of the solution ♦ Perturbation bound “And Now for Something Completely Different”

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Introduction

Sylvester equation AX − XB = W with given A ∈ Rn×n, B ∈ Rm×m, W ∈ Rn×m and unknown X ∈ Rn×m. Lyapunov equation AX + XAT = W with given A ∈ Rn×n, W = W T ∈ Rn×n and unknown X ∈ Rn×n. Overview of problems ♦ Estimating the size of the solution ♦ Perturbation bound “And Now for Something Completely Different” ♦ Estimating the size of the J-unitary matrix

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 3 / 22

Clasicall approach to estimating problem

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 4 / 22

Clasicall approach to estimating problem

Solution of the Sylvester equation: AX − XB = W

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 4 / 22

Clasicall approach to estimating problem

Solution of the Sylvester equation: AX − XB = W Svec(X) ≡

• (Im ⊗ A) − (BT ⊗ In)
• vec(X) = vec(W )
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 4 / 22

Clasicall approach to estimating problem

Solution of the Sylvester equation: AX − XB = W Svec(X) ≡

• (Im ⊗ A) − (BT ⊗ In)
• vec(X) = vec(W )

upper bound:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 4 / 22

Clasicall approach to estimating problem

Solution of the Sylvester equation: AX − XB = W Svec(X) ≡

• (Im ⊗ A) − (BT ⊗ In)
• vec(X) = vec(W )

upper bound: vec(X) ≤ S−1 · vec(W )

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 4 / 22

Clasicall approach to estimating problem

Solution of the Sylvester equation: AX − XB = W Svec(X) ≡

• (Im ⊗ A) − (BT ⊗ In)
• vec(X) = vec(W )

upper bound: vec(X) ≤ S−1 · vec(W ) If (from any reason) R(W ) is close R(S)

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 4 / 22

Clasicall approach to estimating problem

Solution of the Sylvester equation: AX − XB = W Svec(X) ≡

• (Im ⊗ A) − (BT ⊗ In)
• vec(X) = vec(W )

upper bound: vec(X) ≤ S−1 · vec(W ) If (from any reason) R(W ) is close R(S) SOMETIMES THE BOUND CAN BE USELLES

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 4 / 22

Motivating example

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 5 / 22

Motivating example

Consider:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 5 / 22

Motivating example

Consider: A = a ǫ

• , B =

−b µ

• , W =

1

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 5 / 22

Motivating example

Consider: A = a ǫ

• , B =

−b µ

• , W =

1

• Solution of the Sylvester equation: AX − XB = W :
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 5 / 22

Motivating example

Consider: A = a ǫ

• , B =

−b µ

• , W =

1

• Solution of the Sylvester equation: AX − XB = W :

X =

• 1

a+b

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 5 / 22

Motivating example

Consider: A = a ǫ

• , B =

−b µ

• , W =

1

• Solution of the Sylvester equation: AX − XB = W :

X =

• 1

a+b

• n the other hand
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 5 / 22

Motivating example

Consider: A = a ǫ

• , B =

−b µ

• , W =

1

• Solution of the Sylvester equation: AX − XB = W :

X =

• 1

a+b

• n the other hand
• (Im ⊗ A) − (BT ⊗ In)

−1 ≤ max{ 1 a + b, 1 a + µ, 1 b + ǫ, 1 µ + ǫ}

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 5 / 22

Motivating example

Consider: A = a ǫ

• , B =

−b µ

• , W =

1

• Solution of the Sylvester equation: AX − XB = W :

X =

• 1

a+b

• n the other hand
• (Im ⊗ A) − (BT ⊗ In)

−1 ≤ max{ 1 a + b, 1 a + µ, 1 b + ǫ, 1 µ + ǫ} which can be large ( for small µ, ǫ)

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 5 / 22

Estimating the size of the solution

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗ the main tool;

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗ the main tool; factorized version of ADI iterations:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗ the main tool; factorized version of ADI iterations:

1 solve (A − βiI)Xi+1/2 = Xi(B − βiI) + GF ∗

for Xi+1/2;

2 solve Xi+1(B − αiI) = (A − αiI)Xi+1/2 − GF ∗

for Xi+1,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗ the main tool; factorized version of ADI iterations:

1 solve (A − βiI)Xi+1/2 = Xi(B − βiI) + GF ∗

for Xi+1/2;

2 solve Xi+1(B − αiI) = (A − αiI)Xi+1/2 − GF ∗

for Xi+1, where {αi} = {βi} are given two sets of parameters.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗ the main tool; factorized version of ADI iterations:

1 solve (A − βiI)Xi+1/2 = Xi(B − βiI) + GF ∗

for Xi+1/2;

2 solve Xi+1(B − αiI) = (A − αiI)Xi+1/2 − GF ∗

for Xi+1, where {αi} = {βi} are given two sets of parameters. It holods Xi+1 − X = (A − αiI)(A − βiI)−1(Xi − X)(B − βiI)(B − αiI)−1, =  

i

• j=0

(A − αjI)(A − βjI)−1   (X0 − X)  

i

• j=0

(B − βjI)(B − αjI)−1   ,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗ the main tool; factorized version of ADI iterations:

1 solve (A − βiI)Xi+1/2 = Xi(B − βiI) + GF ∗

for Xi+1/2;

2 solve Xi+1(B − αiI) = (A − αiI)Xi+1/2 − GF ∗

for Xi+1, where {αi} = {βi} are given two sets of parameters. It holods Xi+1 − X = (A − αiI)(A − βiI)−1(Xi − X)(B − βiI)(B − αiI)−1, =  

i

• j=0

(A − αjI)(A − βjI)−1   (X0 − X)  

i

• j=0

(B − βjI)(B − αjI)−1   , In addition, if {αi}n

j=1 = σ(A) or {βj}m j=1 = σ(B),

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗ the main tool; factorized version of ADI iterations:

1 solve (A − βiI)Xi+1/2 = Xi(B − βiI) + GF ∗

for Xi+1/2;

2 solve Xi+1(B − αiI) = (A − αiI)Xi+1/2 − GF ∗

for Xi+1, where {αi} = {βi} are given two sets of parameters. It holods Xi+1 − X = (A − αiI)(A − βiI)−1(Xi − X)(B − βiI)(B − αiI)−1, =  

i

• j=0

(A − αjI)(A − βjI)−1   (X0 − X)  

i

• j=0

(B − βjI)(B − αjI)−1   , In addition, if {αi}n

j=1 = σ(A) or {βj}m j=1 = σ(B), then Xmin{m,n} = X,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Estimating the size of the solution

Consider: AX − XB = GF ∗ the main tool; factorized version of ADI iterations:

1 solve (A − βiI)Xi+1/2 = Xi(B − βiI) + GF ∗

for Xi+1/2;

2 solve Xi+1(B − αiI) = (A − αiI)Xi+1/2 − GF ∗

for Xi+1, where {αi} = {βi} are given two sets of parameters. It holods Xi+1 − X = (A − αiI)(A − βiI)−1(Xi − X)(B − βiI)(B − αiI)−1, =  

i

• j=0

(A − αjI)(A − βjI)−1   (X0 − X)  

i

• j=0

(B − βjI)(B − αjI)−1   , In addition, if {αi}n

j=1 = σ(A) or {βj}m j=1 = σ(B), then Xmin{m,n} = X,

the Hamilton-Cayley theorem, p(A) ≡ 0 for p(λ) def = det(λI − A) and q(B) ≡ 0 for q(λ) def = det(λI − B).

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 6 / 22

Low Rank ADI (LRADI) Method for Sylvester Equations

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 7 / 22

Low Rank ADI (LRADI) Method for Sylvester Equations

For A ∈ Rn×n, B ∈ Rm×m G ∈ Rm×r and F ∈ Rn×r,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 7 / 22

Low Rank ADI (LRADI) Method for Sylvester Equations

For A ∈ Rn×n, B ∈ Rm×m G ∈ Rm×r and F ∈ Rn×r, consider Sylvester equation: AX − XB = GF ∗.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 7 / 22

Low Rank ADI (LRADI) Method for Sylvester Equations

For A ∈ Rn×n, B ∈ Rm×m G ∈ Rm×r and F ∈ Rn×r, consider Sylvester equation: AX − XB = GF ∗. Input: (a) A, B, G, and F; (b) ADI shifts {β1, β2, . . .}, {α1, α2, . . .}; (c) k, the number of ADI steps;

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 7 / 22

Low Rank ADI (LRADI) Method for Sylvester Equations

For A ∈ Rn×n, B ∈ Rm×m G ∈ Rm×r and F ∈ Rn×r, consider Sylvester equation: AX − XB = GF ∗. Input: (a) A, B, G, and F; (b) ADI shifts {β1, β2, . . .}, {α1, α2, . . .}; (c) k, the number of ADI steps; Output: Z (m×kr), D(kr×kr), and Y (n×kr) such that ZDY ∗ ≈ X 1. Z1 = (A − β1I)−1G; (Y ∗)1 = F ∗(B − α1I)−1; 2. for i = 1, 2, . . . , k do 3. Zi = Zi−1 + (βi+1 − αi)(A − βi+1I)−1Zi−1; 4. (Y ∗)i = (Y ∗)i−1 + (αi+1 − βi) (Y ∗)i−1 (B − αi+1I)−1; 5. end for; 6. D = diag ((β1 − α1)Ir, . . . , (βk − αk)Ir).

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 7 / 22

Low Rank ADI (LRADI) Method for Sylvester Equations

For A ∈ Rn×n, B ∈ Rm×m G ∈ Rm×r and F ∈ Rn×r, consider Sylvester equation: AX − XB = GF ∗. Input: (a) A, B, G, and F; (b) ADI shifts {β1, β2, . . .}, {α1, α2, . . .}; (c) k, the number of ADI steps; Output: Z (m×kr), D(kr×kr), and Y (n×kr) such that ZDY ∗ ≈ X 1. Z1 = (A − β1I)−1G; (Y ∗)1 = F ∗(B − α1I)−1; 2. for i = 1, 2, . . . , k do 3. Zi = Zi−1 + (βi+1 − αi)(A − βi+1I)−1Zi−1; 4. (Y ∗)i = (Y ∗)i−1 + (αi+1 − βi) (Y ∗)i−1 (B − αi+1I)−1; 5. end for; 6. D = diag ((β1 − α1)Ir, . . . , (βk − αk)Ir). Recall: if {αi}n

j=1 = σ(A) or {βj}m j=1 = σ(B),

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 7 / 22

Low Rank ADI (LRADI) Method for Sylvester Equations

For A ∈ Rn×n, B ∈ Rm×m G ∈ Rm×r and F ∈ Rn×r, consider Sylvester equation: AX − XB = GF ∗. Input: (a) A, B, G, and F; (b) ADI shifts {β1, β2, . . .}, {α1, α2, . . .}; (c) k, the number of ADI steps; Output: Z (m×kr), D(kr×kr), and Y (n×kr) such that ZDY ∗ ≈ X 1. Z1 = (A − β1I)−1G; (Y ∗)1 = F ∗(B − α1I)−1; 2. for i = 1, 2, . . . , k do 3. Zi = Zi−1 + (βi+1 − αi)(A − βi+1I)−1Zi−1; 4. (Y ∗)i = (Y ∗)i−1 + (αi+1 − βi) (Y ∗)i−1 (B − αi+1I)−1; 5. end for; 6. D = diag ((β1 − α1)Ir, . . . , (βk − αk)Ir). Recall: if {αi}n

j=1 = σ(A) or {βj}m j=1 = σ(B),

then X = Xmin{m,n} =

min{m,n}

• j=1

(βj − αj)Z (j)Y (j)∗

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 7 / 22

Estimating the size of the solution

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 8 / 22

Estimating the size of the solution

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 8 / 22

Estimating the size of the solution

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1,

X ≤ ST −1

min{m,n}

• j=1

|βj − αj|

m

• i=1

|σ(i, j − 1)| · ˆ gi |λi − βj|

n

• ℓ=1

|τ(ℓ, j − 1)| ˆ fℓ |µℓ − αj| ,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 8 / 22

Estimating the size of the solution

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1,

X ≤ ST −1

min{m,n}

• j=1

|βj − αj|

m

• i=1

|σ(i, j − 1)| · ˆ gi |λi − βj|

n

• ℓ=1

|τ(ℓ, j − 1)| ˆ fℓ |µℓ − αj| , σ(i, j − 1) =

j−1

• s=1

λi − αs λi − βs , σ(i, 0) = 1, i = 1, . . . , m,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 8 / 22

Estimating the size of the solution

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1,

X ≤ ST −1

min{m,n}

• j=1

|βj − αj|

m

• i=1

|σ(i, j − 1)| · ˆ gi |λi − βj|

n

• ℓ=1

|τ(ℓ, j − 1)| ˆ fℓ |µℓ − αj| , σ(i, j − 1) =

j−1

• s=1

λi − αs λi − βs , σ(i, 0) = 1, i = 1, . . . , m, τ(i, j − 1) =

j−1

• s=1

µi − βs µi − αs , τ(i, 0) = 1, i = 1, . . . , n.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 8 / 22

Estimating the size of the solution

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1,

X ≤ ST −1

min{m,n}

• j=1

|βj − αj|

m

• i=1

|σ(i, j − 1)| · ˆ gi |λi − βj|

n

• ℓ=1

|τ(ℓ, j − 1)| ˆ fℓ |µℓ − αj| , σ(i, j − 1) =

j−1

• s=1

λi − αs λi − βs , σ(i, 0) = 1, i = 1, . . . , m, τ(i, j − 1) =

j−1

• s=1

µi − βs µi − αs , τ(i, 0) = 1, i = 1, . . . , n. ˆ gi is the ith row of ˆ G = S−1G; ˆ fi is the ith column of ˆ F ∗ = F ∗T

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 8 / 22

Example

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 9 / 22

Example

Let A = S diag(1, 2, 3, 4, 5)S−1 where

S =       5.04 −4.2 0.0013 −0.003 0.002 −3.91 −3.67 0.0028 0.01 0.004 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

Example

Let A = S diag(1, 2, 3, 4, 5)S−1 where

S =       5.04 −4.2 0.0013 −0.003 0.002 −3.91 −3.67 0.0028 0.01 0.004 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 T =     

5.0443 −4.1948 0.0022 0.0062 0.0023 −3.9041 −3.6609 0.0032 0.0105 0.0045 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

Example

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 10 / 22

Example

G = 2 5 5 1 T , F = 2.2 4.5 5.1 1.1 T .

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 10 / 22

Example

G = 2 5 5 1 T , F = 2.2 4.5 5.1 1.1 T . Consider AX − XB = GF ∗,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 10 / 22

Example

G = 2 5 5 1 T , F = 2.2 4.5 5.1 1.1 T . Consider AX − XB = GF ∗, X = 4.4884

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 10 / 22

Example

G = 2 5 5 1 T , F = 2.2 4.5 5.1 1.1 T . Consider AX − XB = GF ∗, X = 4.4884 X ≤ 176.5

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 10 / 22

Example

G = 2 5 5 1 T , F = 2.2 4.5 5.1 1.1 T . Consider AX − XB = GF ∗, X = 4.4884 X ≤ 176.5 X ≤

• (I ⊗ A) − (BT ⊗ I)

−1 vec(W ) ≤ 34469

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 10 / 22

Error Bound for Sylvester equation

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 11 / 22

Error Bound for Sylvester equation

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 11 / 22

Error Bound for Sylvester equation

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1, Let Xk obtained by LRADI, s.t. Xk ≈ X , δX = X − Xk

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 11 / 22

Error Bound for Sylvester equation

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1, Let Xk obtained by LRADI, s.t. Xk ≈ X , δX = X − Xk

δX ≤ ST −1

min{m,n}

• j=k+1

|βj − αj|

m

• i=1

|σ(i, j − 1)| · ˆ gi |λi − βj|

n

• ℓ=1

|τ(ℓ, j − 1)| ˆ fℓ |µℓ − αj| ,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 11 / 22

Error Bound for Sylvester equation

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1, Let Xk obtained by LRADI, s.t. Xk ≈ X , δX = X − Xk

δX ≤ ST −1

min{m,n}

• j=k+1

|βj − αj|

m

• i=1

|σ(i, j − 1)| · ˆ gi |λi − βj|

n

• ℓ=1

|τ(ℓ, j − 1)| ˆ fℓ |µℓ − αj| , σ(i, j − 1) =

j−1

• s=1

λi − αs λi − βs , σ(i, 0) = 1, i = 1, . . . , m,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 11 / 22

Error Bound for Sylvester equation

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1, Let Xk obtained by LRADI, s.t. Xk ≈ X , δX = X − Xk

δX ≤ ST −1

min{m,n}

• j=k+1

|βj − αj|

m

• i=1

|σ(i, j − 1)| · ˆ gi |λi − βj|

n

• ℓ=1

|τ(ℓ, j − 1)| ˆ fℓ |µℓ − αj| , σ(i, j − 1) =

j−1

• s=1

λi − αs λi − βs , σ(i, 0) = 1, i = 1, . . . , m, τ(i, j − 1) =

j−1

• s=1

µi − βs µi − αs , τ(i, 0) = 1, i = 1, . . . , n.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 11 / 22

Error Bound for Sylvester equation

Consider AX − XB = GF ∗, where A and B are diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1, Let Xk obtained by LRADI, s.t. Xk ≈ X , δX = X − Xk

δX ≤ ST −1

min{m,n}

• j=k+1

|βj − αj|

m

• i=1

|σ(i, j − 1)| · ˆ gi |λi − βj|

n

• ℓ=1

|τ(ℓ, j − 1)| ˆ fℓ |µℓ − αj| , σ(i, j − 1) =

j−1

• s=1

λi − αs λi − βs , σ(i, 0) = 1, i = 1, . . . , m, τ(i, j − 1) =

j−1

• s=1

µi − βs µi − αs , τ(i, 0) = 1, i = 1, . . . , n. ˆ gi is the ith row of ˆ G = S−1G; ˆ fi is the ith column of ˆ F ∗ = F ∗T

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 11 / 22

Error Bound for Lyapunov equation

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 12 / 22

Error Bound for Lyapunov equation

Let Xk be obtained by LRADI, s.t. Xk ≈ X

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 12 / 22

Error Bound for Lyapunov equation

Let Xk be obtained by LRADI, s.t. Xk ≈ X How close is Xk to X ?

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 12 / 22

Error Bound for Lyapunov equation

Let Xk be obtained by LRADI, s.t. Xk ≈ X How close is Xk to X ? For the case of Lyapunov equation: AX − XA∗ = GG ∗, A = SΛS−1 diagonalizable

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 12 / 22

Error Bound for Lyapunov equation

Let Xk be obtained by LRADI, s.t. Xk ≈ X How close is Xk to X ? For the case of Lyapunov equation: AX − XA∗ = GG ∗, A = SΛS−1 diagonalizable ( T. and Veseli´ c 2007): X − Xk ≤ S2

m

• j=k+1

(−2Re(αj))

m

• k=1

σ(j, k)2 · ˆ gk2

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 12 / 22

Error Bound for Lyapunov equation

Let Xk be obtained by LRADI, s.t. Xk ≈ X How close is Xk to X ? For the case of Lyapunov equation: AX − XA∗ = GG ∗, A = SΛS−1 diagonalizable ( T. and Veseli´ c 2007): X − Xk ≤ S2

m

• j=k+1

(−2Re(αj))

m

• k=1

σ(j, k)2 · ˆ gk2 σ(1, k) = 1 λk + α1 , and σ(j, k) = 1 λk + αj

j−1

• t=1

λk − ¯ αt λk + αt for j > 1 .

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 12 / 22

Error Bound for Lyapunov equation

Let Xk be obtained by LRADI, s.t. Xk ≈ X How close is Xk to X ? For the case of Lyapunov equation: AX − XA∗ = GG ∗, A = SΛS−1 diagonalizable ( T. and Veseli´ c 2007): X − Xk ≤ S2

m

• j=k+1

(−2Re(αj))

m

• k=1

σ(j, k)2 · ˆ gk2 σ(1, k) = 1 λk + α1 , and σ(j, k) = 1 λk + αj

j−1

• t=1

λk − ¯ αt λk + αt for j > 1 . ˆ gk is the kth row of ˆ G = S−1G

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 12 / 22

Perturbation Bound for Sylvester equation

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 13 / 22

Perturbation Bound for Sylvester equation

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (G + δG)(F + δF)∗,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 13 / 22

Perturbation Bound for Sylvester equation

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (G + δG)(F + δF)∗, Using

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 13 / 22

Perturbation Bound for Sylvester equation

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (G + δG)(F + δF)∗, Using A δX − δX B ≈ G δF ∗ + δG F ∗ − δA X + X δB,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 13 / 22

Perturbation Bound for Sylvester equation

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (G + δG)(F + δF)∗, Using A δX − δX B ≈ G δF ∗ + δG F ∗ − δA X + X δB,

• ne gets: δX ≈ δX1 + δX2 + δX3
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 13 / 22

Perturbation Bound for Sylvester equation

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (G + δG)(F + δF)∗, Using A δX − δX B ≈ G δF ∗ + δG F ∗ − δA X + X δB,

• ne gets: δX ≈ δX1 + δX2 + δX3

A δX1 − δX1 B = −δA X, A δX2 − δX2 B = −Xδ B, A δX3 − δX3 B = (G δF ∗ + δG F ∗).

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 13 / 22

Perturbation Bound for Sylvester equation

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 14 / 22

Perturbation Bound for Sylvester equation

The following perturbation bound holds:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 14 / 22

Perturbation Bound for Sylvester equation

The following perturbation bound holds: for A and B diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 14 / 22

Perturbation Bound for Sylvester equation

The following perturbation bound holds: for A and B diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1. and sufficiently small η = max{δA, δB, δG, δF},

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 14 / 22

Perturbation Bound for Sylvester equation

The following perturbation bound holds: for A and B diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1. and sufficiently small η = max{δA, δB, δG, δF}, δX X ≤

• κ(T)SS−1δA + κ(S)T −1δBT

+ST −1S−1(GδF ∗ + δGF ∗)T X

• γ + O(η2),
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 14 / 22

Perturbation Bound for Sylvester equation

The following perturbation bound holds: for A and B diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1. and sufficiently small η = max{δA, δB, δG, δF}, δX X ≤

• κ(T)SS−1δA + κ(S)T −1δBT

+ST −1S−1(GδF ∗ + δGF ∗)T X

• γ + O(η2),

γ =

n0

• j=1

|µj − λj|τ (j)

max σ(j) max

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 14 / 22

Perturbation Bound for Sylvester equation

The following perturbation bound holds: for A and B diagonalizable; A = S diag(λi) S−1, B = T diag(µi)T −1. and sufficiently small η = max{δA, δB, δG, δF}, δX X ≤

• κ(T)SS−1δA + κ(S)T −1δBT

+ST −1S−1(GδF ∗ + δGF ∗)T X

• γ + O(η2),

γ =

n0

• j=1

|µj − λj|τ (j)

max σ(j) max where:

σ(j)

max = ∆Φ(j) = max i

|σ(i, j − 1)| |λi − µj| , τ (j)

max = ∆Ψ(j) = max i

|τ(i, j − 1)| |µi − λj| .

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 14 / 22

Illustration of new perturbation bound

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 15 / 22

Illustration of new perturbation bound

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (C + δC),

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 15 / 22

Illustration of new perturbation bound

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (C + δC), The standard bound (N. J. Higham, 1996): δXF XF ≤ √ 3Ψǫ + O(ǫ2),

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 15 / 22

Illustration of new perturbation bound

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (C + δC), The standard bound (N. J. Higham, 1996): δXF XF ≤ √ 3Ψǫ + O(ǫ2), where Ψ = P−1[α(X T⊗Im) −β(In⊗X) −δImn]/XF, P = In⊗A−BT⊗Im

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 15 / 22

Illustration of new perturbation bound

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (C + δC), The standard bound (N. J. Higham, 1996): δXF XF ≤ √ 3Ψǫ + O(ǫ2), where Ψ = P−1[α(X T⊗Im) −β(In⊗X) −δImn]/XF, P = In⊗A−BT⊗Im ǫ = max

• δAF

α , δBF β , δCF δ

• ,
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 15 / 22

Illustration of new perturbation bound

Consider perturbed Sylvester equation (A + δA)(X + δX) − (X + δX)(B + δB) = (C + δC), The standard bound (N. J. Higham, 1996): δXF XF ≤ √ 3Ψǫ + O(ǫ2), where Ψ = P−1[α(X T⊗Im) −β(In⊗X) −δImn]/XF, P = In⊗A−BT⊗Im ǫ = max

• δAF

α , δBF β , δCF δ

• , α = AF, β = BF and δ = CF
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 15 / 22

Illustration of new perturbation bound

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 16 / 22

Illustration of new perturbation bound

Let A = SΛS−1 with Λ = diag(30, 40, 60, 70, 90)

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 16 / 22

Illustration of new perturbation bound

Let A = SΛS−1 with Λ = diag(30, 40, 60, 70, 90) S =       0.052 0.494 0.935 0.592 0.827 0.264 0.094 0.220 0.388 0.387 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

Illustration of new perturbation bound

Let A = SΛS−1 with Λ = diag(30, 40, 60, 70, 90) S =       0.052 0.494 0.935 0.592 0.827 0.264 0.094 0.220 0.388 0.387 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 = TΩT −1 with Ω = diag(100, 200, 300, 400, 450)

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 16 / 22

Illustration of new perturbation bound

Let A = SΛS−1 with Λ = diag(30, 40, 60, 70, 90) S =       0.052 0.494 0.935 0.592 0.827 0.264 0.094 0.220 0.388 0.387 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 = TΩT −1 with Ω = diag(100, 200, 300, 400, 450) T =       85610 89420 42300 31670 97380 9709 64180 89230 36420 28030 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

Illustration of new perturbation bound

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 17 / 22

Illustration of new perturbation bound

Further, G = −10 −60 −60 400 T , F = G.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 17 / 22

Illustration of new perturbation bound

Further, G = −10 −60 −60 400 T , F = G. Perturbations are δB = 10−9 · diag

• 10−5, 10−5, 1, 1, 1
• ,

δA = 0, δF = δG = 0.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 17 / 22

Illustration of new perturbation bound

Further, G = −10 −60 −60 400 T , F = G. Perturbations are δB = 10−9 · diag

• 10−5, 10−5, 1, 1, 1
• ,

δA = 0, δF = δG = 0. “Exact error” δXF XF = 5.05 · 10−11,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 17 / 22

Illustration of new perturbation bound

Further, G = −10 −60 −60 400 T , F = G. Perturbations are δB = 10−9 · diag

• 10−5, 10−5, 1, 1, 1
• ,

δA = 0, δF = δG = 0. “Exact error” δXF XF = 5.05 · 10−11, “Higham bound”

δXF XF ≤ 5.42 · 10−5

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 17 / 22

Illustration of new perturbation bound

Further, G = −10 −60 −60 400 T , F = G. Perturbations are δB = 10−9 · diag

• 10−5, 10−5, 1, 1, 1
• ,

δA = 0, δF = δG = 0. “Exact error” δXF XF = 5.05 · 10−11, “Higham bound” “The new bound”

δXF XF ≤ 5.42 · 10−5 δXF XF ≤ 9.19 · 10−9

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 17 / 22

The second part: Bound for condition of J-unitary matrices

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 18 / 22

The second part: Bound for condition of J-unitary matrices

Consider “quasi-definite” H

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 18 / 22

The second part: Bound for condition of J-unitary matrices

Consider “quasi-definite” H H = H11 H12 H∗

12

−H22

• ,
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 18 / 22

The second part: Bound for condition of J-unitary matrices

Consider “quasi-definite” H H = H11 H12 H∗

12

−H22

• ,

where H11, H22 > 0.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 18 / 22

The second part: Bound for condition of J-unitary matrices

Consider “quasi-definite” H H = H11 H12 H∗

12

−H22

• ,

where H11, H22 > 0. H can be written as H = H1/2 (J + δJ)H1/2 , where

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 18 / 22

The second part: Bound for condition of J-unitary matrices

Consider “quasi-definite” H H = H11 H12 H∗

12

−H22

• ,

where H11, H22 > 0. H can be written as H = H1/2 (J + δJ)H1/2 , where H1/2 =

• H1/2

11

H1/2

22

• ,

J = I −I

• ,

δJ =

• H−1/2

11

H12H−1/2

22

H−1/2

22

H∗

12H−1/2 11

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 18 / 22

The second part: Bound for condition of J-unitary matrices

Consider “quasi-definite” H H = H11 H12 H∗

12

−H22

• ,

where H11, H22 > 0. H can be written as H = H1/2 (J + δJ)H1/2 , where H1/2 =

• H1/2

11

H1/2

22

• ,

J = I −I

• ,

δJ =

• H−1/2

11

H12H−1/2

22

H−1/2

22

H∗

12H−1/2 11

• Let X be J-unitary, such that
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 18 / 22

The second part: Bound for condition of J-unitary matrices

Consider “quasi-definite” H H = H11 H12 H∗

12

−H22

• ,

where H11, H22 > 0. H can be written as H = H1/2 (J + δJ)H1/2 , where H1/2 =

• H1/2

11

H1/2

22

• ,

J = I −I

• ,

δJ =

• H−1/2

11

H12H−1/2

22

H−1/2

22

H∗

12H−1/2 11

• Let X be J-unitary, such that

X ∗H0X = |Λ+| |Λ−|

• ,

X ∗JX = I −I

• ≡ J .
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 18 / 22

The second part: Bound for condition of J-unitary matrices

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is if H−1/2

11

H12H−1/2

22

F < 2/(4 √ 2 + 3), then:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is if H−1/2

11

H12H−1/2

22

F < 2/(4 √ 2 + 3), then: X2 ≤ 1 1 − 4γqd , where γqd = δJF/(2 − 3δJ).

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is if H−1/2

11

H12H−1/2

22

F < 2/(4 √ 2 + 3), then: X2 ≤ 1 1 − 4γqd , where γqd = δJF/(2 − 3δJ). The similar holds for block scaled diagonally dominant matrices

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is if H−1/2

11

H12H−1/2

22

F < 2/(4 √ 2 + 3), then: X2 ≤ 1 1 − 4γqd , where γqd = δJF/(2 − 3δJ). The similar holds for block scaled diagonally dominant matrices H = D∗

b(Jb + Nb)Db,

Db = D1 ⊕ D2 ⊕ . . . ⊕ Dk

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is if H−1/2

11

H12H−1/2

22

F < 2/(4 √ 2 + 3), then: X2 ≤ 1 1 − 4γqd , where γqd = δJF/(2 − 3δJ). The similar holds for block scaled diagonally dominant matrices H = D∗

b(Jb + Nb)Db,

Db = D1 ⊕ D2 ⊕ . . . ⊕ Dk Di is non-singular, Jb = J1 ⊕ J2 ⊕ . . . ⊕ Jk, Ji = I or Ji = −I.

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is if H−1/2

11

H12H−1/2

22

F < 2/(4 √ 2 + 3), then: X2 ≤ 1 1 − 4γqd , where γqd = δJF/(2 − 3δJ). The similar holds for block scaled diagonally dominant matrices H = D∗

b(Jb + Nb)Db,

Db = D1 ⊕ D2 ⊕ . . . ⊕ Dk Di is non-singular, Jb = J1 ⊕ J2 ⊕ . . . ⊕ Jk, Ji = I or Ji = −I. If NbF < 2/7, then:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is if H−1/2

11

H12H−1/2

22

F < 2/(4 √ 2 + 3), then: X2 ≤ 1 1 − 4γqd , where γqd = δJF/(2 − 3δJ). The similar holds for block scaled diagonally dominant matrices H = D∗

b(Jb + Nb)Db,

Db = D1 ⊕ D2 ⊕ . . . ⊕ Dk Di is non-singular, Jb = J1 ⊕ J2 ⊕ . . . ⊕ Jk, Ji = I or Ji = −I. If NbF < 2/7, then: X2 ≤ 1 1 − 4γb ,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Bound for condition of J-unitary matrices

The existing bounds on XF need small H−1/2

11

H12H−1/2

22

F, that is if H−1/2

11

H12H−1/2

22

F < 2/(4 √ 2 + 3), then: X2 ≤ 1 1 − 4γqd , where γqd = δJF/(2 − 3δJ). The similar holds for block scaled diagonally dominant matrices H = D∗

b(Jb + Nb)Db,

Db = D1 ⊕ D2 ⊕ . . . ⊕ Dk Di is non-singular, Jb = J1 ⊕ J2 ⊕ . . . ⊕ Jk, Ji = I or Ji = −I. If NbF < 2/7, then: X2 ≤ 1 1 − 4γb , where γb = NbF/(2 − 3Nb).

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 19 / 22

The second part: Different view

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 20 / 22

The second part: Different view

Instead H = H1/2 (J + δJ)H1/2

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 20 / 22

The second part: Different view

Instead H = H1/2 (J + δJ)H1/2 write H = (I + Γ)H1/2 JH1/2 (I + Γ)∗,

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 20 / 22

The second part: Different view

Instead H = H1/2 (J + δJ)H1/2 write H = (I + Γ)H1/2 JH1/2 (I + Γ)∗, where Γ is solution of “Riccati eq.”:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 20 / 22

The second part: Different view

Instead H = H1/2 (J + δJ)H1/2 write H = (I + Γ)H1/2 JH1/2 (I + Γ)∗, where Γ is solution of “Riccati eq.”: Γˆ H + ˆ HΓ∗ + Γˆ HΓ∗ =

• H−1/2

11

H12H−1/2

22

H−1/2

22

H∗

12H−1/2 11

• ,

ˆ H ≡ H1/2 JH1/2 = H11 −H22

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 20 / 22

The second part: Different view

Instead H = H1/2 (J + δJ)H1/2 write H = (I + Γ)H1/2 JH1/2 (I + Γ)∗, where Γ is solution of “Riccati eq.”: Γˆ H + ˆ HΓ∗ + Γˆ HΓ∗ =

• H−1/2

11

H12H−1/2

22

H−1/2

22

H∗

12H−1/2 11

• ,

ˆ H ≡ H1/2 JH1/2 = H11 −H22

• we use HSVD:

(I + Γ)H1/2 = ˜ U|˜ Λ|1/2Q−1 , H1/2 = U|Λ|1/2X −1

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 20 / 22

The second part: Different view

Instead H = H1/2 (J + δJ)H1/2 write H = (I + Γ)H1/2 JH1/2 (I + Γ)∗, where Γ is solution of “Riccati eq.”: Γˆ H + ˆ HΓ∗ + Γˆ HΓ∗ =

• H−1/2

11

H12H−1/2

22

H−1/2

22

H∗

12H−1/2 11

• ,

ˆ H ≡ H1/2 JH1/2 = H11 −H22

• we use HSVD:

(I + Γ)H1/2 = ˜ U|˜ Λ|1/2Q−1 , H1/2 = U|Λ|1/2X −1 where: Q∗JQ = J, and Q∗Q = QQ∗ = I

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 20 / 22

The second part: Different view

Instead H = H1/2 (J + δJ)H1/2 write H = (I + Γ)H1/2 JH1/2 (I + Γ)∗, where Γ is solution of “Riccati eq.”: Γˆ H + ˆ HΓ∗ + Γˆ HΓ∗ =

• H−1/2

11

H12H−1/2

22

H−1/2

22

H∗

12H−1/2 11

• ,

ˆ H ≡ H1/2 JH1/2 = H11 −H22

• we use HSVD:

(I + Γ)H1/2 = ˜ U|˜ Λ|1/2Q−1 , H1/2 = U|Λ|1/2X −1 where: Q∗JQ = J, and Q∗Q = QQ∗ = I we consider ∆ = Q∗ H1/2 (I + Γ∗)(I + Γ)H1/2 − H0

• X
• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 20 / 22

The second part: Different view

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 21 / 22

The second part: Different view

it can bee shown that

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 21 / 22

The second part: Different view

it can bee shown that ∆ = ˜ ΛQ∗JX − Q∗JXΛ ∆ = |˜ Λ|1/2 ˜ UδΓU|Λ|, δΓ = (I + Γ) − (I + Γ)−∗

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 21 / 22

The second part: Different view

it can bee shown that ∆ = ˜ ΛQ∗JX − Q∗JXΛ ∆ = |˜ Λ|1/2 ˜ UδΓU|Λ|, δΓ = (I + Γ) − (I + Γ)−∗ Pointwise interpretation gives:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 21 / 22

The second part: Different view

it can bee shown that ∆ = ˜ ΛQ∗JX − Q∗JXΛ ∆ = |˜ Λ|1/2 ˜ UδΓU|Λ|, δΓ = (I + Γ) − (I + Γ)−∗ Pointwise interpretation gives: (Q∗JX)i,j = ˜ U(:,i)δΓU(:,j)

˜ λi−λj

√

˜ λi√ λj

. All together:

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 21 / 22

The second part: Different view

it can bee shown that ∆ = ˜ ΛQ∗JX − Q∗JXΛ ∆ = |˜ Λ|1/2 ˜ UδΓU|Λ|, δΓ = (I + Γ) − (I + Γ)−∗ Pointwise interpretation gives: (Q∗JX)i,j = ˜ U(:,i)δΓU(:,j)

˜ λi−λj

√

˜ λi√ λj

. All together: XF ≤ Ψ +

• 1 + Ψ2,

Ψ = δΓF rg(Λ+, ˜ Λ−)

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 21 / 22

The second part: Different view

it can bee shown that ∆ = ˜ ΛQ∗JX − Q∗JXΛ ∆ = |˜ Λ|1/2 ˜ UδΓU|Λ|, δΓ = (I + Γ) − (I + Γ)−∗ Pointwise interpretation gives: (Q∗JX)i,j = ˜ U(:,i)δΓU(:,j)

˜ λi−λj

√

˜ λi√ λj

. All together: XF ≤ Ψ +

• 1 + Ψ2,

Ψ = δΓF rg(Λ+, ˜ Λ−) where rg(Λ+, ˜ Λ−) = min

˜ λi∈˜ Λ−,λj∈Λ+

|˜ λi − λj|

• |˜

λi| |λj| .

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 21 / 22

Summary

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22

Summary

we present a new bound for estimating the size of the solution of Sylvester equation AX − XB = GF ∗

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22

Summary

we present a new bound for estimating the size of the solution of Sylvester equation AX − XB = GF ∗ we present corresponding perturbation bound

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22

Summary

we present a new bound for estimating the size of the solution of Sylvester equation AX − XB = GF ∗ we present corresponding perturbation bound the all above results hold for diagonalizable A and B, as well as for A and B with Jordan structure

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22

Summary

we present a new bound for estimating the size of the solution of Sylvester equation AX − XB = GF ∗ we present corresponding perturbation bound the all above results hold for diagonalizable A and B, as well as for A and B with Jordan structure the presented bounds include the influence of the right hand side GF ∗

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22

Summary

we present a new bound for estimating the size of the solution of Sylvester equation AX − XB = GF ∗ we present corresponding perturbation bound the all above results hold for diagonalizable A and B, as well as for A and B with Jordan structure the presented bounds include the influence of the right hand side GF ∗ Finally, we show the new bound for XF of J-unitary matrix X

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22

Summary

we present a new bound for estimating the size of the solution of Sylvester equation AX − XB = GF ∗ we present corresponding perturbation bound the all above results hold for diagonalizable A and B, as well as for A and B with Jordan structure the presented bounds include the influence of the right hand side GF ∗ Finally, we show the new bound for XF of J-unitary matrix X which is connected with “quasi-definite” H

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22

Summary

we present a new bound for estimating the size of the solution of Sylvester equation AX − XB = GF ∗ we present corresponding perturbation bound the all above results hold for diagonalizable A and B, as well as for A and B with Jordan structure the presented bounds include the influence of the right hand side GF ∗ Finally, we show the new bound for XF of J-unitary matrix X which is connected with “quasi-definite” H for the new bound there is no limits on the magnitude on “off-diagonal” block of H

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22

Summary

we present a new bound for estimating the size of the solution of Sylvester equation AX − XB = GF ∗ we present corresponding perturbation bound the all above results hold for diagonalizable A and B, as well as for A and B with Jordan structure the presented bounds include the influence of the right hand side GF ∗ Finally, we show the new bound for XF of J-unitary matrix X which is connected with “quasi-definite” H for the new bound there is no limits on the magnitude on “off-diagonal” block of H it depends on magnitude of solution of corresponding “Riccati equation”

• N. Truhar (Osijek, 2009)

Sylvester equation 24.04.2009 22 / 22