Introduction Approach 1 Linearizations Example Robot Conclusions Higher Order Linear Time-Invariant Systems Standard approach: Linearization Consider associated matrix polynomial P ( λ ) = λ ℓ P ℓ + λ ℓ − 1 P ℓ − 1 + · · · + λ P 1 + P 0 ∈ Π n ℓ and convert it into λ E + A ∈ Π ℓ n 1 with the same eigenvalues. Outline Illustrative examples Approach 1: MOR for higher order system by Freund (2005) (Approach 2: MOR for higher order system by Li, Bao, Lin, Wei (2011)) New developments in linearization of matrix polynomials Generalization of companion form linearization L 1 Block Kronecker linearizations G r + 1 Higher order LTI systems and block Kronecker linearizations H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Higher Order Linear Time-Invariant Systems Standard approach: Linearization Consider associated matrix polynomial P ( λ ) = λ ℓ P ℓ + λ ℓ − 1 P ℓ − 1 + · · · + λ P 1 + P 0 ∈ Π n ℓ and convert it into λ E + A ∈ Π ℓ n 1 with the same eigenvalues. Outline Illustrative examples Approach 1: MOR for higher order system by Freund (2005) (Approach 2: MOR for higher order system by Li, Bao, Lin, Wei (2011)) New developments in linearization of matrix polynomials Generalization of companion form linearization L 1 Block Kronecker linearizations G r + 1 Higher order LTI systems and block Kronecker linearizations H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Illustrative examples Gyroscopic system P ( λ ) ∈ Π n 2 P ( λ ) = λ 2 M + λ G + K , M = M T , G = − G T , K = K T , M , G , K ∈ R n × n . Such problems arise, for example, in finite element discretization in structural analysis and in the elastic deformation of anisotropic materials. They are used to model vibrations of spinning structures such as the simulation of tire noise, helicopter rotor blades, or spin-stabilized satellites with appended solar panels or antennas. Robot P ( λ ) ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , Such problems arise, e.g, from the model of a robot with electric motors in the joints. T-even matrix polynomials For both examples: P ( λ ) = P (− λ ) T . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Illustrative examples Gyroscopic system P ( λ ) ∈ Π n 2 P ( λ ) = λ 2 M + λ G + K , M = M T , G = − G T , K = K T , M , G , K ∈ R n × n . Such problems arise, for example, in finite element discretization in structural analysis and in the elastic deformation of anisotropic materials. They are used to model vibrations of spinning structures such as the simulation of tire noise, helicopter rotor blades, or spin-stabilized satellites with appended solar panels or antennas. Robot P ( λ ) ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , Such problems arise, e.g, from the model of a robot with electric motors in the joints. T-even matrix polynomials For both examples: P ( λ ) = P (− λ ) T . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Illustrative examples Gyroscopic system P ( λ ) ∈ Π n 2 P ( λ ) = λ 2 M + λ G + K , M = M T , G = − G T , K = K T , M , G , K ∈ R n × n . Such problems arise, for example, in finite element discretization in structural analysis and in the elastic deformation of anisotropic materials. They are used to model vibrations of spinning structures such as the simulation of tire noise, helicopter rotor blades, or spin-stabilized satellites with appended solar panels or antennas. Robot P ( λ ) ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , Such problems arise, e.g, from the model of a robot with electric motors in the joints. T-even matrix polynomials For both examples: P ( λ ) = P (− λ ) T . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Illustrative examples Gyroscopic system P ( λ ) ∈ Π n 2 P ( λ ) = λ 2 M + λ G + K , M = M T , G = − G T , K = K T , M , G , K ∈ R n × n . Such problems arise, for example, in finite element discretization in structural analysis and in the elastic deformation of anisotropic materials. They are used to model vibrations of spinning structures such as the simulation of tire noise, helicopter rotor blades, or spin-stabilized satellites with appended solar panels or antennas. Robot P ( λ ) ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , Such problems arise, e.g, from the model of a robot with electric motors in the joints. T-even matrix polynomials For both examples: P ( λ ) = P (− λ ) T . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Illustrative examples Gyroscopic system P ( λ ) ∈ Π n 2 P ( λ ) = λ 2 M + λ G + K , M = M T , G = − G T , K = K T , M , G , K ∈ R n × n . Such problems arise, for example, in finite element discretization in structural analysis and in the elastic deformation of anisotropic materials. They are used to model vibrations of spinning structures such as the simulation of tire noise, helicopter rotor blades, or spin-stabilized satellites with appended solar panels or antennas. Robot P ( λ ) ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , Such problems arise, e.g, from the model of a robot with electric motors in the joints. T-even matrix polynomials For both examples: P ( λ ) = P (− λ ) T . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Higher Order Linear Time-Invariant Systems Back to Higher Order Linear Time-Invariant Systems d ℓ d ℓ − 1 d dt ℓ x ( t ) + P ℓ − 1 dt ℓ − 1 x ( t ) + · · · + P 1 dt x ( t ) + P 0 x ( t ) = Bu ( t ) P ℓ d ℓ − 1 d Du ( t ) + C ℓ − 1 dt ℓ − 1 x ( t ) + · · · + C 1 dt x ( t ) + C 0 x ( t ) = y ( t ) Let 0 − I n 0 · · · 0 x ( t ) 0 . ... . d 0 0 − I n . dt x ( t ) . . . . z ( t ) = . , B F = , A F = ... ... ... , . . . 0 . 0 d ℓ − 1 0 · · · 0 0 − I n B dt ℓ − 1 x ( t ) P 0 P 1 P 2 · · · P ℓ − 1 � � I ( ℓ − 1 ) n E F = , C F = [ C 0 C 1 · · · C ℓ − 1 ] , D F = D . P ℓ H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Higher Order Linear Time-Invariant Systems Back to Higher Order Linear Time-Invariant Systems d ℓ d ℓ − 1 d dt ℓ x ( t ) + P ℓ − 1 dt ℓ − 1 x ( t ) + · · · + P 1 dt x ( t ) + P 0 x ( t ) = Bu ( t ) P ℓ d ℓ − 1 d Du ( t ) + C ℓ − 1 dt ℓ − 1 x ( t ) + · · · + C 1 dt x ( t ) + C 0 x ( t ) = y ( t ) Let 0 − I n 0 · · · 0 x ( t ) 0 . ... . d 0 0 − I n . dt x ( t ) . . . . z ( t ) = . , B F = , A F = ... ... ... , . . . 0 . 0 d ℓ − 1 0 · · · 0 0 − I n B dt ℓ − 1 x ( t ) P 0 P 1 P 2 · · · P ℓ − 1 � � I ( ℓ − 1 ) n E F = , C F = [ C 0 C 1 · · · C ℓ − 1 ] , D F = D . P ℓ H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Approach 1: Linearization via the first companion form The higher order system is equivalent to the first order system d E F dt z ( t ) + A F z ( t ) = B F u ( t ) y ( t ) = D F u ( t ) + C F z ( t ) z ( 0 ) = z 0 where − I n 0 0 0 · · · x ( 0 ) x ( t ) 0 . ... 0 . d x ( 1 ) dt x ( t ) 0 0 − I n . . . 0 . . z ( t ) = . , z 0 = , B F = , A F = ... ... ... , . . . . . . 0 0 . d ℓ − 1 0 0 0 − I n dt ℓ − 1 x ( t ) x ( ℓ − 1 ) B · · · 0 P 0 P 1 P 2 P ℓ − 1 · · · � � I ( ℓ − 1 ) n E F = , C F = [ C 0 C 1 · · · C ℓ − 1 ] , D F = D . P ℓ H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Approach 1: Linearization via the first companion form The higher order system is equivalent to the first order system d E F dt z ( t ) + A F z ( t ) = B F u ( t ) y ( t ) = D F u ( t ) + C F z ( t ) z ( 0 ) = z 0 Transfer function G ( s ) = D F + C F ( s E F + A F ) − 1 B F = D + � ℓ − 1 j = 0 C j ( P ( s )) − 1 B ∈ C [ s ] p × m . E F , A F ∈ R ℓ n × ℓ n , B F ∈ R ℓ n × m are large and (block-) sparse. λ E F + A F does not inherit any structure from P ( λ ) , that is, e.g., P ( λ ) = P ( λ ) T does not imply that ( λ E F + A F ) T = λ E F + A F . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Approach 1: Linearization via the first companion form The higher order system is equivalent to the first order system d E F dt z ( t ) + A F z ( t ) = B F u ( t ) y ( t ) = D F u ( t ) + C F z ( t ) z ( 0 ) = z 0 Transfer function G ( s ) = D F + C F ( s E F + A F ) − 1 B F = D + � ℓ − 1 j = 0 C j ( P ( s )) − 1 B ∈ C [ s ] p × m . E F , A F ∈ R ℓ n × ℓ n , B F ∈ R ℓ n × m are large and (block-) sparse. λ E F + A F does not inherit any structure from P ( λ ) , that is, e.g., P ( λ ) = P ( λ ) T does not imply that ( λ E F + A F ) T = λ E F + A F . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Approach 1: Linearization via the first companion form The higher order system is equivalent to the first order system d E F dt z ( t ) + A F z ( t ) = B F u ( t ) y ( t ) = D F u ( t ) + C F z ( t ) z ( 0 ) = z 0 Transfer function G ( s ) = D F + C F ( s E F + A F ) − 1 B F = D + � ℓ − 1 j = 0 C j ( P ( s )) − 1 B ∈ C [ s ] p × m . E F , A F ∈ R ℓ n × ℓ n , B F ∈ R ℓ n × m are large and (block-) sparse. λ E F + A F does not inherit any structure from P ( λ ) , that is, e.g., P ( λ ) = P ( λ ) T does not imply that ( λ E F + A F ) T = λ E F + A F . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Approach 1: Linearization via the first companion form The higher order system is equivalent to the first order system d E F dt z ( t ) + A F z ( t ) = B F u ( t ) y ( t ) = D F u ( t ) + C F z ( t ) z ( 0 ) = z 0 Transfer function G ( s ) = D F + C F ( s E F + A F ) − 1 B F = D + � ℓ − 1 j = 0 C j ( P ( s )) − 1 B ∈ C [ s ] p × m . E F , A F ∈ R ℓ n × ℓ n , B F ∈ R ℓ n × m are large and (block-) sparse. λ E F + A F does not inherit any structure from P ( λ ) , that is, e.g., P ( λ ) = P ( λ ) T does not imply that ( λ E F + A F ) T = λ E F + A F . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Rewrite G ( s ) = D F + C F ( s E F + A F ) − 1 B F for s 0 ∈ C such that s 0 E F + A F is nonsingular as G ( s ) = D F + C F ( I + ( s − s 0 ) M F ) − 1 R F with M F = ( s 0 E F + A F ) − 1 E F ∈ C ℓ n × ℓ n , R F = ( s 0 E F + A F ) − 1 B F ∈ C ℓ n × m . Compute orthonormal basis of K s ( M F , R F ) = span {R F , M F R F , . . . , M s − 1 R F } . F Let W be the matrix representing the basis. Generate reduced order system E d ˆ z ( t ) + ˆ z ( t ) = ˆ dt ˆ A ˆ B u ( t ) y ( t ) = D u ( t ) + ˆ ˆ C ˆ z ( t ) with ˆ E = W T EW , ˆ A = W T AW ∈ C r × r , ˆ B = W T B ∈ C r × m , ˆ C = CW ∈ C p × r . It seems as if no ℓ th order ODE can be extracted. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Rewrite G ( s ) = D F + C F ( s E F + A F ) − 1 B F for s 0 ∈ C such that s 0 E F + A F is nonsingular as G ( s ) = D F + C F ( I + ( s − s 0 ) M F ) − 1 R F with M F = ( s 0 E F + A F ) − 1 E F ∈ C ℓ n × ℓ n , R F = ( s 0 E F + A F ) − 1 B F ∈ C ℓ n × m . Compute orthonormal basis of K s ( M F , R F ) = span {R F , M F R F , . . . , M s − 1 R F } . F Let W be the matrix representing the basis. Generate reduced order system E d ˆ z ( t ) + ˆ z ( t ) = ˆ dt ˆ A ˆ B u ( t ) y ( t ) = D u ( t ) + ˆ ˆ C ˆ z ( t ) with ˆ E = W T EW , ˆ A = W T AW ∈ C r × r , ˆ B = W T B ∈ C r × m , ˆ C = CW ∈ C p × r . It seems as if no ℓ th order ODE can be extracted. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Rewrite G ( s ) = D F + C F ( s E F + A F ) − 1 B F for s 0 ∈ C such that s 0 E F + A F is nonsingular as G ( s ) = D F + C F ( I + ( s − s 0 ) M F ) − 1 R F with M F = ( s 0 E F + A F ) − 1 E F ∈ C ℓ n × ℓ n , R F = ( s 0 E F + A F ) − 1 B F ∈ C ℓ n × m . Compute orthonormal basis of K s ( M F , R F ) = span {R F , M F R F , . . . , M s − 1 R F } . F Let W be the matrix representing the basis. Generate reduced order system E d ˆ z ( t ) + ˆ z ( t ) = ˆ dt ˆ A ˆ B u ( t ) y ( t ) = D u ( t ) + ˆ ˆ C ˆ z ( t ) with ˆ E = W T EW , ˆ A = W T AW ∈ C r × r , ˆ B = W T B ∈ C r × m , ˆ C = CW ∈ C p × r . It seems as if no ℓ th order ODE can be extracted. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Rewrite G ( s ) = D F + C F ( s E F + A F ) − 1 B F for s 0 ∈ C such that s 0 E F + A F is nonsingular as G ( s ) = D F + C F ( I + ( s − s 0 ) M F ) − 1 R F with M F = ( s 0 E F + A F ) − 1 E F ∈ C ℓ n × ℓ n , R F = ( s 0 E F + A F ) − 1 B F ∈ C ℓ n × m . Compute orthonormal basis of K s ( M F , R F ) = span {R F , M F R F , . . . , M s − 1 R F } . F Let W be the matrix representing the basis. Generate reduced order system E d ˆ z ( t ) + ˆ z ( t ) = ˆ dt ˆ A ˆ B u ( t ) y ( t ) = D u ( t ) + ˆ ˆ C ˆ z ( t ) with ˆ E = W T EW , ˆ A = W T AW ∈ C r × r , ˆ B = W T B ∈ C r × m , ˆ C = CW ∈ C p × r . It seems as if no ℓ th order ODE can be extracted. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Rewrite G ( s ) = D F + C F ( s E F + A F ) − 1 B F for s 0 ∈ C such that s 0 E F + A F is nonsingular as G ( s ) = D F + C F ( I + ( s − s 0 ) M F ) − 1 R F with M F = ( s 0 E F + A F ) − 1 E F ∈ C ℓ n × ℓ n , R F = ( s 0 E F + A F ) − 1 B F ∈ C ℓ n × m . Compute orthonormal basis of K s ( M F , R F ) = span {R F , M F R F , . . . , M s − 1 R F } . F Let W be the matrix representing the basis. Generate reduced order system E d ˆ z ( t ) + ˆ z ( t ) = ˆ dt ˆ A ˆ B u ( t ) y ( t ) = D u ( t ) + ˆ ˆ C ˆ z ( t ) with ˆ E = W T EW , ˆ A = W T AW ∈ C r × r , ˆ B = W T B ∈ C r × m , ˆ C = CW ∈ C p × r . It seems as if no ℓ th order ODE can be extracted. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] The matrices M F and R F have a particular structure � M ( ℓ ) � M F = ( s 0 E F + A F ) − 1 E F = ( c ⊗ I n ) M ( 1 ) M ( 2 ) M ( 3 ) · · · + Σ ⊗ I n , R F = ( s 0 E F + A F ) − 1 B F = c ⊗ R , where ℓ − i � M ( i ) = ( P ( s 0 )) − 1 s j 0 P i + j ∈ C n × n , i = 1 , . . . , ℓ j = 0 R = ( P ( s 0 )) − 1 B ∈ C n × m , 0 0 · · · · · · 0 1 . ... . s 0 1 0 . s 2 . ... ∈ C ℓ × ℓ . c = , Σ = . 0 s 0 1 0 . . . . . . ... ... ... . . . . s ℓ − 1 s ℓ − 2 0 · · · s 0 1 0 0 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Theorem ( Freund 2005) � M ( ℓ ) � M ( 1 ) M ( 2 ) M ( 3 ) Let M F = ( c ⊗ I n ) · · · + Σ ⊗ I n , and R F = c ⊗ R with c ∈ C ℓ , c j � = 0 , j = 1 , . . . , ℓ , R ∈ C n × m , M ( i ) ∈ C n × n , i = 1 , . . . , ℓ , Σ ∈ C ℓ × ℓ . Let W ∈ C ℓ n × r be any basis of the block-Krylov subspace K s ( M F , R F ) , r � sm . Then W can be represented in the form WU ( 1 ) WU ( 2 ) where W ∈ C n × r and, for each i = 1 , 2 , . . . , ℓ , . U ( i ) ∈ C r × r is nonsingular and upper triangular. . . WU ( ℓ ) K s ( M F , R F ) ⊂ C ℓ n consists of ℓ ’copies’ of the subspace S r = span { W } ⊂ C n . Let V be the matrix representing an orthonormal basis of span { W } . Choose V = diag ( V , V , . . . , V ) ∈ C ℓ n × ℓ r , V H V = I r . Then K s ( M F , R F ) ⊆ range V . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Theorem ( Freund 2005) � M ( ℓ ) � M ( 1 ) M ( 2 ) M ( 3 ) Let M F = ( c ⊗ I n ) · · · + Σ ⊗ I n , and R F = c ⊗ R with c ∈ C ℓ , c j � = 0 , j = 1 , . . . , ℓ , R ∈ C n × m , M ( i ) ∈ C n × n , i = 1 , . . . , ℓ , Σ ∈ C ℓ × ℓ . Let W ∈ C ℓ n × r be any basis of the block-Krylov subspace K s ( M F , R F ) , r � sm . Then W can be represented in the form WU ( 1 ) WU ( 2 ) where W ∈ C n × r and, for each i = 1 , 2 , . . . , ℓ , . U ( i ) ∈ C r × r is nonsingular and upper triangular. . . WU ( ℓ ) K s ( M F , R F ) ⊂ C ℓ n consists of ℓ ’copies’ of the subspace S r = span { W } ⊂ C n . Let V be the matrix representing an orthonormal basis of span { W } . Choose V = diag ( V , V , . . . , V ) ∈ C ℓ n × ℓ r , V H V = I r . Then K s ( M F , R F ) ⊆ range V . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Theorem ( Freund 2005) � M ( ℓ ) � M ( 1 ) M ( 2 ) M ( 3 ) Let M F = ( c ⊗ I n ) · · · + Σ ⊗ I n , and R F = c ⊗ R with c ∈ C ℓ , c j � = 0 , j = 1 , . . . , ℓ , R ∈ C n × m , M ( i ) ∈ C n × n , i = 1 , . . . , ℓ , Σ ∈ C ℓ × ℓ . Let W ∈ C ℓ n × r be any basis of the block-Krylov subspace K s ( M F , R F ) , r � sm . Then W can be represented in the form WU ( 1 ) WU ( 2 ) where W ∈ C n × r and, for each i = 1 , 2 , . . . , ℓ , . U ( i ) ∈ C r × r is nonsingular and upper triangular. . . WU ( ℓ ) K s ( M F , R F ) ⊂ C ℓ n consists of ℓ ’copies’ of the subspace S r = span { W } ⊂ C n . Let V be the matrix representing an orthonormal basis of span { W } . Choose V = diag ( V , V , . . . , V ) ∈ C ℓ n × ℓ r , V H V = I r . Then K s ( M F , R F ) ⊆ range V . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Theorem ( Freund 2005) � M ( ℓ ) � M ( 1 ) M ( 2 ) M ( 3 ) Let M F = ( c ⊗ I n ) · · · + Σ ⊗ I n , and R F = c ⊗ R with c ∈ C ℓ , c j � = 0 , j = 1 , . . . , ℓ , R ∈ C n × m , M ( i ) ∈ C n × n , i = 1 , . . . , ℓ , Σ ∈ C ℓ × ℓ . Let W ∈ C ℓ n × r be any basis of the block-Krylov subspace K s ( M F , R F ) , r � sm . Then W can be represented in the form WU ( 1 ) WU ( 2 ) where W ∈ C n × r and, for each i = 1 , 2 , . . . , ℓ , . U ( i ) ∈ C r × r is nonsingular and upper triangular. . . WU ( ℓ ) K s ( M F , R F ) ⊂ C ℓ n consists of ℓ ’copies’ of the subspace S r = span { W } ⊂ C n . Let V be the matrix representing an orthonormal basis of span { W } . Choose V = diag ( V , V , . . . , V ) ∈ C ℓ n × ℓ r , V H V = I r . Then K s ( M F , R F ) ⊆ range V . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Theorem ( Freund 2005) � M ( ℓ ) � M ( 1 ) M ( 2 ) M ( 3 ) Let M F = ( c ⊗ I n ) · · · + Σ ⊗ I n , and R F = c ⊗ R with c ∈ C ℓ , c j � = 0 , j = 1 , . . . , ℓ , R ∈ C n × m , M ( i ) ∈ C n × n , i = 1 , . . . , ℓ , Σ ∈ C ℓ × ℓ . Let W ∈ C ℓ n × r be any basis of the block-Krylov subspace K s ( M F , R F ) , r � sm . Then W can be represented in the form WU ( 1 ) WU ( 2 ) where W ∈ C n × r and, for each i = 1 , 2 , . . . , ℓ , . U ( i ) ∈ C r × r is nonsingular and upper triangular. . . WU ( ℓ ) K s ( M F , R F ) ⊂ C ℓ n consists of ℓ ’copies’ of the subspace S r = span { W } ⊂ C n . Let V be the matrix representing an orthonormal basis of span { W } . Choose V = diag ( V , V , . . . , V ) ∈ C ℓ n × ℓ r , V H V = I r . Then K s ( M F , R F ) ⊆ range V . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Project the first order system using V � � � � � � V H d V H E F V V H A F V V H z ( t ) = V H B F dt z ( t ) + u ( t ) y ( t ) = D F u ( t ) + ( C F V ) V H z ( t ) with − I n 0 0 · · · 0 0 . ... . − I n 0 0 . . . . V H B F = , V H A F V = . ... ... ... , . . 0 0 · · · − I n 0 0 0 V H B VP 0 V H VP 1 V H VP 2 V H VP ℓ − 1 V H · · · � � I ( ℓ − 1 ) n V H E F V = , C F V = [ C 0 V C 1 V · · · C ℓ − 1 V ] , D F = D . V H P ℓ V An ℓ th order reduced order system can be read off immediately. The first moments of the reduced order system match those of the original system. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Project the first order system using V � � � � � � V H d V H E F V V H A F V V H z ( t ) = V H B F dt z ( t ) + u ( t ) y ( t ) = D F u ( t ) + ( C F V ) V H z ( t ) with − I n 0 0 · · · 0 0 . ... . − I n 0 0 . . . . V H B F = , V H A F V = . ... ... ... , . . 0 0 · · · − I n 0 0 0 V H B VP 0 V H VP 1 V H VP 2 V H VP ℓ − 1 V H · · · � � I ( ℓ − 1 ) n V H E F V = , C F V = [ C 0 V C 1 V · · · C ℓ − 1 V ] , D F = D . V H P ℓ V An ℓ th order reduced order system can be read off immediately. The first moments of the reduced order system match those of the original system. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 [Freund 2005] Project the first order system using V � � � � � � V H d V H E F V V H A F V V H z ( t ) = V H B F dt z ( t ) + u ( t ) y ( t ) = D F u ( t ) + ( C F V ) V H z ( t ) with − I n 0 0 · · · 0 0 . ... . − I n 0 0 . . . . V H B F = , V H A F V = . ... ... ... , . . 0 0 · · · − I n 0 0 0 V H B VP 0 V H VP 1 V H VP 2 V H VP ℓ − 1 V H · · · � � I ( ℓ − 1 ) n V H E F V = , C F V = [ C 0 V C 1 V · · · C ℓ − 1 V ] , D F = D . V H P ℓ V An ℓ th order reduced order system can be read off immediately. The first moments of the reduced order system match those of the original system. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 and 2 Approach 1 and 2 use companion form linearization. Approach 1 uses block-Krylov subspace K s ( M F , R F ) with M F = ( s 0 E F + A F ) − 1 E F and R F = ( s 0 E F + A F ) − 1 B F . Approach 2 uses block-Krylov subspace K s ( M B , R B ) with M B = A − 1 B E B and R B = A − 1 B B B . Neither λ E F + A F nor λ E B + A B is structure-preserving, e.g., (− λ E F + A F ) T � = λ E F + A F and (− λ E B + A B ) T � = λ E B + A B . There are numerous other linearizations. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 and 2 Approach 1 and 2 use companion form linearization. Approach 1 uses block-Krylov subspace K s ( M F , R F ) with M F = ( s 0 E F + A F ) − 1 E F and R F = ( s 0 E F + A F ) − 1 B F . Approach 2 uses block-Krylov subspace K s ( M B , R B ) with M B = A − 1 B E B and R B = A − 1 B B B . Neither λ E F + A F nor λ E B + A B is structure-preserving, e.g., (− λ E F + A F ) T � = λ E F + A F and (− λ E B + A B ) T � = λ E B + A B . There are numerous other linearizations. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 and 2 Approach 1 and 2 use companion form linearization. Approach 1 uses block-Krylov subspace K s ( M F , R F ) with M F = ( s 0 E F + A F ) − 1 E F and R F = ( s 0 E F + A F ) − 1 B F . Approach 2 uses block-Krylov subspace K s ( M B , R B ) with M B = A − 1 B E B and R B = A − 1 B B B . Neither λ E F + A F nor λ E B + A B is structure-preserving, e.g., (− λ E F + A F ) T � = λ E F + A F and (− λ E B + A B ) T � = λ E B + A B . There are numerous other linearizations. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 and 2 Approach 1 and 2 use companion form linearization. Approach 1 uses block-Krylov subspace K s ( M F , R F ) with M F = ( s 0 E F + A F ) − 1 E F and R F = ( s 0 E F + A F ) − 1 B F . Approach 2 uses block-Krylov subspace K s ( M B , R B ) with M B = A − 1 B E B and R B = A − 1 B B B . Neither λ E F + A F nor λ E B + A B is structure-preserving, e.g., (− λ E F + A F ) T � = λ E F + A F and (− λ E B + A B ) T � = λ E B + A B . There are numerous other linearizations. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Approach 1 and 2 Approach 1 and 2 use companion form linearization. Approach 1 uses block-Krylov subspace K s ( M F , R F ) with M F = ( s 0 E F + A F ) − 1 E F and R F = ( s 0 E F + A F ) − 1 B F . Approach 2 uses block-Krylov subspace K s ( M B , R B ) with M B = A − 1 B E B and R B = A − 1 B B B . Neither λ E F + A F nor λ E B + A B is structure-preserving, e.g., (− λ E F + A F ) T � = λ E F + A F and (− λ E B + A B ) T � = λ E B + A B . There are numerous other linearizations. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations – Motivation Systematic way to construct linearizations that allow for the preservation of structure and/or are better conditioned than the companion forms. [Mackey, Mackey, Mehl, Mehrmann, SIMAX 2006] = [4M] ℓ � λ i P i x P ( λ ) x = i = 0 ⇒ linearization of size ℓ n × ℓ n = λ ℓ − 1 x P ℓ 0 0 0 P ℓ − 1 P ℓ − 2 P 1 P 0 · · · · · · P ( λ ) x λ ℓ − 2 x 0 I n 0 0 − I n 0 0 0 · · · · · · 0 . 0 0 I n 0 0 − I n 0 0 · · · · · · . λ + = . . . . . . . . . . . ... ... ... . . . . . . . . . . . . . . . λ x 0 − I n 0 0 0 I n 0 0 0 · · · · · · x � �� � L 1 ( λ ) H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations – Motivation Systematic way to construct linearizations that allow for the preservation of structure and/or are better conditioned than the companion forms. [Mackey, Mackey, Mehl, Mehrmann, SIMAX 2006] = [4M] ℓ � λ i P i x P ( λ ) x = i = 0 ⇒ linearization of size ℓ n × ℓ n = λ ℓ − 1 x P ℓ 0 0 0 P ℓ − 1 P ℓ − 2 P 1 P 0 · · · · · · P ( λ ) x λ ℓ − 2 x 0 I n 0 0 − I n 0 0 0 · · · · · · 0 . 0 0 I n 0 0 − I n 0 0 · · · · · · . λ + = . . . . . . . . . . . ... ... ... . . . . . . . . . . . . . . . λ x 0 − I n 0 0 0 I n 0 0 0 · · · · · · x � �� � L 1 ( λ ) H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations – Motivation Systematic way to construct linearizations that allow for the preservation of structure and/or are better conditioned than the companion forms. [Mackey, Mackey, Mehl, Mehrmann, SIMAX 2006] = [4M] ℓ � λ i P i x P ( λ ) x = i = 0 ⇒ linearization of size ℓ n × ℓ n = λ ℓ − 1 x P ℓ 0 0 0 P ℓ − 1 P ℓ − 2 P 1 P 0 · · · · · · P ( λ ) x λ ℓ − 2 x 0 I n 0 0 − I n 0 0 0 · · · · · · 0 . 0 0 I n 0 0 − I n 0 0 · · · · · · . λ + = . . . . . . . . . . . ... ... ... . . . . . . . . . . . . . . . λ x 0 − I n 0 0 0 I n 0 0 0 · · · · · · x � �� � L 1 ( λ ) H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations – Motivation Thus λ ℓ − 1 x P ( λ ) x λ ℓ − 2 x 0 . . L 1 ( λ ) = ⇐ ⇒ L 1 ( λ ) · ( Λ ℓ ⊗ I n ) x = e 1 ⊗ P ( λ ) x . . . . λ x 0 x as λ ℓ − 1 x λ ℓ − 1 P ( λ ) x λ ℓ − 2 x λ ℓ − 2 0 . . . = . ⊗ I n x = ( Λ ℓ ⊗ I n ) x = e 1 ⊗ P ( λ ) x . and . . . . . λ x λ 0 x 1 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations – Motivation Thus λ ℓ − 1 x P ( λ ) x λ ℓ − 2 x 0 . . L 1 ( λ ) = ⇐ ⇒ L 1 ( λ ) · ( Λ ℓ ⊗ I n ) x = e 1 ⊗ P ( λ ) x . . . . λ x 0 x as λ ℓ − 1 x λ ℓ − 1 P ( λ ) x λ ℓ − 2 x λ ℓ − 2 0 . . . = . ⊗ I n x = ( Λ ℓ ⊗ I n ) x = e 1 ⊗ P ( λ ) x . and . . . . . λ x λ 0 x 1 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations – Motivation Thus λ ℓ − 1 x P ( λ ) x λ ℓ − 2 x 0 . . L 1 ( λ ) = ⇐ ⇒ L 1 ( λ ) · ( Λ ℓ ⊗ I n ) x = e 1 ⊗ P ( λ ) x . . . . λ x 0 x as λ ℓ − 1 x λ ℓ − 1 P ( λ ) x λ ℓ − 2 x λ ℓ − 2 0 . . . = . ⊗ I n x = ( Λ ℓ ⊗ I n ) x = e 1 ⊗ P ( λ ) x . and . . . . . λ x λ 0 x 1 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations Generalize L 1 ( λ ) · ( Λ ℓ ⊗ I n ) = e 1 ⊗ P ( λ ) to L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for L ( λ ) = λ E + A . Definition [Ansatz space] [4M] L 1 ( P ) = {L ( λ ) = λ E + A | E , A ∈ R ℓ n × ℓ n , L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for some ansatz vector v ∈ R ℓ } . Theorem [4M],[FS-1] L 1 ( P ) is a vector space over R with dim L 1 ( P ) = ℓ ( ℓ − 1 ) n 2 + ℓ . Almost all pencils in L 1 ( P ) are strong linearizations of P ( λ ) . L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) for v � = 0 and an arbitrary W ∈ R ℓ n × ( ℓ − 1 ) n is a strong linearization of P ( λ ) , if [ v ⊗ I n W ] is nonsingular. Similar derivation for second companion form L 2 ( λ ) gives L 2 ( P ) . There do exist linearizations that are not in L 1 ( P ) or L 2 ( P ) . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations Generalize L 1 ( λ ) · ( Λ ℓ ⊗ I n ) = e 1 ⊗ P ( λ ) to L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for L ( λ ) = λ E + A . Definition [Ansatz space] [4M] L 1 ( P ) = {L ( λ ) = λ E + A | E , A ∈ R ℓ n × ℓ n , L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for some ansatz vector v ∈ R ℓ } . Theorem [4M],[FS-1] L 1 ( P ) is a vector space over R with dim L 1 ( P ) = ℓ ( ℓ − 1 ) n 2 + ℓ . Almost all pencils in L 1 ( P ) are strong linearizations of P ( λ ) . L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) for v � = 0 and an arbitrary W ∈ R ℓ n × ( ℓ − 1 ) n is a strong linearization of P ( λ ) , if [ v ⊗ I n W ] is nonsingular. Similar derivation for second companion form L 2 ( λ ) gives L 2 ( P ) . There do exist linearizations that are not in L 1 ( P ) or L 2 ( P ) . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations Generalize L 1 ( λ ) · ( Λ ℓ ⊗ I n ) = e 1 ⊗ P ( λ ) to L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for L ( λ ) = λ E + A . Definition [Ansatz space] [4M] L 1 ( P ) = {L ( λ ) = λ E + A | E , A ∈ R ℓ n × ℓ n , L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for some ansatz vector v ∈ R ℓ } . Theorem [4M],[FS-1] L 1 ( P ) is a vector space over R with dim L 1 ( P ) = ℓ ( ℓ − 1 ) n 2 + ℓ . Almost all pencils in L 1 ( P ) are strong linearizations of P ( λ ) . L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) for v � = 0 and an arbitrary W ∈ R ℓ n × ( ℓ − 1 ) n is a strong linearization of P ( λ ) , if [ v ⊗ I n W ] is nonsingular. Similar derivation for second companion form L 2 ( λ ) gives L 2 ( P ) . There do exist linearizations that are not in L 1 ( P ) or L 2 ( P ) . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations Generalize L 1 ( λ ) · ( Λ ℓ ⊗ I n ) = e 1 ⊗ P ( λ ) to L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for L ( λ ) = λ E + A . Definition [Ansatz space] [4M] L 1 ( P ) = {L ( λ ) = λ E + A | E , A ∈ R ℓ n × ℓ n , L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for some ansatz vector v ∈ R ℓ } . Theorem [4M],[FS-1] L 1 ( P ) is a vector space over R with dim L 1 ( P ) = ℓ ( ℓ − 1 ) n 2 + ℓ . Almost all pencils in L 1 ( P ) are strong linearizations of P ( λ ) . L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) for v � = 0 and an arbitrary W ∈ R ℓ n × ( ℓ − 1 ) n is a strong linearization of P ( λ ) , if [ v ⊗ I n W ] is nonsingular. Similar derivation for second companion form L 2 ( λ ) gives L 2 ( P ) . There do exist linearizations that are not in L 1 ( P ) or L 2 ( P ) . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations Generalize L 1 ( λ ) · ( Λ ℓ ⊗ I n ) = e 1 ⊗ P ( λ ) to L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for L ( λ ) = λ E + A . Definition [Ansatz space] [4M] L 1 ( P ) = {L ( λ ) = λ E + A | E , A ∈ R ℓ n × ℓ n , L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for some ansatz vector v ∈ R ℓ } . Theorem [4M],[FS-1] L 1 ( P ) is a vector space over R with dim L 1 ( P ) = ℓ ( ℓ − 1 ) n 2 + ℓ . Almost all pencils in L 1 ( P ) are strong linearizations of P ( λ ) . L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) for v � = 0 and an arbitrary W ∈ R ℓ n × ( ℓ − 1 ) n is a strong linearization of P ( λ ) , if [ v ⊗ I n W ] is nonsingular. Similar derivation for second companion form L 2 ( λ ) gives L 2 ( P ) . There do exist linearizations that are not in L 1 ( P ) or L 2 ( P ) . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations Generalize L 1 ( λ ) · ( Λ ℓ ⊗ I n ) = e 1 ⊗ P ( λ ) to L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for L ( λ ) = λ E + A . Definition [Ansatz space] [4M] L 1 ( P ) = {L ( λ ) = λ E + A | E , A ∈ R ℓ n × ℓ n , L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for some ansatz vector v ∈ R ℓ } . Theorem [4M],[FS-1] L 1 ( P ) is a vector space over R with dim L 1 ( P ) = ℓ ( ℓ − 1 ) n 2 + ℓ . Almost all pencils in L 1 ( P ) are strong linearizations of P ( λ ) . L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) for v � = 0 and an arbitrary W ∈ R ℓ n × ( ℓ − 1 ) n is a strong linearization of P ( λ ) , if [ v ⊗ I n W ] is nonsingular. Similar derivation for second companion form L 2 ( λ ) gives L 2 ( P ) . There do exist linearizations that are not in L 1 ( P ) or L 2 ( P ) . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations Generalize L 1 ( λ ) · ( Λ ℓ ⊗ I n ) = e 1 ⊗ P ( λ ) to L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for L ( λ ) = λ E + A . Definition [Ansatz space] [4M] L 1 ( P ) = {L ( λ ) = λ E + A | E , A ∈ R ℓ n × ℓ n , L ( λ ) · ( Λ ℓ ⊗ I n ) = v ⊗ P ( λ ) for some ansatz vector v ∈ R ℓ } . Theorem [4M],[FS-1] L 1 ( P ) is a vector space over R with dim L 1 ( P ) = ℓ ( ℓ − 1 ) n 2 + ℓ . Almost all pencils in L 1 ( P ) are strong linearizations of P ( λ ) . L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) for v � = 0 and an arbitrary W ∈ R ℓ n × ( ℓ − 1 ) n is a strong linearization of P ( λ ) , if [ v ⊗ I n W ] is nonsingular. Similar derivation for second companion form L 2 ( λ ) gives L 2 ( P ) . There do exist linearizations that are not in L 1 ( P ) or L 2 ( P ) . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Matrix Polynomials – (Strong) Linearization Definition (Linearization) A pencil L ( λ ) = λ E + A with E , A ∈ R kn × kn is called a linearization of P ( λ ) ∈ Π n ℓ if there exist unimodular matrix polynomials E ( λ ) , F ( λ ) such that � � P ( λ ) 0 E ( λ ) L ( λ ) F ( λ ) = 0 I ( k − 1 ) n for some k ∈ N . A matrix polynomial E ( λ ) is unimodular if det E ( λ ) is a nonzero constant. Theorem [Lancaster, Psarrakos Report 2005] For regular polynomials P ( λ ) : any linearization: the Jordan structure of all finite eigenvalues is preserved. strong linearization: the Jordan structure of the eigenvalue ∞ is preserved. Example � � � � 4 5 1 2 λ 1 = 1 4 , λ 2 = 3 λ P 1 + P 0 = λ − ⇒ 0 = ∞ . = 0 0 0 3 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Matrix Polynomials – (Strong) Linearization Definition (Linearization) A pencil L ( λ ) = λ E + A with E , A ∈ R kn × kn is called a linearization of P ( λ ) ∈ Π n ℓ if there exist unimodular matrix polynomials E ( λ ) , F ( λ ) such that � � P ( λ ) 0 E ( λ ) L ( λ ) F ( λ ) = 0 I ( k − 1 ) n for some k ∈ N . A matrix polynomial E ( λ ) is unimodular if det E ( λ ) is a nonzero constant. Theorem [Lancaster, Psarrakos Report 2005] For regular polynomials P ( λ ) : any linearization: the Jordan structure of all finite eigenvalues is preserved. strong linearization: the Jordan structure of the eigenvalue ∞ is preserved. Example � � � � 4 5 1 2 λ 1 = 1 4 , λ 2 = 3 λ P 1 + P 0 = λ − ⇒ 0 = ∞ . = 0 0 0 3 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Matrix Polynomials – (Strong) Linearization Definition (Linearization) A pencil L ( λ ) = λ E + A with E , A ∈ R kn × kn is called a linearization of P ( λ ) ∈ Π n ℓ if there exist unimodular matrix polynomials E ( λ ) , F ( λ ) such that � � P ( λ ) 0 E ( λ ) L ( λ ) F ( λ ) = 0 I ( k − 1 ) n for some k ∈ N . A matrix polynomial E ( λ ) is unimodular if det E ( λ ) is a nonzero constant. Theorem [Lancaster, Psarrakos Report 2005] For regular polynomials P ( λ ) : any linearization: the Jordan structure of all finite eigenvalues is preserved. strong linearization: the Jordan structure of the eigenvalue ∞ is preserved. Example � � � � 4 5 1 2 λ 1 = 1 4 , λ 2 = 3 λ P 1 + P 0 = λ − ⇒ 0 = ∞ . = 0 0 0 3 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations and Approach 1 Freund considers d E F dt z ( t ) + A F z ( t ) = B F u ( t ) y ( t ) = D F u ( t ) + C F z ( t ) . Interpret Freund’s approach in terms of the first companion form L 1 ( λ ) = λ E 1 + A 1 d E 1 dt � z ( t ) + A 1 � z ( t ) = B 1 u ( t ) y ( t ) = D F u ( t ) + C 1 � z ( t ) . with z ( t ) = P T � � z ( t ) B 1 = P T B C 1 = C F P � � I n as L 1 ( λ ) = λ E 1 + A 1 = λ P T E F P + P T A F P with P = ... . I n H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations and Approach 1 Freund considers d E F dt z ( t ) + A F z ( t ) = B F u ( t ) y ( t ) = D F u ( t ) + C F z ( t ) . Interpret Freund’s approach in terms of the first companion form L 1 ( λ ) = λ E 1 + A 1 d E 1 dt � z ( t ) + A 1 � z ( t ) = B 1 u ( t ) y ( t ) = D F u ( t ) + C 1 � z ( t ) . with z ( t ) = P T � � z ( t ) B 1 = P T B C 1 = C F P � � I n as L 1 ( λ ) = λ E 1 + A 1 = λ P T E F P + P T A F P with P = ... . I n H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations and Approach 1 Approach is based on the Krylov subspace induced by M = ( L 1 ( s 0 )) − 1 E 1 and R = ( L 1 ( s 0 )) − 1 B 1 . All linearizations in L 1 can be written as L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) = TL 1 ( λ ) = λ TE 1 + TA 1 with v ∈ R ℓ , W ∈ R ℓ n × ( ℓ − 1 ) n such that T = [ v ⊗ I n W ] is nonsingular. As ( TE 1 ) d dt z ( t ) + ( TA 1 ) z ( t ) = ( TB 1 ) u ( t ) and ( L ( s 0 )) − 1 ( TE 1 ) = ( L 1 ( s 0 )) − 1 E 1 = M , ( L ( s 0 )) − 1 ( TB 1 ) = ( L 1 ( s 0 )) − 1 B 1 = R , all linearization in L 1 will yield (theoretically) the same reduced order system. A similar observation holds for Approach 2. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations and Approach 1 Approach is based on the Krylov subspace induced by M = ( L 1 ( s 0 )) − 1 E 1 and R = ( L 1 ( s 0 )) − 1 B 1 . All linearizations in L 1 can be written as L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) = TL 1 ( λ ) = λ TE 1 + TA 1 with v ∈ R ℓ , W ∈ R ℓ n × ( ℓ − 1 ) n such that T = [ v ⊗ I n W ] is nonsingular. As ( TE 1 ) d dt z ( t ) + ( TA 1 ) z ( t ) = ( TB 1 ) u ( t ) and ( L ( s 0 )) − 1 ( TE 1 ) = ( L 1 ( s 0 )) − 1 E 1 = M , ( L ( s 0 )) − 1 ( TB 1 ) = ( L 1 ( s 0 )) − 1 B 1 = R , all linearization in L 1 will yield (theoretically) the same reduced order system. A similar observation holds for Approach 2. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations and Approach 1 Approach is based on the Krylov subspace induced by M = ( L 1 ( s 0 )) − 1 E 1 and R = ( L 1 ( s 0 )) − 1 B 1 . All linearizations in L 1 can be written as L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) = TL 1 ( λ ) = λ TE 1 + TA 1 with v ∈ R ℓ , W ∈ R ℓ n × ( ℓ − 1 ) n such that T = [ v ⊗ I n W ] is nonsingular. As ( TE 1 ) d dt z ( t ) + ( TA 1 ) z ( t ) = ( TB 1 ) u ( t ) and ( L ( s 0 )) − 1 ( TE 1 ) = ( L 1 ( s 0 )) − 1 E 1 = M , ( L ( s 0 )) − 1 ( TB 1 ) = ( L 1 ( s 0 )) − 1 B 1 = R , all linearization in L 1 will yield (theoretically) the same reduced order system. A similar observation holds for Approach 2. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) of linearizations and Approach 1 Approach is based on the Krylov subspace induced by M = ( L 1 ( s 0 )) − 1 E 1 and R = ( L 1 ( s 0 )) − 1 B 1 . All linearizations in L 1 can be written as L ( λ ) = [ v ⊗ I n W ] L 1 ( λ ) = TL 1 ( λ ) = λ TE 1 + TA 1 with v ∈ R ℓ , W ∈ R ℓ n × ( ℓ − 1 ) n such that T = [ v ⊗ I n W ] is nonsingular. As ( TE 1 ) d dt z ( t ) + ( TA 1 ) z ( t ) = ( TB 1 ) u ( t ) and ( L ( s 0 )) − 1 ( TE 1 ) = ( L 1 ( s 0 )) − 1 E 1 = M , ( L ( s 0 )) − 1 ( TB 1 ) = ( L 1 ( s 0 )) − 1 B 1 = R , all linearization in L 1 will yield (theoretically) the same reduced order system. A similar observation holds for Approach 2. H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations Gyroscopic system P ( λ ) = P (− λ ) T ∈ Π n 2 P ( λ ) = λ 2 M + λ G + K , M = M T , G = − G T , K = K T , M , G , K ∈ R n × n . Companion form in L 1 ( P ) � � � � M 0 G K L 1 ( λ ) = + − I 0 I 0 is not structure preserving as L 1 ( λ ) � = L 1 (− λ ) T . Structured linearization in L 1 ( P ) � � � � 0 − M M 0 L ( λ ) = λ + ∈ L 1 ( P ) M G 0 K is a structure-preserving linearization ( L ( λ ) = L (− λ ) T ). H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations Gyroscopic system P ( λ ) = P (− λ ) T ∈ Π n 2 P ( λ ) = λ 2 M + λ G + K , M = M T , G = − G T , K = K T , M , G , K ∈ R n × n . Companion form in L 1 ( P ) � � � � M 0 G K L 1 ( λ ) = + − I 0 I 0 is not structure preserving as L 1 ( λ ) � = L 1 (− λ ) T . Structured linearization in L 1 ( P ) � � � � 0 − M M 0 L ( λ ) = λ + ∈ L 1 ( P ) M G 0 K is a structure-preserving linearization ( L ( λ ) = L (− λ ) T ). H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations Gyroscopic system P ( λ ) = P (− λ ) T ∈ Π n 2 P ( λ ) = λ 2 M + λ G + K , M = M T , G = − G T , K = K T , M , G , K ∈ R n × n . Companion form in L 1 ( P ) � � � � M 0 G K L 1 ( λ ) = + − I 0 I 0 is not structure preserving as L 1 ( λ ) � = L 1 (− λ ) T . Structured linearization in L 1 ( P ) � � � � 0 − M M 0 L ( λ ) = λ + ∈ L 1 ( P ) M G 0 K is a structure-preserving linearization ( L ( λ ) = L (− λ ) T ). H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations Robot P ( λ ) = P (− λ ) T ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , Companion form in L 1 ( P ) P 4 0 0 0 P 3 P 2 P 1 P 0 0 I n 0 0 − I n 0 0 0 L 1 ( λ ) = λ + − I n 0 0 I n 0 0 0 0 − I n 0 0 0 I n 0 0 0 Structured linearizations in L 1 ( P ) different [4M] − P 4 − P 4 0 0 P 4 0 P 4 0 P 2 − P 4 P 1 − P 3 P 4 P 3 P 4 P 3 0 P 0 L ( λ ) = λ + − P 4 P 1 − P 3 P 0 − P 2 P 3 − P 1 P 2 − P 0 0 P 4 0 P 2 − P 0 P 4 P 3 P 1 0 P 0 0 P 0 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations Robot P ( λ ) = P (− λ ) T ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , Companion form in L 1 ( P ) P 4 0 0 0 P 3 P 2 P 1 P 0 0 I n 0 0 − I n 0 0 0 L 1 ( λ ) = λ + − I n 0 0 I n 0 0 0 0 − I n 0 0 0 I n 0 0 0 Structured linearizations in L 1 ( P ) different [4M] − P 4 − P 4 0 0 P 4 0 P 4 0 P 2 − P 4 P 1 − P 3 P 4 P 3 P 4 P 3 0 P 0 L ( λ ) = λ + − P 4 P 1 − P 3 P 0 − P 2 P 3 − P 1 P 2 − P 0 0 P 4 0 P 2 − P 0 P 4 P 3 P 1 0 P 0 0 P 0 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations Robot P ( λ ) = P (− λ ) T ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , Companion form in L 1 ( P ) P 4 0 0 0 P 3 P 2 P 1 P 0 0 I n 0 0 − I n 0 0 0 L 1 ( λ ) = λ + − I n 0 0 I n 0 0 0 0 − I n 0 0 0 I n 0 0 0 Structured linearizations in L 1 ( P ) different [4M] − P 4 − P 4 0 0 P 4 0 P 4 0 P 2 − P 4 P 1 − P 3 P 4 P 3 P 4 P 3 0 P 0 L ( λ ) = λ + − P 4 P 1 − P 3 P 0 − P 2 P 3 − P 1 P 2 − P 0 0 P 4 0 P 2 − P 0 P 4 P 3 P 1 0 P 0 0 P 0 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations L 1 ( λ ) and L ( λ ) may be very differently conditioned. P✵❂✶✴✶✵✵✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✷❂r❛♥❞♥✭✶✵✵✮❀P✷❂✭P✷✰P✷✬✮✴✸✵❀ P✹❂❡②❡✭♥✮❀ P✶❂r❛♥❞✭✶✵✵✮❀P✶❂P✶✲P✶✬❀ P✸❂r❛♥❞♥✭✶✵✵✮❀P✸❂P✸✲P✸✬❀ H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations L 1 ( λ ) and L ( λ ) may be very differently conditioned. P✵❂✶✴✶✵✵✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✷❂r❛♥❞♥✭✶✵✵✮❀P✷❂✭P✷✰P✷✬✮✴✸✵❀ P✹❂❡②❡✭♥✮❀ P✶❂r❛♥❞✭✶✵✵✮❀P✶❂P✶✲P✶✬❀ P✸❂r❛♥❞♥✭✶✵✵✮❀P✸❂P✸✲P✸✬❀ H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations L 1 ( λ ) and L ( λ ) may be very differently conditioned. L ( λ ) is not (block) sparse, while L 1 ( λ ) is. P✵❂✶✴✶✵✵✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✷❂r❛♥❞♥✭✶✵✵✮❀P✷❂✭P✷✰P✷✬✮✴✺❀ P✹❂✳✺✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✶❂r❛♥❞✭✶✵✵✮❀P✶❂P✶✲P✶✬❀ P✸❂r❛♥❞♥✭✶✵✵✮❀P✸❂P✸✲P✸✬❀ H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Vector space L 1 ( P ) – Structured Linearizations L 1 ( λ ) and L ( λ ) may be very differently conditioned. L ( λ ) is not (block) sparse, while L 1 ( λ ) is. P✵❂✶✴✶✵✵✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✷❂r❛♥❞♥✭✶✵✵✮❀P✷❂✭P✷✰P✷✬✮✴✺❀ P✹❂✳✺✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✶❂r❛♥❞✭✶✵✵✮❀P✶❂P✶✲P✶✬❀ P✸❂r❛♥❞♥✭✶✵✵✮❀P✸❂P✸✲P✸✬❀ H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Structured Linearization not in L 1 ( P ) Robot P ( λ ) = P (− λ ) T ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , (Structured) Linearization not in L 1 ( P ) P 4 0 0 I 0 − P 2 − λ P 3 0 0 λ I I Note+ E , A ∈ R 5 n × 5 n ! L ( λ ) = P 0 + λ P 1 λ I = λ E + A 0 0 0 − λ I I 0 0 0 − λ I 0 I 0 0 as V ( λ ) L ( λ ) U ( λ ) = diag ( I 4 n , P ( λ )) for λ 2 In In 0 0 − P 4 − λ P 4 0 0 In λ In λ 2 P 4 + λ P 3 + P 2 0 0 0 In λ In − λ In In 0 λ P 4 0 0 0 0 In V ( λ ) = 0 0 0 In 0 , U ( λ ) = , − λ 2 P 4 0 0 0 0 In In 0 0 0 λ 2 In − λ 2 P 4 − λ 3 P 4 − λ 2 P 3 − λ P 2 λ 3 P 4 + λ 2 P 3 + λ P 2 − λ In In 0 In 0 0 det U ( λ ) = det V ( λ ) = 1 . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Structured Linearization not in L 1 ( P ) Robot P ( λ ) = P (− λ ) T ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , (Structured) Linearization not in L 1 ( P ) P 4 0 0 I 0 − P 2 − λ P 3 0 0 λ I I Note+ E , A ∈ R 5 n × 5 n ! L ( λ ) = P 0 + λ P 1 λ I = λ E + A 0 0 0 − λ I I 0 0 0 − λ I 0 I 0 0 as V ( λ ) L ( λ ) U ( λ ) = diag ( I 4 n , P ( λ )) for λ 2 In In 0 0 − P 4 − λ P 4 0 0 In λ In λ 2 P 4 + λ P 3 + P 2 0 0 0 In λ In − λ In In 0 λ P 4 0 0 0 0 In V ( λ ) = 0 0 0 In 0 , U ( λ ) = , − λ 2 P 4 0 0 0 0 In In 0 0 0 λ 2 In − λ 2 P 4 − λ 3 P 4 − λ 2 P 3 − λ P 2 λ 3 P 4 + λ 2 P 3 + λ P 2 − λ In In 0 In 0 0 det U ( λ ) = det V ( λ ) = 1 . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Structured Linearization not in L 1 ( P ) Robot P ( λ ) = P (− λ ) T ∈ Π n 4 P ( λ ) = λ 4 P 4 + λ 3 P 3 + λ 2 P 2 + λ P 1 + P 0 , P i = (− 1 ) i P T P i ∈ R n × n , i = 0 , . . . , 4 . i , (Structured) Linearization not in L 1 ( P ) P 4 0 0 I 0 − P 2 − λ P 3 0 0 λ I I Note+ E , A ∈ R 5 n × 5 n ! L ( λ ) = P 0 + λ P 1 λ I = λ E + A 0 0 0 − λ I I 0 0 0 − λ I 0 I 0 0 as V ( λ ) L ( λ ) U ( λ ) = diag ( I 4 n , P ( λ )) for λ 2 In In 0 0 − P 4 − λ P 4 0 0 In λ In λ 2 P 4 + λ P 3 + P 2 0 0 0 In λ In − λ In In 0 λ P 4 0 0 0 0 In V ( λ ) = 0 0 0 In 0 , U ( λ ) = , − λ 2 P 4 0 0 0 0 In In 0 0 0 λ 2 In − λ 2 P 4 − λ 3 P 4 − λ 2 P 3 − λ P 2 λ 3 P 4 + λ 2 P 3 + λ P 2 − λ In In 0 In 0 0 det U ( λ ) = det V ( λ ) = 1 . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Definition [Block Kronecker Ansatz space] [FS-2] Let P ( λ ) ∈ Π n ℓ with ℓ = r + s + 1 . The block Kronecker ansatz space G r + 1 ( P ) is the set of all ℓ n × ℓ n matrix pencils L ( λ ) that satisfy the block Kronecker ansatz equation L ( λ ) λ s I n � �� � � [ λ r I n · · · I n ] � � L 11 ( λ ) � � α P ( λ ) � . . L 12 ( λ ) 0 . = . I s n L 21 ( λ ) L 22 ( λ ) 0 0 I n I r n G r + 1 ( P ) is a vector space over R of dimension ( ℓ − 1 ) ℓ n 2 + 1 . [FS-2] Thus, L 1 ( P ) � = G r + 1 ( P ) . Almost all pencils in G r + 1 ( P ) are strong linearizations of P ( λ ) . [FS-2] H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Definition [Block Kronecker Ansatz space] [FS-2] Let P ( λ ) ∈ Π n ℓ with ℓ = r + s + 1 . The block Kronecker ansatz space G r + 1 ( P ) is the set of all ℓ n × ℓ n matrix pencils L ( λ ) that satisfy the block Kronecker ansatz equation L ( λ ) λ s I n � �� � � [ λ r I n · · · I n ] � � L 11 ( λ ) � � α P ( λ ) � . . L 12 ( λ ) 0 . = . I s n L 21 ( λ ) L 22 ( λ ) 0 0 I n I r n G r + 1 ( P ) is a vector space over R of dimension ( ℓ − 1 ) ℓ n 2 + 1 . [FS-2] Thus, L 1 ( P ) � = G r + 1 ( P ) . Almost all pencils in G r + 1 ( P ) are strong linearizations of P ( λ ) . [FS-2] H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Definition [Block Kronecker Ansatz space] [FS-2] Let P ( λ ) ∈ Π n ℓ with ℓ = r + s + 1 . The block Kronecker ansatz space G r + 1 ( P ) is the set of all ℓ n × ℓ n matrix pencils L ( λ ) that satisfy the block Kronecker ansatz equation L ( λ ) λ s I n � �� � � [ λ r I n · · · I n ] � � L 11 ( λ ) � � α P ( λ ) � . . L 12 ( λ ) 0 . = . I s n L 21 ( λ ) L 22 ( λ ) 0 0 I n I r n G r + 1 ( P ) is a vector space over R of dimension ( ℓ − 1 ) ℓ n 2 + 1 . [FS-2] Thus, L 1 ( P ) � = G r + 1 ( P ) . Almost all pencils in G r + 1 ( P ) are strong linearizations of P ( λ ) . [FS-2] H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Definition [Block Kronecker Ansatz space] [FS-2] Let P ( λ ) ∈ Π n ℓ with ℓ = r + s + 1 . The block Kronecker ansatz space G r + 1 ( P ) is the set of all ℓ n × ℓ n matrix pencils L ( λ ) that satisfy the block Kronecker ansatz equation L ( λ ) λ s I n � �� � � [ λ r I n · · · I n ] � � L 11 ( λ ) � � α P ( λ ) � . . L 12 ( λ ) 0 . = . I s n L 21 ( λ ) L 22 ( λ ) 0 0 I n I r n G r + 1 ( P ) is a vector space over R of dimension ( ℓ − 1 ) ℓ n 2 + 1 . [FS-2] Thus, L 1 ( P ) � = G r + 1 ( P ) . Almost all pencils in G r + 1 ( P ) are strong linearizations of P ( λ ) . [FS-2] H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Higher order system and block Kronecker linearizations Robot P ( λ ) ∈ Π n 4 d 4 d 3 d 2 d P 4 dt 4 x ( t ) + P 3 dt 3 x ( t ) + P 2 dt 2 x ( t ) + P 1 dt x ( t ) + P 0 x ( t ) = Bu ( t ) d 3 d 2 d Du ( t ) + C 3 dt 3 x ( t ) + C 2 dt 2 x ( t ) + C 1 dt x ( t ) + C 0 x ( t ) = y ( t ) The linearization P 4 0 0 I 0 0 − P 2 − λ P 3 0 λ I I L ( λ ) = λ E + A = 0 0 P 0 + λ P 1 0 λ I I − λ I 0 0 0 0 I − λ I 0 0 does not give an equivalent first order ODE of the form E d dt z ( t ) + A z ( t ) = B u ( t ) λ 2 In P 4 0 0 I 0 0 − P 2 − λ P 3 0 λ I I λ In = P ( λ ) . as [ λ 2 In 0 ] 0 0 P 0 + λ P 1 0 λ I − λ In In 0 In − λ I I 0 0 0 0 − λ I 0 I 0 0 0 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Higher order system and block Kronecker linearizations Robot P ( λ ) ∈ Π n 4 d 4 d 3 d 2 d P 4 dt 4 x ( t ) + P 3 dt 3 x ( t ) + P 2 dt 2 x ( t ) + P 1 dt x ( t ) + P 0 x ( t ) = Bu ( t ) d 3 d 2 d Du ( t ) + C 3 dt 3 x ( t ) + C 2 dt 2 x ( t ) + C 1 dt x ( t ) + C 0 x ( t ) = y ( t ) The linearization P 4 0 0 I 0 0 − P 2 − λ P 3 0 λ I I L ( λ ) = λ E + A = 0 0 P 0 + λ P 1 0 λ I I − λ I 0 0 0 0 I − λ I 0 0 does not give an equivalent first order ODE of the form E d dt z ( t ) + A z ( t ) = B u ( t ) λ 2 In P 4 0 0 I 0 0 − P 2 − λ P 3 0 λ I I λ In = P ( λ ) . as [ λ 2 In 0 ] 0 0 P 0 + λ P 1 0 λ I − λ In In 0 In − λ I I 0 0 0 0 − λ I 0 I 0 0 0 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Higher order system and block Kronecker linearizations Robot P ( λ ) ∈ Π n 4 d 4 d 3 d 2 d P 4 dt 4 x ( t ) + P 3 dt 3 x ( t ) + P 2 dt 2 x ( t ) + P 1 dt x ( t ) + P 0 x ( t ) = Bu ( t ) d 3 d 2 d Du ( t ) + C 3 dt 3 x ( t ) + C 2 dt 2 x ( t ) + C 1 dt x ( t ) + C 0 x ( t ) = y ( t ) The linearization P 4 0 0 I 0 0 − P 2 − λ P 3 0 λ I I L ( λ ) = λ E + A = 0 0 P 0 + λ P 1 0 λ I I − λ I 0 0 0 0 I − λ I 0 0 does not give an equivalent first order ODE of the form E d dt z ( t ) + A z ( t ) = B u ( t ) λ 2 In P 4 0 0 I 0 0 − P 2 − λ P 3 0 λ I I λ In = P ( λ ) . as [ λ 2 In 0 ] 0 0 P 0 + λ P 1 0 λ I − λ In In 0 In − λ I I 0 0 0 0 − λ I 0 I 0 0 0 H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 In L 1 all linearizations are based on L 1 ( λ ) , the linearizations in G r + 1 are based on L K ( λ ) = λ E K + A K λα P ℓ + α P ℓ − 1 α P ℓ − 2 · · · α P r − I n ... α P r − 1 λ I n . ... . . − I n = α P 0 λ I n − I n λ I n ... ... 0 − I n λ I n � Σ r ( λ ) � L T r ( λ ) = L s ( λ ) 0 with ℓ = r + s + 1 , Σ r ( λ ) ∈ C ( r + 1 ) n × sn , and L j ( λ ) ∈ C jn × ( j + 1 ) n . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 We can find B K , C K such that ℓ − 1 � C j (( P ( s )) − 1 B = D K + C K ( L K ( s )) − 1 B K . G ( s ) = D + j = 0 Introduce shift s 0 ∈ C such that L K ( s 0 ) = s 0 E K + A K is nonsingular. Then G ( s ) = D K + C K ( L K ( s )) − 1 B K = D K + C K ( I + ( s − s 0 ) M K ) − 1 R K with M K = ( L K ( s 0 )) − 1 E K , R K = ( L K ( s 0 )) − 1 B K . Compute basis of K s ( M K , R K ) . Represent the basis in block form � W 1 � W 2 W j ∈ C n × r . , . . . W ℓ Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 We can find B K , C K such that ℓ − 1 � C j (( P ( s )) − 1 B = D K + C K ( L K ( s )) − 1 B K . G ( s ) = D + j = 0 Introduce shift s 0 ∈ C such that L K ( s 0 ) = s 0 E K + A K is nonsingular. Then G ( s ) = D K + C K ( L K ( s )) − 1 B K = D K + C K ( I + ( s − s 0 ) M K ) − 1 R K with M K = ( L K ( s 0 )) − 1 E K , R K = ( L K ( s 0 )) − 1 B K . Compute basis of K s ( M K , R K ) . Represent the basis in block form � W 1 � W 2 W j ∈ C n × r . , . . . W ℓ Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 We can find B K , C K such that ℓ − 1 � C j (( P ( s )) − 1 B = D K + C K ( L K ( s )) − 1 B K . G ( s ) = D + j = 0 Introduce shift s 0 ∈ C such that L K ( s 0 ) = s 0 E K + A K is nonsingular. Then G ( s ) = D K + C K ( L K ( s )) − 1 B K = D K + C K ( I + ( s − s 0 ) M K ) − 1 R K with M K = ( L K ( s 0 )) − 1 E K , R K = ( L K ( s 0 )) − 1 B K . Compute basis of K s ( M K , R K ) . Represent the basis in block form � W 1 � W 2 W j ∈ C n × r . , . . . W ℓ Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 We can find B K , C K such that ℓ − 1 � C j (( P ( s )) − 1 B = D K + C K ( L K ( s )) − 1 B K . G ( s ) = D + j = 0 Introduce shift s 0 ∈ C such that L K ( s 0 ) = s 0 E K + A K is nonsingular. Then G ( s ) = D K + C K ( L K ( s )) − 1 B K = D K + C K ( I + ( s − s 0 ) M K ) − 1 R K with M K = ( L K ( s 0 )) − 1 E K , R K = ( L K ( s 0 )) − 1 B K . Compute basis of K s ( M K , R K ) . Represent the basis in block form � W 1 � W 2 W j ∈ C n × r . , . . . W ℓ Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Any linearization in G r + 1 can be expressed as � � � � I ( r + 1 ) n B 1 I ( s + 1 ) n 0 � L K ( λ ) = T 1 L K ( λ ) T 2 with T 1 = T 2 = , 0 C 1 B 2 C 2 and B 1 ∈ R ( r + 1 ) n × s n , B 2 ∈ R r n × ( s + 1 ) n , C 1 ∈ R sn × sn , C 2 ∈ R r n × r n . L K ( s )) − 1 � G ( s ) = D K + � C K ( � B K with � C K = C K T 2 , � B K = T 1 B K . G ( s ) = D K + � C K ( I + ( s − s 0 ) � M K ) − 1 � R K with L K ( s 0 )) − 1 � M K = ( � � R K = ( � � L K ( s 0 )) − 1 T 1 E K T 2 , B K , = T − 1 = T − 1 2 M K T 2 , 2 R K . Thus, K ( � M K , � R k ) = T − 1 2 K ( M K , R k ) . As before: Compute basis of K s ( � M K , � R K ) . Represent it in block form with blocks W j ∈ C n × r , j = 1 , . . . , ℓ . Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Any linearization in G r + 1 can be expressed as � � � � I ( r + 1 ) n B 1 I ( s + 1 ) n 0 � L K ( λ ) = T 1 L K ( λ ) T 2 with T 1 = T 2 = , 0 C 1 B 2 C 2 and B 1 ∈ R ( r + 1 ) n × s n , B 2 ∈ R r n × ( s + 1 ) n , C 1 ∈ R sn × sn , C 2 ∈ R r n × r n . L K ( s )) − 1 � G ( s ) = D K + � C K ( � B K with � C K = C K T 2 , � B K = T 1 B K . G ( s ) = D K + � C K ( I + ( s − s 0 ) � M K ) − 1 � R K with L K ( s 0 )) − 1 � M K = ( � � R K = ( � � L K ( s 0 )) − 1 T 1 E K T 2 , B K , = T − 1 = T − 1 2 M K T 2 , 2 R K . Thus, K ( � M K , � R k ) = T − 1 2 K ( M K , R k ) . As before: Compute basis of K s ( � M K , � R K ) . Represent it in block form with blocks W j ∈ C n × r , j = 1 , . . . , ℓ . Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Any linearization in G r + 1 can be expressed as � � � � I ( r + 1 ) n B 1 I ( s + 1 ) n 0 � L K ( λ ) = T 1 L K ( λ ) T 2 with T 1 = T 2 = , 0 C 1 B 2 C 2 and B 1 ∈ R ( r + 1 ) n × s n , B 2 ∈ R r n × ( s + 1 ) n , C 1 ∈ R sn × sn , C 2 ∈ R r n × r n . L K ( s )) − 1 � G ( s ) = D K + � C K ( � B K with � C K = C K T 2 , � B K = T 1 B K . G ( s ) = D K + � C K ( I + ( s − s 0 ) � M K ) − 1 � R K with L K ( s 0 )) − 1 � M K = ( � � R K = ( � � L K ( s 0 )) − 1 T 1 E K T 2 , B K , = T − 1 = T − 1 2 M K T 2 , 2 R K . Thus, K ( � M K , � R k ) = T − 1 2 K ( M K , R k ) . As before: Compute basis of K s ( � M K , � R K ) . Represent it in block form with blocks W j ∈ C n × r , j = 1 , . . . , ℓ . Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Any linearization in G r + 1 can be expressed as � � � � I ( r + 1 ) n B 1 I ( s + 1 ) n 0 � L K ( λ ) = T 1 L K ( λ ) T 2 with T 1 = T 2 = , 0 C 1 B 2 C 2 and B 1 ∈ R ( r + 1 ) n × s n , B 2 ∈ R r n × ( s + 1 ) n , C 1 ∈ R sn × sn , C 2 ∈ R r n × r n . L K ( s )) − 1 � G ( s ) = D K + � C K ( � B K with � C K = C K T 2 , � B K = T 1 B K . G ( s ) = D K + � C K ( I + ( s − s 0 ) � M K ) − 1 � R K with L K ( s 0 )) − 1 � M K = ( � � R K = ( � � L K ( s 0 )) − 1 T 1 E K T 2 , B K , = T − 1 = T − 1 2 M K T 2 , 2 R K . Thus, K ( � M K , � R k ) = T − 1 2 K ( M K , R k ) . As before: Compute basis of K s ( � M K , � R K ) . Represent it in block form with blocks W j ∈ C n × r , j = 1 , . . . , ℓ . Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Block Kronecker Ansatz space G r + 1 Any linearization in G r + 1 can be expressed as � � � � I ( r + 1 ) n B 1 I ( s + 1 ) n 0 � L K ( λ ) = T 1 L K ( λ ) T 2 with T 1 = T 2 = , 0 C 1 B 2 C 2 and B 1 ∈ R ( r + 1 ) n × s n , B 2 ∈ R r n × ( s + 1 ) n , C 1 ∈ R sn × sn , C 2 ∈ R r n × r n . L K ( s )) − 1 � G ( s ) = D K + � C K ( � B K with � C K = C K T 2 , � B K = T 1 B K . G ( s ) = D K + � C K ( I + ( s − s 0 ) � M K ) − 1 � R K with L K ( s 0 )) − 1 � M K = ( � � R K = ( � � L K ( s 0 )) − 1 T 1 E K T 2 , B K , = T − 1 = T − 1 2 M K T 2 , 2 R K . Thus, K ( � M K , � R k ) = T − 1 2 K ( M K , R k ) . As before: Compute basis of K s ( � M K , � R K ) . Represent it in block form with blocks W j ∈ C n × r , j = 1 , . . . , ℓ . Generate reduced order higher order system via projection with V , the matrix representing an orthonormal basis of span { W r + 1 } . H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Four different Linearizations for Robot Example Robot P ( λ ) ∈ Π n 4 d 4 d 3 d 2 d P i = (− 1 ) i P T P 4 dt 4 x ( t ) + P 3 dt 3 x ( t ) + P 2 dt 2 x ( t ) + P 1 dt x ( t ) + P 0 x ( t ) = Bu ( t ) , i d 3 d 2 d Du ( t ) + C 3 dt 3 x ( t ) + C 2 dt 2 x ( t ) + C 1 dt x ( t ) + C 0 x ( t ) = y ( t ) P✵❂✶✴✶✵✵✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✷❂r❛♥❞♥✭✶✵✵✮❀P✷❂✭P✷✰P✷✬✮✴✺❀ P✹❂✳✺✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✶❂r❛♥❞✭✶✵✵✮❀P✶❂P✶✲P✶✬❀ P✸❂r❛♥❞♥✭✶✵✵✮❀P✸❂P✸✲P✸✬❀ H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Vector space L 1 ( P ) Vector space G r + 1 Four different Linearizations for Robot Example H. Faßbender MOR of Higher Order Systems
Introduction Approach 1 Linearizations Example Robot Conclusions Four different Linearizations for Robot Example Robot P ( λ ) ∈ Π n 4 d 4 d 3 d 2 d P i = (− 1 ) i P T P 4 dt 4 x ( t ) + P 3 dt 3 x ( t ) + P 2 dt 2 x ( t ) + P 1 dt x ( t ) + P 0 x ( t ) = Bu ( t ) , i d 3 d 2 d Du ( t ) + C 3 dt 3 x ( t ) + C 2 dt 2 x ( t ) + C 1 dt x ( t ) + C 0 x ( t ) = y ( t ) P✵❂✶✴✶✵✵✯❣❛❧❧❡r②✭✬♣♦✐ss♦♥✬✱✶✵✮❀ P✷❂r❛♥❞♥✭✶✵✵✮❀P✷❂✭P✷✰P✷✬✮✴✸✵❀ P✹❂❡②❡✭♥✮❀ P✶❂r❛♥❞✭✶✵✵✮❀P✶❂P✶✲P✶✬❀ P✸❂r❛♥❞♥✭✶✵✵✮❀P✸❂P✸✲P✸✬❀ H. Faßbender MOR of Higher Order Systems
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