Permuted Graph Bases for Verified Computation of Invariant Subspaces Federico Poloni 1 Tayyebe Haqiri 2 1 U Pisa, Italy, Dept of Computer Sciences 2 U Kerman, Iran, Dept of Mathematics and Computer Sciences SWIM 2015 9 June, Prague F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 1/34
Invariant subspaces Definition A subspace U ∈ C n × k is called invariant under H ∈ C n × n if Hu is in U for all u in U . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 2/34
Invariant subspaces Definition A subspace U ∈ C n × k is called invariant under H ∈ C n × n if Hu is in U for all u in U . Equivalent problem Find U ∈ C n × k and R ∈ C k × k s.t. HU = UR . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 2/34
Related non-Hermitian algebraic Riccati equation � � I k × k Assumption HU = UR , U = , X ( n − k ) × k Solve the non-Hermitian algebraic Riccati equation (NARE) F ( X ) := Q + XA + ˜ AX − XGX = 0 , (1) instead of finding invariant subspaces for � � A k × k − G k × ( n − k ) H = . − ˜ − Q ( n − k ) × k A ( n − k ) × ( n − k ) • R = A − GX is the closed loop matrix associated to 1. • A solution X of 1 is called stabilizing if the closed loop matrix R is stable. F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 3/34
Graph matrix and graph subspace Definition � � I k × k Graph matrix G ( X ) := . X ( n − k ) × k F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 4/34
Graph matrix and graph subspace Definition � � I k × k Graph matrix G ( X ) := . X ( n − k ) × k Definition Graph subspace := Im ( G ( X )) . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 4/34
Graph matrix and graph subspace Definition � � I k × k Graph matrix G ( X ) := . X ( n − k ) × k Definition Graph subspace := Im ( G ( X )) . Almost every subspace is a graph subspace: � � E k × k If U = full column rank, E invertible then A ( n − k ) × k U = G ( AE − 1 ) E . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 4/34
Graph basis matrix Definition U and V full column rank matrices. U ∼ V for a square invertible matrix E , U = VE ⇐ ⇒ same column space. F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 5/34
Graph basis matrix Definition U and V full column rank matrices. U ∼ V for a square invertible matrix E , U = VE ⇐ ⇒ same column space. � E � � � I • If U = , with E square invertible, U ∼ graph AE − 1 A basis. • E − 1 − → danger: can be ill conditioned. F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 5/34
Permuted graph matrix If E any square invertible submatrix of U , we can post–multiply by E − 1 to enforce an identity in a subset of rows. Example 1 2 3 1 0 0 4 5 6 0 . 5 0 . 5 0 U = 7 8 9 ∼ 0 1 0 . 1 1 2 0 0 1 3 5 8 2 0 1 � I � We can write this as U ∼ P , P permutation matrix. X F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 6/34
Permuted graph matrix If E any square invertible submatrix of U , we can post–multiply by E − 1 to enforce an identity in a subset of rows. Example 1 2 3 1 0 0 4 5 6 0 . 5 0 . 5 0 U = 7 8 9 ∼ 0 1 0 . 1 1 2 0 0 1 3 5 8 2 0 1 � I � We can write this as U ∼ P , P permutation matrix. X Theorem (Knuth, ‘80 or earlier, Mehrmann and Poloni, ‘12) � I � Each full column rank matrix U has a permuted graph basis P X with | x ij | ≤ 1 . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 6/34
Important formulas and notation ( A ⊗ B )( C ⊗ D ) = AC ⊗ BD , vec( ABC ) = ( C T ⊗ A ) vec( B ) , vec(“ uppercase ”) = “ lowercase ” , ⊗ Kronecker product of matrices, vec Stacks columns of a matrix into a long vector. F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 7/34
Frechet derivative of the function F echet derivative of F at X in the direction E ∈ C ( n − k ) × k is The Fr` given as F ′ ( X ) E = E ( A − GX ) + ( ˜ A − XG ) E , so, A − XG ) + ( A − GX ) T ⊗ I n − k ∈ C k ( n − k ) × k ( n − k ) . f ′ ( x ) = I k ⊗ ( ˜ F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 8/34
Standard Krawczyk operator k (˜ x , x ) = ˜ x − Rf ( x ) + ( I − R S )( x − ˜ x ) A − X G ) + ( A − G X ) T ⊗ I )]( x − ˜ x − Rf ( x ) + [ I − R ( I ⊗ ( ˜ = ˜ x ) , • S An interval matrix containing all slopes S for x , y ∈ x , • Standard choice for S f ′ ( x ), • f ′ ( x ) The interval arithmetic evaluation of f ′ ( x ), • R A computed inverse of f ′ ( x ) by using the standard floating point arithmetic. F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 9/34
Aspects of complexity • For obtaining the matrix R , one should invert a matrix of size k ( n − k ) × k ( n − k ) cost = O ( n 6 ) • The product R S with R full and S containing at least O ( n ) non-zeros per column cost = O ( n 5 )! Therefore The number of arithmetic operations needed to implement the classical Krawczyk operator is at-least O ( n 5 )! Challenge Reduce this cost to cubic. F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 10/34
Used tricks Previous works involved: • A. Frommer and B. Hashemi: Verified computation of square roots of a matrix, 2009 affine transformation for reducing wrapping effect (loses uniqueness), • B. Hashemi: Verified computation of Hermitian (Symmetric) solutions to continuous-time algebraic Riccati matrix equation, 2012 spectral decomposition. New work involved: • V. Mehrmann and F. Poloni: Doubling algorithms with permuted Lagrangian graph bases, 2012 permuted graph bases (loses uniqueness). F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 11/34
Modified Krawczyk operator Theorem (Rum, ‘83, Frommer and Hashemi, ‘09) Assume that f : D ⊆ C n → C n is continuous in D. Let ˜ x ∈ D and z ∈ IC n be such that ˜ x + z ⊆ D. Moreover, assume that S ⊆ C n × n is a set of matrices containing all slopes S (˜ x , y ) for y ∈ ˜ x + z := x . Finally, let R ∈ C n × n . Denote by K f (˜ x , R , z , S ) the set K f (˜ x , R , z , S ) := {− Rf (˜ x ) + ( I − RS ) z : S ∈ S , z ∈ z } . Then, if K f (˜ x , R , z , S ) ⊆ int z , (2) the function f has a zero x ∗ in ˜ x + K f (˜ x , R , z , S ) ⊆ x . Moreover, if S also contains all slope matrices S ( x , y ) for x , y ∈ x , then this zero is unique in x . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 12/34
The relation between slopes and derivative Theorem Consider NARE (1). Then, the interval arithmetic evaluation of the derivative of f ( x ) , i.e. the interval matrix A − X G ) + ( A − G X ) T ⊗ I contains slopes S ( x , y ) for all I ⊗ ( ˜ x , y ∈ x . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 13/34
Evaluate the Krawczyk operator ( K f (˜ x , R , z , S ) := − Rf (˜ x ) + ( I − R S ) z , Then, the enclosure property of interval arithmetic displays that K f (˜ x , R , z , S ) ⊂ int z = ⇒ K f (˜ x , R , z , S ) ⊆ int z . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 14/34
Fundamental assumptions Existence of spectral decompositions for V 1 , W 1 , Λ 1 ∈ C k × k , A − GX = V 1 Λ 1 W 1 , Λ 1 = Diag ( λ 11 , . . . , λ k 1 ) , V 1 W 1 = I k , A ∗ − G ∗ X ∗ = V 2 Λ 2 W 2 , ˜ V 2 , W 2 , Λ 2 ∈ C ( n − k ) × ( n − k ) , Λ 2 = Diag ( λ 12 , . . . , λ ( n − k )2 ) , V 2 W 2 = I n − k . F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 15/34
Outcomes of these eigenvalue decompositions A − XG ) + ( A − GX ) T ⊗ I converted to [Frommer, • f ′ ( x ) = I ⊗ ( ˜ Hashemi] f ′ ( x ) = ( V − T ⊗ W ∗ 2 ) · 1 T ∗ W 2 ( ˜ A − XG ) ∗ W − 1 V − 1 I ⊗ + ( A − GX ) V 1 ⊗ I · 2 1 � �� � � �� � =Λ 2 ∼ =Λ 1 ∼ ( V T 1 ⊗ W −∗ ) , 2 2 ) · ∆ − 1 · ( V T • R = ( V − T 2 + Λ T ⊗ W ∗ 1 ⊗ W −∗ ), ∆ = I ⊗ Λ ∗ 1 ⊗ I 1 2 diagonal, 2 ) · ∆ − 1 · • I − Rf ′ ( x ) = ( V − T ⊗ W ∗ 1 � � ] ∗ − [ V − 1 ( A − GX ) V 1 ] T ⊗ I ∆ − I ⊗ [ W 2 ( ˜ A − XG ) ∗ W − 1 2 1 ( V T 1 ⊗ W −∗ ) . 2 F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 16/34
Reducing wrapping effect New issue The problematic wrapping effect of interval arithmetic appears in several lines of the modified Krawczyk algorithm. Solution Use ˆ f as a linearly transformed function instead of f : � � � � ˆ V T ( V − T 1 ⊗ W −∗ ⊗ W ∗ f (ˆ x ) := f 2 )ˆ x , 2 1 ( V − T ⊗ W ∗ 2 )ˆ x := x , X a solution for NARE (1). [Frommer, 1 Hashemi] F. Poloni and T. Haqiri SWIM 2015 Permuted Graph Bases 17/34
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