Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Reliable Evaluation of the Worst-Case Peak Gain Matrix in Multiple Precision Anastasia Volkova, Thibault Hilaire, Christoph Lauter Sorbonne Universit´ es, UPMC Univ Paris 06, UMR 7606, LIP6, F-75005, Paris, France 22nd IEEE Symposium on Computer Arithmetic June 23, 2015 1/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Digital filters u ( k ) y ( k ) H 2/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Digital filters u ( k ) y ( k ) H Linear Time-Invariant filter in state-space representation: � x ( k + 1) = Ax ( k ) + Bu ( k ) H y ( k ) = Cx ( k ) + Du ( k ) where A ∈ R n × n , B ∈ R n × q , C ∈ R p × n , D ∈ R p × q 2/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Digital filters u ( k ) y ( k ) H Linear Time-Invariant filter in state-space representation: � x ( k + 1) = Ax ( k ) + Bu ( k ) H y ( k ) = Cx ( k ) + Du ( k ) where A ∈ R n × n , B ∈ R n × q , C ∈ R p × n , D ∈ R p × q Bounded-Input Bounded-Output (BIBO) stability: ρ ( A ) < 1 2/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Worst-Case Peak Gain: Definitions Definition Worst-case peak gain (WCPG) W is the largest possible peak value of the output y ( k ) over all possible inputs u ( k ): ∞ � � � � CA k B W := | D | + � � � k =0 H 3/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Worst-Case Peak Gain: Motivation WCPG is required: To measure how the computational errors in the implemented filter are propagated to the output To measure the magnitude of each variable for implementations in fixed-point arithmetic 4/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Worst-Case Peak Gain: Motivation WCPG is required: To measure how the computational errors in the implemented filter are propagated to the output To measure the magnitude of each variable for implementations in fixed-point arithmetic Goal: Given a small ε > 0 compute a floating-point approximation S on the WCPG such that element-by-element | W − S | < ε 4/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Outline Problem statement 1 Algorithm of WCPG evaluation 2 Basic bricks 3 Numerical Examples 4 Conclusion 5 5/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Worst-Case Peak Gain ∞ � � � � CA k B W = | D | + � � � k =0 6/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Worst-Case Peak Gain ∞ � � � � CA k B W = | D | + � � � k =0 Cannot sum infinitely = ⇒ need to truncate the sum 6/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Worst-Case Peak Gain ∞ � � � � CA k B W = | D | + � � � k =0 Cannot sum infinitely = ⇒ need to truncate the sum 6 sources of errors = ⇒ allocate 6 ”buckets” ε i out of the error budget ε 6/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 1 ∞ � � � CA k B � � k =0 7/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 1 ∞ N � � � � CA k B � � CA k B � � − → � � k =0 k =0 7/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 1 ∞ � � � CA k B � � � � � ∞ � k =0 N ↓ � � � � � CA k B � � � CA k B � � − � ≤ ε 1 � � N � � CA k B � � � � � � � � � � k =0 k =0 k =0 ↓ N � CV T k V − 1 B � � � � � � k =0 Step 1 Compute an approximate lower bound on truncation ↓ N � C ′ T k B ′ � � order N such that the truncation error is smaller than � � � � k =0 ε 1 . ↓ N � � C ′ P k B ′ � � � k =0 ↓ N � | L k | k =0 ↓ S N ↓ ↓ ↓ ↓ ↓ 7/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 1 ∞ � � � CA k B � � � � � ∞ � k =0 N ↓ � � � � � CA k B � � � � CA k B � − � ≤ ε 1 � � N � CA k B � � � � � � � � � � � k =0 k =0 k =0 ↓ N � � CV T k V − 1 B � � � � � k =0 Step 1 Compute an approximate lower bound on truncation ↓ N � � C ′ T k B ′ � order N such that the truncation error is smaller than � � � � k =0 ε 1 . ↓ N � � C ′ P k B ′ � � � k =0 ↓ Lower bound on truncation order N N � | L k | k =0 ↓ � log � n ε 1 | R l | | λ l | S N � M � min � N ≥ with M := ↓ ↓ log ρ ( A ) 1 − | λ l | ρ ( A ) ↓ l =1 ↓ ↓ 7/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 2 ∞ � � � CA k B � � � � k =0 N ↓ � � CA k B � � N � CA k B � � � � � � � k =0 k =0 ↓ N � CV T k V − 1 B � � � � � � k =0 ↓ N � C ′ T k B ′ � � � � � � k =0 ↓ N � � C ′ P k B ′ � � � k =0 ↓ N � | L k | k =0 ↓ S N ↓ ↓ 8/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 2 ∞ � � � CA k B � � � � k =0 N N ↓ � � � CA k B � � � CV T k V − 1 B � � − → � ( ≤ ε 2 N � � CA k B � � � � � � k =0 k =0 k =0 ↓ N � � CV T k V − 1 B � � � � � k =0 ↓ × = cancellation N � � C ′ T k B ′ � � � � � k =0 ↓ N � C ′ P k B ′ � � � � k =0 ↓ N � | L k | k =0 ↓ S N ↓ ↓ 8/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 2 ∞ � � � CA k B � � � � k =0 N N ↓ � � � CA k B � � � � CV T k V − 1 B � ( − � ( ≤ ε 2 N � CA k B � � � � � � � k =0 k =0 k =0 ↓ N � CV T k V − 1 B � � � � � � k =0 ↓ × = cancellation N � � C ′ T k B ′ � � � � � k =0 ↓ N × = less cancellation � � C ′ P k B ′ � � � k =0 ↓ N � | L k | k =0 ↓ S N ↓ ↓ 8/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 2 ∞ � � � CA k B � � � � k =0 N N ↓ � � � CA k B � � � CV T k V − 1 B � � ( − � ( ≤ ε 2 N � CA k B � � � � � � � k =0 k =0 k =0 ↓ N � CV T k V − 1 B � � � � � � k =0 ↓ × = cancellation N � C ′ T k B ′ � � � � � � k =0 ↓ N × = less cancellation � � C ′ P k B ′ � � � k =0 ↓ N A = XEX − 1 � | L k | k =0 ↓ S N ↓ ↓ 8/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 2 ∞ � � � CA k B � � � � k =0 N N ↓ � � � � CA k B � � CV T k V − 1 B � � ( − � ( ≤ ε 2 N � � CA k B � � � � � � k =0 k =0 k =0 ↓ N � CV T k V − 1 B � � � � � � k =0 ↓ × = cancellation N � � C ′ T k B ′ � � � � � k =0 ↓ N × = less cancellation � � C ′ P k B ′ � � � k =0 ↓ N A = XEX − 1 � | L k | V ≈ X and T ≈ E k =0 ↓ S N ↓ ↓ 8/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 2 ∞ � � � CA k B � � � � k =0 N N ↓ � � � � CA k B � � CV T k V − 1 B � � ( − � ( ≤ ε 2 N � � CA k B � � � � � � k =0 k =0 k =0 ↓ N � CV T k V − 1 B � � � � � � k =0 ↓ × = cancellation N � C ′ T k B ′ � � � � � � k =0 ↓ N × = less cancellation � C ′ P k B ′ � � � � k =0 ↓ N A = XEX − 1 � | L k | V ≈ X and T ≈ E k =0 ↓ S N ↓ T ≈ V − 1 × A × V ↓ A k ≈ V × T k × V − 1 8/24
Problem statement Algorithm of WCPG evaluation Basic bricks Numerical Examples Conclusion Step 2 ∞ � � � CA k B � � � � k =0 N ↓ � � CA k B � � N � � CA k B � � � � � � k =0 k =0 ↓ ↓ N � � C ′ T k B ′ � � � � � k =0 ↓ N � C ′ P k B ′ � � � � k =0 ↓ N � | L k | k =0 ↓ S N ↓ ↓ ↓ ↓ ↓ 9/24
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