Factorization of Indefinite Systems Associated with RLC Circuits. Patricio Rosen Wil Schilders, Joost Rommes

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1 Factorization of Indefinite Systems Associated with RLC Circuits Patricio Rosen Wil Schilders, Joost Rommes

2 Outline Circuit Equations Solution Methods (Schilders Factorization Incidence Matrix Decomposition Schilders Type Factorizations for RL and RLC Performance as Direct Solver Conclusions /computer science and mathematics department PAGE

3 Motivation Small Size Complex Circuits Increasing Complexity Expensive Testing Circuit Simulation MOR is Necessary Solve the System /computer science and mathematics department PAGE 2

4 Circuit Equations System Formulation KVL: Av n =v b A = KCL: ATi b =0 A i A g A c A l,v b = v i v g v c v l,i b = i i i g ic il, i i =I t (t, i g = G v g, i c = Cd v, v c l = ( Ld +R i l, dt dt ( R Al ( ( il L ( 0 d il A T A TG + Ac v } l {{ g }}{{ n 0 A TC Ac }}{{ c dt v }}{{ n } G z(t C d z(t dt = ( 0 I AT t (t, }{{ i } B /computer science and mathematics department PAGE 3

5 Circuit Equations AC Analysis Complex System [( ˆR ˆP ˆPT Ĝ +iω ( ˆL 0 0 Ĉ ]( ir +ii i v r +iv i = ( 0 AT i Θ. Real System R ˆR ωˆl ωl PˆP 0 ωˆl ˆR 0 ˆP ˆPT 0 Ĝ ωĉ 0 ˆPT ωĉ Ĝ i r 0 ii 0 v = AT r i v i 0 Θ non symmetric and ( I 0 = PTA I ( A P PT D indefinite ( A 0 0 S ( I A P 0 I S= (D+PTA P /computer science and mathematics department PAGE 4

6 Circuit used for Eigenvalues /computer science and mathematics department PAGE 5

7 Circuit Equations Spectral Properties Complex Form Complex Stable Form /computer science and mathematics department PAGE 6

8 Circuit Equations Spectral Properties Real Form Real Stable Form /computer science and mathematics department PAGE 7

9 Solution Method

10 Solution Method Saddle Point Problem Indefinite Non symmetric ( ( A B x BT C y = ( b c Solution Methods: Direct Solvers Iterative Solvers expensive for large systems delay of convergence Paper by Greenbaum: Any nonincreasing convergence curve is possible for GMRES /computer science and mathematics department PAGE 9

11 Delay of Convergence Cx=r C= Rx=r R=r ij = N (0, /computer science and mathematics department PAGE 0

12 Schilders Factorization Invertible Symmetric Saddle Point A = ( Â ˆB ˆBT 0 ( x y = ( a b, Perform LQ Rearrange Matrix ΠˆB=BQ QAQ T =( A B BT 0 Q = ( Π 0 0 Q Schilders Factorization A = B 0 L B 2 I n m +L 2 M D 0 I m 0 D I m I m 0 0 A = LDL T B T BT I n m +LT 2 0 LT MT I m /computer science and mathematics department PAGE

13 Reference /computer science and mathematics department PAGE 2

14 RL Factorization

15 Incidence Matrix In general LQ decomposition ΠˆB=BQ We need only permutations Π ˆP =PΠ 2 Algorithm Idea: Π rˆpπc = ( x 0 v P Already Added Directly connected with Added Other Nodes Time Complexity: O(n2 /computer science and mathematics department PAGE 4

16 Remarks Lower Trapezoidal Form P P =( P 2 Inverse of top exists P = Calculation takes P in O(m2 Inverse is exact, it consistof 0,, /computer science and mathematics department PAGE 5

17 RL Factorization A = ˆR ωˆl ˆP 0 ωˆl ˆR 0 ˆP ˆPT ˆPT 0 0 QAQ T = P P = =( P 2 R ωl P 0 ωl R 0 P PT PT 0 0 Π 3 QAQ T ΠT 3 = R ωl R ωl2 2 P 0 ωl R ωl 2 R 2 0 P R ωl2 2 R ωl22 22 P 2 0 ωl 2 R 2 ωl 22 R 22 0 P 2 PT 0 PT PT 0 PT /computer science and mathematics department PAGE 6

18 RL Factorization Theorem: A A 2 B A 2 A 22 B 2 = B 0 L B 2 L 2 M D 0 I 2m 0 D 2 0 B T BT U 2 0 BT BT I 2 2m I 2m 0 0 U F I 2m Sketch of proof: B D BT +B U +L BT =A ( B D BT 2 +B F+L BT 2 =A 2 (2 B 2 D BT +B 2 U +MBT =A 2 (3 L 2 D 2 U 2 +B 2 D BT 2 +B 2 F+MBT 2 =A 22 (4 /computer science and mathematics department PAGE 7

19 Sketch of the proof From( D +U B T +B L = B A B T From(2 and (3 F =B ( A2 B D BT 2 L BT 2, M = ( A 2 B2 D BT B2 U B T. From (4 L 2 D 2 U 2 =A 22 B2 D BT 2 B2 F MBT 2 :=Ŵ If A is sym. pos.def. Computed with Cholesky RL Case: find LDU decomposition of Ŵ =(Ŵ Ŵ 2 Ŵ 2 Ŵ 22 /computer science and mathematics department PAGE 8

20 Sketch of the proof Lemma: Ŵ = ( L2, 0 ωl 2,2 L 2,3 (ω ( D2, ( L T ωl T 2, 2,2 D 2,2 (ω 0 LT (ω 2,3 Proof: PT 2 Ŵ =Ŵ ( ( ( R R P T W 2 =W 22 = P 2 P I 2 R 2 R 22 I Ŵ2 =Ŵ2 = ω ( P2 P I ( ( L L P T 2 L 2 L 22 I PT 2 Symmetric Positive Definite L 2, D 2, Cholesky L 2,2 =Ŵ2 L T 2, D 2, Solving L 2,3 D 2,2 LT 2,3 =Ŵ +ω 2Ŵ 2 Ŵ Ŵ 2 Cholesky /computer science and mathematics department PAGE 9

21 RL Factorization Final RL Factorization A= Π L DŨΠ T, L= I 0 0 2m L B 0, Ũ = M B 2 L 2 Frequency Dependencies: D,L,U,F,M,L 2,,D 2,,L 2,2 I U F 2m 0 BT BT U 2 ω D= 0 I 0 2m I 2m D 0, 0 0 D 2 independent or linearly dependent For different ω need to recompute only L 2,3 (ω,d 2,3 (ω /computer science and mathematics department PAGE 20

22 RLC FACTORIZATION

23 RLC Factorization Invertibility ( proof: A A B =, A= B T C ( ˆR Full rank ωˆl (ˆP 0, B= ωˆl ˆR 0 ˆP, C= (Ĝ ωĉ ωĉ Ĝ H= 2 (A+A T,D= 2 (C+C T Pos.Def 0=vTA v =xtax+xtbyt y TBTx+yTCy=xTHx+yTDy. G Rewrite Circuit Equations Ag i C g 0 R ic 0 Ac Al il + L 0 d dt A T g A T c A T l v n i g ic il v n = AT i I t (t /computer science and mathematics department PAGE 22

24 RLC Factorization ω>0 G 0 Ag 0 Ac R ωc ω L Al G 0 Ag C 0 Ac ω ω L R A l A T A T A T 0 0 g c l A T A T A T 0 0 g c l i gr i cr i lr i gi i ci i li v nr v ni A = ˆX Î(ωŶ ˆP 0 Î(ωŶ ˆX 0 ˆP ˆPT 0 0 0, 0 ˆPT 0 0 Î(ω= I I ω ωi /computer science and mathematics department PAGE 23

25 RLC Factorization Theorem: A A 2 B A 2 A 22 B 2 = B 0 L B 2 I 2(n m M D 0 I 2m 0 Ŵ 0 B T BT I 2(n m 0 BT BT I 2m I 2m 0 0 U F I 2m ŴW is invertible D 0 I 2m 0 Ŵ 0 I 2m 0 0 = 0 0 I 2m 0 Ŵ 0 I 2m 0 D Finish factorization with LDU Π e L 2 D 2 U 2 =Ŵ /computer science and mathematics department PAGE 24

26 RLC Factorization LDU decomposition Π e L 2 D 2 U 2 =Ŵ Theorem (Final RLC Factorization: A = Q T ΠT ΠT ΠT 3 E 4 L DŨΠ 4 Π Q 3 L= I 0 0 2m L B 0, Ũ = ΠTM ΠTB e e 2 L 2 I U F 2m 0 BT BT U 2 D= 0 I 0 2m I 2m D D 2 /computer science and mathematics department PAGE 25

27 Performance as Direct Solver

28 Performance as Direct Solver n ω : Frequencies m+ : Nodes n l : Resistor Inductor Branches n g : Conductances n c : Capacitors /computer science and mathematics department PAGE 27

29 Complexity of RL Algorithm RL Algorithm Rearrange Incidence Matrix P Find and D,L,U { O(m2 O(m3 best case worst case Perform Cholesky twice Ŵ W,Ŵ +ω2ŵ Ŵ,W W 2 W ŴW 2 2 O(n 3 l m 3 O Solve resulting systems ( 6 n ω 3 (n +m 3 l > O (( n ω O (( n ω + 5 (nl m 3 6 (nl m 3+3m3 best case worst case RL IS ALWAYS BETTER THAN LU DECOMPOSITION /computer science and mathematics department PAGE 28

30 RL example L = I N pi N pi N I N M3 M3= p p p p N: number of blocks p: coupling factor /computer science and mathematics department PAGE 29

31 Running times No Coupling Coupling of 0% /computer science and mathematics department PAGE 30

32 RL performs better It does not perform that nice due to: P P P P T We are in the worst case complexity /computer science and mathematics department PAGE 3

33 Running times modified circuit Modified Circuit No Coupling Coupling of 0% /computer science and mathematics department PAGE 32

34 RLC Algorithm

35 Complexity of RLC Algorithm RLC Algorithm Find G, C Perform LU decomposition Ŵ RLC Fact O((n w +4m3+n w 6 3 (n l +n g +n c O ( n w 6 3 (n l +n g +n c m 3 m 3 worst case best case LU O ( n 6 (n +m 3 ω 3 l if n g +n c 2m Conditionally Better than LU /computer science and mathematics department PAGE 34

36 RLC Ladder Example N: number of blocks /computer science and mathematics department PAGE 35

37 Running times LU is better than RLC factorization /computer science and mathematics department PAGE 36

38 Modified RLC Ladder RLC factorization is still usefull Modified RLC Ladder circuit: 0 p pn Remove: capacitor branches from left to right pn and conductances from right to left /computer science and mathematics department PAGE 37

39 P P P P T /computer science and mathematics department PAGE 38

40 Running Times Modified RLC Ladder 50% Conductances and Capacitors /computer science and mathematics department PAGE 39

41 Running Times Modified Circuit 20% Conductances and Capacitors /computer science and mathematics department PAGE 40

42 Running Times Modified Circuit RLC is better than LU if: n g +n c 2m /computer science and mathematics department PAGE 4

43 Conclusions Results: Explicit Factorizations RL/RLC system Frequency dependencies founded RL algorithm always better than LU decomposition RLC algorihtm conditionally better than LU LU needs to recompute all again RLC only need to recompute some parts Future work: Control Fill-in Study non-linear frequency dependencies /computer science and mathematics department PAGE 42

44 THANKS FOR YOUR ATTENTION

45 L = Frequency Dependencies RL Case D = U = F = ( diag(p R P T diag(p R P T ( P strlow(p R P T 0 ωl P T P strlow(p R P T ( strupp(p R P T PT ωp L 0 strupp(p R P TPT ( P R 2 low(p R P T PT ωp 2 L 2 P L P TPT P R 2 2 low(p ω(p L 2 R P T PT 2 M = ( R2 P T P2 upp(p ωl 2 P T R P T ω(l2 P T R 2 P T +P 2 P L P T R P T P2 upp(p /computer science and mathematics department PAGE 44

46 F = M = Frequency Dependencies RLC Case ( diag(p X D = P T 0 0 diag(p X P, T ( P strlow(p X L = P T 0 I (ωy P T P strlow(p X P, T ( strupp(p X U = P TPT P Y I (ω 0 strupp(p X P. TPT ( P X 2 low(p X P TPT P Y 2 2 I 2 (ω P Y 2 I 2 (ω+p I (ωy P TPT P X 2 2 low(p ( X2 P T P2 upp(p X P T I 2 (ωy 2 P T P2 P I2 (ωy 2 P T X 2 P T P2 upp(p X P, TPT 2 Y I (ωp T X P. T /computer science and mathematics department PAGE 45

47 Circuit Equations Incidence Matrix Kirchhoff s Current Law Kirchhoff s Voltage Law Branch Constitutive Relations A= A i A g A c A l Av i n =v, i, A g v n =v g, A c v n =v c, Av l n =v l A, i b, v b, v n v i v g v c v l,v b =,i b = i i i g ic il AT i i i +AT g i g +AT c i c +AT l i l =0 ATi b =0 Av n =v b G, C, R diagonal pos. def. L symmetricpos.def. i i =I t (t, i g = G v g, i c = Cd dt v c, v l = ( Ld dt +R i l, /computer science and mathematics department PAGE 46

48 Circuit Equations Alternate Current Analysis ( R Al A T A TG Ac } l {{ g } G ( il v n Consider Wave Input ( L A TC Ac }{{ c } C d dt ( il v n G(Z(weiωt+C d dt (Z(ωe iωt=bθeiωt, = ( 0 I AT t (t }{{ i } B GZ(w+iωCZ(w=BΘ [( ˆR ˆP ˆPT Ĝ +iω ( ˆL 0 0 Ĉ ]( ir +ii i v r +iv i = ( 0 AT i Θ. /computer science and mathematics department PAGE 47

49 Schilders Factorization Permuting Useful for: Direct Solver Preconditioner A = QLDL T Q T, ( Q 0 Π T = QT 0 L = I 0 0 m L B 0 M B I +L 2 n m 2 D = 0 I 0 m I m D D 2 Goal: develop Schilders type factorizations for the Circuit Equations /computer science and mathematics department PAGE 48

50 Simulation for N=2 Magnitude Phase /computer science and mathematics department PAGE 49

51 Circuit Equations A circuit is a network of interconnected components Circuit topology Incidence Matrix Set ground node g A= N N 2 N {}} 3 { B B 2 B 3 /computer science and mathematics department PAGE 50

52 Introduction Electronics Industry Circuit Design Circuit Simulation /computer science and mathematics department PAGE 5

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