Factorization of Indefinite Systems Associated with RLC Circuits Patricio Rosen Wil Schilders, Joost Rommes
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 25-9-2009 PAGE
Motivation Small Size Complex Circuits Increasing Complexity Expensive Testing Circuit Simulation MOR is Necessary Solve the System /computer science and mathematics department 25-9-2009 PAGE 2
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 25-9-2009 PAGE 3
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 25-9-2009 PAGE 4
Circuit used for Eigenvalues /computer science and mathematics department 25-9-2009 PAGE 5
Circuit Equations Spectral Properties Complex Form Complex Stable Form /computer science and mathematics department 25-9-2009 PAGE 6
Circuit Equations Spectral Properties Real Form Real Stable Form /computer science and mathematics department 25-9-2009 PAGE 7
Solution Method
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 25-9-2009 PAGE 9
Delay of Convergence Cx=r 0 0 0 C= 0 0 0....... 0... 0 Rx=r R=r ij = N (0, /computer science and mathematics department 25-9-2009 PAGE 0
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 2 0 0 0 I m I m 0 0 A = LDL T B T BT 2 0 0 I n m +LT 2 0 LT MT I m /computer science and mathematics department 25-9-2009 PAGE
Reference /computer science and mathematics department 25-9-2009 PAGE 2
RL Factorization
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 25-9-2009 PAGE 4
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 25-9-2009 PAGE 5
RL Factorization A = ˆR ωˆl ˆP 0 ωˆl ˆR 0 ˆP ˆPT 0 0 0 0 ˆPT 0 0 QAQ T = P P = =( P 2 R ωl P 0 ωl R 0 P PT 0 0 0 0 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 0 0 0 2 0 PT 0 PT 0 0 2 /computer science and mathematics department 25-9-2009 PAGE 6
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 0 2 0 U 2 0 BT BT 0 0 0 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 25-9-2009 PAGE 7
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 25-9-2009 PAGE 8
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 25-9-2009 PAGE 9
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 2 0 0 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 25-9-2009 PAGE 20
RLC FACTORIZATION
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 0 0 0 0 0 v n i g ic il v n = 0 0 0 AT i I t (t /computer science and mathematics department 25-9-2009 PAGE 22
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 25-9-2009 PAGE 23
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 2 0 0 I 2(n m 0 BT BT 2 0 0 0 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 25-9-2009 PAGE 24
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 2 0 0 U 2 D= 0 I 0 2m I 2m D 0 0 0 D 2 /computer science and mathematics department 25-9-2009 PAGE 25
Performance as Direct Solver
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 25-9-2009 PAGE 27
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 + 5 6 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 25-9-2009 PAGE 28
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 25-9-2009 PAGE 29
Running times No Coupling Coupling of 0% /computer science and mathematics department 25-9-2009 PAGE 30
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 25-9-2009 PAGE 3
Running times modified circuit Modified Circuit No Coupling Coupling of 0% /computer science and mathematics department 25-9-2009 PAGE 32
RLC Algorithm
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 25-9-2009 PAGE 34
RLC Ladder Example N: number of blocks /computer science and mathematics department 25-9-2009 PAGE 35
Running times LU is better than RLC factorization /computer science and mathematics department 25-9-2009 PAGE 36
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 25-9-2009 PAGE 37
P P P P T /computer science and mathematics department 25-9-2009 PAGE 38
Running Times Modified RLC Ladder 50% Conductances and Capacitors /computer science and mathematics department 25-9-2009 PAGE 39
Running Times Modified Circuit 20% Conductances and Capacitors /computer science and mathematics department 25-9-2009 PAGE 40
Running Times Modified Circuit RLC is better than LU if: n g +n c 2m /computer science and mathematics department 25-9-2009 PAGE 4
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 25-9-2009 PAGE 42
THANKS FOR YOUR ATTENTION
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 25-9-2009 PAGE 44
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 25-9-2009 PAGE 45
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 25-9-2009 PAGE 46
Circuit Equations Alternate Current Analysis ( R Al A T A TG Ac } l {{ g } G ( il v n Consider Wave Input ( L 0 + 0 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 25-9-2009 PAGE 47
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 0 0 0 D 2 Goal: develop Schilders type factorizations for the Circuit Equations /computer science and mathematics department 25-9-2009 PAGE 48
Simulation for N=2 Magnitude Phase /computer science and mathematics department 25-9-2009 PAGE 49
Circuit Equations A circuit is a network of interconnected components Circuit topology Incidence Matrix Set ground node g A= N N 2 N {}} 3 { 0 0 0 B B 2 B 3 /computer science and mathematics department 25-9-2009 PAGE 50
Introduction Electronics Industry Circuit Design Circuit Simulation /computer science and mathematics department 25-9-2009 PAGE 5