J. Roychowdhury, University of California at Berkeley Slide 1

Quiescent Steady State (DC) Analysis

The Newton-Raphson Method

J. Roychowdhury, University of California at Berkeley Slide 2

Solving the System's DAEs

● DAEs: many types of solutions useful● DC steady stateDC steady state: no time variations● transienttransient: ckt. waveforms changing with time● periodic steady state: changes periodic w time

➔ linear(ized): all sinusoidal waveforms: AC analysisAC analysis➔ nonlinear steady state: shootingshooting, harmonic balanceharmonic balance

● noise analysisnoise analysis: random/stochastic waveforms● sensitivity analysissensitivity analysis: effects of changes in circuit

parameters

d

dt~q (~x(t)) + ~f (~x(t)) +~b(t) = ~0

J. Roychowdhury, University of California at Berkeley Slide 3

QSS: Quiescent Steady State(“DC”) Analysis

● Assumption: nothing changes with timeAssumption: nothing changes with time● x, b are constant vectors; d/dt term vanishes

d

dt~q (~x(t)) + ~f (~x(t)) +~b(t) = ~0

~g(~x)z }| {~f (~x) +~b = ~0

● Why do QSS?➔ quiescent operation: first step in verifying functionality➔ stepping stone to other analyses: AC, transient, noise, ...

● Nonlinear system of equationsNonlinear system of equations➔ the problem: solving them numerically➔ most common/useful technique: Newton-RaphsonNewton-Raphson method

J. Roychowdhury, University of California at Berkeley Slide 4

The Newton Raphson Method● IterativeIterative numerical algorithm to solve

1 start with some guess for the solution2 repeat

a check if current guess solves equationi if yes: done!ii if no: do something to update/improve the guess

● Newton-Raphson algorithm● start with initial guess ; i=0● repeat until “convergence” (or max #iterations)

➔ compute Jacobian matrix:

➔ solve for update :➔

➔ update guess: ➔ i++;

~g(~x) = ~0

~x0

Ji =d~g(~xi)

d~x

Ji ±~x = ¡~g(~xi)±~x

~xi+1 = ~xi + ±~x

J. Roychowdhury, University of California at Berkeley Slide 5

Newton-Raphson Graphically

● Scalar case above● Key property: generalizes to vector case

g(x)

J. Roychowdhury, University of California at Berkeley Slide 6

Newton Raphson (contd.)● Does it always work? No.

● Conditions for NR to converge reliably➔ g(x) must be “smooth”: continuous, differentiable➔ starting guess “close enough” to solution

● practical NR: needs application-specific heuristics

J. Roychowdhury, University of California at Berkeley Slide 7

NR: Convergence Rate

● Key property of NR: quadratic convergence● Suppose is the exact solution of ● At the NR iteration, define the error● meaning of quadratic convergence:

● (where c is a constant)

● NR's quadratic convergence properties➔ if is smooth (at least continuous 1st and 2nd derivatives)➔ and ➔ and is small enough, then:

➔ NR features quadratic convergence

ithx¤ g(x) = 0

²i = xi ¡ x¤²i+1 < c²

2i

kxi ¡ x¤k

g(x)

g0(x¤) 6= 0

J. Roychowdhury, University of California at Berkeley Slide 8

Convergence Rate in Digits of Accuracy

Quadratic convergence Linear convergence

J. Roychowdhury, University of California at Berkeley Slide 9

NR: Convergence Strategies

● reltol-abstol on deltax● stop if norm(deltax) <= tolerancenorm(deltax) <= tolerance

➔ tolerance = abstol + reltol*xtolerance = abstol + reltol*x● reltol ~ 1e-3 to 1e-6● abstol ~ 1e-9 to 1e-12

● better➔ apply to individual vector entries (and AND)➔ organize x in variable groups: e.g., voltages, currents, …➔ (scale DAE equations/unknowns first)

● more sophisticated possible➔ e.g., use sequence of x values to estimate conv. rate

● residual convergence criterion● stop if

● Combinations of deltax and residual● ultimately: heuristics, tuned to application

k~g(~x)k < ²residual

J. Roychowdhury, University of California at Berkeley Slide 10

Newton Raphson Update Step● Need to solve linear matrix equation

● : Ax = b problem

● : Jacobian matrixJacobian matrix

● Derivatives of vector functionsDerivatives of vector functions

● If

● … then

J =d~g(~x)

d~x

J ¢~x = ¡~g(~x)

~x(t) =

264x1...xn

375 ; ~g(~x) =

264g1(x1; ¢ ¢ ¢ ; xn)

...g1(x1; ¢ ¢ ¢ ; xn)

375

d~g

d~x,

266666664

dg1dx1

dg1dx2

¢ ¢ ¢ dg1dxn¡1

dg1dxn

dg2dx1

dg2dx2

¢ ¢ ¢ dg2dxn¡1

dg2dxn

...... ¢ ¢ ¢

......

dgn¡1

dx1

dgn¡1

dx2¢ ¢ ¢ dgn¡1

dxn¡1

dgn¡1

dxndgndx1

dgndx2

¢ ¢ ¢ dgndxn¡1

dgndxn

377777775

J. Roychowdhury, University of California at Berkeley Slide 11

DAE Jacobian Matrices

● Ckt DAE:iL

2°1°

iE

d

dt~q (~x(t)) + ~f (~x(t)) +~b(t) = ~0

~x(t) =

2664

e1(t)e2(t)iL(t)iE(t)

3775 ~q(~x) =

2664

0Ce20

¡LiL

3775 ~f(~x) =

2664

¡diode(¡e1; IS ; Vt)¡ iEiE + iL +

e2R

e2 ¡ e1e2

3775~b(t) =

2664

00

¡E(t)0

3775

Jf ,d~f

d~x=

2664

ddiodedv (¡e1) 0 0 ¡10 1

R 1 1¡1 1 0 00 1 0 0

3775Jq ,

d~q

d~x=

2664

0 0 0 00 C 0 00 0 0 00 0 ¡L 0

3775

J. Roychowdhury, University of California at Berkeley Slide 12

Newton Raphson: Computation

● Need to solve linear matrix equation● : Ax = b problem

● Ax=b: where much of the computation liesAx=b: where much of the computation lies● large circuits (many nodes): large DAE systems,

large Jacobian matrices● in general (for arbitrary matrices of size n)

➔ solving Ax = b requires● O(n2) memory● O(n3) computation!● (using, e.g., Gaussian Elimination)

➔ but for most circuit Jacobian matrices● O(n) memory, ~O(n1.4) computation● … because circuit Jacobians are typically sparsesparse

J ¢~x = ¡~g(~x)

J. Roychowdhury, University of California at Berkeley Slide 13

Dense vs Sparse Matrices

● Sparse Jacobians: typically 3N-4N non-zeros● compare against N2 for dense