1. Quiescent Steady State (DC) Analysis
The Newton-Raphson Method
J. Roychowdhury, University of California at Berkeley Slide 1

2. Solving the System's DAEs
d
~ (~ (t)) + f (~ (t)) + ~ = ~
q x ~ x b(t) 0
dt
● DAEs: many types of solutions useful
● DC steady state: no time variations
state
● transient: ckt. waveforms changing with time
transient
● periodic steady state: changes periodic w time
➔ linear(ized): all sinusoidal waveforms: AC analysis
➔ nonlinear steady state: shooting, harmonic balance
shooting
● noise analysis: random/stochastic waveforms
analysis
● sensitivity analysis: effects of changes in circuit
analysis
parameters
J. Roychowdhury, University of California at Berkeley Slide 2

3. QSS: Quiescent Steady State
(“DC”) Analysis
d
~ (~ (t)) + f (~ (t)) + ~ = ~
q x ~ x b(t) 0
dt
● Assumption: nothing changes with time
● x, b are constant vectors; d/dt term vanishes
~ (~ )
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 equations
➔ the problem: solving them numerically
➔ most common/useful technique: Newton-Raphson method
J. Roychowdhury, University of California at Berkeley Slide 3

4. The Newton Raphson Method
● Iterative numerical algorithm to solve ~ (~ ) = ~
g x 0
1 start with some guess for the solution
2 repeat
a check if current guess solves equation
i if yes: done!
ii if no: do something to update/improve the guess
● Newton-Raphson algorithm
● start with initial guess ~ 0 ; i=0
x
● repeat until “convergence” (or max #iterations)
d~ (~ i )
g x
➔ compute Jacobian matrix: Ji =
d~x
➔ solve for update ±~ : Ji ±~ = ¡~ (~ i )
x x g x
➔
➔ update guess: ~ i+1 = ~ i + ±~
x x x
➔ i++;
J. Roychowdhury, University of California at Berkeley Slide 4

5. Newton-Raphson Graphically
g(x)
● Scalar case above
● Key property: generalizes to vector case
J. Roychowdhury, University of California at Berkeley Slide 5

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 6

7. NR: Convergence Rate
● Key property of NR: quadratic convergence
¤
● Suppose x is the exact solution of g(x) = 0
● At the i
th
NR iteration, define the error ²i = xi ¡ x¤
● meaning of quadratic convergence: ²i+1 < c²2
i
●
(where c is a constant)
● NR's quadratic convergence properties
➔
if g(x) is smooth (at least continuous 1st and 2nd derivatives)
➔ and g 0 (x¤ ) 6= 0
➔ and kxi ¡ x¤ k is small enough, then:
➔ NR features quadratic convergence
J. Roychowdhury, University of California at Berkeley Slide 7

8. Convergence Rate in Digits of
Accuracy
Quadratic convergence Linear convergence
J. Roychowdhury, University of California at Berkeley Slide 8

9. NR: Convergence Strategies
● reltol-abstol on deltax
● stop if norm(deltax) <= tolerance
➔ tolerance = 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 k~ (~ )k < ²residual
g x
● Combinations of deltax and residual
● ultimately: heuristics, tuned to application
J. Roychowdhury, University of California at Berkeley Slide 9

10. Newton Raphson Update Step
● Need to solve linear matrix equation
● J ¢~ = ¡~ (~ ) : Ax = b problem
x g x
d~ (~ )
g x
● J= : Jacobian matrix
d~x
● Derivatives of vector functions
2 3 2 3
x1 g1 (x1 ; ¢ ¢ ¢ ; xn )
6 . 7 6 7
● If ~ (t) = 4 . 5 ;
x . ~ (~ ) = 4
g x .
.
. 5
xn g1 (x1 ; ¢ ¢ ¢ ; xn )
2 dg1 dg1 dg1 dg1
3
dx1 dx2 ¢¢¢ dxn¡1 dxn
6 dg2 dg2 dg2 dg2 7
6 ¢¢¢ 7
d~
g 6 dx1 dx2 dxn¡1 dxn 7
6 . . . . 7
● … then ,6 . . ¢¢¢ . . 7
x 6 dg .
d~ . . . 7
6 n¡1 dgn¡1
¢¢¢ dgn¡1 dgn¡1 7
4 dx1 dx2 dxn¡1 dxn 5
dgn dgn dgn dgn
dx1 dx2 ¢¢¢ dxn¡1 dxn
J. Roychowdhury, University of California at Berkeley Slide 10

11. DAE Jacobian Matrices
1
° 2
°
d
● Ckt DAE: ~ (~ (t)) + f (~ (t)) + ~ = ~
q x ~ x b(t) 0 iE
dt iL
2 3 2 2 3 3 2 3
e1 (t) 0 ¡diode(¡e1 ; IS ; Vt ) ¡ iE 0
6 e2 (t) 7 6 Ce2 7 6 0 7
~ (t) = 6 7 ~(~ ) = 6 7 f (~ ) = 6 iE + iL + e2 7~
7 b(t) = 6 7
x 4 iL (t) 5 q x 4 0 5 ~x 6
4 e2 ¡ e1
R
5 4¡E(t)5
iE (t) ¡LiL e2 0
2 3 2 ddiode 3
0 0 0 0 dv (¡e1 ) 0 0 ¡1
d~
q 60 C 0 07 ~ 6
df 0 1
1 17
Jq , =6
40
7 Jf , =6 R 7
d~
x 0 0 05 d~
x 4 ¡1 1 0 05
0 0 ¡L 0 0 1 0 0
J. Roychowdhury, University of California at Berkeley Slide 11

12. Newton Raphson: Computation
● Need to solve linear matrix equation
● J ¢~ = ¡~ (~ ) : Ax = b problem
x g x
● Ax=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 sparse
J. Roychowdhury, University of California at Berkeley Slide 12

13. Dense vs Sparse Matrices
● Sparse Jacobians: typically 3N-4N non-zeros
● compare against N2 for dense
J. Roychowdhury, University of California at Berkeley Slide 13