8/3/2019 02 Newton Raphson

1/13

J. Roychowdhury, University of California at Berkeley Slide 1

Quiescent Steady State (DC) Analysis

The Newton-Raphson Method

8/3/2019 02 Newton Raphson

2/13

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 analy

sis

nonlinear steady state: shootingshooting

, harmonic balanceharmonic balance noise analysisnoise analy

sis: random/stochastic waveforms sensitivity analysissensitivity

analysis: effects of changes in circuitparameters

ddt

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

8/3/2019 02 Newton Raphson

3/13

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-Rap

hson method

8/3/2019 02 Newton Raphson

4/13J. Roychowdhury, University of California at Berkeley Slide 4

The Newton Raphson Method

IterativeIterative numerical algorithm to solve1 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~xJi ~x = ~g(~xi)~x

~xi+1 = ~xi + ~x

8/3/2019 02 Newton Raphson

5/13J. Roychowdhury, University of California at Berkeley Slide 5

Newton-Raphson Graphically

Scalar case above Key property: generalizes to vector case

g(x)

8/3/2019 02 Newton Raphson

6/13J. 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

8/3/2019 02 Newton Raphson

7/13J. Roychowdhury, University of California at Berkeley Slide 7

NR: Convergence Rate

Key property of NR: quadratic convergence

Suppose is the exactsolution 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

i = xi x

i+1 < c2i

kxi xk

g(x)

g0(x) 6= 0

8/3/2019 02 Newton Raphson

8/13J. Roychowdhury, University of California at Berkeley Slide 8

Convergence Rate in Digits ofAccuracy

Quadratic convergence Linear convergence

8/3/2019 02 Newton Raphson

9/13J. Roychowdhury, University of California at Berkeley Slide 9

NR: Convergence Strategies

reltol-abstol on deltax stop ifnorm(deltax)

8/3/2019 02 Newton Raphson

10/13

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) =

264

x1...

xn

375 ; ~g(~x) =

264

g1(x1; ; xn)...

g1(x1; ; xn)

375

d~g

d~x,

266666664

dg1dx1

dg1dx2

dg1dxn1

dg1dxn

dg2dx1

dg2dx2

dg2dxn1

dg2dxn

......

......

dgn1dx1

dgn1dx2

dgn1dxn1

dgn1dxn

dgn

dx1

dgn

dx2 dgn

dxn1

dgn

dxn

377777775

8/3/2019 02 Newton Raphson

11/13

J. Roychowdhury, University of California at Berkeley Slide 11

DAE Jacobian Matrices

Ckt DAE:iL

21

iE

d

dt

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

~x(t) =

266

4

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

377

5~q(~x) =

2664

0Ce2

0LiL

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=

2664ddiodedv

(e1) 0 0 10 1

R1 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

8/3/2019 02 Newton Raphson

12/13

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)

8/3/2019 02 Newton Raphson

13/13

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