CSE 245: Computer Aided Circuit Simulation and Verification Fall 2004, Nov Nonlinear Equation.
CSE 245: Computer Aided Circuit Simulation and Verification
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Transcript of CSE 245: Computer Aided Circuit Simulation and Verification
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CSE 245: Computer Aided Circuit Simulation and Verification
Spring 2010
Nonlinear Equation
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April 19, 2023 2 courtesy Alessandra Nardi UCB
Outline Nonlinear problems Iterative Methods Newton’s Method
Derivation of Newton Quadratic Convergence Examples Convergence Testing
Multidimensonal Newton Method Basic Algorithm Quadratic convergence Application to circuits
Improve Convergence Limiting Schemes
Direction Corrupting Non corrupting (Damped Newton)
Continuation Schemes Source stepping
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April 19, 2023 3 courtesy Alessandra Nardi UCB
Need to Solve
( 1) 0d
t
VV
d sI I e
Nonlinear Problems - Example
0
1
IrI1 Id
0)1(1
0
11
1
1
IeIeR
III
tV
e
s
dr
11)( Ieg
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Nonlinear Equations Given g(V)=I It can be expressed as: f(V)=g(V)-I
Solve g(V)=I equivalent to solve f(V)=0
Hard to find analytical solution for f(x)=0
Solve iteratively
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Nonlinear Equations – Iterative Methods
Start from an initial value x0
Generate a sequence of iterate xn-1, xn, xn+1 which hopefully converges to the solution x*
Iterates are generated according to an iteration function F: xn+1=F(xn)
Ask• When does it converge to correct solution ?• What is the convergence rate ?
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April 19, 2023 6 courtesy Alessandra Nardi UCB
Newton-Raphson (NR) Method
Consists of linearizing the system.Want to solve f(x)=0 Replace f(x) with its linearized version and solve.
Note: at each step need to evaluate f and f’
functionIterationxfdx
xdfxx
xxdx
xdfxfxf
iesTaylor Serxxdx
xdfxfxf
kk
kk
kkk
kk
)()(
)()(
)()(
)()(
)()(
1
1
11
**
*
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Newton-Raphson Method – Graphical View
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Newton-Raphson Method – Algorithm
Define iteration
Do k = 0 to ….
until convergence
How about convergence? An iteration {x(k)} is said to converge with order q if
there exists a vector norm such that for each k N:
qkk xxxx ˆˆ1
)()(
1
1 kk
kk xfdx
xdfxx
)(
)(1
1 kk
kk xfdx
xdfxx
)(
)(1
1 kk
kk xfdx
xdfxx
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April 19, 2023 9 courtesy Alessandra Nardi UCB
Mean Value theoremtruncates Taylor seriesBut
by Newtondefinition
Newton-Raphson Method – Convergence
2*2
2** )(
)~()(
)()()(0 kk
kk xx
dx
xfdxx
dx
xdfxfxf
)()(
)(0 1 kkk
k xxdx
xdfxf
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April 19, 2023 10 courtesy Alessandra Nardi UCB
Subtracting
21 * 1 * 2
2 ( ) [ ( )] ( )( )k k kdf d f
x x x x x xdx d x
Convergence is quadratic
Dividing through
21
2
21 * *
Let [ ( )] ( )
then
k k
k k k
df d fx x K
dx d x
x x K x x
Newton-Raphson Method – Convergence
2*2
2*1 )(
)~()(
)(xx
dx
xfdxx
dx
xdf kkk
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Local Convergence Theorem
If
Then Newton’s method converges given a sufficiently close initial guess (and
convergence is quadratic)
boundedisK
boundeddx
fdb
roay from zebounded awdx
dfa
)
)
2
2
Newton-Raphson Method – Convergence
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April 19, 2023 12 courtesy Alessandra Nardi UCB
21
2 21 * * *
* 2
2
1 *
*
( ) 2
( ) 1 0,
2 ( ) 1
2 ( ) 2 ( )
1( ) (
( 1
)2
)
k k
k k k k
k k k k k
k kk
dfx x
dx
x x x x
x x x x x x x
f x x fin
x
d x x
or x x x xx
Convergence is quadratic
Newton-Raphson Method – Convergence
Example 1
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1 2
1 *
* *1
2 *
( ) 2
2 ( 0) ( 0)
10 0
( ) 0,
for 02
1( ) ( )
2
0
k k
k k k
k k k
k k
dfx x
dx
x x x
x x x x
or x x x
f x
x
x x
Convergence is linear
Note : not bounded
away from zero
1df
dx
Newton-Raphson Method – Convergence
Example 2
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Newton-Raphson Method – Convergence
Example 1,2
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0 Initial Guess,= 0x k
Repeat { 1
k
k k kf x
x x f xx
} Until ?
1 1? ?k k kf x thresholdx x threshold
1k k
Newton-Raphson Method – Convergence
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Need a "delta-x" check to avoid false convergence
X
f(x)
1kx kx *x
1 a
kff x
1 1 a r
k k kx xx x x
Newton-Raphson Method – Convergence
Convergence Check
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April 19, 2023 17 courtesy Alessandra Nardi UCB
Also need an " " check to avoid false convergencef x
X
f(x)
1kx kx
*x
1 a
kff x
1 1 a r
k k kx xx x x
Newton-Raphson Method – Convergence
Convergence Check
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demo2
Newton-Raphson Method – Convergence
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X
f(x)
Convergence Depends on a Good Initial Guess
0x
1x2x 0x
1x
Newton-Raphson Method – Convergence
Local Convergence
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Convergence Depends on a Good Initial Guess
Newton-Raphson Method – Convergence
Local Convergence
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Nodal Analysis
1 2At Node 1: 0i i
1bv
+
-
1i2i 2
bv
3i+ -
3bv
+
-
NonlinearResistors
i g v
1v2v
1 1 2 0g v g v v
3 2At Node 2: 0i i
3 1 2 0g v g v v
Two coupled nonlinear equations in two unknowns
Nonlinear Problems – Multidimensional Example
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April 19, 2023 22 courtesy Alessandra Nardi UCB
Outline Nonlinear problems Iterative Methods Newton’s Method
Derivation of Newton Quadratic Convergence Examples Convergence Testing
Multidimensonal Newton Method Basic Algorithm Quadratic convergence Application to circuits
Improve Convergence Limiting Schemes
Direction Corrupting Non corrupting (Damped Newton)
Continuation Schemes Source stepping
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* *Problem: Find such tha 0t x F x * N N N and : x F
Multidimensional Newton Method
functionIterationxFxJxx
atrixJacobian M
x
xF
x
xF
x
xF
x
xF
xJ
iesTaylor SerxxxJxFxF
kkkk
N
NN
N
)()(
)()(
)()(
)(
))(()()(
11
1
1
1
1
***
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Multidimensional Newton Method
)()()( :solve Instead
sparse).not is(it )( computenot Do
)()(:
1
1
11
kkkk
k
kkkk
xFxxxJ
xJ
xFxJxxIteration
Each iteration requires:1. Evaluation of F(xk)2. Computation of J(xk) 3. Solution of a linear system of algebraic
equations whose coefficient matrix is J(xk) and whose RHS is -F(xk)
Computational Aspects
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0 Initial Guess,= 0x k
Repeat {
1 1Solve for k k k k kFJ x x x F x x
} Until 11 , small en ug o hkk kx x f x
1k k
Compute ,k kFF x J x
Multidimensional Newton Method
Algorithm
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If
1) Inverse is boundedkFa J x
) Derivative is Lipschitz ContF Fb J x J y x y
Then Newton’s method converges given a sufficiently close initial guess (and
convergence is quadratic)
Multidimensional Newton Method
Local Convergence TheoremConvergence
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Application of NR to Circuit Equations
Applying NR to the system of equations we find that at iteration k+1: all the coefficients of KCL, KVL and of BCE of the
linear elements remain unchanged with respect to iteration k
Nonlinear elements are represented by a linearization of BCE around iteration k
This system of equations can be interpreted as the STA of a linear circuit (companion network) whose elements are specified by the linearized BCE.
Companion Network
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Application of NR to Circuit Equations
General procedure: the NR method applied to a nonlinear circuit whose eqns are formulated in the STA form produces at each iteration the STA eqns of a linear resistive circuit obtained by linearizing the BCE of the nonlinear elements and leaving all the other BCE unmodified
After the linear circuit is produced, there is no need to stick to STA, but other methods (such as MNA) may be used to assemble the circuit eqns
Companion Network
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Note: G0 and Id depend on the iteration count k G0=G0(k) and Id=Id(k)
Application of NR to Circuit Equations
Companion Network – MNA templates
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Application of NR to Circuit Equations
Companion Network – MNA templates
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Modeling a MOSFET (MOS Level 1, linear regime)
d
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Modeling a MOSFET (MOS Level 1, linear regime)
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DC Analysis Flow Diagram
For each state variable in the system
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Implications Device model equations must be continuous
with continuous derivatives (not all models do this - - be sure models are decent - beware of user-supplied models)
Watch out for floating nodes (If a node becomes disconnected, then J(x) is singular)
Give good initial guess for x(0)
Most model computations produce errors in function values and derivatives. Want to have convergence criteria || x(k+1) - x(k) || < such that > than model errors.
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April 19, 2023 35 courtesy Alessandra Nardi UCB
Outline Nonlinear problems Iterative Methods Newton’s Method
Derivation of Newton Quadratic Convergence Examples Convergence Testing
Multidimensonal Newton Method Basic Algorithm Quadratic convergence Application to circuits
Improve Convergence Limiting Schemes
Direction Corrupting Non corrupting (Damped Newton)
Continuation Schemes Source stepping
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Improving convergence Improve Models (80% of problems) Improve Algorithms (20% of
problems)
Focus on new algorithms:Limiting SchemesContinuations Schemes
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Improve Convergence
Limiting Schemes Direction Corrupting Non corrupting (Damped Newton)
Globally Convergent if Jacobian is Nonsingular
Difficulty with Singular Jacobians
Continuation Schemes Source stepping
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At a local minimum, 0f
x
LocalMinimum
Multidimensional Case: is singularFJ x
Multidimensional Newton Method
Convergence Problems – Local Minimum
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f(x)
X0x
1x
Must Somehow Limit the changes in X
Multidimensional Newton Method
Convergence Problems – Nearly singular
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Multidimensional Newton Method
f(x)
X0x 1x
Must Somehow Limit the changes in X
Convergence Problems - Overflow
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0 Initial Guess,= 0x k
Repeat {
1 1Solve for k k k kFJ x x F x x
} Until 1 1 , small enoughkkx F x 1k k
Compute ,k kFF x J x
Newton Algorithm for Solving 0F x
1 1limitedk k kx x x
Newton Method with Limiting
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1limited k
ix
• NonCorrupting
1 1 if k ki ix x
1 otherwisekisign x
1kx 1limited kx • Direction Corrupting
1 1limited k kx x
1min 1,
kx
1kx 1limited kx
Heuristics, No Guarantee of Global Convergence
Newton Method with Limiting
Limiting Methods
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General Damping Scheme
Key Idea: Line Search
21
2Pick to minimize k k k kF x x
Method Performs a one-dimensional search in Newton Direction
21 1 1
2
Tk k k k k k k k kF x x F x x F x x
1 1k k k kx x x
1 1Solve for k k k kFJ x x F x x
Newton Method with LimitingDamped Newton Scheme
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If
1) Inverse is boundedkFa J x
) Derivative is Lipschitz ContF Fb J x J y x y
Then
There exists a set of ' 0,1 such thatk s
1 1 with <1k k k k kF x F x x F x
Every Step reduces F-- Global Convergence!
Newton Method with Limiting
Damped Newton – Convergence Theorem
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April 19, 2023 45 courtesy Alessandra Nardi UCB
0 Initial Guess,= 0x k Repeat {
1 1Solve for k k k kFJ x x F x x
} Until 1 1 , small enoughkkx F x
1k k
Compute ,k kFF x J x
1Find 0,1 such that is minimi e z dk k kk F x x 1 1k k k kx x x
Newton Method with Limiting
Damped Newton – Nested Iteration
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X0x1x
2x1Dx
Damped Newton Methods “push” iterates to local minimumsFinds the points where Jacobian is Singular
Newton Method with LimitingDamped Newton – Singular Jacobian Problem
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Solve , 0 where:F x
a) 0 ,0 0 is easy to solveF x
b) 1 ,1F x F x
c) is sufficiently smoothx
Starts the continuation
Ends the continuation
Hard to insure!
Newton with Continuation schemes
Newton converges given a close initial guess Idea: Generate a sequence of problems, s.t.
a problem is a good initial guess for the following one
0)(xF
Basic Concepts - General setting
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Solve 0 ,0 , 0prevF x x x
While 1 {
}
=0.01,
0prevx x
Try to Solve , 0 with NewtonF x
If Newton Converged
, 2,prevx x Else
,1
2 prev
Newton with Continuation schemes
Basic Concepts – Template Algorithm
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Newton with Continuation schemes
Basic Concepts – Source Stepping Example
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+-Vs
R
Diode
1, 0diode sf v i v v V
R
v
, 1Not dependent!diodef v i v
v v R
Source Stepping Does Not Alter Jacobian
Newton with Continuation schemes
Basic Concepts – Source Stepping Example
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Transient Analysis Flow Diagram Predict values of variables at tl
Replace C and L with resistive elements via integration formula
Replace nonlinear elements with G and indep. sources via NR
Assemble linear circuit equations
Solve linear circuit equations
Did NR converge?
Test solution accuracy
Save solution if acceptable
Select new t and compute new integration formula coeff.
Done?
YES
NO
NO