1. CSE 245: Computer Aided Circuit Simulation and Verification
Nonlinear Equation

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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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Nonlinear Problems - Example 1 Ir I1 Id
Need to Solve
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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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5. 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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6. 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’
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7. Newton-Raphson Method – Graphical View
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8. 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:
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Newton-Raphson Method – Convergence
Mean Value theorem
truncates Taylor series
But
by Newton
definition
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Newton-Raphson Method – Convergence
Subtracting
Dividing through
Convergence is quadratic
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Newton-Raphson Method – Convergence
Local Convergence Theorem If
Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic)
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Newton-Raphson Method – Convergence
Example 1
Convergence is quadratic
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Newton-Raphson Method – Convergence
Example 2
Note : not bounded
away from zero
Convergence is linear
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Newton-Raphson Method – Convergence
Example 1,2
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Newton-Raphson Method – Convergence
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Newton-Raphson Method – Convergence
Convergence Check
f(x) X
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Newton-Raphson Method – Convergence
Convergence Check X
f(x)
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Newton-Raphson Method – Convergence
demo2
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Newton-Raphson Method – Convergence
Local Convergence
Convergence Depends on a Good Initial Guess
f(x) X
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Newton-Raphson Method – Convergence
Local Convergence
Convergence Depends on a Good Initial Guess
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21. Nonlinear Problems – Multidimensional Example
Nodal Analysis + - + + - -
Nonlinear
Resistors
Two coupled nonlinear equations in two unknowns
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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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Multidimensional Newton Method
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Multidimensional Newton Method
Computational Aspects
Each iteration requires:
Evaluation of F(xk)
Computation of J(xk)
Solution of a linear system of algebraic equations whose coefficient matrix is J(xk) and whose RHS is -F(xk)
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Multidimensional Newton Method
Algorithm
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Multidimensional Newton Method
Convergence
Local Convergence Theorem If
Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic)
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27. Application of NR to Circuit Equations
Companion Network
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.
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28. Application of NR to Circuit Equations
Companion Network
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
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29. Application of NR to Circuit Equations
Companion Network – MNA templates
Note: G0 and Id depend on the iteration count k
 G0=G0(k) and Id=Id(k)
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30. Application of NR to Circuit Equations
Companion Network – MNA templates
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31. Modeling a MOSFET (MOS Level 1, linear regime)
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32. Modeling a MOSFET (MOS Level 1, linear regime)
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33. 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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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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36. Improving convergence
Improve Models (80% of problems)
Improve Algorithms (20% of problems)
Focus on new algorithms:
Limiting Schemes
Continuations 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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Multidimensional Newton Method
Convergence Problems – Local Minimum
Local
Minimum
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Multidimensional Newton Method
Convergence Problems – Nearly singular
f(x) X
Must Somehow Limit the changes in X
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40. Multidimensional Newton Method
Convergence Problems - Overflow
f(x) X
Must Somehow Limit the changes in X
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Newton Method with Limiting
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Newton Method with Limiting
Limiting Methods
Direction Corrupting
NonCorrupting
Heuristics, No Guarantee of Global Convergence
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43. Newton Method with Limiting
Damped Newton Scheme
General Damping Scheme
Key Idea: Line Search
Method Performs a one-dimensional search in Newton Direction
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Newton Method with Limiting
Damped Newton – Convergence Theorem If
Then
Every Step reduces F-- Global Convergence!
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Newton Method with Limiting
Damped Newton – Nested Iteration
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46. Newton Method with Limiting
Damped Newton – Singular Jacobian Problem X
Damped Newton Methods “push” iterates to local minimums
Finds the points where Jacobian is Singular
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Newton with Continuation schemes
Basic Concepts - General setting
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
 Starts the continuation
Ends the continuation
Hard to insure!
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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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Newton with Continuation schemes
Basic Concepts – Source Stepping Example R Vs + -
Diode
Source Stepping Does Not Alter Jacobian
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51. 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 NO
Did NR converge?
YES
Test solution accuracy
Save solution if acceptable
Select new Dt and compute new integration formula coeff. NO
Done?
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