CSE 245: Computer Aided Circuit Simulation and Verification

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CSE 245: Computer Aided Circuit Simulation and Verification. Fall 2004, Nov Nonlinear Equation. Outline. Nonlinear problems Iterative Methods Newton ’ s Method Derivation of Newton Quadratic Convergence Examples Convergence Testing Multidimensonal Newton Method Basic Algorithm

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CSE 245: Computer Aided Circuit Simulation and Verification

Fall 2004, Nov

Nonlinear Equation

Outline
• Nonlinear problems
• Iterative Methods
• Newton’s Method
• Derivation of Newton
• Examples
• Convergence Testing
• Multidimensonal Newton Method
• Basic Algorithm
• Application to circuits
• Improve Convergence
• Limiting Schemes
• Direction Corrupting
• Non corrupting (Damped Newton)
• Continuation Schemes
• Source stepping

courtesy Alessandra Nardi UCB

1

I1

Id

Ir

0

Nonlinear Problems - Example

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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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)
• When does it converge to correct solution ?
• What is the convergence rate ?

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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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Newton-Raphson Method – Graphical View

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Newton-Raphson Method – Algorithm
• An iteration {x(k)} is said to converge with order q if there exists a vector norm such that for each k  N:

Define iteration

Dok = 0 to ….

until convergence

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Mean Value theorem

truncates Taylor series

Newton-Raphson Method – Convergence

But

by Newton

definition

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Newton-Raphson Method – Convergence

Subtracting

Dividing through

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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

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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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f(x)

X

Newton-Raphson Method – Convergence

Convergence Check

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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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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
• Examples
• Convergence Testing
• Multidimensonal Newton Method
• Basic Algorithm
• 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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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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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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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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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
• Examples
• Convergence Testing
• Multidimensonal Newton Method
• Basic Algorithm
• 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 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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X

Multidimensional Newton Method

Convergence Problems – Nearly singular

f(x)

Must Somehow Limit the changes in X

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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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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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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

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

Basic Concepts - General setting

 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

Diode

Vs

Source Stepping Does Not Alter Jacobian

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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

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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