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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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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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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Define iteration
Dok = 0 to ….
until convergence
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truncates Taylor series
NewtonRaphson Method – Convergence
But
by Newton
definition
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NewtonRaphson Method – Convergence
Subtracting
Dividing through
Convergence is quadratic
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NewtonRaphson 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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NewtonRaphson Method – Convergence
Example 1
Convergence is quadratic
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NewtonRaphson Method – Convergence
Example 2
Note : not bounded
away from zero
Convergence is linear
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NewtonRaphson Method – Convergence
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NewtonRaphson Method – Convergence
Local Convergence
Convergence Depends on a Good Initial Guess
f(x)
X
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NewtonRaphson Method – Convergence
Local Convergence
Convergence Depends on a Good Initial Guess
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Nonlinear Problems– Multidimensional Example
Nodal Analysis
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Nonlinear
Resistors
Two coupled nonlinear equations in two unknowns
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Multidimensional Newton Method
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Multidimensional Newton Method
Computational Aspects
Each iteration requires:
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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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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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Companion Network
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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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Companion Network – MNA templates
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Focus on new algorithms:
Limiting Schemes
Continuations Schemes
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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)
Must Somehow Limit the changes in X
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Convergence Problems  Overflow
f(x)
X
Must Somehow Limit the changes in X
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Limiting Methods
Heuristics, No Guarantee of Global Convergence
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Damped Newton Scheme
General Damping Scheme
Key Idea: Line Search
Method Performs a onedimensional search in Newton Direction
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Damped Newton – Convergence Theorem
If
Then
Every Step reduces F Global Convergence!
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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 – 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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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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