Loading in 2 Seconds...
Loading in 2 Seconds...
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
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
courtesy Alessandra Nardi UCB
Solve g(V)=I equivalent to solve f(V)=0
Hard to find analytical solution for f(x)=0
Solve iteratively
courtesy Alessandra Nardi UCB
courtesy Alessandra Nardi UCB
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’
courtesy Alessandra Nardi UCB
courtesy Alessandra Nardi UCB
Define iteration
Dok = 0 to ….
until convergence
courtesy Alessandra Nardi UCB
truncates Taylor series
NewtonRaphson Method – Convergence
But
by Newton
definition
courtesy Alessandra Nardi UCB
NewtonRaphson Method – Convergence
Subtracting
Dividing through
Convergence is quadratic
courtesy Alessandra Nardi UCB
NewtonRaphson Method – Convergence
Local Convergence Theorem
If
Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic)
courtesy Alessandra Nardi UCB
NewtonRaphson Method – Convergence
Example 2
Note : not bounded
away from zero
Convergence is linear
courtesy Alessandra Nardi UCB
NewtonRaphson Method – Convergence
courtesy Alessandra Nardi UCB
NewtonRaphson Method – Convergence
Local Convergence
Convergence Depends on a Good Initial Guess
f(x)
X
courtesy Alessandra Nardi UCB
NewtonRaphson Method – Convergence
Local Convergence
Convergence Depends on a Good Initial Guess
courtesy Alessandra Nardi UCB
Nonlinear Problems– Multidimensional Example
Nodal Analysis
+

+
+


Nonlinear
Resistors
Two coupled nonlinear equations in two unknowns
courtesy Alessandra Nardi UCB
courtesy Alessandra Nardi UCB
Multidimensional Newton Method
courtesy Alessandra Nardi UCB
Multidimensional Newton Method
Computational Aspects
Each iteration requires:
courtesy Alessandra Nardi UCB
Multidimensional Newton Method
Convergence
Local Convergence Theorem
If
Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic)
courtesy Alessandra Nardi UCB
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
courtesy Alessandra Nardi UCB
Companion Network
courtesy Alessandra Nardi UCB
Companion Network – MNA templates
Note: G0 and Id depend on the iteration count k
G0=G0(k) and Id=Id(k)
courtesy Alessandra Nardi UCB
Companion Network – MNA templates
courtesy Alessandra Nardi UCB
courtesy Alessandra Nardi UCB
courtesy Alessandra Nardi UCB
courtesy Alessandra Nardi UCB
Focus on new algorithms:
Limiting Schemes
Continuations Schemes
courtesy Alessandra Nardi UCB
courtesy Alessandra Nardi UCB
Multidimensional Newton Method
Convergence Problems – Local Minimum
Local
Minimum
courtesy Alessandra Nardi UCB
Multidimensional Newton Method
Convergence Problems – Nearly singular
f(x)
Must Somehow Limit the changes in X
courtesy Alessandra Nardi UCB
Convergence Problems  Overflow
f(x)
X
Must Somehow Limit the changes in X
courtesy Alessandra Nardi UCB
courtesy Alessandra Nardi UCB
Limiting Methods
Heuristics, No Guarantee of Global Convergence
courtesy Alessandra Nardi UCB
Damped Newton Scheme
General Damping Scheme
Key Idea: Line Search
Method Performs a onedimensional search in Newton Direction
courtesy Alessandra Nardi UCB
Damped Newton – Convergence Theorem
If
Then
Every Step reduces F Global Convergence!
courtesy Alessandra Nardi UCB
Damped Newton – Singular Jacobian Problem
X
Damped Newton Methods “push” iterates to local minimums
Finds the points where Jacobian is Singular
courtesy Alessandra Nardi UCB
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!
courtesy Alessandra Nardi UCB
Newton with Continuation schemes
Basic Concepts – Source Stepping Example
courtesy Alessandra Nardi UCB

Newton with Continuation schemes
Basic Concepts – Source Stepping Example
R
Diode
Vs
Source Stepping Does Not Alter Jacobian
courtesy Alessandra Nardi UCB
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?
courtesy Alessandra Nardi UCB