1 EE 616 Computer Aided Analysis of Electronic Networks Lecture 5 Instructor: Dr. J. A. Starzyk, Professor School of EECS Ohio University Athens, OH, /19/2005 Note: materials in this lecture are from the notes of EE219A UC-berkeley cad.eecs.berkeley.edu/~nardi/EE219A/contents.html

2 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

3 Need to Solve 0 1 IrIr I1I1 IdId DC Analysis of Nonlinear Circuits - Example

4 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

5 Nonlinear Equations – Iterative Methods Start from an initial value x 0 Generate a sequence of iterate x n-1, x n, x n+1 which hopefully converges to the solution x* Iterates are generated according to an iteration function F: x n+1 =F(x n ) Ask When does it converge to correct solution ? What is the convergence rate ?

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’

7 Newton-Raphson Method – Graphical View

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:

9 Mean Value theorem truncates Taylor series But by Newton definition Newton-Raphson Method – Convergence

10 Subtracting Convergence is quadratic Dividing Newton-Raphson Method – Convergence

11 Local Convergence Theorem If Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic) Newton-Raphson Method – Convergence

12 Newton-Raphson Method – Convergence

13 X f(x) Newton-Raphson Method – Convergence Check False convergence since increment in f small but with large increment in X

14 X f(x) Newton-Raphson Method – Convergence Check False convergence since increment in x small but with large increment in f

15 Newton-Raphson Method – Convergence

16 Convergence Depends on a Good Initial Guess X f(x) Newton-Raphson Method – Local Convergence

17 Convergence Depends on a Good Initial Guess Newton-Raphson Method – Local Convergence

18 Nodal Analysis Nonlinear Resistors Two coupled nonlinear equations in two unknowns Nonlinear Problems – Multidimensional Example

19 Multidimensional Newton Method

20 Each iteration requires: 1. Evaluation of F(x k ) 2. Computation of J(x k ) 3. Solution of a linear system of algebraic equations whose coefficient matrix is J(x k ) and whose RHS is -F(x k ) Multidimensional Newton Method – Computational Aspects

21 Multidimensional Newton Method – Algorithm

22 If Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic) Local Convergence Theorem Multidimensional Newton Method – Convergence

23 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.

24 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 Application of NR to Circuit Equations Companion Network

25 Note: G 0 and I d depend on the iteration count k  G 0 =G 0 (k) and I d =I d (k) Application of NR to Circuit Equations Companion Network – MNA templates

26 Application of NR to Circuit Equations Companion Network – MNA templates

27 Modeling a MOSFET (MOS Level 1, linear regime) d

28 Modeling a MOSFET (MOS Level 1, linear regime) g ds

29 DC Analysis Flow Diagram For each state variable in the system

30 Implications Device model equations must be continuous with continuous derivatives and derivative calculation must be accurate derivative of function 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) || than model errors.

31 Summary Nonlinear problems Iterative Methods Newton’s Method – Derivation of Newton – Quadratic Convergence – Examples – Convergence Testing Multidimensional Newton Method – Basic Algorithm – Quadratic convergence – Application to circuits

32 Improving convergence Improve Models (80% of problems) Improve Algorithms (20% of problems) Focus on new algorithms: Limiting Schemes Continuations Schemes

33 Outline Limiting Schemes – Direction Corrupting – Non corrupting (Damped Newton) Globally Convergent if Jacobian is Nonsingular Difficulty with Singular Jacobians Continuation Schemes – Source stepping – More General Continuation Scheme – Improving Efficiency Better first guess for each continuation step

34 Local Minimum Multidimensional Newton Method Convergence Problems – Local Minimum

35 f(x) X Must Somehow Limit the changes in X Multidimensional Newton Method Convergence Problems – Nearly singular

36 Multidimensional Newton Method Convergence Problems - Overflow f(x) X Must Somehow Limit the changes in f(x)

37 Newton Method with Limiting

38 NonCorrupting Direction Corrupting Heuristics, No Guarantee of Global Convergence Newton Method with Limiting Limiting Methods Scales proportionally In all dimensions Scales differently different dimensions

39 General Damping Scheme Key Idea: Line Search Method Performs a one-dimensional search in Newton Direction Newton Method with Limiting Damped Newton Scheme

40 If Then Every Step reduces F-- Global Convergence! Newton Method with Limiting Damped Newton – Convergence Theorem

41 X Damped Newton Methods “push” iterates to local minimums Finds the points where Jacobian is Singular Newton Method with Limiting Damped Newton – Singular Jacobian Problem

42  Starts the continuation  Ends the continuation  Hard to insure! Newton with Continuation schemes Basic Concepts - General setting Newton converges given a close initial guess  Idea: Generate a sequence of problems, such that a problem is a good initial guess for the following one

43 Newton with Continuation schemes Basic Concepts – Template Algorithm

44 Newton with Continuation schemes Basic Concepts – Source Stepping Example

45 Diode +-+- VsVs R Source Stepping Does Not Alter Jacobian Newton with Continuation schemes Basic Concepts – Source Stepping Example

46 Observations Problem is easy to solve and Jacobian definitely nonsingular. Back to the original problem and original Jacobian Newton with Continuation schemes Jacobian Altering Scheme (1),1)

47 Summary Newton’s Method works fine: – given a close enough initial guess In case Newton does not converge: – Limiting Schemes Direction Corrupting Non corrupting (Damped Newton) – Globally Convergent if Jacobian is Nonsingular – Difficulty with Singular Jacobians – Continuation Schemes Source stepping More General Continuation Scheme Improving Efficiency – Better first guess for each continuation step