1. CSE 245: Computer Aided Circuit Simulation and Verification Fall 2004, Nov Nonlinear Equation

2. June 9, 20152courtesy Alessandra Nardi UCB 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

3. June 9, 20153courtesy Alessandra Nardi UCB Need to Solve Nonlinear Problems - Example 0 1 IrIr I1I1 IdId

4. June 9, 20154courtesy Alessandra Nardi UCB 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. June 9, 20155courtesy Alessandra Nardi UCB 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. June 9, 20156courtesy Alessandra Nardi UCB 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. June 9, 20157courtesy Alessandra Nardi UCB Newton-Raphson Method – Graphical View

8. June 9, 20158courtesy Alessandra Nardi UCB 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. June 9, 20159courtesy Alessandra Nardi UCB Mean Value theorem truncates Taylor series But by Newton definition Newton-Raphson Method – Convergence

10. June 9, 201510courtesy Alessandra Nardi UCB Subtracting Convergence is quadratic Dividing through Newton-Raphson Method – Convergence

11. June 9, 201511courtesy Alessandra Nardi UCB Local Convergence Theorem If Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic) Newton-Raphson Method – Convergence

12. June 9, 201512courtesy Alessandra Nardi UCB Convergence is quadratic Newton-Raphson Method – Convergence Example 1

13. June 9, 201513courtesy Alessandra Nardi UCB Convergence is linear Note : not bounded away from zero Newton-Raphson Method – Convergence Example 2

14. June 9, 201514courtesy Alessandra Nardi UCB Newton-Raphson Method – Convergence Example 1,2

15. June 9, 201515courtesy Alessandra Nardi UCB Newton-Raphson Method – Convergence

16. June 9, 201516courtesy Alessandra Nardi UCB X f(x) Newton-Raphson Method – Convergence Convergence Check

17. June 9, 201517courtesy Alessandra Nardi UCB X f(x) Newton-Raphson Method – Convergence Convergence Check

18. June 9, 201518courtesy Alessandra Nardi UCB demo2 Newton-Raphson Method – Convergence

19. June 9, 201519courtesy Alessandra Nardi UCB X f(x) Convergence Depends on a Good Initial Guess Newton-Raphson Method – Convergence Local Convergence

20. June 9, 201520courtesy Alessandra Nardi UCB Convergence Depends on a Good Initial Guess Newton-Raphson Method – Convergence Local Convergence

21. June 9, 201521courtesy Alessandra Nardi UCB Nodal Analysis + - +- + - Nonlinear Resistors Two coupled nonlinear equations in two unknowns Nonlinear Problems – Multidimensional Example

22. June 9, 201522courtesy Alessandra Nardi UCB 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

23. June 9, 201523courtesy Alessandra Nardi UCB Multidimensional Newton Method

24. June 9, 201524courtesy Alessandra Nardi UCB Multidimensional Newton Method 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 ) Computational Aspects

25. June 9, 201525courtesy Alessandra Nardi UCB Multidimensional Newton Method Algorithm

26. June 9, 201526courtesy Alessandra Nardi UCB If Then Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic) Multidimensional Newton Method Local Convergence Theorem Convergence

27. June 9, 201527courtesy Alessandra Nardi UCB 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

28. June 9, 201528courtesy Alessandra Nardi UCB 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

29. June 9, 201529courtesy Alessandra Nardi UCB 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

30. June 9, 201530courtesy Alessandra Nardi UCB Application of NR to Circuit Equations Companion Network – MNA templates

31. June 9, 201531courtesy Alessandra Nardi UCB Modeling a MOSFET (MOS Level 1, linear regime) d

32. June 9, 201532courtesy Alessandra Nardi UCB Modeling a MOSFET (MOS Level 1, linear regime)

33. June 9, 201533courtesy Alessandra Nardi UCB DC Analysis Flow Diagram For each state variable in the system

34. June 9, 201534courtesy Alessandra Nardi UCB 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) || than model errors.

35. June 9, 201535courtesy Alessandra Nardi UCB 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

36. June 9, 201536courtesy Alessandra Nardi UCB Improving convergence  Improve Models (80% of problems)  Improve Algorithms (20% of problems) Focus on new algorithms: Limiting Schemes Continuations Schemes

37. June 9, 201537courtesy Alessandra Nardi UCB Improve Convergence  Limiting Schemes Direction Corrupting Non corrupting (Damped Newton)  Globally Convergent if Jacobian is Nonsingular  Difficulty with Singular Jacobians  Continuation Schemes Source stepping

38. June 9, 201538courtesy Alessandra Nardi UCB Local Minimum Multidimensional Newton Method Convergence Problems – Local Minimum

39. June 9, 201539courtesy Alessandra Nardi UCB f(x) X Must Somehow Limit the changes in X Multidimensional Newton Method Convergence Problems – Nearly singular

40. June 9, 201540courtesy Alessandra Nardi UCB Multidimensional Newton Method f(x) X Must Somehow Limit the changes in X Convergence Problems - Overflow

41. June 9, 201541courtesy Alessandra Nardi UCB Newton Method with Limiting

42. June 9, 201542courtesy Alessandra Nardi UCB NonCorrupting Direction Corrupting Heuristics, No Guarantee of Global Convergence Newton Method with Limiting Limiting Methods

43. June 9, 201543courtesy Alessandra Nardi UCB General Damping Scheme Key Idea: Line Search Method Performs a one-dimensional search in Newton Direction Newton Method with Limiting Damped Newton Scheme

44. June 9, 201544courtesy Alessandra Nardi UCB If Then Every Step reduces F-- Global Convergence! Newton Method with Limiting Damped Newton – Convergence Theorem

45. June 9, 201545courtesy Alessandra Nardi UCB Newton Method with Limiting Damped Newton – Nested Iteration

46. June 9, 201546courtesy Alessandra Nardi UCB 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

47. June 9, 201547courtesy Alessandra Nardi UCB  Starts the continuation  Ends the continuation  Hard to insure! 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

48. June 9, 201548courtesy Alessandra Nardi UCB Newton with Continuation schemes Basic Concepts – Template Algorithm

49. June 9, 201549courtesy Alessandra Nardi UCB Newton with Continuation schemes Basic Concepts – Source Stepping Example

50. June 9, 201550courtesy Alessandra Nardi UCB +-+- VsVs R Diode Source Stepping Does Not Alter Jacobian Newton with Continuation schemes Basic Concepts – Source Stepping Example

51. June 9, 201551courtesy Alessandra Nardi UCB Transient Analysis Flow Diagram Predict values of variables at t l 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 Did NR converge? Test solution accuracy Save solution if acceptable Select new  t and compute new integration formula coeff. Done? YES NO