CSE 245: Computer Aided Circuit Simulation and Verification

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

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 • Improve Convergence • Limiting Schemes • Direction Corrupting • Non corrupting (Damped Newton) • Continuation Schemes • Source stepping courtesy Alessandra Nardi UCB

3. 1 I1 Id Ir 0 Nonlinear Problems - Example Need to Solve courtesy Alessandra Nardi UCB

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 courtesy Alessandra Nardi UCB

5. 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) • Ask • When does it converge to correct solution ? • What is the convergence rate ? courtesy Alessandra Nardi UCB

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’ courtesy Alessandra Nardi UCB

7. Newton-Raphson Method – Graphical View courtesy Alessandra Nardi UCB

8. Newton-Raphson Method – Algorithm • 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: Define iteration Dok = 0 to …. until convergence courtesy Alessandra Nardi UCB

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

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

11. Newton-Raphson 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

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

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

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

15. Newton-Raphson Method – Convergence courtesy Alessandra Nardi UCB

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

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

18. Newton-Raphson Method – Convergence demo2 courtesy Alessandra Nardi UCB

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

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

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

22. 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 courtesy Alessandra Nardi UCB

23. Multidimensional Newton Method courtesy Alessandra Nardi UCB

24. 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) courtesy Alessandra Nardi UCB

25. Multidimensional Newton Method Algorithm courtesy Alessandra Nardi UCB

26. 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

27. 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 courtesy Alessandra Nardi UCB

28. 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 courtesy Alessandra Nardi UCB

29. 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) courtesy Alessandra Nardi UCB

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

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

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

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

34. 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. courtesy Alessandra Nardi UCB

35. 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 courtesy Alessandra Nardi UCB

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

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

38. Multidimensional Newton Method Convergence Problems – Local Minimum Local Minimum courtesy Alessandra Nardi UCB

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

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

41. Newton Method with Limiting courtesy Alessandra Nardi UCB

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

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

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

45. Newton Method with Limiting Damped Newton – Nested Iteration courtesy Alessandra Nardi UCB

46. 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 courtesy Alessandra Nardi UCB

47. 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

48. Newton with Continuation schemes Basic Concepts – Template Algorithm courtesy Alessandra Nardi UCB

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

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