Nonlinear Equation February 24 2015 2 courtesy Alessandra Nardi UCB Outline Nonlinear problems Iterative Methods Newton s Method ID: 1020931
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1. CSE 245: Computer Aided Circuit Simulation and Verification Nonlinear Equation
2. February 24, 20152courtesy Alessandra Nardi UCB OutlineNonlinear problemsIterative MethodsNewton’s MethodDerivation of NewtonQuadratic ConvergenceExamplesConvergence TestingMultidimensonal Newton MethodBasic Algorithm Quadratic convergenceApplication to circuitsImprove Convergence Limiting SchemesDirection CorruptingNon corrupting (Damped Newton)Continuation SchemesSource stepping
3. February 24, 20153courtesy Alessandra Nardi UCB Need to SolveNonlinear Problems - Example01IrI1Id
4. February 24, 20154courtesy Alessandra Nardi UCB Nonlinear EquationsGiven g(V)=IIt can be expressed as: f(V)=g(V)-I Solve g(V)=I equivalent to solve f(V)=0Hard to find analytical solution for f(x)=0Solve iteratively
5. February 24, 20155courtesy Alessandra Nardi UCB Nonlinear Equations – Iterative MethodsStart from an initial value x0Generate 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 ?
6. February 24, 20156courtesy Alessandra Nardi UCB Newton-Raphson (NR) MethodConsists 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. February 24, 20157courtesy Alessandra Nardi UCB Newton-Raphson Method – Graphical View
8. February 24, 20158courtesy Alessandra Nardi UCB Newton-Raphson Method – AlgorithmDefine iterationDo k = 0 to ….until convergenceHow 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. February 24, 20159courtesy Alessandra Nardi UCB Mean Value theoremtruncates Taylor seriesButby NewtondefinitionNewton-Raphson Method – Convergence
10. February 24, 201510courtesy Alessandra Nardi UCB Subtracting Convergence is quadraticDividing through Newton-Raphson Method – Convergence-=[df()/df()/d Thus, we have |-|where K[df()/df()/d
11. February 24, 201511courtesy Alessandra Nardi UCB Local Convergence TheoremIfThen Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic)Newton-Raphson Method – Convergence
12. February 24, 201512courtesy Alessandra Nardi UCB Convergence is quadraticNewton-Raphson Method – ConvergenceExample 1
13. February 24, 201513courtesy Alessandra Nardi UCB Convergence is linearNote : not boundedaway from zeroNewton-Raphson Method – ConvergenceExample 2
14. February 24, 201514courtesy Alessandra Nardi UCB Newton-Raphson Method – ConvergenceExample 1,2
15. February 24, 201515courtesy Alessandra Nardi UCB Newton-Raphson Method – Convergence
16. February 24, 201516courtesy Alessandra Nardi UCB Xf(x)Newton-Raphson Method – ConvergenceConvergence Check
17. February 24, 201517courtesy Alessandra Nardi UCB Xf(x)Newton-Raphson Method – ConvergenceConvergence Check
18. February 24, 201518courtesy Alessandra Nardi UCB demo2Newton-Raphson Method – Convergence
19. February 24, 201519courtesy Alessandra Nardi UCB Xf(x)Convergence Depends on a Good Initial GuessNewton-Raphson Method – ConvergenceLocal Convergence
20. February 24, 201520courtesy Alessandra Nardi UCB Convergence Depends on a Good Initial GuessNewton-Raphson Method – ConvergenceLocal Convergence
21. February 24, 201521courtesy Alessandra Nardi UCB Nodal Analysis+-+-+-NonlinearResistorsTwo coupled nonlinear equations in two unknownsNonlinear Problems – Multidimensional Example
22. February 24, 201522courtesy Alessandra Nardi UCB OutlineNonlinear problemsIterative MethodsNewton’s MethodDerivation of NewtonQuadratic ConvergenceExamplesConvergence TestingMultidimensonal Newton MethodBasic Algorithm Quadratic convergenceApplication to circuitsImprove Convergence Limiting SchemesDirection CorruptingNon corrupting (Damped Newton)Continuation SchemesSource stepping
23. February 24, 201523courtesy Alessandra Nardi UCB Multidimensional Newton Method
24. February 24, 201524courtesy Alessandra Nardi UCB Multidimensional Newton MethodEach 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)Computational Aspects
25. February 24, 201525courtesy Alessandra Nardi UCB Multidimensional Newton MethodAlgorithm
26. February 24, 201526courtesy Alessandra Nardi UCB IfThen Newton’s method converges given a sufficiently close initial guess (and convergence is quadratic)Multidimensional Newton MethodLocal Convergence TheoremConvergence
27. February 24, 201527courtesy Alessandra Nardi UCB Application of NR to Circuit EquationsApplying 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 kNonlinear 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. February 24, 201528courtesy Alessandra Nardi UCB Application of NR to Circuit EquationsGeneral 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 unmodifiedAfter 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 eqnsCompanion Network
29. February 24, 201529courtesy Alessandra Nardi UCB Note: G0 and Id depend on the iteration count k G0=G0(k) and Id=Id(k)Application of NR to Circuit EquationsCompanion Network – MNA templates
30. February 24, 201530courtesy Alessandra Nardi UCB Application of NR to Circuit EquationsCompanion Network – MNA templates
31. February 24, 201531courtesy Alessandra Nardi UCB Application of NR to Circuit EquationsTrapezoidal Integration (assuming u=0)x(t+h)=x(t)+h/2(f(t+h)+f(t))x(t+h)=x(t)-h/2(C(t+h)-1G(t+h)x(t+h)+C(t)-1G(t)x(t))Thus, we have(C(t+h)+G(t+h))x(t+h)= (C(t+h)-C(t+h)C(t)-1G(t))x(t)For each capacitor, c, the equivalent circuit hascurrent source ieq= c(t+h)vc(t) +c(t+h)/c(t) ic(t) in parallel with conductance geq= c(t+h)
32. February 24, 201532courtesy Alessandra Nardi UCB Modeling a MOSFET (MOS Level 1, linear regime)d
33. February 24, 201533courtesy Alessandra Nardi UCB Modeling a MOSFET (MOS Level 1, linear regime)
34. February 24, 201534courtesy Alessandra Nardi UCB DC Analysis Flow DiagramFor each state variable in the system
35. February 24, 201535courtesy Alessandra Nardi UCB ImplicationsDevice 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.
36. February 24, 201536courtesy Alessandra Nardi UCB OutlineNonlinear problemsIterative MethodsNewton’s MethodDerivation of NewtonQuadratic ConvergenceExamplesConvergence TestingMultidimensonal Newton MethodBasic Algorithm Quadratic convergenceApplication to circuitsImprove Convergence Limiting SchemesDirection CorruptingNon corrupting (Damped Newton)Continuation SchemesSource stepping
37. February 24, 201537courtesy Alessandra Nardi UCB Improving convergenceImprove Models (80% of problems)Improve Algorithms (20% of problems) Focus on new algorithms: Limiting Schemes Continuations Schemes
38. February 24, 201538courtesy Alessandra Nardi UCB Improve ConvergenceLimiting SchemesDirection CorruptingNon corrupting (Damped Newton)Globally Convergent if Jacobian is NonsingularDifficulty with Singular JacobiansContinuation SchemesSource stepping
39. February 24, 201539courtesy Alessandra Nardi UCB LocalMinimumMultidimensional Newton MethodConvergence Problems – Local Minimum
40. February 24, 201540courtesy Alessandra Nardi UCB f(x)XMust Somehow Limit the changes in XMultidimensional Newton MethodConvergence Problems – Nearly singular
41. February 24, 201541courtesy Alessandra Nardi UCB Multidimensional Newton Methodf(x)XMust Somehow Limit the changes in XConvergence Problems - Overflow
42. February 24, 201542courtesy Alessandra Nardi UCB Newton Method with Limiting
43. February 24, 201543courtesy Alessandra Nardi UCB NonCorrupting Direction CorruptingHeuristics, No Guarantee of Global ConvergenceNewton Method with LimitingLimiting Methods
44. February 24, 201544courtesy Alessandra Nardi UCB General Damping SchemeKey Idea: Line SearchMethod Performs a one-dimensional search in Newton DirectionNewton Method with LimitingDamped Newton Scheme
45. February 24, 201545courtesy Alessandra Nardi UCB IfThenEvery Step reduces F-- Global Convergence!Newton Method with LimitingDamped Newton – Convergence Theorem
46. February 24, 201546courtesy Alessandra Nardi UCB Newton Method with LimitingDamped Newton – Nested Iteration
47. February 24, 201547courtesy Alessandra Nardi UCB XDamped Newton Methods “push” iterates to local minimumsFinds the points where Jacobian is SingularNewton Method with LimitingDamped Newton – Singular Jacobian Problem
48. February 24, 201548courtesy Alessandra Nardi UCB Starts the continuationEnds the continuationHard 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 oneBasic Concepts - General setting
49. February 24, 201549courtesy Alessandra Nardi UCB Newton with Continuation schemes Basic Concepts – Template Algorithm
50. February 24, 201550courtesy Alessandra Nardi UCB Newton with Continuation schemes Basic Concepts – Source Stepping Example
51. February 24, 201551courtesy Alessandra Nardi UCB +-VsRDiodeSource Stepping Does Not Alter JacobianNewton with Continuation schemes Basic Concepts – Source Stepping Example
52. February 24, 201552courtesy Alessandra Nardi UCB Transient Analysis Flow Diagram Predict values of variables at tlReplace C and L with resistive elements via integration formulaReplace nonlinear elements with G and indep. sources via NRAssemble linear circuit equationsSolve linear circuit equationsDid NR converge?Test solution accuracySave solution if acceptableSelect new Dt and compute new integration formula coeff.Done?YESNONO