Solutions for Nonlinear Equations
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Transcript of Solutions for Nonlinear Equations
Solutions for Nonlinear Equations
Lecture 8
Alessandra Nardi
Thanks to Prof. Newton, Prof. Sangiovanni, Prof. White, Jaime Peraire, Deepak Ramaswamy, Michal Rewienski, and Karen Veroy
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
Need to Solve
( 1) 0d
t
VV
d sI I e
Nonlinear Problems - Example
0
1
IrI1 Id
0)1(1
0
11
1
1
IeIeR
III
tV
e
s
dr
11)( Ieg
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
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 ?
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’
functionIterationxfxdx
dfxx
xxxdx
dfxfxf
iesTaylor Serxxxdx
dfxfxf
kkkk
kkkkk
)()(
))(()()(
))(()()(
11
11
***
Newton-Raphson Method – Graphical View
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:
qkk xxxx ˆˆ1
)()(1
1 kkkk xfxdx
dfxx
2* * * 2
2
*
0 ( ) ( ) ( )( ) ( )( )
some [ , ]
k k k k
k
df d ff x f x x x x x x x
dx dx
x x x
Mean Value theoremtruncates Taylor series
But10 ( ) ( )( )k k k kdf
f x x x xdx
by Newtondefinition
Newton-Raphson Method – Convergence
Subtracting
21 * 1 * 2
2 ( ) [ ( )] ( )( )k k kdf d f
x x x x x xdx d x
Convergence is quadratic
21 * * 2
2( )( ) ( )( )k k kdf d fx x x x x x
dx d x
Dividing through
21
2
21 * *
Let [ ( )] ( )
then
k k
k k k
df d fx x K
dx d x
x x K x x
Newton-Raphson Method – Convergence
Newton-Raphson Method – Convergence
Local Convergence Theorem
If
Then Newton’s method converges given a sufficiently close initial guess (and
convergence is quadratic)
boundedisK
boundeddx
fdb
roay from zebounded awdx
dfa
)
)
2
2
21
2 21 * * *
* 2
2
1 *
*
( ) 2
( ) 1 0,
2 ( ) 1
2 ( ) 2 ( )
1( ) (
( 1
)2
)
k k
k k k k
k k k k k
k kk
dfx x
dx
x x x x
x x x x x x x
f x x fin
x
d x x
or x x x xx
Convergence is quadratic
Newton-Raphson Method – ConvergenceExample 1
1 2
1 *
* *1
2 *
( ) 2
2 ( 0) ( 0)
10 0
( ) 0,
for 02
1( ) ( )
2
0
k k
k k k
k k k
k k
dfx x
dx
x x x
x x x x
or x x x
f x
x
x x
Convergence is linear
Note : not bounded
away from zero
1df
dx
Newton-Raphson Method – ConvergenceExample 2
Newton-Raphson Method – ConvergenceExample 1, 2
0 Initial Guess,= 0x k
Repeat { 1
k
k k kf x
x x f xx
} Until ?
1 1? ?k k kf x thresholdx x threshold
1k k
Newton-Raphson Method – Convergence
Need a "delta-x" check to avoid false convergence
X
f(x)
1kx kx *x
1 a
kff x
1 1 a r
k k kx xx x x
Newton-Raphson Method – ConvergenceConvergence Checks
Also need an " " check to avoid false convergencef x
X
f(x)
1kx kx
*x
1 a
kff x
1 1 a r
k k kx xx x x
Newton-Raphson Method – ConvergenceConvergence Checks
demo2
Newton-Raphson Method – Convergence
X
f(x)
Convergence Depends on a Good Initial Guess
0x
1x2x 0x
1x
Newton-Raphson Method – ConvergenceLocal Convergence
Convergence Depends on a Good Initial Guess
Newton-Raphson Method – ConvergenceLocal Convergence
Nodal Analysis
1 2At Node 1: 0i i
1bv
+
-
1i2i 2
bv
3i+ -
3bv
+
-
NonlinearResistors
i g v
1v2v
1 1 2 0g v g v v
3 2At Node 2: 0i i
3 1 2 0g v g v v
Two coupled nonlinear equations in two unknowns
Nonlinear Problems – Multidimensional Example
* *Problem: Find such tha 0t x F x * N N N and : x F
Multidimensional Newton Method
functionIterationxFxJxx
atrixJacobian M
x
xF
x
xF
x
xF
x
xF
xJ
iesTaylor SerxxxJxFxF
kkkk
N
NN
N
)()(
)()(
)()(
)(
))(()()(
11
1
1
1
1
***
Multidimensional Newton MethodComputational Aspects
)()()( :solve Instead
sparse).not is(it )( computenot Do
)()(:
1
1
11
kkkk
k
kkkk
xFxxxJ
xJ
xFxJxxIteration
Each iteration requires:
1. Evaluation of F(xk)
2. Computation of J(xk)
3. Solution of a linear system of algebraic equations whose coefficient matrix is J(xk) and whose RHS is -F(xk)
0 Initial Guess,= 0x k
Repeat {
1 1Solve for k k k k kFJ x x x F x x
} Until 11 , small en ug o hkk kx x f x
1k k
Compute ,k kFF x J x
Multidimensional Newton MethodAlgorithm
If
1) Inverse is boundedkFa J x
) Derivative is Lipschitz ContF Fb J x J y x y
Then Newton’s method converges given a sufficiently close initial guess (and
convergence is quadratic)
Multidimensional Newton Method Convergence
Local Convergence Theorem
Application of NR to Circuit EquationsCompanion 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.
Application of NR to Circuit EquationsCompanion Network
• 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
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
Application of NR to Circuit EquationsCompanion Network – MNA templates
Modeling a MOSFET(MOS Level 1, linear regime)
d
Modeling a MOSFET(MOS Level 1, linear regime)
DC Analysis Flow DiagramFor each state variable in the system
Implications• Device model equations must be continuous with continuous
derivatives and derivative calculation must be accurate derivative of function (not all models do this - Poor diode models and breakdown models don’t - 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.
Summary
• 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