Lecture 5Software engineering

8/18/2019 Lecture 5Software engineering

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1

EE 616

Computer Aided Analysis of Electronic Networks

Lecture 5

Instructor: Dr. J. A. Starzyk, Professor

School of EECS

Ohio University

Athens, O, !"#$%

09/19/2005

Note materials in t!is lecture are from

t!e notes of EE219A "C#$erkeley

!ttp//www#cad%eecs%$erkeley%edu/&nardi/EE219A/contents%!tml



&

'utline

Nonlinear pro$lems (terati)e *et!ods Newton+s *et!od

, -eri)ation of Newton

, .uadratic Con)erence

, Eamples

, Con)erence estin

*ultidimensonal Newton *et!od , asic Alorit!m

, .uadratic con)erence

, Application to circuits



'

Need to Solve

( %) $d

t

V V

d s I I e− − =

0

1

Ir I% I

*

$)%(%

$

%%

%

%

=−−+

=−+

I e I e R

I I I

t V

e

s

d r

%%)( I e g =

-C Analysis of Nonlinear Circuits # Eample



!

Nonlinear E3uations

4i)en g(V)=I

(t can $e epressed as f(V)=g(V)-I

⇒ ol)e g(V)=I e3ui)alent to sol)e f(V)=0

Hard to find analytical solution for f(x)=0

Solve iteratively



"

Nonlinear E3uations , (terati)e *et!ods

tart from an initial )alue x 0

4enerate a se3uence of iterate x n-1, x n, x n+1

w!ic! !opefully con)eres to t!e solution x* (terates are enerated accordin to an

iteration function F: x n+1=F(x n )

Ask

• When does it converge to correct solution ?

• What is the convergence rate ?



+

Newton#ap!son 7N8 *et!od

Consists of lineariin t!e system%:ant to sol)e f78;0 → eplace f78 wit! its linearied )ersion

and sol)e%

Note at eac! step need to e)aluate f and f’

function Iteration x f xdxdf x x

x x xdx

df x f x f

iesTaylor Ser x x xdx

df x f x f

k k k k

k k k k k

)()(

))(()()(

))(()()(

%

%

%%

−

+

++

−=⇒

−+=

−+=



#

Newton#ap!son *et!od , 4rap!ical <iew



-

Newton#ap!son *et!od , Alorit!m

Define iteration

Dok = 0 to ….?

until convergence

=ow a$out con)erence> An iteration {x (k) } is said to con)ere wit! order q if t!ere eists a

)ector norm suc! t!at for eac! k ≥ N

qk k

x x x x

%

−≤−

+

)()(

%

% k k k k

x f xdx

df

x x

−

+

−=



/

& &

&

$ ( ) ( ) ( )( ) ( )( )

so0e 1 , 2

k k k k

k

df d f f x f x x x x x x x

dx dx

x x x

= = + − + −

∈

%

%

Mean Value theoremtruncates Taylor series

But

%$ ( ) ( )( )k k k k df f x x x x

dx

+= + − by Newtondefinition

Newton#ap!son *et!od , Con)erence



%$

Subtracting

&

% % &&( ) 1 ( )2 ( )( )k k k df d f x x x x x x

dx d x+ −− = −%

Convergence is quadratic

&% &

&( )( ) ( )( )k k k df d f x x x x x x

dx d x

+ − = −%

Dividing

&%

&

&%

3et 1 ( )2 ( )

then

k k

k k k

df d f x x K

dx d x

x x K x x

−

+

=

− ≤ −

%




%%

Local Convergence Theorem

If

Then Newton’s method converges given asufficiently close initial guess(and convergence isquadratic)

bounded is K

bounded dx

f d b

roay from zebounded awdx

df a

)

)

&

&




%&

( )( )

( ) ( )( )

&%

& &%

&

&

%

( ) &

( ) % $,

& ( ) %

& ( ) & ( )

%( ) (

( %

)&

)

k k

k k k k

k k k k k

k k

k

df

x xdx

x x x x

x x x x x x x

f x x fin

x

d x x

or x x x x x

+

+

+

=− = − −

− + − = −

=

−

−

−

= −

= =

Convergence is quadratic


Eample 1



%'

( )

% &

%

%

&

( ) &

& ( $) ( $)

%$ $

( ) $,

for $

&%

( ) ( )&

$

k k

k k k

k k k

k k

df x 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

%df

dx

−


Eample 2



%!


Eample 1? 2



%"

$ Initial 4uess,5 $ x k =

6e7eat 8

( ) ( ) ( )%

k

k k k

f x x x f x

x+

∂− = −∂

9 Until

( )% % k k k f x tresold x x tresold ++ <− <

%k k = +




%+

;ee* a <*elta=>< check to avoi* false conver?ence

X

f(x)

%k

x

+ k

x

x

( )% a

k

f f x ε + <

% % a r

k k k x x x x xε ε + +− > +

Newton#ap!son *et!od , Con)erence C!eck



%#

( )Also nee* an < < check to avoi* false conver?ence f x

X

f(x)

%k x + k x

x

( )% a

k

f f x ε + >

% % a r

k k k

x x x x xε ε + +− < +

Newton#ap!son *et!od , Con)erence C!eck



%-




%/

Convergence Depends on a Good Initial Guess

X

f(x)

$ x

% x

& x $ x

% x

Newton#ap!son *et!od , Local Con)erence



&$

Convergence Depends on a Good Initial Guess

Newton#ap!son *et!od , Local Con)erence



&%

Nodal Analysis

% &At ;o*e %: $i i+ =

NonlinearResistors

( )i g !=

%

b

!

+

-

%i

&i&

b!

'i+ -

'

b

!

+

-

%!&!

( ) ( )% % & $ g ! g ! !⇒ + − =

' &At ;o*e &: $i i− =

( ) ( )' % & $ g ! g ! !⇒ − − =

Two couplednonlinear equationsin two unknowns

Nonlinear @ro$lems , *ultidimensional Eample



&&

( ) Pro@le0: in* such tha $t x " x = ; ; ; an* : x " ∈ →¡ ¡ ¡

*ultidimensional Newton *et!od

function Iteration x " x # x x

atrix #acobian $

x

x "

x

x "

x

x "

x

x "

x #

iesTaylor Ser x x x # x " x "

k k k k

%

% %

%

)()(

)()(

)()(

)(

))(()()(

%%

%

%

%

%

−+ −=⇒

∂∂

∂∂

∂∂

∂∂

=

−+=



&'

)()()( :solveInstea*

s7arse).notis(it)(co07utenotDo

)()(:

%

%

%%

k k k k

k

k k k k

x " x x x #

x #

x " x # x x Iteration

−=−

−=

+

−

−+

Eac! iteration re3uires

1% E)aluation of F(x k )

2% Computation of J(x k

) % olution of a linear system of ale$raic

e3uations w!ose coefficient matri is J(x k ) and

w!ose = is -F(x k )

*ultidimensional Newton *et!od , Computational Aspects



&!


6e7eat 8

( ) ( ) ( )% %Solve fork k k k k

" # x x x " x x+ +− = −

9 Until ( )%% , s0all en u?o hk k k x x f x ++ −

%k k = +

( ) ( )Co07ute ,k k

" " x # x

*ultidimensional Newton *et!od , Algorithm



&"

If

( ) ( )%) Inverse is @oun*e*k " a # x β − ≤

( ) ( ) ( )) Derivative is 3i7schitz Cont " " b # x # y x y− ≤ −l

Then Newton’s method converges given asufficiently close initial guess(and convergenceis quadratic)

Local Convergence Theorem

*ultidimensional Newton *et!od , Convergence



&+

Application of N to Circuit E3uationsCompanion Network

Applyin N to t!e system of e3uations we findt!at at iteration kB1 , all t!e coefficients of CL? <L and of CE of t!e linear

elements remain unc!aned wit! respect to iteration k , Nonlinear elements are represented $y a lineariationof CE around iteration k

⇒ !is system of e3uations can $e interpreted ast!e A of a linear circuit 7companion network8

w!ose elements are specified $y t!e lineariedCE%



&#

4eneral procedure t!e N met!od applied to anonlinear circuit 7w!ose e3ns are formulated int!e A form8 produces at eac! iteration t!eA e3ns of a linear resisti)e circuit o$tained $y

lineariin t!e CE of t!e nonlinear elementsand lea)in all t!e ot!er CE unmodified

After t!e linear circuit is produced? t!ere is noneed to stick to A? $ut ot!er met!ods 7suc! as

*NA8 may $e used to assem$le t!e circuit e3ns

Application of N to Circuit E3uationsCompanion Network



&-

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 N to Circuit E3uationsCompanion Network , *NA templates



&/

Application of N to Circuit E3uationsCompanion Network , *NA templates



'$

*odelin a *'DE7*' Le)el 1? linear reime8

d



'%

*odelin a *'DE7*' Le)el 1? linear reime8



'&

-C Analysis Dlow -iaram

For each state variable in the system



''

(mplications

-e)ice model e3uations must $e continuous wit!continuous deri)ati)es and deri)ati)e calculation must $eaccurate deri)ati)e of function , 7not all models do t!is # @oor diode models and $reakdown

models don+t # $e sure models are decent # $eware of user#

supplied models8 :atc! out for floatin nodes 7(f a node $ecomes

disconnected? t!en 78 is sinular8 4i)e ood initial uess for 708

*ost model computations produce errors in function)alues and deri)ati)es% , :ant to !a)e con)erence criteria FF 7kB18 # 7k8 FF G ε suc! t!at ε H

t!an model errors%



'!

ummary

Nonlinear pro$lems (terati)e *et!ods

Newton+s *et!od

, -eri)ation of Newton , .uadratic Con)erence

, Eamples

, Con)erence estin

*ultidimensonal Newton *et!od , asic Alorit!m

, .uadratic con)erence

, Application to circuits



'"

(mpro)in con)erence

(mpro)e *odels 7I0J of pro$lems8

(mpro)e Alorit!ms 720J of pro$lems8

Docus on new alorit!msLimitin c!emes

Continuations c!emes



'+

'utline

Limitin c!emes , -irection Corruptin

, Non corruptin 7-amped Newton84lo$ally Con)erent if aco$ian is Nonsinular

-ifficulty wit! inular aco$ians

Continuation c!emes , ource steppin

, *ore 4eneral Continuation c!eme , (mpro)in Efficiency

etter first uess for eac! continuation step



'#

At a local 0ini0u0, $ f

x

∂=

∂

LocalMinimum

( )Bulti*i0ensional Case: is sin?ular " # x

*ultidimensional Newton *et!odCon)erence @ro$lems , Local *inimum



'-

f(>)

$

x

% x

Bust So0eho 3i0it the chan?es in

*ultidimensional Newton *et!odCon)erence @ro$lems , Nearly sinular



'/

*ultidimensional Newton *et!odCon)erence @ro$lems # ')erflow

f(>)

$ x %

x

Bust So0eho 3i0it the chan?es in



!$


6e7eat 8

( ) ( )% %Solve fork k k k

" # x x " x x+ +∆ = − ∆

9 Until ( )% % , s0all enou?hk k x " x ++∆

%k k = +

( ) ( )Co07ute ,k k

" " x # x

( ) ;eton Al?orith0 for Solvin? $ " x =

( )% %li0ite*k k k x x x+ += + ∆

Newton *et!od wit! Limitin



!%

( )%li0ite* k

i x +∆ =

;onCorru7tin?

% % ifk k

i i x x γ + +∆ ∆ <

( )% otherisek

i sign xγ +∆

Direction Corru7tin?

( )% %li0ite* k k x xα + +∆ = ∆

%0in %,

k x

γ α +

= ∆

γ

%k x +∆( )%li0ite*

k x

+∆

γ

%k x +∆( )%li0ite* k x +∆

euristics, ;o 4uarantee of 4lo@al Conver?ence

Newton *et!od wit! LimitinLimitin *et!ods



!&

4eneral Da07in? Sche0e

Fey I*ea: 3ine Search

( )&

%

&Pick to 0ini0izek k k k " x xα α ++ ∆

Betho* Perfor0s a one=*i0ensional search in

;eton Direction

( ) ( ) ( )&% % %

&

T k k k k k k k k k " x x " x x " x xα α α + + ++ ∆ ≡ + ∆ + ∆

% %k k k k x x xα + += + ∆


" # x x " x x+ +∆ = − ∆

Newton *et!od wit! Limitin-amped Newton c!eme



!'

If

( ) ( )%) Inverse is @oun*e*k

" a # x β − ≤

( ) ( ) ( )) Derivative is 3i7schitz Cont " " b # x # y x y− ≤ −l

Then

( ]Ghere e>ists a set of H $,% such thatk sα ∈

( ) ( ) ( )% % ith %k k k k k " x " x x " xα γ γ + += + ∆ <

Every Ste7 re*uces == 4lo@al Conver?ence

Newton *et!od wit! Limitin-amped Newton , Con)erence !eorem



!!


6e7eat 8


" # x x " x x+ +∆ = − ∆

9 Until ( )% % , s0all enou?hk k x " x ++∆

%k k = +

( ) ( )Co07ute ,k k

" " x # x

( ] ( )%

in* $,% such that is 0ini0i ez *

k k k k

" x xα α

+

+ ∆∈% %k k k k x x xα + += + ∆

Newton *et!od wit! Limitin-amped Newton , Nested (teration



!"

X$ x% x

& x%

& x

Damped Newton Methods “push” iterates to local minimumsFinds the points where Jacobian is Singular

Newton *et!od wit! Limitin-amped Newton , inular aco$ian @ro$lem



!+

( )( )Solve , $ here: " x λ λ =%

( )( )a) $ ,$ $ is easy to solve " x =%

( )( ) ( ) @) % ,% " x " x=%

( )c) is sufficiently s0ooth x λ

Starts the continuation

Ends the continuation

Hard to insure!

Newton wit! Continuation sc!emesasic Concepts # 4eneral settin

Newton con)eres i)en a close initial uess

⇒ (dea 4enerate a se3uence of pro$lems? s%t% a pro$lem

is a ood initial uess for t!e followin one

→= $)( x "



!#

( )( ) ( ) ( )Solve $ ,$ , $ 're! " x x xλ =%

Khile % 8λ <

9

( ) ( )$

're! x xλ λ =( )( )Gry to Solve , $ ith ;eton " x λ λ =%

If ;eton Conver?e*

( ) ( ), &,

're!

x xλ λ λ λ λ δλ δλ δ = = =+

Else

,%

& 're!λ δλ λ δ δλ λ = +=

Newton wit! Continuation sc!emesasic Concepts , emplate Alorit!m



!-

Newton wit! Continuation sc!emesasic Concepts , ource teppin Eample



!/

Diode

( )( ) ( ) ( )%

, $diode s f ! i ! ! V R

λ λ λ = + − =%

B

#Vs

R !

( ) ( ), % ;ot *e7en*entJ

diode f ! i !

! ! R

λ λ ∂ ∂= + ←∂ ∂

%

Source Stepping Does Not Alter Jacobian

Newton wit! Continuation sc!emesasic Concepts , ource teppin Eample



"$

( )( ) ( ) $ ,$5 $ $$ " x xλ = =%

Observations

( )( )$ ,$ " x I

x

∂=

∂

%

( )( ) ( )( ) 5% % ,% % " x " xλ =%

( )( ) ( )( )$ ,$ % " x " x

x x

∂ ∂=

∂ ∂

%

Problem is easy to solve andJacobian definitely nonsingular.

Back to the original problemand original Jacobian

( )( ) ( )( ) ( ) ( ), % " x " x xλ λ λ λ λ λ = + −%

Newton wit! Continuation sc!emesaco$ian Alterin c!eme

718?18



ummary

Newton+s *et!od works fine , i)en a close enou! initial uess

(n case Newton does not con)ere , Limitin c!emes

-irection Corruptin

Non corruptin 7-amped Newton8

, 4lo$ally Con)erent if aco$ian is Nonsinular

, -ifficulty wit! inular aco$ians

, Continuation c!emesource steppin

*ore 4eneral Continuation c!eme

(mpro)in Efficiency

,

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Lecture 5Software engineering