Numerical integration
Download
1 / 23

Numerical Integration - PowerPoint PPT Presentation


  • 99 Views
  • Uploaded on

Numerical Integration. CSE245 Lecture Notes. Content. Introduction Linear Multistep Formulae Local Error and The Order of Integration Time Domain Solution of Linear Networks. Transient analysis is to obtain the transient response of the circuits.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Numerical Integration' - lana-kane


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Numerical integration

Numerical Integration

CSE245 Lecture Notes


Content
Content

  • Introduction

  • Linear Multistep Formulae

  • Local Error and The Order of Integration

  • Time Domain Solution of Linear Networks


Introduction

Transient analysis is to obtain the transient response of the circuits.

Equations for transient analysis are usually differential equations.

Numerical integration: calculate the approximate solutions Xn.

Linear multistep formulae are the primary numerical integration method.

Introduction


Linear multistep formulae

k the circuits.

k

iXn-i + h iXn-i = 0

i=0

i=0

Linear Multistep Formulae

  • Differential equations are

    X = F(X)

  • Assume values Xn-1, Xn-2, … , Xn-k and derivatives Xn-1, Xn-2, … , Xn-k are known, the solution Xn and Xn can be approximated by a polynomial of these values:


Linear multistep formulae1
Linear Multistep Formulae the circuits.

  • There are two distinct classes LMS:

  • Explicit predictors

    --- 0 = 0

    --- Xn is the only unknown variable

  • Implicit

    --- 0 0

    --- Xn, Xn are all unknown variables.


Linear multistep formulae2
Linear Multistep Formulae the circuits.

  • Three simplest LMS formulae:

  • The forward Euler

  • The backward Euler

  • Trapezoidal


Linear multistep formulae3

X(t) the circuits.

Xn

X(tn)

Xn-1

tn

tn-1

t

Linear Multistep Formulae

  • The forward Euler

    Xn– Xn-1– h Xn-1 = 0

    where 0 = 1, 1 = -1, 0 = 0, 1 = -1


Linear multistep formulae4
Linear Multistep Formulae the circuits.

  • The backward Euler

    Xn– Xn-1– h Xn = 0

    where 0 = 1, 1 = -1, 0 = -1, 1 = 0

  • It is an implicit representation. We may assume some initial value for Xn and iterate to approximate the solution Xn and Xn.


Linear multistep formulae5
Linear Multistep Formulae the circuits.

  • Trapezoidal

    Xn– Xn-1– h (Xn + Xn-1 )/2= 0

    where 0 = 1, 1 = -1, 0 = -1/2, 1 = -1/2

  • It is also an implicit representation. Xn, Xn can be obtained through some iterative procedure.


Local error
Local Error the circuits.

  • Two crucial concepts

  • Local error --- the error introduced in a single step of the integration routine.

  • Global error--- the overall error caused by repeated application of the integration formula.


Local error1

Diverging flow the circuits.

X(t)

Converging flow

t

Global error and local error

Local Error


Local error2
Local Error the circuits.

  • Two types of error in each step:

  • Round-offerror --- due to the finite-precision (floating-point) arithmetic.

  • Truncationerror --- caused by truncation of the infinite Taylor series, present even with infinite-precision arithmetic.


Local error and order of integration

k the circuits.

k

iX(tn-i) + h iX(tn-i)

i=1

i=0

Local Error and Order of Integration

  • Local error Ek for LMS

    Ek = X(tn) +

  • Ek can be expanded into Taylor series. If the coefficients of the first pth derivatives are zero, the order of integration is p.


Order of integration

k the circuits.

k

i((tn-tn-i)/h)l + h (-l/h)i((tn-tn-i)/h)l-1

i=0

i=0

k

i = 0

i=0

k

k

(ii - i) = 0

[(ii - pi)ip-1] = 0

i=0

i=0

Order of Integration

  • Let X(t) = ((tn-t)/h)l and tn– tn-i = ih,

  • Ek =

  • For pth order integration, the first p+1 elements (l = 0, 1, … , p) will all be zeros:

    • l = 0

    • l = 1

    • l = p


Order of integration1
Order of Integration the circuits.

  • The forward Euler

    0 = 1, 1 = -1, 0 = 0, 1 = -1

    So l = 0 0 + 1 = 1 + (-1) = 0;

    l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - 0 – (-1) = 0;

    l = 2 (11 - 21)1 = ((-1)1 - 2(-1))1 = 1  0;

    The forward Euler is 1th order.


Order of integration2
Order of Integration the circuits.

  • The backward Euler

    0 = 1, 1 = -1, 0 = -1, 1 = 0

    So l = 0 0 + 1 = 1 + (-1) = 0;

    l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - (-1) - 0 = 0;

    l = 2 (11 - 21)1 = ((-1)1 - 20)1 = -1  0;

    The backward Euler is 1th order.


Order of integration3
Order of Integration the circuits.

  • Trapezoidal

    0 = 1, 1 = -1, 0 = -1/2, 1 = -1/2

    So l = 0 0 + 1 = 1 + (-1) = 0;

    l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - (-1/2) – (-1/2) = 0;

    l = 2 (11 - 21)1 = ((-1)1 - 2(-1/2))1 = 0;

    l = 3 (11 - 31)12 = ((-1)1 - 3(-1/2))1 = 1/2  0;

    The trapezoidal method is 2th order


Order of integration4
Order of Integration the circuits.

  • The algorithm for defining  and  :

    --- Choose p, the order of the numerical integration method needed;

    --- Choose k, the number of previous values needed;

    --- Write down the (p+1) equations of pth order accuracy;

    --- Choose other (2k-p) constrains of the coefficients  and ;

    --- Combine and solve above (2k+1) equations;

    --- Get the result coefficients  and .


Solution of linear networks

k the circuits.

k

iXn-i + h iXn-i = 0

i=0

i=0

Solution of Linear Networks

  • Combine the differential equations for linear networks and the numerical integration equations:

    MX = -GX + Pu

(1)

(2)


Solution of linear networks1

k the circuits.

k

iXn-i + h iXn-i = 0

i=1

i=1

Solution of Linear Networks

(1)  Xn + h0Xn +

 Xn + h0Xn + b = 0

 Xn = (-1/h0)( Xn + b)

(2)+(3)M[(-1/h0)( Xn + b)] = -GXn + Pu

 (-1/h0) Xn = -GXn + Pu + (M/h0)b

(3)


Solution of linear networks2

i the circuits.c

(-C/h0)

– (C/h0) bc

vc

ic

vc

Solution of Linear Networks

  • For capacitance

    C vc = ic

     C [(-1/h0)( vc + bc)] = ic

     (-C/h0) vc– (C/h0) bc = ic


Solution of linear networks3

i the circuits.l

– (L/h0) bl

+

-

vl

(-L/h0)

il

vl

Solution of Linear Networks

  • For inductance

    L il = vl

     L [(-1/h0)( il + bl)] = vl

     (-L/h0) il– (L/h0) bl = vl


References
References the circuits.

  • CK. Cheng, John Lillis, Shen Lin and Norman Chang

    “Interconnect Analysis and Synthesis”, Wiley and Sons, 2000

  • Jiri Vlach and Kishore Singhal

    “Computer Methods for Circuit Analysis and Design”, 1983


ad