Numerical Integration

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Numerical Integration

• CSE245 Lecture Notes

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Content

• Introduction
• Linear Multistep Formulae
• Local Error and The Order of Integration
• Time Domain Solution of Linear Networks

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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.

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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

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Linear Multistep Formulae

• 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.

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Linear Multistep Formulae

• Three simplest LMS formulae
• The forward Euler
• The backward Euler
• Trapezoidal

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Linear Multistep Formulae

• The forward Euler
• Xn Xn-1 h X?n-1 0
• where ?0 1, ?1 -1, ?0 0, ?1 -1

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Linear Multistep Formulae

• 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.

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Linear Multistep Formulae

• 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.

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Local Error

• 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.

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Local Error
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Local Error

• Two types of error in each step
• Round-off error --- due to the finite-precision
  (floating-point) arithmetic.
• Truncation error --- caused by truncation of the
  infinite Taylor series, present even with
  infinite-precision arithmetic.

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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.

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Order of Integration

• Let X(t) ((tn-t)/h)l and tn tn-i ih,
• Ek
• For pth order integration, the first p1 elements
  (l 0, 1, , p) will all be zeros
• l 0
• l 1
• l p

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Order of Integration

• 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.

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Order of Integration

• 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.

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Order of Integration

• 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

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Order of Integration

• 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 (p1) equations of pth order
  accuracy
• --- Choose other (2k-p) constrains of the
  coefficients ? and ?
• --- Combine and solve above (2k1) equations
• --- Get the result coefficients ? and ?.

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Solution of Linear Networks

• Combine the differential equations for linear
  networks and the numerical integration equations
• MX? -GX Pu

(1)
(2)
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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)
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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

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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

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References

• 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