1. Numerical Integration CSE245 Lecture Notes

2. Content Introduction Linear Multistep Formulae Local Error and The Order of Integration Time Domain Solution of Linear Networks

3. 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 X n. Linear multistep formulae are the primary numerical integration method.

4. Linear Multistep Formulae Differential equations are X = F(X) Assume values X n-1, X n-2, …, X n-k and derivatives X n-1, X n-2, …, X n-k are known, the solution X n and X n can be approximated by a polynomial of these values:   i X n-i + h   i X n-i = 0 i=0 k k

5. Linear Multistep Formulae There are two distinct classes LMS:  Explicit predictors ---  0 = 0 --- Xn is the only unknown variable  Implicit ---  0  0 --- X n, X n are all unknown variables.

6. Linear Multistep Formulae Three simplest LMS formulae:  The forward Euler  The backward Euler  Trapezoidal

7. Linear Multistep Formulae The forward Euler X n – X n-1 – h X n-1 = 0 where  0 = 1,  1 = -1,  0 = 0,  1 = -1 t n-1 tntn X n-1 XnXn X(t n ) X(t) t

8. Linear Multistep Formulae The backward Euler X n – X n-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 X n and iterate to approximate the solution X n and X n.

9. Linear Multistep Formulae Trapezoidal X n – X n-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. X n, X n can be obtained through some iterative procedure.

10. 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.

11. Local Error X(t) t Global error and local error Converging flow Diverging flow

12. 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.

13. Local Error and Order of Integration Local error E k for LMS E k = X(t n ) + E k can be expanded into Taylor series. If the coefficients of the first pth derivatives are zero, the order of integration is p.   i X(t n-i ) + h   i X(t n-i ) i=1 k i=0 k

14. Order of Integration Let X(t) = ((t n -t)/h) l and t n – t n-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   i ((t n -t n-i )/h) l + h (-l/h)   i ((tn-tn-i)/h) l-1 i=0 k k   i = 0 i=0 k  (  i i -  i ) = 0 i=0 k  [(  i i - p  i )i p-1 ] = 0 i=0 k

15. Order of Integration The forward Euler  0 = 1,  1 = -1,  0 = 0,  1 = -1 Sol = 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.

16. Order of Integration The backward Euler  0 = 1,  1 = -1,  0 = -1,  1 = 0 Sol = 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.

17. Order of Integration Trapezoidal  0 = 1,  1 = -1,  0 = -1/2,  1 = -1/2 Sol = 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

18. 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 (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 .

19. Solution of Linear Networks Combine the differential equations for linear networks and the numerical integration equations: MX = -GX + Pu   i X n-i + h   i X n-i = 0 i=0 k k (1) (2)

20. Solution of Linear Networks (1)  X n + h  0 X n +  X n + h  0 X n + b = 0  X n = (-1/h  0 )( X n + b) (2)+(3)  M[(-1/h  0 )( X n + b)] = -GX n + Pu  (-1/h  0 ) X n = -GX n + Pu + (M/h  0 )b   i X n-i + h   i X n-i = 0 i=1 k k (3)

21. Solution of Linear Networks For capacitance C v c = i c  C [(-1/h  0 )( v c + b c )] = i c  (-C/h  0 ) v c – (C/h  0 ) b c = i c icic vcvc vcvc icic – (C/h  0 ) b c (-C/h  0 ) 

22. Solution of Linear Networks For inductance L i l = v l  L [(-1/h  0 )( i l + b l )] = v l  (-L/h  0 ) i l – (L/h  0 ) b l = v l ilil vlvl + - ilil – (L/h  0 ) b l (-L/h  0 ) vlvl 

23. 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