Numerical Integration
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Transcript of Numerical Integration
Numerical Integration
CSE245 Lecture Notes
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.
Linear Multistep Formulae Differential equations are
X = F(X) Assume values Xn-1, Xn-2, … , Xn-k and
derivatives Xn-1, Xn-2, … , Xn-k are known, the solution Xn and Xn can be approximated by a polynomial of these values: iXn-i + h iXn-i = 0
i=0
k
i=0
k
Linear Multistep Formulae
There are two distinct classes LMS: Explicit predictors
--- 0 = 0
--- Xn is the only unknown variable Implicit
--- 0 0
--- Xn, Xn are all unknown variables.
Linear Multistep Formulae
Three simplest LMS formulae: The forward Euler The backward Euler Trapezoidal
Linear Multistep Formulae The forward Euler
Xn – Xn-1 – h Xn-1 = 0
where 0 = 1, 1 = -1, 0 = 0, 1 = -1
tn-1 tn
Xn-1
Xn X(tn)
X(t)
t
Linear Multistep Formulae
The backward EulerXn – Xn-1 – h Xn = 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 Xn.
Linear Multistep Formulae
TrapezoidalXn – Xn-1 – h (Xn + Xn-1 )/2= 0
where 0 = 1, 1 = -1, 0 = -1/2, 1 = -1/2
It is also an implicit representation. Xn, Xn can be obtained through some iterative procedure.
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.
Local Error
X(t)
t
Global error and local error
Converging flow
Diverging flow
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.
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.
iX(tn-i) + h iX(tn-i) i=1
k
i=0
k
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
i((tn-tn-i)/h)l + h (-l/h) i ((tn-tn-i)/h)l-1 i=0
k
i=0
k
i = 0i=0
k
(ii - i) = 0i=0
k
[(ii - pi)ip-1] = 0i=0
k
Order of Integration
The forward Euler0 = 1, 1 = -1, 0 = 0, 1 = -1 So l = 0 0 + 1 = 1 + (-1) = 0;
l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - 0 – (-1) = 0;
l = 2 (11 - 21)1 = ((-1)1 - 2(-1))1 = 1 0;
The forward Euler is 1th order.
Order of Integration
The backward Euler0 = 1, 1 = -1, 0 = -1, 1 = 0
So l = 0 0 + 1 = 1 + (-1) = 0;l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 -
(-1) - 0 = 0;l = 2 (11 - 21)1 = ((-1)1 - 20)1 = -1
0;
The backward Euler is 1th order.
Order of Integration
Trapezoidal0 = 1, 1 = -1, 0 = -1/2, 1 = -1/2So l = 0 0 + 1 = 1 + (-1) = 0;
l = 1 00 + 11 - 0 - 1 = 10 + (-1)1 - (-1/2) – (-1/2) = 0;
l = 2 (11 - 21)1 = ((-1)1 - 2(-1/2))1 = 0;l = 3 (11 - 31)12 = ((-1)1 - 3(-1/2))1 = 1/2 0;
The trapezoidal method is 2th order
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 .
Solution of Linear Networks
Combine the differential equations for linear networks and the numerical integration equations:
MX = -GX + Pu
iXn-i + h iXn-i = 0 i=0
k
i=0
k(1)
(2)
Solution of Linear Networks
(1) Xn + h0Xn +
Xn + h0Xn + b = 0
Xn = (-1/h0)( Xn + b)
(2)+(3) M[(-1/h0)( Xn + b)] = -GXn + Pu
(-1/h0) Xn = -GXn + Pu +
(M/h0)b
iXn-i + h iXn-i = 0 i=1
k
i=1
k
(3)
Solution of Linear Networks For capacitance
C vc = ic
C [(-1/h0)( vc + bc)] = ic
(-C/h0) vc – (C/h0) bc = ic
ic
vc
vc
ic
– (C/h0) bc (-C/h0)
Solution of Linear Networks For inductance
L il = vl
L [(-1/h0)( il + bl)] = vl
(-L/h0) il – (L/h0) bl = vl
il
vl
+-
il
– (L/h0) bl
(-L/h0)
vl
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