Re: Can we predict simulation?

Dilips wrote:
Hello,

This question may seem strange to you, but it is of very huge
significance for me.
Can we predict, if a circuit will simulate or not looking at the matrix

formed?
The AX=B which the iterative or direct solvers in SPICE use, I want to
know if given A and B. Is there some mathematical operation that can be

performed to know that
a) a solution will exist?
b) if a solution exists, will it converge to the correct answer?
c) will it converge to an answer at all?


If we can predict only for linear system of equations and cannot do the
same for non-linear system of equations. How good is the result? Is it
of any use at all?
These questions mean a lot to my thesis work and I shall be really
really thankful if someone can answer this


Thank you, Dilip

To obtain answers to questions at this basic level, you would be better off reading a good textbook on numerical analysis. A newsgroup is not the place for obtaining basic instruction.

If A is "given", and constant, the equations are linear. If A is also square, as you seem to imply, and A is not singular, the solution is unique -- simple explanation: suppose X1 and X2 are two distinct solutions. Then A(X1-X2) = B - B = 0; multiply by A^{-1}, which exists if A is not singular; you see that X1-X2=0, and X1 and X2 cannot be distinct.

For direct methods, the question of convergence has no meaning. The question of accuracy, however, is valid, and depends on the "condition" of A (see a text on numerical analysis).

For nonlinear equations, in general there need be NO solution at all, or multiple or even an infinite number of solutions, real and complex, may exist. In general, direct solutions are impossible for nonlinear equations even when solutions are known to exist. There are many iterative methods, some of which may converge under certain conditions. The properties of local convergence and global convergence are of interest.

For a beautiful example of the possibilities, see these picture of the Julia sets associated with applying Newton's method to Z^n + c = 0:

http://mcgoodwin.net/julia/juliajewels.html

Whether a solution is "good" and "of use" is for the user, to decide. Such questions fall outside the realm of numerical analysis/mathematics. For example, one user may state that the equation cosh z = 0 has no root (assuming that only real roots have any meaning), whereas another user may find the root i\pi/2 perfectly acceptable.

-- mecej4
.

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