Re: Can we predict Simulation?

Numerical Methods in Engineering with Python by Jaan Kiusalaas provides
the following information.

On page 28 the book says "A system on n linear equations in n unknowns
has a unique solution, provided the determinant of the coefficient
matrix is nonsingular; that is |A| is not equal to 0.".

Page 85 in the book provides the following guidance when using
iterative methods to solve a system of linear equations. "A serious
drawback of iterative methods is that they do not always converge to
the solution. It can be shown that convergence is guaranteed only if
the coefficient matrix is diagonally dominant. The initial guess for X
plays no role in determining whether convergence takes place--if the
procedure converges for one starting vector, it would do so for any
starting vector. The initial guess affects only the number of
iterations that are required for convergence."

Page 158 and 159 in the book provides the following guidance when using
the Newton-Raphson method to solve n simultaneous nonlinear equations,
"The simplest and the most effective method of computing X is the
Newton-Raphson method. It works well with simultaneous equations,
provided that it is supplied a good starting point. There are other
methods that have better global convergence characteristics, but all of
them are variants of the Newton-Raphson method. As in the one
dimensional case, suceess of the Newton-Raphson method procedure
depends entirely on the initial estimate of X. If a good starting
point is used, convergence to the solution is very rapid. Otherwise,
the results are unpredictable."

Hence, I conclude the answer is yes if you have a system of linear
equations but the answer is no if you have a system of nonlinear
equations.

You also might want to post your questions in the Sci-math.num.analysis
group to see what the numerical method pros there say.

Howard

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?

These questions mean a lot to my thesis work and I shall be really
really thankful if someone can answer this

Thank you,
Dilip

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