Page 100 - Fundamentals of Magnetic Thermonuclear Reactor Design

84 Fundamentals of Magnetic Thermonuclear Reactor Design

n
of a synthesised magnet system, u = {u } ∈ R is a preset vector of the desired
i
n
m
system parameters, R and R are the function spaces of n and m dimension,
respectively.
Any numerical solution requires a preliminary discretisation of the prob-
lem in question. A continuous medium is approximated using a discrete
model with a fixed number of degrees of freedom. Integration is substituted
with summation, and partial differential equations, describing a medium be-
haviour, are reduced to systems of algebraic equations. Therefore, one has
to solve matrix equations (4.11) to be able to handle a wide range of applied
problems.
A SLAE may have a unique solution, an infinite number of solutions, or no
solutions. General-case SLAE is solved based on a pseudo-solution, or a vector
m
that minimises the discrepancy ||Az ─ u|| in the space R . A pseudo-solution for
system (4.11) is the solution of matrix equation
*
*
A*Az=A*u, AAz = A u, (4.12)
2
−
Az−uE2 which is the Euler equation for the residual functional Az u . Here,
E
A* = {a } is a matrix of size n × m transposed relative to the A matrix. We note
ji
*
that A A is a symmetric square matrix of size n × n, and that system (4.12) is
consistent, no matter what the A matrix and the right-hand side of u are. If the
initial SLAE (4.11) is consistent, then it is equivalent to (4.12).
The SLAE solution may be treated as some function of the matrix A and
vector u. Because machine-readable data involve rounding errors, the calculated
matrix A and vector u appear perturbed. Something similar takes place in the
problems associated with physical measurements and interpretation of physical
experimental data due to unavoidable measurement uncertainty. In fact, the fol-
lowing system needs to be solved:

+
)
⋅ =+ b
(A+B)⋅z∇=u+b ( AB z u (4.13)
Let A be a non-degenerate square matrix, and let the perturbation B be gov-
−1
B<A−1−1 erned by B < A −1 −1 . Here, A is the matrix inverse to A. In this case, the
A + B matrix is known to be non-degenerate. Using expressions δA = ||B||/||A||,
δz = ||ž − z||/||z||, and δu = ||b||/||u|| and assuming the matrix norm to be consistent
−1
with the vector norm and ||A ||·||B|| < 1, we obtain
cond A
z
δz≤cond A1−cond A δA(δA+δu), δ ≤ ( δ A + δu), (4.14)
−
δ
1cond AA
−1
where cond A = ||A||·||A || is the matrix A conditionality number. Expression
(4.14) estimates the relative error of the (4.11) solution, caused by errors in
the input. The obtained solution accuracy is very much dependent upon the
matrix conditionality number. According to (4.14), small changes in the input