Nonlinear Equations - UFRJ

108 [CH. 8: CONDITION NUMBER THEORY
Σ is a m × n matrix. It is possible to rewrite this in an ‘economical’
formulation with Σ an r × r matrix, U and V orthogonal
(resp. unitary) m × r and n × r matrices. The numbers σ 1 , . . . , σ r
are called singular values of A. They may be computed by extracting
the positive square root of the non-zero eigenvalues of A ∗ A or AA ∗ ,
whatever matrix is smaller. The operator and Frobenius norm of A
may be written in terms of the σ i ’s:
√
‖A‖ 2 = σ 1 ‖A‖ F = σ1 2 + · · · + σ2 r.
The discussion and the results above hold when A is a linear operator
between finite dimensional inner product spaces. It suffices to
choose an orthonormal basis, and apply Theorem 8.1 to the corresponding
matrix.
When m = n = r, ‖A −1 ‖ 2 = σ n . In this case, the condition
number of A for linear solving is defined as
κ(A) = ‖A‖ ∗ ‖A −1 ‖ ∗∗ .
The choice of norms is arbitrary, as long as operator and vector norms
are consistent. Two canonical choices are
κ 2 (A) = ‖A‖ 2 ‖A −1 ‖ 2 and κ D (A) = ‖A‖ F ‖A −1 ‖ 2 .
The second choice was suggested by Demmel [35]. Using that
definition he obtained bounds on the probability that a matrix is
poorly conditioned. The exact probability distribution for the most
usual probability measures in matrix space was computed in [38].
Assume that A(t)x(t) ≡ b(t) is a family of problems and solutions
depending smoothly on a parameter t. Differentiating implicitly,
which amounts to
˙ Ax + Aẋ = ḃ
ẋ = A −1 ḃ − A −1 ˙ Ax.
Passing to norms and to relative errors, we quickly obtain
(
‖ẋ‖
‖ẋ‖ ≤ κ ‖ A‖
D(A)
˙ )
F
+ ‖ḃ‖ .
‖A‖ F ‖b‖