scipy tutorial - Baustatik-Info-Server

For matrix A the only valid values for norm are ±2, ±1, ± inf, and ‘fro’ (or ‘f’) Thus,
⎧ �
maxi j
⎪⎨
�A� =
⎪⎩
|aij| ord = inf
�
mini j |aij| ord = −inf
�
maxj i |aij|
�
ord = 1
minj i |aij| ord = −1
max σi ord = 2
min σi �
ord = −2
trace (AHA) ord = ’fro’
where σi are the singular values of A .
Solving linear least-squares problems and pseudo-inverses
SciPy Reference Guide, Release 0.8.dev
Linear least-squares problems occur in many branches of applied mathematics. In this problem a set of linear scaling
coefficients is sought that allow a model to fit data. In particular it is assumed that data yi is related to data xi through
a set of coefficients cj and model functions fj (xi) via the model
yi = �
cjfj (xi) + ɛi
j
where ɛi represents uncertainty in the data. The strategy of least squares is to pick the coefficients cj to minimize
J (c) = �
�
�
�
�
�
� yi − �
Theoretically, a global minimum will occur when
∂J
= 0 = �
⎛
⎝yi − �
or
where
When A H A is invertible, then
∂c ∗ n
�
j
cj
�
i
i
i
j
j
cjfj (xi)
⎞
�2
�
�
�
� .
�
cjfj (xi) ⎠ (−f ∗ n (xi))
fj (xi) f ∗ n (xi) = �
yif ∗ n (xi)
A H Ac = A H y
{A} ij = fj (xi) .
c = � A H A �−1 A H y = A † y
where A † is called the pseudo-inverse of A. Notice that using this definition of A the model can be written
y = Ac + ɛ.
The command linalg.lstsq will solve the linear least squares problem for c given A and y . In addition
linalg.pinv or linalg.pinv2 (uses a different method based on singular value decomposition) will find A †
given A.
The following example and figure demonstrate the use of linalg.lstsq and linalg.pinv for solving a datafitting
problem. The data shown below were generated using the model:
yi = c1e −xi + c2xi
where xi = 0.1i for i = 1 . . . 10 , c1 = 5 , and c2 = 4. Noise is added to yi and the coefficients c1 and c2 are estimated
using linear least squares.
1.8. Linear Algebra 41
i