A sensor reads (in mV) at the times

PHYSICS · Multi-part · Fermi University Physics · Scientific Practice & Experimental Methods · Mathematical Methods for Physics · Data analysis and model fitting in the laboratory

Error, Chance and the Machine, Ch Linear systems, least squares and the singular value decomposition

Problem

A sensor reads

y

(in mV) at the times

t = 1000.0

1000.5

and

1001.0\ \mathrm{s}

, and the three readings lie exactly on a line

y = c_0 + c_1t

. The line is fitted by least squares with the design matrix

\mathsf{A} = [\mathbf{1}\ \ \mathbf{t}]

, whose columns hold the number 1 and the value of

t

in seconds. Use binary64, with

u = 2^{-53} = 1.11\times10^{-16}

, and the first-order error estimates of Section 2.5. ((a)) Show that

\det(\mathsf{A}^{\rm T}\mathsf{A}) = 1.5

and

\mathrm{tr}(\mathsf{A}^{\rm T}\mathsf{A}) = 3\,003\,004.25

. Use

\sigma_1^2\sigma_2^2 = \det

and

\sigma_1^2 + \sigma_2^2 = \mathrm{tr}

to find

\kappa(\mathsf{A})

and

\kappa(\mathsf{A}^{\rm T}\mathsf{A})

. ((b)) The data are exact, so the residual is zero. Estimate the relative error of

(c_0, c_1)

when the fit is solved by QR and when it is solved by the normal equations, and the number of correct significant digits each keeps, taken as

-\log_{10}

of the error. ((c)) Now use the variable

t' = t - 1000.5\ \mathrm{s}

. Show that the columns of the new design matrix are orthogonal, find its condition number, and repeat the estimates of part 2. Say what the new intercept

c_0'

means and how

c_0

follows from it. ((d)) The three readings are taken instead at

t = T

T + 0.5

and

T + 1\ \mathrm{s}

. Show that

\det(\mathsf{A}^{\rm T}\mathsf{A})

is still

1.5

, that

\kappa(\mathsf{A}) \approx 2.45\,T^2

for

T \gg 1\ \mathrm{s}

, and find the values of

T

at which the estimated error of the normal equations, and that of QR, reach 1 (no correct digit left).

Browse the problem library