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Ordinary least-squares fit

Objective
Fit a linear combination of given functions, ∑ ckfk(x)|k=1..m, to a (measured) data set {xi, yi, δyi}i=1..n, where δyi are the uncertainties of the measurement (error-bars).
Hints
You can pass around a set of functions fi(x) as an array of functions in a variable of the type Func<double,double>[] for example, 
var fs = new Func<double,double>[] { z => 1.0, z => z, z => z*z };
Tasks
  1. (6 points) Ordinary least-squares fit by QR-decomposition
    • Make sure that your QR-decomposition routines work for tall matrices.
    • Implement a function that makes a least-squares fit—using your QR-decomposition routines—of a given data-set, 
    {xi, yi, δyi}i=1...n ,
    with a linear combination 
    Fc(x) ≐ ∑k=1..m ck fk(x)
    of given functions fk(x)|k=1..m .
    • The arguments to your least-squares function should be the data to fit, {xi, yi, δyi}, and the set of functions, {fk}, the linear combination of which should fit the data. The function must calculate the vector of the coefficients, {ck}.
    • In 1902 [Rutherford and Soddy] measured the radioactivity of the (then not well explored) element, called ThX at the time, and obtained the following results, 
    Time t (days)                     : 1,  2,  3, 4, 6, 9,   10,  13,  15
Activity y of ThX (relative units): 117,100,88,72,53,29.5,25.2,15.2,11.1

    From this data they correctly deduced that radioactive decay follows exponentil law, y(t)=ae-λt (equation (1) in the article).

    Now, assume that the incertainty δy of the measurement was determined by the last-but-one digit of the measurement, 

    δy: 5,5,5,5,5,5,1,1,1,1
    and fit the data with exponential function in the usual logarithmic way, ln(y)=ln(a)-λt. The uncertainty of the logarithm should be probably taken as δln(y)=δy/y.

    Plot the experimental data (with error-bars) and your best fit.

    From your fit find out the half-life time of ThX.

    ThX is today known as 224Ra – compare your result with the modern value.

  2. (3 points) Uncertainties of the fitting coefficients
    • Modify you least-squares fitting function such that it also calculates the covariance matrix and the uncertainties of the fitting coefficients.
    • Calculate the uncertainty of the half-life value for ThX from the given data.
    • Does it agree with the modern value within the estimated uncertainty?

  3. (1 points) Evaluation of the quality of the uncertainties on the fit coefficients
    • Plot your best fit,
    ∑ k=1..m ck fk(x),
    together with the fits where you change the fit coefficients by the estimated δc, that is,
    ∑ k=1..m (ck±δck) fk(x).