I am quite surprised that [a variant of linear regression](https://codegolf.stackexchange.com/questions/106260/linear-regression-on-a-string) has been proposed for a challenge, whereas an estimation via ordinary least squares regression has not! [![OLS visualisation][1]][1] For details, check out the [Wikipedia page on OLS](https://en.wikipedia.org/wiki/Ordinary_least_squares). To keep things concise, suppose one has a model: [![Linear regression model][2]][2] where all right-hand side variables—regressors—are linearly independent. Then you estimate this model on a sample of size *n*, given observations [![Data used for estimation][3]][3] The OLS solution to this problem looks like [![enter image description here][4]][4] where **Y** is the first column of the input matrix and **X** is a matrix made of remaining columns. This solution can be obtained via many numerical methods (matrix inversion, QR decomposition, Cholesky decomposition etc.), so pick your favourite! *Of course, econometricians prefer slightly different notation, but let’s just ignore them. — Do you know what econometricians use as a contraceptive? — Their personalities! (Don’t worry, I can say that, I am an econometrician...)* Non other than Gauss himself is watching you from the skies, so do not disappoint one of the greatest mathematicians of all times and write the shortest code possible. ## Task Given the observations in a matrix form as shown above, estimate the coefficients of the linear regression model via OLS. ## Input A matrix of values. The first column is always `Y[1], ..., Y[n]`, the second column is `X1[1], ..., X1[n]`, the next one is `X2[1], ..., X2[n]` etc. **NB.** In statistics, regressing on a constant is widely used. This means that the model is `Y = b0 + U`, and the OLS estimate of `b0` is the sample average of `Y`. In this case, the input is just a matrix with one column, `Y`. You can safely assume that the variables are not exactly collinear, so the matrices above are invertible. In case your language cannot invert matrices with condition numbers larger than a certain threshold, state it explicitly, and provide a unique return value (that cannot be confused with output in case of success) denoting that the system seems to be computationally singular (`S` or any other unambiguous characters). ## Output OLS estimates of `beta_0, ..., beta_k` from a linear regression model in an unambiguous format or an indication that your language could not solve the system. ## Challenge rules - I/O formats are flexible. A matrix can be several lines of space-delimited numbers separated by newlines, or an array of row vectors, or an array of column vectors etc. - This is [tag:code-golf], so shortest answer in bytes wins. - [Standard rules apply](https://codegolf.meta.stackexchange.com/questions/2419/default-for-code-golf-program-function-or-snippet/2422#2422) for your answer, so you are allowed to use STDIN/STDOUT, functions/method with the proper parameters and return-type, full programs. - [Default loopholes](https://codegolf.meta.stackexchange.com/questions/1061/loopholes-that-are-forbidden-by-default) are forbidden. ## Test cases 1. `[[4,5,6],[1,2,3]]` → output: `[3,1]`. 2. `[[5.5,4.1,10.5,7.7,6.6,7.2],[1.9,0.4,5.6,3.3,3.8,1.7],[4.2,2.2,3.2,3.2,2.5,6.6]]` → output: `[2.1171050,1.1351122,0.4539268]`. 3. `[[1,-2,3,-4],[1,2,3,4],[1,2,3,4.000001]]` → output: `[-1.3219977,6657598.3906250,-6657597.3945312]` or `S` (any code for a computationally singular system). 4. `[[1,2,3,4,5]]` → output: `3`. **Bonus points** (to your karma, not to your byte count) if your code can solve very ill-conditioned (quasi-multicollinear) problems (e. g. if you throw in an extra zero in the decimal part of `X2[4]` in Test Case 3) with high precision. [1]: https://i.sstatic.net/hNohR.png [2]: https://i.sstatic.net/4HhZv.png [3]: https://i.sstatic.net/rlAZZ.png [4]: https://i.sstatic.net/tYYlN.png