# High `cv::solve` error if trained on single line

I use `cv::solve`

to solve two dimensional linear regression. If the training data happen to be on single line (that is, `y`

is equal for all training variables, then the extrapolation produces large error.

Say, the training data are:

```
X Y Z
---- ---- -----
4 7 458
5 7 554
7 7 735
8 7 826
```

The calculated coefficients are (notice the last two are very large numbers):

```
{92.8825, 3.74394e+007, -2.62076e+008}
```

If I use these to extrapolate the original values, large error is produced:

```
X Y Z'
---- ---- -----
4 7 427.53
5 7 520.412
7 7 706.177
8 7 799.06
```

All values are smaller by about 26-30. This seems to be an edge case. In my use case, if I have values all on single line (horizontal or vertical), I will predict the values only for that line, turning it effectively into one-dimensional linear regression. But the error is unacceptable.

Here is the code:

```
static void print(float a, float b, float c, int x, int y) {
cout << "x=" << x << ", y=" << y << ", z=" << (a*x + b*y + c) << endl;
}
int main() {
Mat matX(4, 3, CV_32F);
Mat matZ(4, 1, CV_32F);
int idx = 0;
matX.at<float>(idx, 0) = 4;
matX.at<float>(idx, 1) = 7;
matX.at<float>(idx, 2) = 1;
matZ.at<float>(idx++, 0) = 458;
matX.at<float>(idx, 0) = 5;
matX.at<float>(idx, 1) = 7;
matX.at<float>(idx, 2) = 1;
matZ.at<float>(idx++, 0) = 554;
matX.at<float>(idx, 0) = 7;
matX.at<float>(idx, 1) = 7;
matX.at<float>(idx, 2) = 1;
matZ.at<float>(idx++, 0) = 734;
matX.at<float>(idx, 0) = 8;
matX.at<float>(idx, 1) = 7;
matX.at<float>(idx, 2) = 1;
matZ.at<float>(idx++, 0) = 826;
Mat res(3, 1, CV_32F);
cv::solve(matX, matZ, res, DECOMP_QR);
float a = res.at<float>(0);
float b = res.at<float>(1);
float c = res.at<float>(2);
cout << "a=" << a << ", b=" << b << ", c=" << c << endl;
print(a, b, c, 4, 7);
print(a, b, c, 5, 7);
print(a, b, c, 6, 7);
print(a, b, c, 7, 7);
print(a, b, c, 8, 7);
}
```