This routine performs a least-squares fit of a 2d parabola to a set of data. The form of the parabola is:-

$$z=A0+A1\ast {\left(x-X0\right)}^2+A1\ast {\left(y-Y0\right)}^2$$

(37)

The apex of the parabola is at X0,Y0 and its value there is A0.

If status is good on entry the routine will work through the data calculating the various sums required for the least-squares method. Data with bad quality or zero variance are ignored. If no ‘good’ data are found then an error will be reported and the routine will return with bad status.

A matrix equation of the form $M\ast B=Z$ is constructed, where:-

$$M=\left(\begin{array}{cccc}\hfill \sum \frac{1}{var}\hfill & \hfill \sum \frac{x}{var}\hfill & \hfill \sum \frac{y}{var}\hfill & \hfill \sum \frac{x^2+y^2}{var}\hfill \\ \hfill \sum \frac{x}{var}\hfill & \hfill \sum \frac{x^2}{var}\hfill & \hfill \sum \frac{xy}{var}\hfill & \hfill \sum \frac{x^3+x{y}^2}{var}\hfill \\ \hfill \sum \frac{y}{var}\hfill & \hfill \sum \frac{xy}{var}\hfill & \hfill \sum \frac{y^2}{var}\hfill & \hfill \sum \frac{x^{2}y+y^3}{var}\hfill \\ \hfill \sum \frac{x^2+y^2}{var}\hfill & \hfill \sum \frac{x^3+x{y}^2}{var}\hfill & \hfill \sum \frac{x^{2}y+y^3}{var}\hfill & \hfill \sum \frac{{\left(x^2+y^2\right)}^2}{var}\hfill \\ \hfill \hfill \end{array}\right)$$

(38)

$$B=\left(\begin{array}{c}\hfill A0+A1\ast X{0}^2+A1\ast Y{0}^2\hfill \\ \hfill -2\ast A1\ast X0\hfill \\ \hfill -2\ast A1\ast Y0\hfill \\ \hfill A1\hfill \\ \hfill \hfill \end{array}\right)$$

(39)

$$Z=\left(\begin{array}{c}\hfill \sum \frac{z}{var}\hfill \\ \hfill \sum \frac{xz}{var}\hfill \\ \hfill \sum \frac{yz}{var}\hfill \\ \hfill \sum \frac{\left(x^2+y^2\right)z}{var}\hfill \\ \hfill \hfill \end{array}\right)$$

(40)

where x, y, z and var are the x,y coords, value and variance of the measured points and the sums are over all valid points.

SCULIB_INVERT_MATRIX is called to invert the matrix and SCULIB_FIT_MULT to multiply Z by the inverse. This yields the fitted value for B from which the other fit parameters are derived.

CALL SCULIB_FIT_2D_PARABOLA (N, DATA, VARIANCE, QUALITY, X, Y, A0, A1, X0, Y0, Z_PEAK, Z_PEAK_VAR, BADBIT, STATUS)

N = INTEGER (Given)

DATA (N) = REAL (Given)

VARIANCE (N) = REAL (Given)

QUALITY (N) = BYTE (Given)

X (N) = REAL (Given)

Y (N) = REAL (Given)

A0 = REAL (Returned)

A1 = REAL (Returned)

X0 = REAL (Returned)

Y0 = REAL (Returned)

Z_PEAK = REAL (Returned)

Z_PEAK_VAR = REAL (Returned)

BADBIT = BYTE (Given)

STATUS = INTEGER (Given and returned)

If the variances are not a true reflection of the errors on the data then very strange numbers can result.

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