"Information contained herein is available to all without regard to
race, color, sex, or national origin."

Footnote in a paper by
Gale A. Buchanan, Director, Auburn University, Alabama
July 1985

First Things First

I am grateful to all who published emulators, firmware, software, information and what so ever for nothing and contribute by that to democratise evolution, a bit at least. It helped me a lot to prepare this paper, which in turn is also filed for free.
— Thank you.

Contents

I – Theory
II – Programming
III – The Fine Print
and Revision History


HP-41 Curve Fitting –
Three Special Cases of Deming Regression

In users' libraries and program collections you find several curve fitting solutions. Here is another one.

This contribution sets its main focus on how an appropriate Computer Algebra System (CAS) may help to improve old programs. If unassisted, calculus and algebra may be so time consuming (or too error prone for tricky tasks) to be considered as impossible. With CAS I was able to avoid an annoying test selecting the correct one out of two eventual solutions. No need any more to decide on concrete numbers because it is now definite and universal how to compute the only correct solution.

YX- and XY-Regression, source: wikimedia.org perpendicular distances to fitting line, source: wikimedia.org

Deming Regression is about curve fitting considering different quantum of measuring errors:

  1. x-values taken as free of error, all errors are within y-values,
  2. y-values taken as free of error, all errors are within x-values,
  3. errors are similarly within x- and y-values,
  4. anything else in-between.
I regard cases 1..3 as special cases, 1 and 2 denote the limits within a fitting line may stand if drawn by visual judgment only, and 3 is called Orthogonal Regression or at odd times also "isotonic" regression. For it the perpendicular distances from measuring point to fitting line are minimized. The sum of the squared perpendicular distances is minimized, to be precise.

Notes:

Red Neon Hint: Although not yet described in users' manual Reduce may display results with indexed variables, $x_i$ instead of $x(i)$. To get the results as shown throughout this paper enter at first (once per session is enough)
operator (x, y); print_indexed (x, y);

Which CAS to pick?
In case you'd like to redo what I describe in the following check if your CAS is capable to solve $\displaystyle f(\bar{x})=\sum_{i=1}^{n}(x_i-\bar{x})^2$ for ${\mathrm{d}f(\bar{x})}/{\mathrm{d}\bar{x}}=0$ to get the well-known formula of the arithmetic average value $\bar{x}=\displaystyle\frac{1}{n}\sum_{i=1}^n\,x_i$. This approach is the significant LS method (Least Square method, claimed by Carl Gauss to be his idea). Replacing $\bar{x}$ by $a$ for simpler input, Reduce returns as result of
solve(df(sum((x(k)-a)^2,k,1,n),a),a); $\:\Rightarrow\left\{a=\displaystyle\frac{\sum_{k=1}^n\,x_k}{n}\right\}$, what is outstanding compared by looking at some others I've "tested" once — HP-40G, HP-50G, HP-Prime, Wolfram Alpha, and Maxima.
("Tested" set in quotation marks as with no result by few trials I gave up quite quick, so it's highly likely I simply had not found in reasonable time how to make it work.)

"YX Regression" or L.R.

The correct term would be 'regression of y on x'. It's the first of the above-mentioned cases, widely known as the Linear Regression and comes with many calculator models as L.R. function – alas not with the bare HP-41. For this model find it within chapter 'Curve Fitting' of the "HP-41C Standard Applications Handbook", or also within chapter 'Curve Fitting' of "The HP-41 Advantage Advanced Solutions Pac" (ROM included in V41 installation).

The L.R. formulas result from applying the a. m. LS method to find $m$ and $b$ for $\sum_{i=1}^n (y_i - (mx_i +b))^2\:\Rightarrow min$.

In case you like to do it on your own, enter in Reduce
sum((y(k)-(m*x(k)+b))^2,k,1,n); (if requested declare x and y as operators or consider this Red Neon Hint),
next, to get the formula for $m$ without intermediate steps, enter
order n; solve(df(sub(solve(df(ws, b), b), ws), m), m);

Note: Further on $\sum$ in formulas stands for either $\displaystyle \sum_{i=1}^n$ or $\displaystyle \sum_{k=1}^n$

"XY Regression"

The correct term would be 'regression of x on y'. In this case it's about to minimize the horizontal distances $\displaystyle\sum (x_i - \frac{y_i - b}{m})^2\Rightarrow min$. Or to ease the work for Reduce: $\displaystyle\frac{\sum (m x_i - (y_i - b))^2}{m^2}\Rightarrow min$. Solve $\frac{\partial}{\partial b}=0$ for $b$ results in \[\left\{b=\displaystyle\frac{-(\sum x_k)\cdot m+\sum y_k}{n}\right\}\] Substitute with it $b$ in the formula above, then solve $\frac{\partial}{\partial m}=0$ for $m$ and get \[\left\{m=\frac{n\left(\sum y_k^2\right)-\left(\sum y_k\right)^2}{n\left(\sum x_k\cdot y_k\right)-\left(\sum x_k\right)\cdot\left(\sum y_k\right)}\right\}\]

A side glance to Wikipedia shows, this is correct. Or in concordance at least. Look out! Some publications just swap x and y, what is amazing simple, alas in the solution they do not swing it back again.

So far nothing to write home about, no equivocal formulas. The first two cases, as closely related to the third, are mentioned here for completeness only. But now to the tricky one  —

Orthogonal Regression

Find $m$ and $b$ of the fitting line $y=m x+b$ so the sum of squared perpendicular distances from metered points to that line is smallest possible. In other words: \[\sum \frac{(y_i - (mx_i +b))^2}{1+m^2}\Rightarrow min\]

In Reduce enter
order m, b, n; a := sum((y(k)-(m*x(k)+b))^2,k,1,n)/(1+m^2);

First get $b$ by
solve(df(a, b), b); $\Rightarrow$ $\left\{b=\displaystyle\frac{-m\left(\sum x_k\right)+\sum y_k}{n}\right\}$ what in fact is $b=\bar{y}-m\bar{x}$ and proves, as mentioned before, the line through the barycentre of the dot cloud.

Now replace $b$ in the first formula by this result and save it as new $a$:
a := sub(ws, a); $\Rightarrow$ \[a\mathrm{:=}\left(m^2 n\left(\sum x_k^2\right)-m^2\left(\sum x_k\right)^2-2mn\left(\sum x_ky_k\right)+2m\left(\sum x_k\right)\left(\sum y_k\right)+ n\left(\sum y_k^2\right)-\left(\sum y_k\right)^2\right)/\left(n\left(m^2+1\right)\right)\]

Next get $m$ by the LS method.
solve(df(a, m), m); $\Rightarrow$ \[ \left\{m= \left(-n\sum x_k^2+n\sum y_k^2+(\sum x_k)^2-(\sum y_k)^2 \\ +\mathrm{sqrt}\left((n^2\sum x_k^2)^2-2n^2\sum x_k^2\sum y_k^2-2n\sum x_k^2(\sum x_k)^2+ 2n\sum x_k^2(\sum y_k)^2+n^2(\sum y_k^2)^2+2n\sum y_k^2(\sum x_k)^2-2n\sum y_k^2(\sum y_k)^2+ 4n^2(\sum x_k y_k)^2-8n\sum x_k y_k\sum x_k\sum y_k+(\sum x_k)^4+2(\sum x_k)^2(\sum y_k)^2+ (\sum y_k)^4\right)\right)/\left(2(n\sum x_k y_k-\sum x_k\sum y_k)\right)\,\mathrm{,\,}\, \\ m= \left(-n\sum x_k^2+n\sum y_k^2+(\sum x_k)^2-(\sum y_k)^2 \\ -\mathrm{sqrt}\left((n^2\sum x_k^2)^2-2n^2\sum x_k^2\sum y_k^2-2n\sum x_k^2(\sum x_k)^2+ 2n\sum x_k^2(\sum y_k)^2+n^2(\sum y_k^2)^2+2n\sum y_k^2(\sum x_k)^2-2n\sum y_k^2(\sum y_k)^2+ 4n^2(\sum x_k y_k)^2-8n\sum x_k y_k\sum x_k\sum y_k+(\sum x_k)^4+2(\sum x_k)^2(\sum y_k)^2+ (\sum y_k)^4\right)\right)/\left(2(n\sum x_k y_k-\sum x_k\sum y_k)\right) \right\} \]

Decades ago (skip this blah-blah...) as student w/o CAS I gave up at this point due to lack of paper, pencil and time. Instead of going straight on I made a program that computed both eventual solutions to decide then which one to keep. Similar does William M. Kolb in his "Curve Fitting for Programmable Calculators", 1982 (find a PDF of it on W. Furlow's DVD in directory \Books). See part II, chapter "Isotonic Linear Regression", p. 21f (page 31..32 of the PDF). Cit: "The correct solution for most applications is given by..." End cit.

I do like programs that compute for most correctly. My routine I found again on an old disk is accurate in any case. It computes both, $m_1 = C + \sqrt{1+C^2}$ and $m_2 = C - \sqrt{1+C^2}$ and keeps the one with the same sign as the value in R06. Here the snippet in question (excerpt from a longish program, § stands for $\textstyle\sum$):

Registers content:
R06 = R15 * R14 - R10 * R 12 = n * §(xy) - §x * §y = w
R07 = R15 * R11 - R10^2      = n * §(x^2) - (§x)^2 = u
R08 = R15 * R13 - R12^2      = n * §(y^2) - (§y)^2 = v
121·LBL 07  RCL 08
  RCL 07  -  RCL 06  X=0?  | if w = 0 give up
  GTO 01  RCL 06  +  /     | (v - u) / 2w = C
  ENTER^  ENTER^  X^2  1   | Y: C - sqrt(C^2 + 1) = m2
  +  SQRT  ST- Z  +        | X: C + sqrt(C^2 + 1) = m1
  RCL 06  RCL Y  *  X>0    | (w * m1) > 0? m1 : m2
  R^  MEAN  RCL Z  *  -    | b in X, m in Y, Z u. T
(32 bytes)                 | (here GTO 01 has 3 bytes)
Today (if $u, v$ and $w$ are computed anyway for other reasons) I would calculate $m = C + sign(w)\sqrt{1+C^2}$ what saves few bytes.

But now – please! – forget that. As we all know, either relative or absolute minima and maxima of a function are at the roots of its first derivative, to see if it is a minima or maxima look at the sign of the second derivative at that given spot — full stop. So the task is (due to CAS: was) to determine where will which solution lead to a positive second derivative. With CAS no problem, w/o almost impossible. (Have a look how far Wolfram MathWorld goes. Cit: "After a fair bit of algebra, the result is ...", though in the result still a $\pm$ present, without hint how to proceed.)

To continue where I stopped before, apply some simplifications to the last result and save it as $r$.
let {n*sum(x(k)^2,k,1,n)-sum(x(k),k,1,n)^2=>u,
     n*sum(y(k)^2,k,1,n)-sum(y(k),k,1,n)^2=>v,
     n*sum(x(k)*y(k),k,1,n)-sum(x(k),k,1,n)*sum(y(k),k,1,n)=>w,
     v-u=>p, 2*w=>q};
 r := ws;
$\Rightarrow$ $\; r\mathrm{:=}\left\{m=\displaystyle \frac{\sqrt{p^2+q^2}+p}{q}\,\mathrm{,\,}\,m=\frac{-\sqrt{p^2+q^2}+p}{q}\right\}$

Next, in the 2nd derivative of $a$ substitute $m$ by the first found solution and show output in factored form.

on factor;
 sub(first r, df(a, m, 2));
$\Rightarrow$ $\; \displaystyle \frac{4\left(\sqrt{p^2+q^2}+p\right)\left(p^2+q^2\right)q^4}{n\left(\left(\sqrt{p^2+q^2}+p\right)^2+q^2\right)^3}$

For detecting the sign of this result I hope you agree that the denominator may be disregarded because it is for sure positive. So it is entirely sufficient to check the numerator.
sign(num(ws)); $\Rightarrow$ $\mathrm{sign}\displaystyle\left(\sqrt{p^2+q^2}+p\right)$

Well, seems this term is left as an exercise for the user. If $q=0=2w=\displaystyle 2\left(n\sum x_k y_k-\sum x_k\cdot\sum y_k\right)$ something is questionable anyway. So $\sqrt{p^2+q^2} > |p|$ and therefore – even for $p<0$ – the term $\sqrt{p^2+q^2}+p$ is never $\le 0$.

Hence it is the first solution shown above that will result in a minimum sum of the squared distances.
In any case.

What a finding! No need any more to first compute two choices and pick then the correct solution by "guesswork" or "obscure necromancy". That considerably simplifies and quickens the routine. Some algebraic transformation helps in addition.

— . —

Enough of theory, let's get physical. Some of the following programs are included in accompanying ZIP to be used – in case of RAW files – with an emulator like V41 or Emu42, the HPP file (HP-67/-97 magnetic card) with HP97 or NutEm/PC. (To recap theory read the RED file into REDUCE.)

One 41 Program – Three Solutions

Perhaps you are disappointed by my suggestion as it does not benefit from HP-41's distinctive capabilities to make programs user friendly. It is a compromise between comfort and size with a bias towards smaller size. There are options for improvement in either direction, farther squeezed in size or more conveniently like the following examples.
(Frankly, this program without literals, prompts, printout is my way of Spice series souvenir. The first RPN calculator I came in touch was a girlfriend's HP-32E. Probably one of the reasons why I purchased an HP-41C, an enduring source of pleasure, today again with Orthogonal Regression.)

HP-41 Curve Fitting, YXR, XYR, and OR. Image source: ./LR3K-41.png LBL 02 computes $m$ and $b$ for XY-Regression
$m$ is returned in Y and R04, $b$ in X and R05

LBL 11 is for program internal use only.

LBL 03 computes $m$ and $b$ for Orthogonal Regression
$m$ is returned in Y and R04,
$b$ in X and R05


LBL "$\textstyle\sum$" will clear the current statistics data to start a new run

LBL 01 computes $m, b$, and $r$ for YX-Regression
$r$ is returned in Z only,
$m$ in Y and R04, and
$b$ in X and R05
(66 bytes, END comprised)

How to use it

1 – XEQ "$\textstyle\sum$"
Sets start of statistics registers to R10 and clears statistics registers to prepare entering a new set of data points. The prompt to do so is "DATA ERROR" what is not common practise but saves several bytes. It's up to you to change that (before LBL 01).
2 – data entry/correction
Accumulating data points with $\scriptstyle\sum+$ and possible correction with $\scriptstyle\sum-$ is described in the "HP-41CX Owner's Manual".
Now do any of the following steps, in sequence at will:
— XEQ 01
Compute $m, b$ and $r$ for L.R. Except for $r$ the results are saved in registers 04..05 for later use, see comment for LBL 01 above. They are also put on the stack, use X<>Y and/or RDN to view either value.
— XEQ 02
Computes $m$ and $b$ for XY-Regression. The results are saved in registers for later use, they are also put on the stack, see comment for LBL 02 above.
— XEQ 03
Computes $m$ and $b$ for Orthogonal Regression. The results are saved as usual, see comment for LBL 03 above.
— data manipulation
At all times it is possible to go to step 2 to add or correct data. Anyway to #1 too.
Notes:

Same Program on Other Models

HP41 Predecessor and Successor

HP-97 and HP-67

HP-97 Curve Fitting, YXR, XYR, and OR. Image source: ./OR-97.png This program was developed on a virtual HP-97 using Tony Nixons' stand alone emulator. Results are printed but it runs also on an HP-67 w/o printer.

Usage

1 - Switch on your HP-97 (or start its emulator)
Calculators at that time had no 'continuous memory' so switch off and on is a reset to maidenliness.
Note: Only the sliders for printer mode and prgm/run mode may remain set as left at power down. So before inserting the program card make sure the machine is not in program mode but in run mode. Otherwise (if the card is not secured) the program on the card may be overwritten what is in most cases not what you intended.
2 - read in program from card
Run the card through the card reader and place it in the holder above the keys. So the labels on the card may serve as reminder which function key does now what.
3 - hit R/S or Shift a
After reading the program card the program pointer is on line 1. So R/S and Shift a do the same, prepare registers for data entry and set flag 0 to enable print. This sequence ends with Error in the display, remove it with CLX before moving on to next step.
4 - data entry
Data entry with $\scriptstyle\sum+$ and possible correction with $\scriptstyle\sum-$ is described in the "The HP-97 Programmable Printing Calculator Owner's Handbook and Programming Guide".
5 - now it's completely your choice
If you followed the instructions up to here in sequence you may continue with R/S to run through functions A - b - B - C. Or you may press out of sequence any of the function keys as described below.
A - L.R.
Compute $m$ and $b$ for Linear Regression (YX-R.) The results are saved in registers for later use, $m$ in R1 and $b$ in R0, printed ($m$ first) and also put on the stack, $m$ in Y and $b$ in X. Use X<>Y and/or RDN to view either value.
B - XY-Regression
Computes $m$ and $b$ for XY-Regression. Results are printed ($m$ first), put on the stack, $m$ in Y and $b$ in X, and saved in registers, $m$ in R1 and $b$ in R0.
C - Orthogonal Regression
Computes $m$ and $b$ for Orthogonal Regression. Results are printed ($m$ first), put on the stack, $m$ in Y and $b$ in X, and saved in registers, $m$ in R1 and $b$ in R0.
b - Correlation Coefficient
Computes $r$ and prints it. Result also in X.
any time - data manipulation
At all times it is possible to go to step 4 to add or correct data.
a - clear statistics registers
To start over with a new data set clear the previous one with Shift a.

HP-42S

HP-42S Curve Fitting, YXR, XYR, and OR. Image source: ./LR3K.png This program was developed on a virtual HP-42S using Christoph Giesselink's Emu42. Results are printed (with literals) if possible.

Usage

Step 1 – XEQ "LR3K"
This will start a menu which makes it pretty obvious how to use this program.
Step 2..n – hit any menu key
Only $\scriptstyle\sum+$ and $\scriptstyle\sum-$ expect something useful in X and Y, all others just start the indicated routine.
Step n+1 – hit R/S
With "b = ..." in the display the execution is halted with program pointer at line 77. R/S now shows CORR with literal (and prints if enabled and possible).
Step n+m – EXIT
At any time hit EXIT to quit the menu system.
Note: indicated program size does not include the END.

Other RPN Models

The simplification of Orthogonal Regression makes it now a candidate also for simple calculators formerly considered as unsuitable for this task. Of course, if a machine comes with inbuilt L.R. it is very much advantageous.

HP-10C

This model is very tight in memory. As program space is shared with registers' storage, the longer a program the less registers are available. To keep the statistics data and save results impedes lengthy programs.

Following routine computes coefficients for Orthogonal Regression only and saves them in registers for later use, $m$ in R6, $b$ in R7. Due to rectangular to polar coordinate conversion the routine is now short enough so besides $m$ also $b$ can be stored as well (at last — Whew!). First version I developed on my NutEm Coconut firmware interpreter (running for sure on one, perhaps on two emulated mainframes and – at minimum – on one real iron under z/VM 7.2 at that time when I installed it somewhere in Texas). This youngest update was done with NutEm/PC.

01-  42  3  L.R.       09-     26  EEX         17-  42  0  MEAN
02-     34  x<>y       10-     30  -	       18-   45 6  RCL 6
03-     36  ENTER^     11-  42 22  R->P	       19-     20  *
04-     40  +	       12-  42 36  Last X      20-     30  -
05-   44 6  STO 6      13-     40  +	       21-   44 7  STO 7
06-  42 48  SDEV       14-   45 6  RCL 6       22-  22 00  RTN
07-     10  /	       15-     10  /	       23-  22 00  RTN
08-  42 11  x^2	       16-   44 6  STO 6
Usage
  1. Make sure the program pointer is at top of program memory, quit programming mode if applicable,
  2. enter/modify statistics data as described in "HP-10C Owner's Handbook",
  3. hit R/S to compute Orthogonal Regression, when the program ends w/o error, find $m$ in R6 and Y, $b$ in R7 and X (displayed),
  4. repeat from step 2 (in case of error from step 1).

HP-12C

In my opinion programming the HP-12C is challenging. Amongst other things it lacks the L.R. but comes with linear estimation, correlation coefficient, and sample standard deviation what together makes some kind of "halfway L.R.". Fortunately this machine is not so extreme sparsely equipped with storage as the a. m. HP-10C.

I developed following routine on my Coconut firmware interpreter. It computes the coefficients for L.R., XY- and Orthogonal Regression.

*** L.R. ***		*** Orthogonal Regression ***
01-      35  CLX	11-   43 48  SDEV
02-   43  2  Y(X)	12-      10  /
03-      34  X<>Y	13-      36  ENTER^
04-   43 48  SDEV	14-      20  *
05-      10  /		15-      26  EEX
06-      20  *		16-      30  -
07-   44  7  STO 7	17-      36  ENTER^
08-      34  X<>Y	18-      36  ENTER^
09-   44  8  STO 8	19-      20  *
10-      31  R/S	20-   45  7  RCL 7
			21-      36  ENTER^
			22-      40  +
			23-   44  7  STO 7
*** XY-R. ***		24-      36  ENTER^
38-   43 48  SDEV	25-      20  *
39-      10  /		26-      40  +
40-      34  X<>Y	27-   43 21  SQRT
41-   43  1  X(Y)	28-      40  +
42-      33  RDN	29-   45  7  RCL 7
43-43,33 30  GTO 30 --> 30-      10  /
			31-   44  7  STO 7
			32-   43  0  MEAN
			33-   45  7  RCL 7
			34-      20  *
			35-      30  -
			36-   44  8  STO 8
			37-43,33 00  RTN
Usage
All three routines end with $b$ in X and R8, $m$ in Y and R7.
  1. Make sure the program pointer is at top of program memory, if applicable quit programming mode,
  2. enter/modify statistics data as described in "hp 12c financial calculator user's guide",
  3. hit R/S to compute L.R.,
  4. next R/S will compute Orthogonal Regression,
  5. repeat from step 2 – or...
  6. enter GTO 38 and hit R/S to compute XY-Regression,
  7. repeat from step 2.

Honestly, I found the basic idea for the L.R. part years ago on a forum. What I added besides the two STO is the X<>Y at the end to be in compliance with other HP-RPN calculators' L.R. function (only AOS models return $m$ in display register X).

HP-32S and -32Sii

Both calculators are equipped with L.R., so my program is just for XY-Regression and Orthogonal Regression. I developed it on a virtual HP-32Sii under Christoph Giesselink's Emu42.
B01 LBL B      C01 LBL C      D01 LBL D
B02 m          C02 m          D02 /
B03 r          C03 ENTER      D03 STO W
B04 x^2        C04 +          D04 y_bar
B05 GTO D      C05 STO Z      D05 x_bar
               C06 sig_y      D06 RCL W
               C07 sig_x      D07 *
               C08 /          D08 -
               C09 x^2        D09 STO Z
               C10 1
               C11 -
               C12 R-P
               C13 LASTx
               C14 +
               C15 RCL Z
Note: before adding more routines put a line D10 RTN to the very end.

Usage

  1. Enter statistical data as described in "HP 32SII RPN Scientific Calculator Owner’s Manual",
  2. XEQ B for XY-Regression, $b$ in X and Reg Z, $m$ in Y and Reg W,
  3. XEQ C for Orthogonal Regression, $b$ in X and Reg Z, $m$ in Y and Reg W,
  4. LBL D is for internal use only, alas there are no local labels on this machines.

HP-33C

HP-33C Curve Fitting, XYR and OR. Image source: ./OR on 33C.png My personal victory: both, XY-Regression and Orthogonal Regression in one routine of 26 bytes only. Compared with the lengthy and slow HP41 program of my students days this is a big joy.

Find my routine in adjoined screen shot of Tony Nixons many model running platform HP Classic Calculator Emulator +.

Usage
  1. Quit programming mode, shift right slider to position RUN if applicable,
  2. make sure the program pointer is at top of program memory (an OFF-ON sequence will do),
  3. enter/modify statistics data as described in "The HP-33E Programmable Scientific Calculator Owner's Handbook and Programming Guide",
  4. hit R/S to compute XY-Regression,
  5. hit R/S again to compute Orthogonal Regression,
  6. repeat from step 3.

Non-RPN Models

HP-22S

This de facto not programmable machine evaluates entered equations and offers with this feature a simple way to compute user defined formulas. Alas, it seems there is no option for a "combined result", so you have to get $m$ first to next get $b$ with it. Enter your data as mentioned in the "HP-22S Scientific Calculator Owner's Manual", paragraph "Entering Data for Two-Variable Statistics or Weighted Mean", then apply the following formulas to compute...
XY-Regression:
$\displaystyle m_{XY}=\frac{m_{YX}}{r^2}$; corresponding equation in HP-22S: M=m/SQ(r)
 .
Orthogonal Regression:
$m_{OR}=\displaystyle \frac{\sqrt{\left (2m_{YX}\right )^2 + \left ( \left (s_y/s_x\right )^2-1\right )^2} + \left ( \frac{s_y}{s_x} \right )^2 - 1}{2m_{YX}}$;
as equation in 22S: M=(r(2*m:SQ(sy/sx)-1)+SQ(sy/sx)-1)/(2*m)
 .
Ordinate intercept (for both a. m. cases, it is mandatory to compute $m$ first):
$b=\bar{y}-m\bar{x}$; in 22S: B=$\mathrm{\bar{y}}$-M*$\mathrm{\bar{x}}$

HP-48GX and -50G

To put those models in section "non-RPN models" is based on i) applying power the first time to an HP-50G makes it wakeup in ALG mode, not RPN, ii) size and location of the Enter key on the HP-50G shows obviously: this is not a RPN machine, iii) the LR function returns slope on 1st stack level and y-intercept on 2nd – that's the way AOS models do it. All "straight" RPN calculators of HP return y-intercept in X and slope in Y. This for that.

To compute XY- and Orthogonal Regression I hoped the CAS of the 50G would allow an approach as simple as on the HP-22S. Alas, the built-in LR function and typical statistical values are either not accessible from Equation Writer or I did not find how to do so in the many manuals within reasonable time. So I may only suggest two RPL routines I developed on an HP-48GX (under Emu48), which run unchanged also on an HP-50G and highly likely on a -49G too.

First things first:
Before running those routines set your machine to RPN mode, enter data as you would for the LR function, and set the LR function to solve for the Linear Fit model. The programs base on the slope returned by LR, also on the subroutine 'YINTS', which computes the Y-interception.
 .
Subroutine 'YINTS':
<< ΣY OVER ΣX * - NΣ / >>
 .
XY-Regression:
<< LR SWAP DROP CORR SQ / YINTS "b(XY)" ->TAG SWAP "m(XY)" ->TAG >>
 .
Orthogonal Regression:
<< LR SWAP DROP DUP DUP + SWAP CORR / SQ 1 - DUP2 R->C ABS + SWAP /
YINTS "b(or)" ->TAG SWAP "m(or)" ->TAG >>
Both routines return the results in the same sequence like the LR function and with literals (tagged) as well.

Other Non-RPN Models

I attempted to double the routines from HP-32Sii on HP-20S and -21S. It works, it computes the expected figures, alas the programs have 47 and 49 steps. Thus they lack that lean onward-pace of some of the sleekly RPN counterparts. I don't feel comfy with AOS, I do not publish those programs.

Conclusion

This refurbished Orthogonal Regression is probably only one example of many where CAS helps to quicken and to shrink ancient procedures. So the gain for a single case might be mean to marginal or questionable I have no doubt that CAS is more than just a toy. The benefit is the enhanced knowledge that helps for future tasks what will yield in an advantage on the whole.


For The Inquisitive
How the program listings made it to this HTML? There are several ways because I use virtual calculators only, from really simple to almost simple:

The Fine Print
All programs, routines, code snippets, and similar shown in this paper or contained in adjunct ZIP are published under the following QPL Public License V1.0

THE Q PUBLIC LICENSE version 1.0


Copyright (C) 1999 Trolltech AS, Norway.
Everyone is permitted to copy and
distribute this license document.

The intent of this license is to establish freedom to share and change the software regulated by this license under the open source model.

This license applies to any software containing a notice placed by the copyright holder saying that it may be distributed under the terms of the Q Public License version 1.0. Such software is herein referred to as the Software. This license covers modification and distribution of the Software, use of third-party application programs based on the Software, and development of free software which uses the Software.

Granted Rights

1. You are granted the non-exclusive rights set forth in this license provided you agree to and comply with any and all conditions in this license. Whole or partial distribution of the Software, or software items that link with the Software, in any form signifies acceptance of this license.

2. You may copy and distribute the Software in unmodified form provided that the entire package, including - but not restricted to - copyright, trademark notices and disclaimers, as released by the initial developer of the Software, is distributed.

3. You may make modifications to the Software and distribute your modifications, in a form that is separate from the Software, such as patches. The following restrictions apply to modifications:

a. Modifications must not alter or remove any copyright notices in the Software.

b. When modifications to the Software are released under this license, a non-exclusive royalty-free right is granted to the initial developer of the Software to distribute your modification in future versions of the Software provided such versions remain available under these terms in addition to any other license(s) of the initial developer.

4. You may distribute machine-executable forms of the Software or machine-executable forms of modified versions of the Software, provided that you meet these restrictions:

a. You must include this license document in the distribution.

b. You must ensure that all recipients of the machine-executable forms are also able to receive the complete machine-readable source code to the distributed Software, including all modifications, without any charge beyond the costs of data transfer, and place prominent notices in the distribution explaining this.

c. You must ensure that all modifications included in the machine-executable forms are available under the terms of this license.

5. You may use the original or modified versions of the Software to compile, link and run application programs legally developed by you or by others.

6. You may develop application programs, reusable components and other software items that link with the original or modified versions of the Software. These items, when distributed, are subject to the following requirements:

a. You must ensure that all recipients of machine-executable forms of these items are also able to receive and use the complete machine-readable source code to the items without any charge beyond the costs of data transfer.

b. You must explicitly license all recipients of your items to use and re-distribute original and modified versions of the items in both machine-executable and source code forms. The recipients must be able to do so without any charges whatsoever, and they must be able to re-distribute to anyone they choose.

c. If the items are not available to the general public, and the initial developer of the Software requests a copy of the items, then you must supply one.

Limitations of Liability

In no event shall the initial developers or copyright holders be liable for any damages whatsoever, including - but not restricted to - lost revenue or profits or other direct, indirect, special, incidental or consequential damages, even if they have been advised of the possibility of such damages, except to the extent invariable law, if any, provides otherwise.

No Warranty

The Software and this license document are provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.

Choice of Law

This license is governed by the laws of most of the remaining members of the European Union (if still existing).


The bottom line of this license is (at least for me), if you are able to improve any of the herein shown programs, routines, procedures, codes, methods, younameit, you have to inform me about that amendment. To fulfil this stipulation publish it in an adequate manner just as I did.

And, of course, do not use these programs. All consequences by disobeying this clause are in full your actual fault.

.....Mike, in October November December 2019 June October 2020 September December 2021

Changes since first release in October:
HP41 program improved, no "don't forget to XEQ 01 when..." instruction any more,
HP97 program now saves $m$ and $b$ for all three regression variants,
compacted a few algebraic transformations,
put a big thank you in front of all.

Changes since 2nd release in November:
HP41 program same functionality but three bytes tinier, thus a bit faster.

Changes since 3rd release in December:
link to the bundle of virtual HP calculators adjusted.

Changes since 4th release in June:
Reduce is able to show results with indices, added a hint about it,
HP41 program same functionality but one byte less,
adjusted few links, wordings, notes.

Changes since 5th release last year:
Minor adjustments for HP-32Sii as HP-32S is also available in Emu42 now,
added a paragraph about HP-22S because this machine is now emulated too.

Changes since 6th release in September:
Smaller routine for HP-42S,
added RPL-routines for HP-48GX and -50G,
added two..three links,
added a table of contents (kind of).