Subject: 29.0.50; Calc: finding roots utpn doesn't work as advertised

From: Matt Armstrong <[email-masked]>
Date: Mon, 31 Oct 2022 11:10:22 -0700

Quick facts:
- all of these examples work with emacs -Q
- reproduces in Emacs 29, 27, 25

I think I have found that the calc package's "root" finding command does
not work as documented with calc's probability distribution functions.
Or, at least, not with "utpn", the probability distribution function for
the normal (gaussian) distribution.  I have found similar issues with
"utpt" (the Student's t-distribution), but will focus on "utpn" here.

The Emacs Calc package info page for Probability Distribution Functions
(https://www.gnu.org/software/emacs/manual/html_node/calc/Probability-Distribution-Functions.html)
has this to say about the normal distribution function:

> The ‘utpn(x,m,s)’ function uses a normal (Gaussian) distribution with
> mean ‘m’ and standard deviation ‘s’. It is the probability that such a
> normal-distributed random variable would exceed ‘x’.

I am interested in solving `utpn(x,m,s) = p` for `p` given a fixed `m`
and `s`.  The same info page has this to say about that:

> While Calc does not provide inverses of the probability distribution
> functions, the a R command can be used to solve for the inverse. Since
> the distribution functions are monotonic, a R is guaranteed to be able
> to find a solution given any initial guess.

SUPPOSITION: the above is true, especially this: "a R is guaranteed to be
able to find a solution given any initial guess."

The Root Finding info page
(https://www.gnu.org/software/emacs/manual/html_node/calc/Root-Finding.html)
has this to say about the a R command:

> The a R (calc-find-root) [root] command finds a numerical solution (or
> root) of an equation. (This command treats inequalities the same as
> equations. If the input is any other kind of formula, it is interpreted
> as an equation of the form ‘X = 0’.)
>
> The a R command requires an initial guess on the top of the stack, and a
> formula in the second-to-top position. It prompts for a solution
> variable, which must appear in the formula. All other variables that
> appear in the formula must have assigned values, i.e., when a value is
> assigned to the solution variable and the formula is evaluated with =, it
> should evaluate to a number. Any assigned value for the solution variable
> itself is ignored and unaffected by this command.
>
> When the command completes, the initial guess is replaced on the stack by
> a vector of two numbers: The value of the solution variable that solves
> the equation, and the difference between the lefthand and righthand sides
> of the equation at that value. Ordinarily, the second number will be zero
> or very nearly zero. (Note that Calc uses a slightly higher precision
> while finding the root, and thus the second number may be slightly
> different from the value you would compute from the equation yourself.)

WHAT I HAVE TRIED

I have tried entering the formula `utpn(x, 10, 1) = 0.5` and invoking `a
R` with an initial guess of `10`.  I then solve for `x`, expecting
`x = 10`.

Note that the correct answer of `x = 10` is "obviously" correct.  I am
passing `m = 10` for the "mean" arg to `utpn`, and by definition a
random value from the normal distribution has an `0.5` chance of being
larger than the mean.

To do this I type "C-x * *" to enter the *Calculator* buffer and then
type:

    ` utpn(x, 10, 1) = 0.5 RET
    10 RET                      <-- initial guess
    a R RET                     <-- invoke "root" function
    x RET                       <-- answer to prompt; solve for "x"

and I end up with this in the `*Calculator*` buffer:

    2:  utpn(x, 10, 1) = 0.5
    1:  [10., 0.]

This is correct.  The 'a R' command has returned a vector whose first
element is `x` and the second is an error delta (not important here).

Alternatively, evaluate this lisp to do the same thing:

    ELISP> (calc-eval '("root(utpn(x, 10, 1) = 0.5, x, 10)"))
    "[10., 0.]"

While correct, here I have *given* the answer to calc in the initial
guess (the root function's third arg).  The problem is that the function
doesn't do well when given other guesses.

```
` utpn(x,10,1)=0.5 RET
1.0 RET                  <---- a different guess
a R RET
x RET
```

results in this:

```
2:  utpn(x, 10, 1) = 0.5
1:  root(utpn(x, 10, 1) = 0.5, x, 1)
```

With this in the minibuffer:

> *Newton's method failed to converge: 4.8639203627839e17*

I'm finding that if the initial guess is close enough to the right answer,
in this case `10`, then `a R` works, otherwise not.  E.g. guessing `8.6`
fails but `8.7` works.

This behavior makes `a R` much less useful.  Also, documentation
suggests that any guess is expected to work (re-quoting from above with
my emphasis):

> Since the distribution functions are monotonic, 'a R' is guaranteed to
> be able to find a solution *given any initial guess*.

Have I found a bug here, or am I doing something wrong?

ADDENDUM

This shows the function failing to find a root for guesses not all that
far away from the actual answer:

ELISP> (cl-loop for x from 8.0 to 12.0 by 0.1 collect (list (format "%.1f" x) (calc-eval '("root(utpn(x,10,1)=0.5,x,$)") nil (prin1-to-string x))))
(("8.0" "root(utpn(x, 10, 1) = 0.5, x, 8.)")
 ("8.1" "root(utpn(x, 10, 1) = 0.5, x, 8.1)")
 ("8.2" "root(utpn(x, 10, 1) = 0.5, x, 8.2)")
 ("8.3" "root(utpn(x, 10, 1) = 0.5, x, 8.3)")
 ("8.4" "root(utpn(x, 10, 1) = 0.5, x, 8.4)")
 ("8.5" "root(utpn(x, 10, 1) = 0.5, x, 8.5)")
 ("8.6" "root(utpn(x, 10, 1) = 0.5, x, 8.6)")
 ("8.7" "[10., 0.]")
 ("8.8" "[10., 0.]")
 ("8.9" "[10., 0.]")
 ("9.0" "[10., 0.]")
 ("9.1" "[10., 0.]")
 ("9.2" "[10., 0.]")
 ("9.3" "[10., 0.]")
 ("9.4" "[10., 0.]")
 ("9.5" "[10., 0.]")
 ("9.6" "[10., 0.]")
 ("9.7" "[10., 0.]")
 ("9.8" "[10., 0.]")
 ("9.9" "[10., 0.]")
 ("10.0" "[10., 0.]")
 ("10.1" "[10., 0.]")
 ("10.2" "[10., 0.]")
 ("10.3" "[10., 0.]")
 ("10.4" "[10., 0.]")
 ("10.5" "[10., 0.]")
 ("10.6" "[10., 0.]")
 ("10.7" "[10., 0.]")
 ("10.8" "[10., 0.]")
 ("10.9" "[10., 0.]")
 ("11.0" "[10., 0.]")
 ("11.1" "[10., 0.]")
 ("11.2" "[10., 0.]")
 ("11.3" "[10., 0.]")
 ("11.4" "root(utpn(x, 10, 1) = 0.5, x, 11.4)")
 ("11.5" "root(utpn(x, 10, 1) = 0.5, x, 11.5)")
 ("11.6" "root(utpn(x, 10, 1) = 0.5, x, 11.6)")
 ("11.7" "root(utpn(x, 10, 1) = 0.5, x, 11.7)")
 ("11.8" "root(utpn(x, 10, 1) = 0.5, x, 11.8)")
 ("11.9" "root(utpn(x, 10, 1) = 0.5, x, 11.9)")
 ("12.0" "root(utpn(x, 10, 1) = 0.5, x, 12.)"))

In GNU Emacs 29.0.50 (build 3, x86_64-pc-linux-gnu, GTK+ Version
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