Inverse Harmonic and Inverse Digamma Functions

[Albert Chan]

I don't actually have HP Prime, below code tested with XCas

invPsi_guess(k) based on inverse harmonic function f3, optimized for small k = e^x
For small x, Ψ(x) ≈ -1/x. Flip the order, we have x ≈ -1/Ψ(x)

Guess is improved by Halley's method, 2 times

Code:

invPsi_guess(k) := k - k/(24.*k*k+1.) + 0.5;  // k = e^x

invPsi(x) := {
 local g, f0, f1;
 g := when(x < -1.8, -1./x, invPsi_guess(e^x));
 g -= (f0 := Psi(g)-x) * (f1 := Psi(g,1)) / (f1*f1 - 0.5*f0*Psi(g,2));
 g -= (f0 := Psi(g)-x) * (f1 := Psi(g,1)) / (f1*f1 - 0.5*f0*Psi(g,2));
};

invH(h) := invPsi(h-euler_gamma) - 1;

Test functions for round-trip ...

XCas> map(x -> Psi(invPsi(x)), range(-5,5))
      → [-5.0, -4.0, -3.0, -2.0, -1.0, 3.19839640883e-17, 1.0, 2.0, 3.0, 4.0]

XCas> map(k -> invPsi(Psi(10.^k)), range(-5,5))
      → [1e-05, 0.0001, 0.001, 0.01, 0.1, 1.0, 10.0, 100.0, 1000.0, 10000.0]

XCas> map(n -> invH(harmonic(n)), range(1,10))
      → [1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0]

Update: replaced ifte by when, to avoided the ifte bug.

[Albert Chan]

I try to plot invPsi(x), but failed. Why ?

XCas> plot(invPsi(x), x=-1 .. 1)

"Ifte: Unable to check test Error: Bad Argument Value"

[Albert Chan]

Compare these expressions, which should be equivalent:

XCas> subst(x>0, x=1)                              → true
XCas> subst(ifte(x>0, true, false), x=1)      → "Ifte: Unable to check test Error: Bad Argument Value"

ifte() tried to evaluate the expression, before substitution. However, "x>0", cannot be evaluated.

XCas> subst(ifte(x-x, true, false), x=1)       → false
XCas> subst(0/x, x=0)                              → 0

Again, expressions were evaluated first, before substitution.

This is branchless invPsi(x), to avoid above ifte() issues.
This version is able to do "plot(invPsi(x), x=-1 .. 1)"

Code:

invPsi(x) := {
 local g, f0, f1;
 g := [x >= -1.8, x < -1.8] * [invPsi_guess(e^x), -1/(x+not(x))];
 g -= (f0 := Psi(g)-x) * (f1 := Psi(g,1)) / (f1*f1 - 0.5*f0*Psi(g,2));
 g -= (f0 := Psi(g)-x) * (f1 := Psi(g,1)) / (f1*f1 - 0.5*f0*Psi(g,2));
};

Both "then" and "else" branch were evaluated, which is not ideal.
Also, -1/x guess replaced by -1/(x+not(x)), to patch against divide-by-zero.

XCas> subst(invPsi(x), x = 0)                          → 1.46163214497
XCas> subst(invPsi(x), x = -euler_gamma)       → 1.0

[Albert Chan]

Something unexpected when I try to test invPsi() accuracy.
Note: Psi(1) = -euler_gamma = -0.57721566490153286 ...

XCas> 1 - invPsi(-0.57721566490153286)       → 5.55111512313e-16
XCas> 1 - invPsi(-euler_gamma)                     → 1.713074127e-13 ???

Above were done on XCas 1.5.0-63 (win32), with 12-digits display float format.
Raising to 17-digits, expressions now agreed, error = 5.5511151231258e-16

Rounding constant with changing float display format is likely a XCas bug.

This bug seems to affect only euler_gamma.
Example, setting float display format to 6 digits:

XCas> 3.1415926535897931 - float(pi)                          → 0.0
XCas> 0.57721566490153286 - float(euler_gamma)       → 0.470199e-6 ???

[Albert Chan]

(09-02-2020 11:03 AM)Albert Chan Wrote:  I try to plot invPsi(x), but failed. Why ?

XCas> plot(invPsi(x), x=-1 .. 1)

"Ifte: Unable to check test Error: Bad Argument Value"

I found the fix !
Replace Ifte by when (piecewise also work)