BUG: missing complex lqmn derivatives when |z|<1.001

[jorenham]

Closes scipy/scipy#23589

[jorenham]

My VSCode autoformatter was a bit too enthusiastic, but it should be fixed now

[lucascolley]

I promise I haven't hacked your machine to include my changes in your commits

[jorenham]

Erm, I can't seem to find the lqmn tests; are there any?

No xsref tables either 🤔

[steppi]

Thanks @jorenham. That's a good catch. As for missing tests and tables, we're actually missing tests for all gufunc kernels. The process I followed to autogenerate a test suite based on the scipy.special tests only worked for ufunc kernels. I had intended to follow up adding tests for gufunc kernels, but ran out of funding for such work, and until recently, out of bandwidth for unfunded work. I'm trying to triage tasks based on the bandwidth I have, and tests for gufunc kernels is a very important one. I'll try to get that through over the next few weeks. In the meantime, we can just add a few hand coded tests here.

[jorenham]

I wrote smoke tests for lqmn. I based two on the python tests in scipy (https://github.com/scipy/scipy/blob/bdd3b0e77a3813c22c038c908d992b6de23ffcda/scipy/special/tests/test_legendre.py#L693-L708), and the other is the reproducer (real vs complex) for this issue.

[jorenham]

And FWIW, this fix makes it look more like the original fortran code (src):

        SUBROUTINE CLQMN(MM,M,N,X,Y,CQM,CQD)
C
C       =======================================================
C       Purpose: Compute the associated Legendre functions of
C                the second kind, Qmn(z) and Qmn'(z), for a
C                complex argument
C       Input :  x  --- Real part of z
C                y  --- Imaginary part of z
C                m  --- Order of Qmn(z)  ( m = 0,1,2,… )
C                n  --- Degree of Qmn(z) ( n = 0,1,2,… )
C                mm --- Physical dimension of CQM and CQD
C       Output:  CQM(m,n) --- Qmn(z)
C                CQD(m,n) --- Qmn'(z)
C       =======================================================
C
        IMPLICIT DOUBLE PRECISION (X,Y)
        IMPLICIT COMPLEX*16 (C,Z)
        DIMENSION CQM(0:MM,0:N),CQD(0:MM,0:N)
        Z = DCMPLX(X, Y)
        IF (DABS(X).EQ.1.0D0.AND.Y.EQ.0.0D0) THEN
           DO 10 I=0,M
           DO 10 J=0,N
              CQM(I,J)=(1.0D+300,0.0D0)
              CQD(I,J)=(1.0D+300,0.0D0)
10         CONTINUE
           RETURN
        ENDIF
        XC=ABS(Z)
        LS=0
        IF (DIMAG(Z).EQ.0.0D0.OR.XC.LT.1.0D0) LS=1
        IF (XC.GT.1.0D0) LS=-1
        ZQ=SQRT(LS*(1.0D0-Z*Z))
        ZS=LS*(1.0D0-Z*Z)
        CQ0=0.5D0*LOG(LS*(1.0D0+Z)/(1.0D0-Z))
        IF (XC.LT.1.0001D0) THEN
           CQM(0,0)=CQ0
           CQM(0,1)=Z*CQ0-1.0D0
           CQM(1,0)=-1.0D0/ZQ
           CQM(1,1)=-ZQ*(CQ0+Z/(1.0D0-Z*Z))
           DO 15 I=0,1
           DO 15 J=2,N
              CQM(I,J)=((2.0D0*J-1.0D0)*Z*CQM(I,J-1)
     &                -(J+I-1.0D0)*CQM(I,J-2))/(J-I)
15         CONTINUE
           DO 20 J=0,N
           DO 20 I=2,M
              CQM(I,J)=-2.0D0*(I-1.0D0)*Z/ZQ*CQM(I-1,J)-LS*
     &                 (J+I-1.0D0)*(J-I+2.0D0)*CQM(I-2,J)
20         CONTINUE
        ELSE
           IF (XC.GT.1.1) THEN
              KM=40+M+N
           ELSE
              KM=(40+M+N)*INT(-1.0-1.8*LOG(XC-1.0))
           ENDIF
           CQF2=(0.0D0,0.0D0)
           CQF1=(1.0D0,0.0D0)
           DO 25 K=KM,0,-1
              CQF0=((2*K+3.0D0)*Z*CQF1-(K+2.0D0)*CQF2)/(K+1.0D0)
              IF (K.LE.N) CQM(0,K)=CQF0
              CQF2=CQF1
25            CQF1=CQF0
           DO 30 K=0,N
30            CQM(0,K)=CQ0*CQM(0,K)/CQF0
           CQF2=0.0D0
           CQF1=1.0D0
           DO 35 K=KM,0,-1
              CQF0=((2*K+3.0D0)*Z*CQF1-(K+1.0D0)*CQF2)/(K+2.0D0)
              IF (K.LE.N) CQM(1,K)=CQF0
              CQF2=CQF1
35            CQF1=CQF0
           CQ10=-1.0D0/ZQ
           DO 40 K=0,N
40            CQM(1,K)=CQ10*CQM(1,K)/CQF0
           DO 45 J=0,N
              CQ0=CQM(0,J)
              CQ1=CQM(1,J)
              DO 45 I=0,M-2
                 CQF=-2.0D0*(I+1)*Z/ZQ*CQ1+(J-I)*(J+I+1.0D0)*CQ0
                 CQM(I+2,J)=CQF
                 CQ0=CQ1
                 CQ1=CQF
45         CONTINUE
        ENDIF
        CQD(0,0)=LS/ZS
        DO 50 J=1,N
50         CQD(0,J)=LS*J*(CQM(0,J-1)-Z*CQM(0,J))/ZS
        DO 55 J=0,N
        DO 55 I=1,M
           CQD(I,J)=LS*I*Z/ZS*CQM(I,J)+(I+J)*(J-I+1.0D0)
     &              /ZQ*CQM(I-1,J)
55      CONTINUE
        RETURN
        END

[jorenham]

I have no idea what the macos and windows test failures are about though (first time writing non-trivial C++) 😅. Could it be a caching thing, or did I mess up some obvious C pointer thingy or something? Is there a way for me to repro this on my linux (ubuntu) machine?

[steppi]

I have no idea what the macos and windows test failures are about though (first time writing non-trivial C++) 😅. Could it be a caching thing, or did I mess up some obvious C pointer thingy or something? Is there a way for me to repro this on my linux (ubuntu) machine?

Thanks @jorenham for writing the tests. I'll take a look at the failures in the next couple days.