PTC_Twiss 2x2 coupling matrix is incorrect

[ldeniau]

First, the alpha functions should not be scaled by onedp here:

MAD-X/src/madx_ptc_twiss.f90

Line 2179 in 88211ac

ax=kx*tw%alfa(1,1) * onedp -tw%alfa(1,2) * onedp /kx;

Second, the algebra is wrong here:

MAD-X/src/madx_ptc_twiss.f90

Line 2215 in 88211ac

cosvp = (ax*ay-bx*cy)/(ay*ay+cy*cy)

cosvp = (ax*ay-bx*cy)/(ay*ay+cy*cy) ! ok
sinvp = (ax*cy+ay*bx)/(ay*ay+cy*cy) ! ok
cosvm = (ax*ay+by*cx)/(ax*ax+cx*cx) ! bug: (ax*ay + cx*by)/(ay*ay + by*by)
sinvm = (ax*by-ay*cx)/(ax*ax+cx*cx) ! bug: (ay*cx - ax*cy)/(ay*ay + by*by)

cosv2 =  sqrt((1 + cosvp*cosvm - sinvp*sinvm)/2) ! bug: ± sqrt((1 + cosvp*cosvm + sinvp*sinvm)/2)
sinv2 = -sqrt((1 - cosvp*cosvm + sinvp*sinvm)/2) ! bug: ± sqrt((1 - cosvp*cosvm - sinvp*sinvm)/2)
cosv1 = -sqrt((1 + cosvp*cosvm + sinvp*sinvm)/2) ! bug: ± sqrt((1 + cosvp*cosvm - sinvp*sinvm)/2)
sinv1 = -sqrt((1 - cosvp*cosvm - sinvp*sinvm)/2) ! bug: ± sqrt((1 - cosvp*cosvm + sinvp*sinvm)/2)

Third, the disambiguation of the signs for the four square roots is complicated, it would be better to let the complex functions do the job (usually by going back and forth to polar coordinates).

[rdemaria]

see https://iopscience.iop.org/article/10.1088/1748-0221/5/10/P10010/pdf for reference 4.10

https://github.com/MethodicalAcceleratorDesign/MAD-NG/blob/559e0e5d2724fd8262924e5babb70f052c234140/src/madl_gphys.mad#L1800