light tracing iterations don't converge

[zpelgrims]

For example, the double_gauss_angenieux_1953_49mm out of the box has significant areas where the newton iterations do not converge.

Why is this the case?

• I already tried implementing a more robust numerical method (W4), as opposed to newton-raphson. That didn't make a difference.
• When replacing the jacobian (domega2_dx0) with the jacobian of another lens (fisheye from the lenses.js demo), IT WORKS. I did not expect this to happen.

Erik said I could see if there's a singularity in the jacobian if the determinant is 0. I should try that, maybe it is the case for the 'bad' lenses.

The magical jacobian from the fisheye is the following:

    domega2_dx0(0, 0) =  + -0.0841526  + 0.0180182 *lens_ipow(begin_dy, 2) + 0.0627794 *lens_ipow(begin_dx, 2) + 0.0014138 *begin_y*begin_dy + 6.7031e-05 *lens_ipow(begin_y, 2) + 0.00539724 *begin_x*begin_dx + 0.000241426 *lens_ipow(begin_x, 2) + -1.91925e-05 *begin_x*begin_y*begin_dx*begin_dy + -6.95202e-06 *lens_ipow(begin_y, 3)*lens_ipow(begin_dy, 3) + 0.128222 *begin_x*lens_ipow(begin_dx, 3)*lens_ipow(begin_dy, 2) + -6.09124e-06 *begin_x*lens_ipow(begin_y, 2)*begin_dx*lens_ipow(begin_lambda, 2) + 1.94672e-06 *lens_ipow(begin_x, 2)*lens_ipow(begin_y, 2)*lens_ipow(begin_dy, 2) + -0.00176443 *lens_ipow(begin_lambda, 8) + -0.0310725 *begin_x*begin_y*begin_dx*lens_ipow(begin_dy, 5) + 0.000176266 *begin_x*lens_ipow(begin_y, 2)*lens_ipow(begin_dx, 5) + -0.0150674 *lens_ipow(begin_x, 2)*lens_ipow(begin_dx, 6) + -1.52378e-11 *lens_ipow(begin_x, 2)*lens_ipow(begin_y, 6) + -1.64675e-07 *lens_ipow(begin_x, 5)*begin_dx*lens_ipow(begin_lambda, 2) + -1.11794 *lens_ipow(begin_dx, 4)*lens_ipow(begin_lambda, 6) + -3.65549e-15 *lens_ipow(begin_y, 10) + 7.65407e-08 *lens_ipow(begin_x, 5)*begin_y*begin_dx*begin_dy*lens_ipow(begin_lambda, 2) + -3.05221e-09 *lens_ipow(begin_x, 6)*lens_ipow(begin_lambda, 4) + -2.93555e-13 *lens_ipow(begin_x, 6)*lens_ipow(begin_y, 4) + -1.60463e-13 *lens_ipow(begin_x, 8)*lens_ipow(begin_y, 2) + -3.61904e-14 *lens_ipow(begin_x, 10);
    domega2_dx0(0, 1) =  + 0.0445784 *begin_dx*begin_dy + 0.00270553 *begin_y*begin_dx + 0.0014138 *begin_x*begin_dy + 0.000134062 *begin_x*begin_y + 0.00458532 *begin_y*lens_ipow(begin_dx, 3) + -9.59623e-06 *lens_ipow(begin_x, 2)*begin_dx*begin_dy + -0.0028004 *lens_ipow(begin_y, 2)*begin_dx*lens_ipow(begin_dy, 3) + -2.08561e-05 *begin_x*lens_ipow(begin_y, 2)*lens_ipow(begin_dy, 3) + -6.09124e-06 *lens_ipow(begin_x, 2)*begin_y*begin_dx*lens_ipow(begin_lambda, 2) + 1.29781e-06 *lens_ipow(begin_x, 3)*begin_y*lens_ipow(begin_dy, 2) + -2.57134e-05 *lens_ipow(begin_y, 4)*lens_ipow(begin_dx, 3)*begin_dy + -0.0155363 *lens_ipow(begin_x, 2)*begin_dx*lens_ipow(begin_dy, 5) + 0.000176266 *lens_ipow(begin_x, 2)*begin_y*lens_ipow(begin_dx, 5) + -3.04756e-11 *lens_ipow(begin_x, 3)*lens_ipow(begin_y, 5) + -0.121145 *begin_dx*begin_dy*lens_ipow(begin_lambda, 8) + -3.65549e-14 *begin_x*lens_ipow(begin_y, 9) + 1.27568e-08 *lens_ipow(begin_x, 6)*begin_dx*begin_dy*lens_ipow(begin_lambda, 2) + -1.67746e-13 *lens_ipow(begin_x, 7)*lens_ipow(begin_y, 3) + -3.56585e-14 *lens_ipow(begin_x, 9)*begin_y;
    domega2_dx0(1, 0) =  + 1.58352e-06  + 0.037641 *begin_dx*begin_dy + 0.00114241 *begin_y*begin_dx + 0.00232168 *begin_x*begin_dy + 0.000185005 *begin_x*begin_y + -0.0122638 *lens_ipow(begin_dx, 2)*lens_ipow(begin_dy, 2) + -0.000485762 *begin_y*begin_dx*lens_ipow(begin_dy, 2) + -8.08543e-06 *begin_x*lens_ipow(begin_y, 2)*begin_dy*begin_lambda + -0.000863194 *lens_ipow(begin_y, 2)*lens_ipow(begin_dx, 3)*begin_dy + 0.061474 *begin_x*lens_ipow(begin_dy, 5) + 0.0636202 *begin_x*lens_ipow(begin_dx, 4)*begin_dy + -1.23491e-09 *begin_x*lens_ipow(begin_y, 5) + -4.62094e-05 *lens_ipow(begin_x, 2)*begin_y*lens_ipow(begin_dx, 3) + 2.4501e-07 *lens_ipow(begin_x, 5)*lens_ipow(begin_dy, 3) + 8.40323e-08 *lens_ipow(begin_x, 2)*lens_ipow(begin_y, 4)*begin_dx*begin_dy*begin_lambda + 1.86325e-11 *lens_ipow(begin_y, 8)*begin_dx*begin_dy + 5.56914e-05 *lens_ipow(begin_x, 3)*lens_ipow(begin_y, 2)*lens_ipow(begin_dx, 2)*lens_ipow(begin_dy, 3) + 8.51035e-08 *lens_ipow(begin_x, 3)*lens_ipow(begin_y, 3)*lens_ipow(begin_dx, 2)*lens_ipow(begin_lambda, 2) + -2.29075e-13 *lens_ipow(begin_x, 7)*lens_ipow(begin_y, 3);
    domega2_dx0(1, 1) =  + -0.0841902  + 0.0628713 *lens_ipow(begin_dy, 2) + 0.000197341 *begin_dx*begin_dy + 0.0140972 *lens_ipow(begin_dx, 2) + 0.00543763 *begin_y*begin_dy + 0.000245926 *lens_ipow(begin_y, 2) + 0.00114241 *begin_x*begin_dx + 9.25026e-05 *lens_ipow(begin_x, 2) + 5.93229e-05 *lens_ipow(begin_y, 2)*lens_ipow(begin_dx, 2) + -0.000485762 *begin_x*begin_dx*lens_ipow(begin_dy, 2) + -8.08543e-06 *lens_ipow(begin_x, 2)*begin_y*begin_dy*begin_lambda + 0.134065 *begin_y*lens_ipow(begin_dx, 2)*lens_ipow(begin_dy, 3) + -0.00172639 *begin_x*begin_y*lens_ipow(begin_dx, 3)*begin_dy + -3.08727e-09 *lens_ipow(begin_x, 2)*lens_ipow(begin_y, 4) + -1.54031e-05 *lens_ipow(begin_x, 3)*lens_ipow(begin_dx, 3) + -0.000968929 *lens_ipow(begin_lambda, 7) + 0.00053879 *lens_ipow(begin_y, 3)*lens_ipow(begin_dy, 5) + -2.07699e-07 *lens_ipow(begin_y, 5)*begin_dy*lens_ipow(begin_lambda, 2) + -7.57489e-12 *lens_ipow(begin_y, 8) + 1.12043e-07 *lens_ipow(begin_x, 3)*lens_ipow(begin_y, 3)*begin_dx*begin_dy*begin_lambda + -1.13772 *lens_ipow(begin_dy, 4)*lens_ipow(begin_lambda, 6) + -3.63686e-09 *lens_ipow(begin_y, 6)*lens_ipow(begin_lambda, 4) + 1.4906e-10 *begin_x*lens_ipow(begin_y, 7)*begin_dx*begin_dy + 2.78457e-05 *lens_ipow(begin_x, 4)*begin_y*lens_ipow(begin_dx, 2)*lens_ipow(begin_dy, 3) + 6.38276e-08 *lens_ipow(begin_x, 4)*lens_ipow(begin_y, 2)*lens_ipow(begin_dx, 2)*lens_ipow(begin_lambda, 2) + -8.59032e-14 *lens_ipow(begin_x, 8)*lens_ipow(begin_y, 2);
    Eigen::Matrix2d invJ = domega2_dx0.inverse().eval();