Selected documents by Peter Jäckel

NL2SOL is a highly regarded nonlinear least-squares implementation with many advanced features. For example, it keeps the second-derivative term we dropped in the Levenberg-Marquardt method whenever it would be better to do so, a so-called full Newton-type method.

1. The first identified-and-fixed problem one is a simple code typo where what was meant to be V(XMSAVE) was accidentally coded up as X(XMSAVE) (lines 2328 and 2335 in the original ACM TOMS 573 code).
   
   
2. The second one is related to a rare occurrence of a mathematical failure when an estimate of an upper bound happens to be too low for the algorithm to succeed. This was discovered by a colleague of mine nearly 20 years ago who at the time consulted with the original NL2SOL authors by email to confirm his findings. I wish to preserve his anonymity but declare that it was not me who identified and solved the issue.
   
   
3. The last potential pitfall was when the input specification of the false convergence threshold is not greater than DBL_EPSILON since the algorithm in function ASSESS does essentially not allow for this setting. However, since we often want to suppress spurious false convergence assessments, we typically have V[XFTOL] == 0.0, i.e., we often set the false convergence threshold to zero. This led to NL2SOL then reporting that it is unable to compute the Jacobian at a location where it was in fact able to compute the function value, which, when we provide the Jacobian analytically, should not be possible. I fixed this by restoring the the stored last good point and returning with FALSE CONVERGENCE (IV[1] = 8) which is in most situations an acceptable form of convergence.

• John Burkardt maintains a site dedicated to NL2SOL and provides a FORTRAN90 version.
• The Decision Tree for Optimization Software provides a link to what it describes as f90-translations of newer versions.

The NL2SOL algorithm [DGW81] is a secant-based least-squares algorithm that is q-superlinearly convergent. It adaptively chooses between the Gauss-Newton Hessian approximation and this approximation augmented by a correction term from a secant update. NL2SOL tends to be more robust (than conventional Gauss-Newton approaches) for nonlinear functions and “large residual” problems, i.e., least-squares problems for which the residuals do not tend towards zero at the solution.

    if (alphak <= 0.) {
      alphak = uk * .5;
    }
    if (alphak <= 0.) {
      alphak = uk;
    }

1. Introduction: the case for multi-dimensional HJB calculations of moderate accuracy
2. Scattered (cluster) data interpolation
3. Meshless backward induction
4. There’s more to life than Crank-Nicolson

• 40 years of evolution of smile parametrisation
• Commitment to Spatial Discreteness and the importance of being Metzler
• Stencil Viability, Boundary Conditions, Continuous-Time stability, Ito's lemma of pure jump processes for Exact Martingales, Algebraic Splitting, Anti-Diffusion Limiting, and other lessons to learn from 60 years of Computational Fluid Dynamics.
• From Continuous Time to Numerical Integration
• Box Transition Probability Translation
• Cash dividends without arbitrage or approximations

• Monte Carlo simulation (Peter Jäckel and Eckhard Platen) - this is the introduction to the Encyclopedia's chapter on Monte Carlo simulation co-edited by Eckhard Platen and Peter Jäckel.
• Weighted Monte Carlo (Marco Avellaneda and Peter Jäckel).
• Simulation of square-root processes (Leif Andersen, Christian Kahl, and Peter Jäckel).
• Exercise boundary optimization methods (Peter Jäckel and Leif Andersen).
• Inflation products and inflation models (Peter Jäckel and Jerome Bonneton).