Q&A with Dr. Lawrence Shampine - Part I

Prof. Lawrence Shampine of Southern Methodist University is a world renowned expert in Numerical Integration Methods, and has researched extensively on the Rosenbrock techniques. I recently had the privilege of corresponding with Dr. Shampine to get a few topics clarified and answered on the Rosenbrock techniques, specifically related to the definition of the Jacobians. Dr. Shampine graciously agreed to me posting these Q&A’s on my blog. Here is part I of the correspondence.

Q1:

How does the definition of the Jacobian differ for Rosenbrock and W-methods?

A:

Rosenbrock methods are one-step methods that use a fixed number of values of the Jacobian matrix at each step.  Many can be viewed as coming from the approximate evaluation of an implicit formula by a fixed number of iterations of Newton's method.  When evaluating an implicit formula, the Jacobian is used only to enhance convergence of the iteration, so it need not be evaluated exactly. This is important in practice because the default in many programs is to approximate it by finite differences and often some of the components are not at all accurate.  Moreover, many popular programs based on implicit formulas save approximate Jacobians and reuse them as long as the rate of convergence is acceptable.  A Rosenbrock method is much simpler to implement, but its order depends on having exact values of the Jacobian, which might be computed from an analytical expression or numerically with automatic differentiation or complex-step approximation.  Wolfbrandt devised methods, now called W-methods, that look just like Rosenbrock methods, but the order does not require the matrix to be exactly equal to the Jacobian.  Because it has more information at its disposal, a Rosenbrock method of the same number of evaluations will be more accurate, but if you like the convenience of finite difference Jacobians, you do have to worry about whether they are sufficiently accurate that a Rosenbrock method will behave as expected.  A W-method is not sensitive to the quality of the approximate Jacobian.