I am solving a large system of nonlinear equations. The Jacobian for this system of equations is a block tridiagonal matrix. When solved using Newton's method, the equation residuals may keep oscillating around lower values. In this case I have found that a rank one correction to the Jacobian, i.e. the broyden method, converges more quickly. The problem is that the traditional broyden method of correcting the Jacobian destroys its sparse pattern. Is there a way to update the Jacobi's method while maintaining the (block) tridiagonal matrix?
In case of linear system, is common to extend the Thomas algorithm to block tridiagonal matrix. Are you using a linearization?
Yes, the basic idea of Newton's method is to solve for the zeros of a system of equations based on a linear approximation of the function at a point. Thus each Newton step requires solving a system of linear equations Ax=b. A is a Jacobian. In this case, A is block tridiagonal, though not so with the broyden correction. In that case you can't continue with Thomas' algorithm. Instead of continuing to find A(k+1) numerically, which would be time consuming, I want just make some corrections to A(k) to get A(k+1), but keep A block tridiagonal.
Why don’t you solve the block tridiagonal system without correction?
If no corrections are made to the Jacobian in each iteration of Newton's method, then the search direction will become no longer the descent direction of the residuals of the equation. While this method of not updating the Jacobian exists, it does not work for my system of nonlinear equations.
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