HOMPACK90 is a Fortran 90 version of the Fortran 77 package HOMPACK (Algorithm 652), a collection of codes for finding zeros or fixed points of nonlinear systems using globally convergent probability-one homotopy algorithms. Three qualitatively different algorithms— ordinary differential equation based, normal flow, quasi-Newton augmented Jacobian matrix—are provided for tracking homotopy zero curves, as well as separate routines for dense and sparse Jacobian matrices. A high level driver for the special case of polynomial systems is also provided. Changes to HOMPACK include numerous minor improvements, simpler and more elegant interfaces, use of modules, new end games, support for several sparse matrix data structures, and new iterative algorithms for large sparse Jacobian matrices.

https://users.wpi.edu/~walker/Papers/hompack90,ACM-TOMS_23,1997,514-549.pdf

Algorithm 777: HOMPACK90: a suite of Fortran 90 codes for globally convergent homotopy algorithms

In this paper, an analytical technique, namely the homotopy analysis method (HAM) is applied to obtain an approximate analytical solution of the Burgers–Huxley equation. This paper introduces the two theorems which provide us with a simple and convenient way to apply the HAM to the nonlinear PDEs with the power-law nonlinearity. The homotopy analysis method contains the auxiliary parameter ħ , which provides us with a simple way to adjust and control the convergence region of solution series.

The homotopy analysis method to solve the Burgers–Huxley equation

Analytical solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient

In this paper, we present a new implementation of the harmonic balance method which extends its applicability to circuits 2-3 orders of magnitude larger than was previously practical. The results reported here extend our previous work which only considered large circuits operating in a mildly nonlinear regime. The new implementation is based on quadratically convergent Newton methods and is able to simulate general nonlinear circuits. The significant efficiency improvement is achieved by use of Krylov subspace methods and a problem-specific preconditioner for inverting the harmonic balance Jacobian matrix. The analysis of radio-frequency mixers, implemented in integrated circuit technology, is an important application of our new method. We describe the theory behind the method, then report performance results on a complete receiver design using detailed transistor models.

Efficient frequency domain analysis of large nonlinear analog circuits

Finding all possible positions of a Stewart platform from the knowledge of the lengths of its limbs amounts to solving a set of nonlinear equations, the number of real solutions of which is, in the general case, unknown a priori. This paper shows how Hansen's algorithm (1992), based upon interval analysis, can be used to get all solutions in an approximate but guaranteed way even when the platform is nonplanar and the equations are nonpolynomial and with real coefficients. Computation is simple enough to be performed on a personal computer.

T WatsonLayne

Papers

Globally convergent homotopy methods