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A number of basic algorithms for the numerical analysis and control of bifurcation phenomena are described. The emphasis is on algorithms based on pseudoarclength continuation for algebraic equations. Several illustrative examples computed with the AUTO software package are included. Part II of this paper deals with ordinary differential equations and will appear in the next issue.

Abstract. Fixed-point itcrative procedures for solving nonlinear parameter dependent problems can converge for some interval of parameter values and diverge as the parameter changes. The Recursive Projection Method (RPM), which stabilizes such procedures by computing a projection onto the unstable subspace is presented. On this subspace a Newton or special Newton iteration is performed, and the fixed-point iteration is used on the complement. As continuation in the parameter proceeds, the projection is efficiently updated, possibly increasing or decreasing the dimension of the unstable subspace. The method is extremely effective when the dimension of the unstable subspace is small compared to the dimension of the system. Convergence proofs are given and pseudo-arclength continuation on the unstable subspace is introduced to allow continuation past folds. Examples are presented for an important application of the RPM in which a "black-box" time integration scheme is stabilized, enabling it to compute unstable steady states. The RPM can also be used to accelerate iterative procedures when slow convergence is due to a few slowly decaying modes.

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