One-Step Solution in Numerical Analysis

• One-Step Solution is an iterative method that uses a forward-looking update strategy to accelerate fixed point convergence by addressing significant residuals.
• It employs a dual-vector system, combining a history vector and a fluid vector, to anticipate and correct coordinate-wise updates.
• The approach is applied to both linear (D-iteration) and nonlinear systems, offering faster convergence than traditional methods like Jacobi and Gauss–Seidel.

A one-step solution, in the context of computational mathematics and numerical analysis, refers to an approach that advances the solution of a problem by considering only the current state (and possibly the current input) at each update, rather than requiring multiple previous states as in multistep methods or multiple passes as in iterative approaches. In certain contexts—such as the “one step back” iterative method for fixed point problems (Hong, 2013)—the one-step solution is realized through a forward-looking update strategy that anticipates the consequence of each coordinate update and optimizes the update sequence to accelerate convergence. This paradigm is particularly significant for both linear and nonlinear fixed point equations, offering advantages in computational efficiency and convergence properties.

1. Formulation and Core Methodology

The one step back (OSB) approach is introduced to solve fixed point problems of the form

$X = H(X)$

starting from an initial vector $X_0$. Unlike Jacobi or Gauss–Seidel methods, which either update all coordinates simultaneously or in a cyclic sequence, OSB augments the standard process by maintaining two vectors:

• $H_n$: a "history" vector tracking the accumulated coordinate-wise updates
• $F_n$: a residual or “fluid” vector measuring diffusion error that propagates through the system

The iterative system is governed by

$H_0 = X_0$

$H_n = H_{n-1} + J_{(i_n)} F_{n-1}$

$F_0 = H(H_0) - H_0$

$$F_n = (I - J_{(i_n)}) F_{n-1} + H(H_n) - H(H_{n-1})$$

where $J_{(i_n)}$ is a coordinate selector—a diagonal matrix with a 1 in the $(i_n, i_n)$ entry—and $\{i_n\}$ is the update sequence. This system enables the algorithm to leverage the effect of each individual coordinate update on the global residual.

2. Anticipation and Coordinate Optimization

A defining OSB feature is its anticipation of the effect of a coordinate update. The term $H(H_n) - H(H_{n-1})$ in the update for $F_n$ acts as a predictive correction, quantifying how much an update in a single coordinate changes the intended fixed point mapping. This forward-looking correction allows the update sequence $\{i_n\}$ to be chosen explicitly for maximal effect. A typical optimization is to select

$i_n = \arg \max_i |(F_{n-1})_i|$

thus always addressing the most significant current residual.

This coordinate selection strategy sharply contrasts with standard Jacobi or Gauss–Seidel methods, which do not explicitly prioritize coordinates by their impact. In systems where some coordinates are intrinsically harder to converge, this OSB-guided update ordering produces significantly faster overall convergence.

3. State Vector Structure and Step Loss Recovery

A unique aspect of OSB is the need to manage an expanded state, composed of both $H_n$ and $F_n$, rather than a single iterated vector. This dual-vector structure enables the algorithm to carry both accumulated updates and remaining residuals, permitting anticipation and correction at each step.

Notably, the approach incurs a “one-step loss”: the first update cannot use the corrective term $H(H_n) - H(H_{n-1})$ until two iterates are available. Nonetheless, this loss is rectified at convergence, as the final estimate is recovered by summing $H_n + F_n$, ensuring that no information is ultimately lost.

4. Applications to Linear and Nonlinear Fixed Point Problems

The OSB method generalizes across both linear and nonlinear equations.

• Linear equations (D-iteration): When $H$ is linear, denoted $P$, the OSB scheme becomes

$$F_n = (I - J_{(i_n)}) F_{n-1} + P J_{(i_n)} F_{n-1}$$

This is closely related to the D-iteration algorithm, known for high efficiency in large-scale, sparse linear systems.

• Nonlinear equations: For instance, consider

$$H(x,y,z) = (\sqrt{x y} + 1,\, (x+z)/4 + 1,\, (x+y)/4)$$

with a known fixed point near $(4.247, 2.482, 1.682)$. Using the OSB approach, coordinate updates exploit difference increments, and targeted selection for the highest residual can yield convergence rates two orders of magnitude faster than Jacobi or Gauss–Seidel after 10 iterations under heterogeneous initializations.

This flexibility allows OSB to outperform traditional methods, especially in large or inhomogeneous systems where update impact varies sharply across coordinates.

5. Comparative Performance and Computational Considerations

Empirical studies in the referenced work demonstrate that OSB achieves:

• Accelerated convergence compared to Jacobi and Gauss–Seidel, particularly for heterogeneous initial conditions.
• The ability to recover traditional methods as special cases (fixed $\{i_n\}$), situating OSB as a generalization rather than a competitor.

An additional advantage for the linear case is compatibility with distributed and asynchronous computing architectures, as the OSB (D-iteration) update admits independent update scheduling per coordinate.

The chief computational overhead arises from maintaining two vectors and the need to compute corrections at each step, but this is often offset by the dramatic reduction in iteration count needed for convergence.

6. Broader Context and Limitations

The OSB formalism highlights the benefits of exploiting coordinate impact and explicit anticipation in iterative solvers for fixed point problems. While initial overhead is incurred in bookkeeping and in the first step (due to the look-ahead requirement), these are systematically recovered in final convergence. The method is particularly effective for problems where conventional uniform update schemes fail to exploit the structure of residual propagation.

Its principal limitation is the requirement of additional memory for the dual state vectors and potentially more complex update logic compared to vanilla Jacobi or Gauss–Seidel. For very small or homogeneous systems, the gains are less pronounced, but for large-scale or ill-conditioned problems, the OSB approach yields clear and theoretically justified computational benefits.

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A one-step solution in this context thus encapsulates a rigorous, predictive coordinate-wise iterative framework, leveraging residual anticipation and impact-based update scheduling, with demonstrated superior performance for both linear and nonlinear fixed point problems (Hong, 2013).