Dynamic Block-Triangular Maps

• Dynamic block-triangular maps are mappings with block-structured geometry that preserve invariant fibers in discrete dynamical systems and facilitate efficient GPU computations.
• They structure computational domains into lower-triangular segments, reducing wasted resources from O(n²) to O(n) and achieving performance improvements of up to 15% over naive approaches.
• These maps are pivotal for analyzing convergence in difference equations and optimizing recursive algorithms, thereby impacting numerical linear algebra and simulation tasks.

Dynamic block-triangular maps represent a confluence of ideas arising in both dynamical systems theory—where they generalize classical triangular maps in the analysis of discrete-time recurrences—and high-performance computing, where lower-triangular domain decompositions are leveraged for efficient GPU resource utilization. Their formalism encompasses mappings that exploit triangular (block-structured) geometries, preserve natural fibrations of the underlying space, and exhibit properties central to convergence analysis and computational optimization. Block-triangular mappings are fundamental in the study of difference equations, quasi-homogeneous maps, and algorithms requiring data locality in triangular domains.

1. Formal Definitions and Structural Properties

A planar block-triangular (triangular) map is a self-map $F\colon \mathbb{R}^2 \to \mathbb{R}^2$ of the form

$F(x, u) = (f_0(u) + f_1(u) x,\; \phi(u)),$

where $f_0, f_1, \phi$ are continuous functions. This map preserves the natural foliation $\mathcal{F} = \{u = \text{const}\}$ since $u_{n+1} = \phi(u_n)$ is independent of $x_n$. In block notation, this reads: $$F\binom{x}{u} = \begin{pmatrix} f_1(u) & f_0(u) \ 0 & \phi'(u) \end{pmatrix} \binom{x}{1} + \binom{0}{\phi(u) - u}.$$ Such maps arise naturally in discrete dynamics, matrix decompositions, and recursive algorithmic applications where off-diagonal dependencies are suppressed, and the system can be analyzed fiberwise (Cima et al., 2013).

In GPU computation, the block-triangular paradigm appears when mapping thread blocks onto two-dimensional domains with triangular structure. Given a problem size $N(N+1)/2$ (lower-triangular part of an $N \times N$ matrix), it is standard to define a mapping $$g: \{0, 1, \ldots, n(n+1)/2-1\} \to \{(i, j) : 0 \leq j \leq i < n\}$$ by: $$i = \left\lfloor \sqrt{\frac{1}{4} + 2\lambda} - \frac{1}{2} \right\rfloor, \quad j = \lambda - \frac{i(i+1)}{2}$$ for a 1D block index $\lambda$, with $n = \lceil N/\rho \rceil$ and block side $\rho$ (Navarro et al., 2013).

2. Dynamics and Basin Geometry

The orbits of triangular maps are entirely characterized by the interplay between $\phi(u)$ and the fiberwise maps in $x$ (Cima et al., 2013). For the system: $$\begin{cases} x_{n+1} = f_0(u_n) + f_1(u_n)\, x_n,\ u_{n+1} = \phi(u_n), \end{cases}$$ analysis focuses on attracting fibers $\{u = u_*\}$, which are invariant under $\phi$ and stable in the sense that $u_n \to u_*$ for initial $u_0$ in their basin. The classification of asymptotic behavior is determined by $|f_1(u_*)|$:

• If $|f_1(u_*)| < 1$, then every orbit with $u_n \to u_*$ converges to $(x_*, u_*)$ with $x_* = f_0(u_*) / (1 - f_1(u_*))$ (global attractor on the fiber).
• The critical cases $|f_1(u_*)| = 1$ require a "fast–enough convergence" hypothesis: if $f_1(u_n) \to f_1(u_*)$ and $f_0(u_n) \to f_0(u_*)$ summably quickly, then orbits remain bounded and approach fixed or periodic points determined by the data, else divergence is possible.

These phenomena are robust to generalizations in higher dimensions, where the block structure persists and analogous convergence criteria can be posed for matrix coefficients (Cima et al., 2013).

3. Computational Mappings and Algorithmic Efficiency

Block-triangular mappings in GPU computation address the inefficiency of launching a naive bounding-box (BB) grid over triangular domains. The LTM mapping $g(\lambda)$ achieves an $O(1)$ per-block overhead and reduces the number of wasted blocks from $O(n^2)$ (BB) to $O(n)$, approaching the information-theoretic lower bound for such decompositions (Navarro et al., 2013). Comparison of mapping strategies yields:

Strategy	Wasted Blocks	Thread Organization	Relative Speed vs BB (Kepler)
BB	$O(n^2)$	Preserved	Baseline
LTM ($g(\lambda)$)	$O(n)$	Preserved	+12%–15%
RB (Rectangular Box)	$O(n)$	Retiled, less optimal	+14%–16%
UTM	$O(n^2)$	Higher overhead	Slower
REC (Recursive Partition)	Varies ($O(n)$ for $N \gg 1$)	Compromised at small $N$	+5% at large $N$

Mapping $g(\lambda)$ is computed using $\sqrtf$ or $\rsqrtf$ and incurs minimal thread divergence; intra-block divergence is confined to $O(n)$ diagonal blocks. Memory coalescing and intra-block data layouts are preserved, yielding optimal alignment for row-major storage (Navarro et al., 2013).

4. Applications and Typical Examples

Block-triangular and triangular map formalisms appear in a range of applications:

• Matrix computations: LU decomposition algorithms partition work along triangular domains.
• Simulation and collision detection: Many spatial computations over triangular tilings utilize block-triangular mappings for domain traversal.
• Discrete-time dynamics: Difference equations of the form $x_{n+1} = f_0(u_n) + f_1(u_n) x_n$, $u_{n+1} = \phi(u_n)$, including multiplicative and additive second-order recurrences, quasi-homogeneous mappings, and planar systems (Cima et al., 2013).
• Euclidean distance maps: EDM problems reduce to two-dimensional triangular problems suitable for LTM mapping, yielding up to 15% end-to-end speedups in GPU implementations (Navarro et al., 2013).

5. Analysis of Limit Dynamics and Examples

Explicit classification of limit sets on invariant fibers includes:

• Global attractor: For $|f_1(u_*)| < 1$, all orbits in the basin converge to a unique fixed point.
• Fiber by fixed points: For $f_1(u_*) = 1, f_0(u_*) = 0$, the entire fiber consists of fixed points; convergence depends on initial data and the fast-enough hypothesis.
• 2–periodic fibers: For $f_1(u_*) = -1$, periodicity appears; the even and odd subsequences of orbits converge to antipodal points.

Worked examples in (Cima et al., 2013) highlight nontrivial dynamical behaviors, including divergence when $f_1(u_n) \to 1$ too slowly, and parameter-dependent dynamics in quasi-homogeneous maps and higher-order difference equations.

6. Extensions, Limitations, and Future Directions

In computational settings, dynamic block-triangular maps exhibit flexibility for relaunch with varying domain sizes, require no code changes for domain adjustments, and can be extended with small lookup tables to accommodate nonuniform or perforated triangular domains (Navarro et al., 2013). Hardware trends suggest further reduction in mapping overhead as GPU SFU (special function unit) rates improve.

Higher-dimensional extensions—such as $\lambda \mapsto (i, j, k)$ mappings for tetrahedral domains—require solving cubic prefix-sum equations, an open direction for future research. In dynamical systems, the generalization to block-triangular maps in $\mathbb{R}^{m+n}$ is conjectured to maintain the key features of invariant attracting fibers, with matrix norm conditions replacing scalar $|f_1(u)|$ (Cima et al., 2013).

Current limitations occur at extremely large $N$ or very small block sizes (e.g., $1\times1$), where the mapping overhead may become non-negligible. Proposed further improvements include leveraging CUDA dynamic parallelism for row-wise kernel spawning and precomputing row offsets in fast memory, as well as auto-tuning blocksize per architecture (Navarro et al., 2013).

7. Connections and Significance

Dynamic block-triangular maps embody a structurally rich class of object with both theoretical and algorithmic significance. They unify concepts from dynamical systems—foliation, fiber dynamics, and contraction theory—with practical domain partitioning strategies for high-performance algorithms. Their rigorous analysis forms the basis for robust, efficient, and mathematically predictable methods used in simulation, numerical linear algebra, and computational geometry, and their generalizations provide a pathway to further advances in multidimensional discrete dynamics and GPU computing methodologies (Cima et al., 2013, Navarro et al., 2013).