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A hierarchical matrix C/C++ library

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#### Overview 
`hmat-oss` is [hierarchical matrix](http://en.wikipedia.org/wiki/Hierarchical_matrix) library
written in C++. It has a C API. It contains a [LU](http://en.wikipedia.org/wiki/LU_decomposition)
and LLt solver, and a few other things.

#### Do you want to know more ?

This repository contains the **HMat-OSS** (Hierarchical Matrix open source software) library, initially developed by Airbus. It is an advanced and optimized C++ sequential implementation of **Hierarchical Matrices** (H-Matrices). It is released under the terms of GPL v2 license.

Here is a quick analysis of this codebase, structured by logical components:

### **1. Main Objective of the Library**

The purpose of this code is to process very large dense matrices (often generated by integral equations, such as the Boundary Element Method or BEM) by approximating them.

Instead of storing $N \times N$ elements (complexity $\mathcal{O}(N^2)$), the space is partitioned:

* **Near-field** interactions are calculated and stored densely (exactly).
  * **Far-field** interactions are approximated by **low-rank** matrices.

This drastically reduces memory and computational time complexity, approaching $\mathcal{O}(N \log N)$.

### **2. Code Architecture and Components**

**A. Spatial Partitioning (Clustering)**

* `coordinates.hpp/cpp`: Manages the geometric coordinates of the Degrees of Freedom (DoF).
  * `cluster_tree.hpp/cpp` & `clustering.hpp/cpp`: Take these coordinates and build a tree (ClusterTree) by dividing the space. Several algorithms are implemented: geometric bisection, median, hybrid, etc.

**B. Admissibility**

* `admissibility.hpp/cpp`: Evaluates if two nodes of the tree (a matrix block) are sufficiently far apart to be compressed. It notably implements the classical Hackbusch condition ($\min(\text{diam}(\tau), \text{diam}(\sigma)) \le \eta \cdot \text{dist}(\tau, \sigma)$).

**C. Matrix Structures**

* `scalar_array.hpp/cpp`: Low-level class managing contiguous arrays in memory (Row/Col-major) and serving as a direct interface with BLAS/LAPACK.
  * `full_matrix.hpp/cpp`: Represents a standard dense block (tree leaf for near-field interactions).
  * `rk_matrix.hpp/cpp`: Represents a low-rank block in the factorized form $R = A \times B^T$. This file manages addition, recompression (often via modified Gram-Schmidt or SVD), and multiplication of these compressed blocks.
  * `h_matrix.hpp/cpp`: The hierarchical structure itself. It is a tree where each internal node points to sub-blocks (generally 4), and each leaf is either a FullMatrix or an RkMatrix.

**D. Compression Algorithms**

* `compression.hpp/cpp`: Contains the mathematics for generating the matrices $A$ and $B$ from an original dense block. It implements **Truncated SVD**, the **RRQR** (Rank Revealing QR) method, and especially the **ACA** (Adaptive Cross Approximation: Full, Partial, Plus, Random) family of algorithms which allow for finding a low-rank approximation heuristically and very quickly.
  * `fp_compression.hpp/cpp`: Adds a floating-point bit-level compression layer (ZFP, SZ, SZ3 via the optional [composyx](https://gitlab.inria.fr/composyx/composyx) library), further reducing the memory footprint of dense arrays or factors $A$ and $B$.

**E. Hierarchical Arithmetic**

* `recursion.hpp/cpp` & `assembly.hpp/cpp`: Implement matrix construction (assembly) and especially arithmetic on the trees: matrix-vector product (gemv), matrix-matrix product (gemm), factorizations (LU, LLt/Cholesky, LDLt), and triangular system solutions (solve).
  * `hodlr.hpp/cpp`: A specialization for HODLR (Hierarchical Off-Diagonal Low-Rank) type matrices, a variant of H-Matrices optimal for certain types of geometries (e.g., 1D).

**F. Interfaces and Utilities**

* `hmat.h`, `c_wrapping.hpp`, `c_default_interface.cpp`: The pure C API that allows using this C++ library from C, Fortran, or Python.
  * `blas_overloads.hpp`, `lapack_overloads.hpp`, `lapack_operations.hpp`: Clean "wrappers" (adapters) for calling BLAS and LAPACK routines (MKL, OpenBLAS, etc.) independently of type (real, complex, single, or double precision).
  * `common/\*`: Chronometry tools (timeline.cpp), performance tracing, and memory instrumentation (tree export to JSON via json.cpp).

### **3. Technical Design Choices**

* **Strongly-typed C++ Templates**: All matrix code is templated via `<T>` defined in data_types.hpp to support float, double, complex\<float\>, and complex\<double\>.
  * **Performance-oriented**: Intensive use of pointers, in-place routines, and precise control over memory allocations. There is a mechanism (RAII via DisableThreadingInBlock) to forbid native BLAS/OpenMP multi-threading inside sub-blocks in favor of higher-level multi-threading managed by the library (often via StarPU, although not included here, or OpenMP on the tree).

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