MUMPS4.9.2用户指南：大型稀疏矩阵求解器
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"MUMPS用户指南，用于大型稀疏矩阵的解决工具，版本为MUMPS4.9.2，提供了Fortran90和C语言的用户接口，包含数据结构、参数、调用序列和错误诊断的详细描述，并提供示例程序。"
MUMPS（MUltifrontal Massively Parallel Solver）是一款专为处理大型稀疏矩阵问题设计的高效求解器，版本号为4.9.2。这个用户指南主要涵盖了以下几个关键知识点：
1. **输入矩阵结构**：MUMPS处理的是稀疏矩阵，这意味着在大量的零元素中只有少数非零元素。它支持不同的输入矩阵格式，允许用户以高效的方式存储和操作这些矩阵。
2. **预处理**：在实际求解之前，MUMPS提供了预处理功能，包括矩阵的规范化、压缩和分解，以优化后续的计算过程。
3. **后处理设施**：解决系统后，MUMPS提供了后处理工具，例如解的检查、结果的可视化和输出，帮助用户分析和理解计算结果。
4. **求解转置系统**：除了求解原系统外，MUMPS还支持求解矩阵的转置系统，这对于对称或反对称问题特别有用。
5. **界面减少/压缩问题（Schur补）**：MUMPS可以处理接口上的问题，通过Schur补方法来减少或压缩问题规模，这在处理大规模复杂系统时尤其有效。
6. **算术版本**：MUMPS可能有不同的实现版本，针对不同的算术体系，如浮点或复数运算，满足不同应用需求。
7. **工作主机处理器**：指南中提到了工作主机处理器的概念，这可能是指计算过程中主要负责数据处理的处理器配置。
8. **顺序版本与共享内存版本**：MUMPS提供了一个顺序版本，适用于单处理器环境，同时还有一个共享内存版本，用于多处理器或多核系统，利用并行计算提高效率。
9. **离核心（Out-of-core）设施**：对于内存不足以容纳整个矩阵的问题，MUMPS的离核心功能允许将部分数据存储到硬盘上，分批次进行计算，这是处理超大规模矩阵的关键特性。
10. **调用顺序**：用户指南详细阐述了在使用MUMPS时，各函数和子程序的调用顺序，这对于正确实现算法流程至关重要。
11. **输入和输出参数**：MUMPS的接口控制着分析、因子化和求解这三个主要阶段，通过特定的参数设置，用户可以调整算法的行为和性能。此外，还提供了版本号识别和并行计算的控制选项。
通过这个用户指南，开发者和科研人员能够深入了解如何利用MUMPS解决大规模稀疏线性系统的计算问题，以及如何根据具体需求调整和优化其性能。同时，提供的示例程序可以帮助初学者快速上手，更好地理解和运用MUMPS库。
3. Expansion phase :
Given x
2
and y
1
, option ICNTL(26)=2 of the solve phase can be used to compute x
1
. Note
that the package uses y
1
computed (and stored in the mumps structure) during the first step
(ICNTL(26)=1) and that the complete solution x is provided on output.
Note that the Schur complement could be considered as an element contribution to the interface block
in a domain decomposition approach. MUMPS could then be used to solve this interface problem using
the element entry functionality.
2.6 Arithmetic versions
Several versions of the package MUMPS are available: REAL, DOUBLE PRECISION, COMPLEX, and
DOUBLE COMPLEX.
To compile all or any particular version, please refer to the root README of the MUMPS sources.
This document applies to all four arithmetics. In the following we use the conventions below:
1. the term real is used for REAL or DOUBLE PRECISION,
2. the term complex is used for COMPLEX or DOUBLE COMPLEX,
2.7 The working host processor
The analysis phase is performed on the host processor. This processor is the one with rank 0 in the
communicator provided to MUMPS. By setting the variable PAR to 1 (see Section
4.3), MUMPS allows the
host to participate in computations during the factorization and solve phases, just like any other processor.
This allows MUMPS to run on a single processor and prevents the host processor being idle during the
factorization and solve phases (as would be the case for PAR=0). We thus generally recommend using a
working host processor (PAR=1).
The only case where it may be worth using PAR=0 is with a large centralized matrix on a purely
distributed architecture with relatively small local memory: PAR=1 will lead to a memory imbalance
because of the storage related to the initial matrix on the host.
2.8 Sequential version
It is possible to use MUMPS sequentially by limiting the number of processors to one, but the link phase
still requires the MPI, BLACS, and ScaLAPACK libraries and the user program needs to make explicit
calls to MPI
INIT and MPI FINALIZE.
A purely sequential version of MUMPS is also available. For this, a special library is distributed that
provides all external references needed by MUMPS for a sequential environment. MUMPS can thus be
used in a simple sequential program, ignoring everything related to parallelism or MPI. Details on how
to build a purely sequential version of MUMPS are available in the file README available in the MUMPS
distribution. Note that for the sequential version, the component PAR must be set to 1 (see Section
4.3)
and that the calling program should not make use of MPI.
2.9 Shared memory version
On networks of SMP nodes (multiprocessor nodes with a shared memory), a parallel shared memory
BLAS library (also called multithread BLAS) is often provided by the manufacturer. Using shared
memory BLAS (between 2 and 4 threads per MPI process) can be significantly more efficient than running
with only MPI processes. For example on a computer with 2 SMP nodes and 16 processors per node, we
advise to run using 16 MPI processes with 2 threads per MPI process.
2.10 Out-of-core facility
Controlled by
ICNTL(22), a preliminary out-of-core facility is available in both sequential and parallel
environments. In this version only the factors are written to disk during the factorization phase and
will be read each time a solution phase is requested. Our experience is that on a reasonably small
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