Cholesky算法的矩形全压缩格式：分解、求解与反演
Rectangular Full Packed Format for Cholesky's Algorithm: Factorization, Solution and Inversion
摘要：
我们描述了一种新的数据格式，用于存储三角形，对称和埃尔米特矩阵称为RFPF（矩形全封装格式）。Fortran和C的标准二维的数组（也称为全格式）用于表示三角和对称矩阵，浪费了近一半的存储空间，但通过使用Level 3 BLAS提供了高性能。标准打包格式阵列可充分利用存储（阵列空间），但性能较低，因为没有3级打包BLAS。由于RFPF是一种标准的全格式表示，因此我们使用3级BLAS来结合RFPF的打包和全存储的良好特性以获得高性能。此外，RFPF需要与打包格式完全相同的最小存储。基于RFPF的八种可能的数据布局，每个LAPACK完整和/或打包的三角形、对称和埃尔米特例程成为单个新的RFPF例程。这个新的RFPF例程通常包括两个对相应的LAPACK全格式例程的调用和两个对3级BLAS例程的调用。这意味着{ \it no}需要新软件。作为例子，我们提出了LAPACK例程的Cholesky分解，Cholesky解决方案和Cholesky逆计算RFPF来说明这一新的工作，并描述其性能在几个常用的计算机平台上。对于串行和SMP并行处理，使用RFPF的LAPACK完整例程与使用标准格式的LAPACK完整例程的性能大致相同，但使用一半的存储。使用带有RFPF的供应商LAPACK完整例程比使用供应商和/或参考打包例程的性能增益对于串行大约是1到43倍，对于SMP并行大约是1到97倍。
英文摘要：
We describe a new data format for storing triangular, symmetric, and Hermitian matrices called RFPF (Rectangular Full Packed Format). The standard two dimensional arrays of Fortran and C (also known as full format) that are used to represent triangular and symmetric matrices waste nearly half of the storage space but provide high performance via the use of Level 3 BLAS. Standard packed format arrays fully utilize storage (array space) but provide low performance as there is no Level 3 packed BLAS. We combine the good features of packed and full storage using RFPF to obtain high performance via using Level 3 BLAS as RFPF is a standard full format representation. Also, RFPF requires exactly the same minimal storage as packed format. Each LAPACK full and/or packed triangular, symmetric, and Hermitian routine becomes a single new RFPF routine based on eight possible data layouts of RFPF. This new RFPF routine usually consists of two calls to the corresponding LAPACK full format routine and two calls to Level 3 BLAS routines. This means {\it no} new software is required. As examples, we present LAPACK routines for Cholesky factorization, Cholesky solution and Cholesky inverse computation in RFPF to illustrate this new work and to describe its performance on several commonly used computer platforms. Performance of LAPACK full routines using RFPF versus LAPACK full routines using standard format for both serial and SMP parallel processing is about the same while using half the storage. Performance gains are roughly one to a factor of 43 for serial and one to a factor of 97 for SMP parallel times faster using vendor LAPACK full routines with RFPF than with using vendor and/or reference packed routines.
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