高性能线性代数库LAPACK++ V.1.0用户指南

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LAPACK++-V.1.0-High-Performance-Linear-Algebra-Users'-Guide.pdf LAPACK++是一个高性能的线性代数库，它提供了一个广泛的线性代数运算的实现，包括矩阵操作、向量操作、矩阵分解、矩阵乘法、矩阵求逆等。该库是基于BLAS（Basic Linear Algebra Subprograms）和LAPACK（Linear Algebra Package）库的扩展，提供了高性能的线性代数运算解决方案。 LAPACK++的主要特点包括： * 高性能：LAPACK++使用了最新的优化技术和算法，提供了高性能的线性代数运算解决方案。 * 可扩展性：LAPACK++提供了一个灵活的架构，允许用户自定义和扩展库的功能。 * 易用性：LAPACK++提供了一个简单易用的接口，允许用户快速地使用库的功能。 LAPACK++的主要组件包括： * BLAS++：基本线性代数子程序库，提供了基本的矩阵和向量操作。 * LAPACK Matrix Objects：提供了矩阵对象的实现，包括矩阵创建、矩阵操作和矩阵销毁等。 * Fundamental Matrix Operations：提供了基本的矩阵操作，包括矩阵加法、矩阵乘法、矩阵转置等。 * General Rectangular Matrices：提供了矩阵的实现，包括矩阵创建、矩阵操作和矩阵销毁等。 LAPACK++的应用领域包括： * 高性能计算：LAPACK++提供了高性能的线性代数运算解决方案，适合高性能计算领域。 * 科学计算：LAPACK++提供了广泛的线性代数运算解决方案，适合科学计算领域。 * 工程计算：LAPACK++提供了高性能的线性代数运算解决方案，适合工程计算领域。 LAPACK++的优点包括： * 高性能：LAPACK++提供了高性能的线性代数运算解决方案。 * 灵活性：LAPACK++提供了一个灵活的架构，允许用户自定义和扩展库的功能。 * 易用性：LAPACK++提供了一个简单易用的接口，允许用户快速地使用库的功能。 LAPACK++的缺点包括： * 学习成本高：LAPACK++的使用需要一定的数学和编程基础。 * 依赖关系复杂：LAPACK++依赖于BLAS和LAPACK库，需要正确地安装和配置这些库。 LAPACK++是一个功能强大且高性能的线性代数库，适合高性能计算、科学计算和工程计算领域。

Chapter 2

Overview

The underlying philosophy of the LAPACK++ is to provide an interface which is relatively simple, yet

powerful enough to express all complex and subtle tasks within LAPACK, including those which optimize

performance and/or storage. Following the framework of LAPACK, the C++ extension contains

driver

routines

for solving standard types of problems,

computational routines

to p erform a distinct computa-

tional task, and

auxiliary routines

to p erform a certain subtask or common low-level computation. Each

driver routine typically calls a sequence of computational routines. Taken as a whole, the computational

routines can p erform a wider range of tasks than are covered by the driver routines. Currently, dense and

band matrices are supp orted. General sparse matrices are handled in [7].

2.1 Contents of LAPACK++

With over 1,000 subroutines in the original f77 LAPACK, not every routine is implemented in LAPACK++.

Instead, source co de examples in the various ma jor areas are provided, allowing users to easily extend the

package for their particular needs. LAPACK++ provides source co de for



Algorithms

{

LU Factorization

{

Cholesky (

) Factorization

{

QR Factorization

{

Eigenvalue problems



Storage Classes

{

rectangular matrices

{

symmetric and symmetric p ositive denite (SPD)

{

banded matrices

{

tri/bidiagonal matrices



Element Data Types

{

int, long int, oat, double, (double precision) complex,

{

arbitrary

Vector

data types via templates (section B.6.3)

CHAPTER 2. OVERVIEW

2.2 A simple co de example

To provide a rst glimpse at how LAPACK++ simplies the user interface, this section presents a few simple

code fragments. The examples are incomplete and are meant to merely illustrate the interface style. The

next few sections will further discuss the details of matrix classes and their operations.

The rst example illustrates a co de fragment to solve a linear system

using LU factorization:

#include <lapack++.h> // 1

LaGenMatDouble A(N,N); // 2

LaVectorDouble x(N), b(N); // 3

...

LaLinSolve(A,x,b); // 4

Line (1) includes all of the LAPACK++ ob ject and function declarations. Line (2) declares

to b e a

square



coecient matrix, while line (3) declares the right-hand-side and solution vectors. Finally, the

LaLinSolve()

function in line (4) calls the underlying LAPACK driver routine

SGESV()

for solving linear

equations.

Consider now solving a similar system with a tridiagonal co ecient matrix:

#include <lapack++.h>

LaSPDMatDouble A(N,N);

LaVectorDouble x(N), b(N);

...

LaLinSolve(A,x,b);

The only code mo dcation is in the declaration of

. In this case

LaLinSolve()

calls the Cholesky driver

routine for solving symmetric, p ositive-denite linear systems. The

LaLinSolve()

function has been

over-

loaded

to p erform dierent tasks dep ending on the type of the input matrix

. If the matrix types are

known at compile time, as in this example, then there is no runtime overhead associated with this.

2.3 Basic Linear Algebra Subroutines (BLAS++)

The Basic Linear Algebra Subprograms (BLAS) [4] has been the key to obtaining goo d performance on

a wide variety of computer architectures. The BLAS dene a common interface for low-level operations

often found in computational kernels. These operations, such as matrix/matrix multiply and triangular

solves, typically comprise of most of the computatioal workload found in dense and banded linear algebra

algorithms. The Level 3 BLAS obtains goo d p erformance on a wide variety of architectures by keeping data

used most often in the closest level of memory hierarhcy (registers, cache, etc.).

The BLAS++ interface simplies many of the calling sequences to the traditional f77 BLAS interface,

by using the LAPACK++ matrix classes. These routines are called within the LAPACK++ algorithms, or

can be called directly at the user-level within applications.

There are two levels of the BLAS++ interface. The rst is a direct interface, as shown in Table C, are

essentially inlined to call BLAS directly in eliminating any overhead. The other, more elegant interface,

overloads the binary op ertors * and + for simple expressions such as

C=A*B

. Having these two interfaces

gives the users the choice between simplicity and p erformance in their application co des. See Appendix B

for a list of BLAS++ interface routines.

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高性能线性代数库LAPACK++ V.1.0用户指南

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