学习BLAS库 -- LAPACK

BLAS库中Level 3函数是用于矩阵与矩阵之间运算。LAPACK库设计目标是作为BLAS中Level 3函数的扩展库。

Category
应用范围	求解线性方程组，最小二乘问题 特征值问题, 奇异值分解问题
编程语言	Fortran90
矩阵类型	稠密阵和带状阵(不支持稀疏阵)
数据类型	单精度和双精度的实矩阵及复矩阵

LAPACK函数

命名

LAPCAK中函数名由 数据类型+ 矩阵类型 + 矩阵运算 组成。例如:
SGETRF
其中
- S – 数据类型，S代表单精度浮点数
- GE – 矩阵类型，GE代表稠密阵
- TRF – 矩阵运算, TRF代表 三角分解法

数据类型

矩阵内元素的数据类型，有以下几种：
S - 单精度浮点数
D - 双精度浮点数
C - 复数
Z - 16位复数

矩阵类型

矩阵类型有：

缩写代号	矩阵类型
BD	bidiagonal 二对角阵
DI	diagonal 对角阵
GB	general band 普通带状阵
GE	general (i.e., unsymmetric, in some cases rectangular) 普通矩阵(例如非对称阵，矩形
GG	general matrices, generalized problem (i.e., a pair of general matrices) 普通矩阵组，广义问题(例如，一对普通矩阵)
GT	general tridiagonal 普通三角阵
HB	(complex) Hermitian band (复) 埃尔米特带状阵
HE	(complex) Hermitian (复)埃尔米特矩阵
HG	upper Hessenberg matrix, generalized problem (i.e a Hessenberg and a triangular matrix) 上 海森阵，广义问题(例如一个海森阵和一个三角阵)
HP	(complex) Hermitian, packed storage (复)海森阵(压缩存储)
HS	upper Hessenberg 上 海森阵
OP	(real) orthogonal, packed storage (实)正交阵(压缩存储)
OR	(real) orthogonal (实)正交阵
PB	symmetric or Hermitian positive definite band 正定对称(埃尔米特矩阵)带状阵
PO	symmetric or Hermitian positive definite 正定对称(埃尔米特矩阵)阵
PP	symmetric or Hermitian positive definite, packed storage 正定对称(埃尔米特矩阵)阵(压缩存储)
PT	symmetric or Hermitian positive definite tridiagonal 正定对称(埃尔米特矩阵)三角阵
SB	(real) symmetric band (实)对称带状阵
SP	symmetric, packed storage 对称阵(压缩存储)
ST	(real) symmetric tridiagonal (实)对称三对角阵
SY	symmetric 对称阵
TB	triangular band 三角带状阵
TG	triangular matrices, generalized problem (i.e., a pair of triangular matrices) 三角阵组，广义问题(例
TP	triangular, packed storage 三角阵(压缩存储)
TR	triangular (or in some cases quasi-triangular) 三角阵
TZ	trapezoidal 梯形阵
UN	(complex) unitary (复)酉矩阵
UP	(complex) unitary, packed storage (复)酉矩阵(压缩存储)

函数类别

LAPCAK中矩阵运算函数分为三类：driver函数, computational函数 auxiliary函数

Driver函数

Driver Routines for Linear Equations

Name	Element Precision	Matrix Type and Storage Scheme	Description
XGESV	S, C, D, Z	General matrix	Solving linear equations system AX = B

Driver Routines for Linear Least Squares Problem

Name	Element Precision	Matrix Type and Storage Scheme	Description
SGELS	S, C, D, Z	General matrix	Solving the linear least squares problem min||B−AX||2 m i n | | B − A X | | 2 using QR or LQ factorization
SGELSY	S, C, D, Z	General matrix	Solving the linear least squares problem min||B−AX||2 m i n | | B − A X | | 2 using complete orthogonal factorization
SGELSS	S, C, D, Z	General matrix	Solving the linear least squares problem min||B−AX||2 m i n | | B − A X | | 2 using SVD
SGELSD	S, C, D, Z	General matrix	Solving the linear least squares problem min||B−AX||2 m i n | | B − A X | | 2 using divide-and-conquer SVD

Computational Routines

Routines for Linear Equations

Name	Element Precision	Matrix Type and Storage Scheme	Description
XGETRF	S, C, D, Z	General matrix	factorize
XGETRS	S, C, D, Z	General matrix	solve using factorization
XGECON	S, C, D, Z	General matrix	estimate condition number
XGERFS	S, C, D, Z	General matrix	error bounds for solution
XGETRI	S, C, D, Z	General matrix	invert using factorization
XGEEQU	S, C, D, Z	General matrix	equilibrate
XPOTRF	S, C, D, Z	Symmetric positive	factorize
XPOTRS	S, C, D, Z	Symmetric positive	solve using factorization
XPOCON	S, C, D, Z	Symmetric positive	estimate condition number
XPORFS	S, C, D, Z	Symmetric positive	error bounds for solution
XPOTRI	S, C, D, Z	Symmetric positive	invert using factorization
XPOEQU	S, C, D, Z	Symmetric positive	equilibrate
XPPTRF	S, C, D, Z	Symmetric positive definite band	factorize
XPPTRS	S, C, D, Z	Symmetric positive definite band	solve using factorization
XPPCON	S, C, D, Z	Symmetric positive definite band	estimate condition number
XPPRFS	S, C, D, Z	Symmetric positive definite band	error bounds for solution
XPPTRI	S, C, D, Z	Symmetric positive definite band	invert using factorization
XPPEQU	S, C, D, Z	Symmetric positive definite band	equilibrate
XPTTRF	S, C, D, Z	Symmetric tridiagonal definite band	factorize
XPTTRS	S, C, D, Z	Symmetric tridiagonal definite band	solve using factorization
XPTCON	S, C, D, Z	Symmetric tridiagonal definite band	estimate condition number
XPTRFS	S, C, D, Z	Symmetric tridiagonal definite band	error bounds for solution

( … 待续)

在Windows的Visual Studio环境中安装与使用CLAPACK

预备工作: 已经安装好Visual Studio环境，本人使用的是Visual Studio 2015.

1. 从[1] 下载LAPACKE_examples.zip, 此文件中已经准备好Windows环境中所需的动态库以及头文件. 解压缩此文件到自定的文件夹中.
2. 进入\example_DGESV_rowmajor 文件夹, 运行example_DGESV_rowmajor.vcxproj, 即可在Release文件夹中生成测试算例.

使用nmake编译

1. 从[1] 下载LAPACKE_examples.zip, 此文件中已经准备好Windows环境中所需的动态库以及头文件. 解压缩此文件到自定的文件夹中.
2. 在此文件夹中创建名为src文件夹，在src文件夹main.cpp.

#include 
#include 
#include "lapacke.h"

/* Auxiliary routine: printing a matrix */
void print_matrix( char* desc, lapack_int m, lapack_int n, double* a, lapack_int lda ) {
        lapack_int i, j;
        printf( "\n %s\n", desc );
        for( i = 0; i < m; i++ ) {
                for( j = 0; j < n; j++ ) printf( " %6.2f", a[i*lda+j] );
                printf( "\n" );
        }
}

void gesvTest() {
    printf( "GESV Testing \n");
    int n = 5;
    int lda = n;
    int ldb = 1;
    int nrhs = 1;
     /* Pivot indices */
    int ipiv[N];

    double a[LDA*N] = {
        1.00,  -0.0, -0.0,  0.0, -0.0,
        1.0, -3.30,  2.58,  2.71, -5.14,
        0.0, 5.36, -2.70,  4.35, -7.26,
        0.0, -4.44,  0.27, -7.17, 6.08,
        0.0, 1.08,  9.04,  2.14, -6.87
    };

    double b[N] = {2.0, 1.0, 0.0, 0.0, 0.0 };

    /* Print Entry Matrix */
    print_matrix( "Entry Matrix A", n, n, a, lda );
    print_matrix( "RHS vector", n, 1, b, 1 );

    int info = LAPACKE_dgesv( LAPACK_ROW_MAJOR, n, nrhs, a, lda, ipiv,
                            b, ldb );

    print_matrix( "Result vector", n, 1, b, 1 );
}

/* Main program */
int main() { 
        gesvTest();
        exit( 0 );
}

1. 在此文件夹中创建名为tmp文件夹，在tmp文件夹中，创建nmake编译时所需的Makefile文件。Makefile文件如下:

    # Target     
    PROGRAM = test.exe    

    LIBDIRS = "../lib/" 
    inc = "../include/"    
    src = "../src/"      
    INCLUDEDIRS =  /I $(inc)  /I $(LIBDIRS)    

    # Flags    
    CPPOPT = $(INCLUDEDIRS) /w /EHsc /D_CRT_SECURE_NO_DEPRECATE /D ADD_ /D HAVE_LAPACK_CONFIG_H /D LAPACK_COMPLEX_STRUCTURE /D WIN32 /D NDEBUG /D _CONSOLE    

    LIBS =  ../lib/liblapacke.lib  ../lib/liblapack.lib  ../lib/libblas.lib

    # Compiler     
    cc = cl     
    CFLAGS =     

    LIBFLAGS =  /LIBPATH $(LIBDIRS)  

    # list of source files     
    CPPSOURCES =  main.cpp       

    # expands to list of object files            
    CPPOBJECTS = $(CPPSOURCES:.cpp=.obj)     

    all: $(PROGRAM)    

    $(PROGRAM): $(CPPOBJECTS)   
        link.exe /out:$(PROGRAM)  $(CPPOBJECTS)   $(LIBS)      

    main.obj:     
        $(cc) $(CPPOPT) /c ../src/main.cpp         

    clean:      
        del $(CPPOBJECTS) $(PROGRAM)   

4.启动Developer Command Prompt for VS2015，进入tmp文件夹，运行nmake。会生成test.exe 文件。
5. 将压缩包里Release文件夹中的dll库文件拷贝到tmp文件夹中，运行test.exe.
运行结果：

[1] http://icl.cs.utk.edu/lapack-for-windows/clapack/
[2] http://www.netlib.org/lapack/index.html
[3] http://math.nist.gov/lapack++/
[4] http://www.netlib.org/lapack/lug/