前言

前面介紹了BLAS Level 1中向量-向量操作，以及BLAS Level 2中矩陣-向量的操作，就剩下這一篇的BLAS Level 3中的矩陣-矩陣的操作了。對稀疏矩陣的操作以后等要用再看看。對基本的矩陣和向量運算的函數有所了解以后，就進行常用參數分析，以及實現了。

所有函數概覽

函數名稱缺失部分描述

cblas_?gemm	s,d,c,z	一般矩陣的矩陣-矩陣乘法
cblas_?hemm	c,z	某個輸入矩陣是Hermitian，然后計算矩陣-矩陣乘法
cblas_?herk	c,z	Hermitian的k階更新
cblas_?herk2	c,z	Hermitian的2k階更新
cblas_?symm	s,d,c,z	某個輸入矩陣是對稱的，然后計算矩陣-矩陣乘法
cblas_?syrk	s,d,c,z	對稱矩陣的k階更新
cblas_?syr2k	s,d,c,z	對稱矩陣的2k階更新
cblas_?trmm	s,d,c,z	某個輸入矩陣是三角的，就是你矩陣-矩陣的乘法
cblas_?trsm	s,d,c,z	解三角矩陣方程

矩陣運算

cblas_?gemm

• 作用 : 一般矩陣的矩陣-矩陣乘法

• 定義函數

  void cblas_sgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const float alpha, const float *a, const MKL_INT lda, const float *b, const MKL_INT ldb, const float beta, float *c, const MKL_INT ldc);void cblas_dgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const double alpha, const double *a, const MKL_INT lda, const double *b, const MKL_INT ldb, const double beta, double *c, const MKL_INT ldc);void cblas_cgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);void cblas_zgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);

• 運算
  

  C:=α?op(A)?op(B)+β?C
  其中op(x)可以是op(x)=x或者op(x)=xT或者op(x)=XH

  α和β是標量

  A,B,C是矩陣: op(A)是一個m?k的矩陣，op(B)是k?n的矩陣，C是一個m?n的矩陣

• 輸入參數

  Layout: 指定矩陣是行優先(CblasRowMajor)還是列優先(CblasColMajor)

  transa: 指定對矩陣的操作op(A)，如果transa=CblasNoTrans那么op(A)=A；如果transa=CblasTrans那么op(A)=AT；如果transa=CblasConjTrans那么op(A)=AH

  transb: 同上

  m: 矩陣op(A)和C的行數

  n: 矩陣op(B)和C的列數

  k: 矩陣op(A)和op(B)的列數

  alpha: 標量

  a:

transa=CblasNoTranstransa=CblasTrans或者transa=CblasConjTrans

Layout=CblasColMajor	矩陣大小lda?k
Layout=CblasRowMajor	矩陣大小lda?m

lda : 引導維度

transa=CblasNoTranstransa=CblasTrans或者transa=CblasConjTrans

Layout=CblasColMajor	lda至少為max(1,m)
Layout=CblasRowMajor	lda至少為max(1,k)

b:

transa=CblasNoTranstransa=CblasTrans或者transa=CblasConjTrans

Layout=CblasColMajor	矩陣大小ldb?n
Layout=CblasRowMajor	矩陣大小ldb?k

ldb:

transa=CblasNoTranstransa=CblasTrans或者transa=CblasConjTrans

Layout=CblasColMajor	ldb至少為max(1,k)
Layout=CblasRowMajor	ldb至少為max(1,n)

beta: 標量

c: 當Layout=CblasColMajor時候,數組大小為lda?n；當Layout=CblasRowMajor時候,數組大小為lda?m

ldc:當Layout=CblasColMajor的時候，ldc必須至少為max(1,m);當Layout=CblasRowMajor的時候，ldc必須至少為max(1,n)

• 輸出參數: 將計算得到的矩陣寫入到c

cblas_?hemm

• 作用 : 當一個輸入矩陣為Hermitian時，計算矩陣-矩陣的乘法

• 定義函數

  void cblas_chemm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);void cblas_zhemm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);

• 運算
  

  C:=α?A?B+β?CC:=α?B?A+β?C
  其中，α,β是標量，A是Hermitian矩陣，B,C是m?n的矩陣

cblas_?herk

• 作用: Hermitian的k階更新

• 定義函數:

  void cblas_cherk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const float alpha, const void *a, const MKL_INT lda, const float beta, void *c, const MKL_INT ldc);void cblas_zherk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const double alpha, const void *a, const MKL_INT lda, const double beta, void *c, const MKL_INT ldc);

• 運算
  

  C:=α?A?AH+β?CC:=α?AH?A+β?C
  其中，α,β是標量，C是n?n的Hermitian矩陣，第一個式子的A是n?k矩陣，第二個式子的A是k?n的矩陣

cblas_?herk2

• 作用: Hermitian矩陣的2階更新

• 定義函數

  void cblas_cher2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const float beta, void *c, const MKL_INT ldc);void cblas_zher2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const double beta, void *c, const MKL_INT ldc);

• 運算
  

  C:=α?A?BH+conjg(α)?B?AH+β?CC:=α?AH?B+conjg(α)?BH?A+β?C
  其中，α,β是標量，C是n?n是Hermitian矩陣，第一個式子中A,B是n?k矩陣，第二個式子中是k?n的矩陣

cblas_?symm

• 作用: 某個輸入是對稱矩陣時候，計算矩陣-矩陣的乘法

• 定義函數

  void cblas_ssymm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const MKL_INT m, const MKL_INT n, const float alpha, const float *a, const MKL_INT lda, const float *b, const MKL_INT ldb, const float beta, float *c, const MKL_INT ldc);void cblas_dsymm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const MKL_INT m, const MKL_INT n, const double alpha, const double *a, const MKL_INT lda, const double *b, const MKL_INT ldb, const double beta, double *c, const MKL_INT ldc);void cblas_csymm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);void cblas_zsymm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);

• 運算
  

  C:=α?A?b+β?CC:=α?B?A+β?C
  其中α,β是標量，A是對稱陣,B,C是m?n的矩陣

cblas_?syrk

• 作用: 對稱矩陣的k階更新

• 定義函數

  void cblas_ssyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const float alpha, const float *a, const MKL_INT lda, const float beta, float *c, const MKL_INT ldc);void cblas_dsyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const double alpha, const double *a, const MKL_INT lda, const double beta, double *c, const MKL_INT ldc);void cblas_csyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *beta, void *c, const MKL_INT ldc);void cblas_zsyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *beta, void *c, const MKL_INT ldc)

• 運算
  

  C:=α?A?A′+β?CC:=α?A′?A+β?C
  其中,α,β是標量，C是n?n的對稱陣,第一個式子中A是n?n的矩陣，第二個式子中A是k?n的矩陣

cblas_?syr2k

• 作用: 對稱陣的二階更新

• 定義函數

  void cblas_ssyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const float alpha, const float *a, const MKL_INT lda, const float *b, const MKL_INT ldb, const float beta, float *c, const MKL_INT ldc);void cblas_dsyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const double alpha, const double *a, const MKL_INT lda, const double *b, const MKL_INT ldb, const double beta, double *c, const MKL_INT ldc);void cblas_csyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);void cblas_zsyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);

• 運算
  

  C:=α?A?B′+α?B?A′+β?CC:=α?A′?B+α?B′?A+β?C
  其中,α,β是標量，C是n?n的對稱陣，第一個式子中A,B是n?k的矩陣，第二個式子中A,B是k?n的矩陣

cblas_?trmm

• 作用: 某個輸入矩陣為三角陣的時候，計算矩陣-矩陣的乘法

• 定義函數

  void cblas_strmm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const MKL_INT n, const float alpha, const float *a, const MKL_INT lda, float *b, const MKL_INT ldb);void cblas_dtrmm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const MKL_INT n, const double alpha, const double *a, const MKL_INT lda, double *b, const MKL_INT ldb);void cblas_ctrmm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, void *b, const MKL_INT ldb);void cblas_ztrmm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, void *b, const MKL_INT ldb);

• 運算
  

  B:=α?op(A)?BB:=α?B?op(A)
  其中，α是標量，B是m?n的矩陣，op(A)=A或者op(A)=A′或者op(A)=conjg(A′)

cblas_?trsm

• 作用: 解三角矩陣方程

• 定義函數

  void cblas_strsm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const MKL_INT n, const float alpha, const float *a, const MKL_INT lda, float *b, const MKL_INT ldb);void cblas_dtrsm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const MKL_INT n, const double alpha, const double *a, const MKL_INT lda, double *b, const MKL_INT ldb);void cblas_ctrsm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, void *b, const MKL_INT ldb);void cblas_ztrsm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, void *b, const MKL_INT ldb);

• 運算
  

  op(A)?X=α?BX?op(A)=alpha?B
  其中，α是一個標量，X和B是一個m?n的矩陣，A是單位或者非單位，上三角或者下三角矩陣，op(A)=A或者op(A)=A′或者op(A)=conjg(A′)。把等式的解矩陣X沖寫入到B矩陣中。

后續

研究BLAS Level 1 2 3中各種矩陣相關概念。隨后是代碼實現

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