MKL学习——矩阵矩阵操作

前言

前面介绍了BLAS Level 1中向量-向量操作，以及BLAS Level 2中矩阵-向量的操作，就剩下这一篇的BLAS Level 3中的矩阵-矩阵的操作了。对稀疏矩阵的操作以后等要用再看看。对基本的矩阵和向量运算的函数有所了解以后，就进行常用参数分析，以及实现了。

所有函数概览

函数名称 缺失部分 描述 cblas_?gemm s,d,c,z 一般矩阵的矩阵-矩阵乘法 cblas_?hemm c,z 某个输入矩阵是Hermitian，然后计算矩阵-矩阵乘法 cblas_?herk c,z Hermitian的k阶更新 cblas_?herk2 c,z Hermitian的2k阶更新 cblas_?symm s,d,c,z 某个输入矩阵是对称的，然后计算矩阵-矩阵乘法 cblas_?syrk s,d,c,z 对称矩阵的k阶更新 cblas_?syr2k s,d,c,z 对称矩阵的2k阶更新 cblas_?trmm s,d,c,z 某个输入矩阵是三角的，就是你矩阵-矩阵的乘法 cblas_?trsm s,d,c,z 解三角矩阵方程

矩阵运算

cblas_?gemm

• 作用 : 一般矩阵的矩阵-矩阵乘法

• 定义函数

  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);

• 运算
  

  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是标量

  A,B,C是矩阵: op(A)是一个m∗k的矩阵，op(B)是k∗n的矩阵，C是一个m∗n的矩阵

• 输入参数

  Layout: 指定矩阵是行优先(CblasRowMajor)还是列优先(CblasColMajor)

  transa: 指定对矩阵的操作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: 矩阵op(A)和C的行数

  n: 矩阵op(B)和C的列数

  k: 矩阵op(A)和op(B)的列数

  alpha: 标量

  a:

transa=CblasNoTrans transa=CblasTrans或者transa=CblasConjTrans Layout=CblasColMajor 矩阵大小lda∗k Layout=CblasRowMajor 矩阵大小lda∗m

lda : 引导维度

transa=CblasNoTrans transa=CblasTrans或者transa=CblasConjTrans Layout=CblasColMajor lda至少为max(1,m) Layout=CblasRowMajor lda至少为max(1,k)

b:

transa=CblasNoTrans transa=CblasTrans或者transa=CblasConjTrans Layout=CblasColMajor 矩阵大小ldb∗n Layout=CblasRowMajor 矩阵大小ldb∗k

ldb:

transa=CblasNoTrans transa=CblasTrans或者transa=CblasConjTrans Layout=CblasColMajor ldb至少为max(1,k) Layout=CblasRowMajor ldb至少为max(1,n)

beta: 标量

c: 当Layout=CblasColMajor时候,数组大小为lda∗n；当Layout=CblasRowMajor时候,数组大小为lda∗m

ldc:当Layout=CblasColMajor的时候，ldc必须至少为max(1,m);当Layout=CblasRowMajor的时候，ldc必须至少为max(1,n)

• 输出参数: 将计算得到的矩阵写入到c

cblas_?hemm

• 作用 : 当一个输入矩阵为Hermitian时，计算矩阵-矩阵的乘法

• 定义函数

  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);

• 运算
  

  C:=α∗A∗B+β∗CC:=α∗B∗A+β∗C
  
  其中，α,β是标量，A是Hermitian矩阵，B,C是m∗n的矩阵

cblas_?herk

• 作用: Hermitian的k阶更新

• 定义函数:

  void cblas_cherk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, constCBLAS_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, constCBLAS_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);

• 运算
  

  C:=α∗A∗AH+β∗CC:=α∗AH∗A+β∗C
  
  其中，α,β是标量，C是n∗n的Hermitian矩阵，第一个式子的A是n∗k矩阵，第二个式子的A是k∗n的矩阵

cblas_?herk2

• 作用: Hermitian矩阵的2阶更新

• 定义函数

  void cblas_cher2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, constCBLAS_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, constCBLAS_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);

• 运算
  

  C:=α∗A∗BH+conjg(α)∗B∗AH+β∗CC:=α∗AH∗B+conjg(α)∗BH∗A+β∗C
  
  其中，α,β是标量，C是n∗n是Hermitian矩阵，第一个式子中A,B是n∗k矩阵，第二个式子中是k∗n的矩阵

cblas_?symm

• 作用: 某个输入是对称矩阵时候，计算矩阵-矩阵的乘法

• 定义函数

  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);

• 运算
  

  C:=α∗A∗b+β∗CC:=α∗B∗A+β∗C
  
  其中α,β是标量，A是对称阵,B,C是m∗n的矩阵

cblas_?syrk

• 作用: 对称矩阵的k阶更新

• 定义函数

  void cblas_ssyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, constCBLAS_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, constCBLAS_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, constCBLAS_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, constCBLAS_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)

• 运算
  

  C:=α∗A∗A′+β∗CC:=α∗A′∗A+β∗C
  
  其中,α,β是标量，C是n∗n的对称阵,第一个式子中A是n∗n的矩阵，第二个式子中A是k∗n的矩阵

cblas_?syr2k

• 作用: 对称阵的二阶更新

• 定义函数

  void cblas_ssyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, constCBLAS_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, constCBLAS_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, constCBLAS_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, constCBLAS_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);

• 运算
  

  C:=α∗A∗B′+α∗B∗A′+β∗CC:=α∗A′∗B+α∗B′∗A+β∗C
  
  其中,α,β是标量，C是n∗n的对称阵，第一个式子中A,B是n∗k的矩阵，第二个式子中A,B是k∗n的矩阵

cblas_?trmm

• 作用: 某个输入矩阵为三角阵的时候，计算矩阵-矩阵的乘法

• 定义函数

  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);

• 运算
  

  B:=α∗op(A)∗BB:=α∗B∗op(A)
  
  其中，α是标量，B是m∗n的矩阵，op(A)=A或者op(A)=A′或者op(A)=conjg(A′)

cblas_?trsm

• 作用: 解三角矩阵方程

• 定义函数

  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);

• 运算
  

  op(A)∗X=α∗BX∗op(A)=alpha∗B
  
  其中，α是一个标量，X和B是一个m∗n的矩阵，A是单位或者非单位，上三角或者下三角矩阵，op(A)=A或者op(A)=A′或者op(A)=conjg(A′)。把等式的解矩阵X冲写入到B矩阵中。

后续

研究BLAS Level 1 2 3中各种矩阵相关概念。随后是代码实现