ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation.
pm_matrixMulAdd Module Reference

This module contains procedures and generic interfaces relevant to combined matrix-matrix or matrix-vector multiplication and addition. More...

Data Types

interface  setMatMulAdd
 Return the result of the multiplication of the input matrices/vector matA and matB in the user-specified form.
More...
 

Variables

character(*, SK), parameter MODULE_NAME = "@pm_matrixMulAdd"
 

Detailed Description

This module contains procedures and generic interfaces relevant to combined matrix-matrix or matrix-vector multiplication and addition.

The procedures under the generic interface setMatMulAdd of this module return the result of the multiplication of the input matrices matA and matB in one of the following forms,

\begin{align*} & \ms{matC} \leftarrow \alpha \ms{matA}~~ \ms{matB} + \beta \ms{matC} && \ms{matC} \leftarrow \alpha \ms{matA}~~ \ms{matB}^T + \beta \ms{matC} && \ms{matC} \leftarrow \alpha \ms{matA}~~ \ms{matB}^H + \beta \ms{matC} \\ & \ms{matC} \leftarrow \alpha \ms{matA}^T \ms{matB} + \beta \ms{matC} && \ms{matC} \leftarrow \alpha \ms{matA}^T \ms{matB}^T + \beta \ms{matC} && \ms{matC} \leftarrow \alpha \ms{matA}^T \ms{matB}^H + \beta \ms{matC} \\ & \ms{matC} \leftarrow \alpha \ms{matA}^H \ms{matB} + \beta \ms{matC} && \ms{matC} \leftarrow \alpha \ms{matA}^H \ms{matB}^T + \beta \ms{matC} && \ms{matC} \leftarrow \alpha \ms{matA}^H \ms{matB}^H + \beta \ms{matC} \end{align*}

where \(\cdot^T\) represents a Symmetric transpose, \(\cdot^H\) represents a Hermitian transpose, and matA or matB can be also specified as Symmetric/Hermitian upper/lower triangular matrices.

The following figure illustrates the form of the general matrix-matrix or matrix-vector multiplication depending on the input values.

  1. Default Multiplication (no transposition involved).

  2. Same as the default but with operationA set to transSymm or transHerm.

  3. Same as the default but with operationB set to transSymm or transHerm.

For symmetric or Hermitian matrices, only the upper or lower triangles of the corresponding matrices are required and referenced.
Although the implementation is custom-defined for Symmetric/Hermitian matrices, the multiplication is defined as in the figure of case 1 above.

Note
For triangular matrix-matrix or matrix-vector multiplications, see pm_matrixMulTri.
BLAS/LAPACK equivalent:
The procedures under discussion combine, modernize, and extend the interface and functionalities of Version 3.11 of BLAS/LAPACK routine(s): SAXPY, DAXPY, CAXPY, ZAXPY SGEMV, DGEMV, CGEMV, ZGEMV SSPMV, DSPMV, CHPMV, ZHPMV, SSYMV, DSYMV, CHEMV, ZHEMV, SGEMM, DGEMM, CGEMM, ZGEMM, SSYMM, DSYMM, CSYMM, ZSYMM, CHEMM, ZHEMM.
In particular multiplications of matrices of type integer of arbitrary kinds are also possible.
See also
lowDia
uppDia
symmetric
hermitian
transSymm
transHerm
setMatMulTri
Benchmarks:


Benchmark :: The runtime performance of setMatMulAdd vs. other other approaches.

1! Test the performance of `setMatMulAdd()` vs. optimized BLAS.
2program benchmark
3
4 use pm_matrixInit, only: getMatInit, uppLowDia
6 use pm_distUnif, only: setUnifRand
7 use iso_fortran_env, only: error_unit
8 use pm_kind, only: SK, IK, RKG => RK
9 use pm_bench, only: bench_type
10
11 implicit none
12
13 integer(IK) :: itry, ntry
14 integer(IK) :: ibench
15 integer(IK) :: irank
16 integer(IK) :: rank
17 integer(IK) :: fileUnit
18 integer(IK) , parameter :: NUM_RANK = 10_IK
19 integer(IK) , parameter :: MAX_RANK = 2**NUM_RANK
20 integer(IK) , parameter :: MAX_ITER = 10000
21 real(RKG) , parameter :: ALPHA = 1._RKG
22 real(RKG) , parameter :: BETA = 1._RKG
23 real(RKG) :: dummySum = 0._RKG
24 real(RKG) , dimension(:,:), allocatable :: matA, matB, matC
25 type(bench_type) , allocatable :: bench(:)
26
27 bench = [ bench_type(name = SK_"setMatMulAddExplicit", exec = setMatMulAddExplicit, overhead = setOverhead) &
28 , bench_type(name = SK_"setMatMulAddAssumed", exec = setMatMulAddAssumed, overhead = setOverhead) &
29 , bench_type(name = SK_"matmul", exec = setMatMulTri, overhead = setOverhead) &
30#if BLAS_ENABLED
31 , bench_type(name = SK_"GEMM", exec = setBlasGEMM, overhead = setOverhead) &
32#endif
33 ]
34
35 write(*,"(*(g0,:,' '))")
36 write(*,"(*(g0,:,' '))") "setMatMulAdd() vs. matmul() vs. GEMM()"
37 write(*,"(*(g0,:,' '))")
38
39 open(newunit = fileUnit, file = "main.out", status = "replace")
40 write(fileUnit, "(*(g0,:,', '))") "Matrix Rank", (bench(ibench)%name, ibench = 1, size(bench)) !, (bench(ibench)%name, ibench = 1, size(bench))
41 loopOverMatDiaRank: do irank = 1, NUM_RANK
42
43 rank = 2**irank
44 ntry = MAX_ITER / rank
45 write(*,"(*(g0,:,' '))") "Benchmarking with rank:", rank
46 matA = getMatInit([rank, rank], uppLowDia, 0._RKG, 0._RKG, 0._RKG); !call setUnifRand(matA)
47 matB = getMatInit([rank, rank], uppLowDia, 0._RKG, 0._RKG, 0._RKG); !call setUnifRand(matB)
48 matC = getMatInit([rank, rank], uppLowDia, 0._RKG, 0._RKG, 0._RKG); !call setUnifRand(matC)
49
50 ! warmup
51 call setMatMulAddExplicit()
52 call setMatMulAddAssumed()
53 call setMatMulTri()
54#if BLAS_ENABLED
55 call setBlasGEMM()
56#endif
57
58 do ibench = 1, size(bench)
59 bench(ibench)%timing = bench(ibench)%getTiming(minsec = 0.07_RKG)
60 end do
61 write(fileUnit, "(*(g0,:,', '))") rank &
62 , (bench(ibench)%timing%mean / ntry, ibench = 1, size(bench)) !&
63 !, (bench(1)%timing%mean/bench(ibench)%timing%mean, ibench = 1, NBENCH)
64 end do loopOverMatDiaRank
65 write(*,"(*(g0,:,' '))") sum(matC)
66 write(*,"(*(g0,:,' '))")
67 close(fileUnit)
68
69contains
70
71 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
72 ! procedure wrappers.
73 !%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
74
75 subroutine setOverhead()
76 do itry = 1, ntry
77 call getDummy()
78 end do
79 end subroutine
80
81 subroutine getDummy()
82 dummySum = dummySum + matC(1,1)
83 end subroutine
84
85 subroutine setMatMulAddExplicit()
86 do itry = 1, ntry
87 call setMatMulAdd(matA, matB, matC, alpha, beta, rank, rank, rank, 0_IK, 0_IK, 0_IK, 0_IK, 0_IK, 0_IK)
88 call getDummy()
89 end do
90 end subroutine
91
92 subroutine setMatMulAddAssumed()
93 do itry = 1, ntry
94 call setMatMulAdd(matA, matB, matC)
95 call getDummy()
96 end do
97 end subroutine
98
99 subroutine setMatMulTri()
100 do itry = 1, ntry
101 matC = alpha * matmul(matA(1:rank, 1:rank), matB(1:rank, 1:rank)) + beta * matC(1:rank, 1:rank)
102 call getDummy()
103 end do
104 end subroutine
105
106#if BLAS_ENABLED
107 subroutine setBlasGEMM()
108 do itry = 1, ntry
109 call DGEMM( "N" & ! transa
110 , "N" & ! transb
111 , rank & ! m
112 , rank & ! n
113 , rank & ! k
114 , alpha & ! alpha
115 , matA & ! a
116 , rank & ! lda
117 , matB & ! b
118 , rank & ! ldb
119 , beta & ! beta
120 , matC & ! c
121 , rank & ! ldc
122 )
123 call getDummy()
124 end do
125 end subroutine
126#endif
127
128end program benchmark
Generate and return an object of type timing_type containing the benchmark timing information and sta...
Definition: pm_bench.F90:574
Return a uniform random scalar or contiguous array of arbitrary rank of randomly uniformly distribute...
Generate and return a matrix of shape (shape(1), shape(2)) with the upper/lower triangle and the diag...
Return the result of the multiplication of the input matrices/vector matA and matB in the user-specif...
This module contains abstract interfaces and types that facilitate benchmarking of different procedur...
Definition: pm_bench.F90:41
This module contains classes and procedures for computing various statistical quantities related to t...
This module defines the relevant Fortran kind type-parameters frequently used in the ParaMonte librar...
Definition: pm_kind.F90:268
integer, parameter RK
The default real kind in the ParaMonte library: real64 in Fortran, c_double in C-Fortran Interoperati...
Definition: pm_kind.F90:543
integer, parameter IK
The default integer kind in the ParaMonte library: int32 in Fortran, c_int32_t in C-Fortran Interoper...
Definition: pm_kind.F90:540
integer, parameter SK
The default character kind in the ParaMonte library: kind("a") in Fortran, c_char in C-Fortran Intero...
Definition: pm_kind.F90:539
This module contains procedures and generic interfaces relevant to generating and initializing matric...
This module contains procedures and generic interfaces relevant to combined matrix-matrix or matrix-v...
This is the class for creating benchmark and performance-profiling objects.
Definition: pm_bench.F90:386

Example Unix compile command via Intel ifort compiler
1#!/usr/bin/env sh
2rm main.exe
3ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Example Windows Batch compile command via Intel ifort compiler
1del main.exe
2set PATH=..\..\..\lib;%PATH%
3ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
4main.exe

Example Unix / MinGW compile command via GNU gfortran compiler
1#!/usr/bin/env sh
2rm main.exe
3gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
4./main.exe

Postprocessing of the benchmark output
1#!/usr/bin/env python
2
3import matplotlib.pyplot as plt
4import pandas as pd
5import numpy as np
6
7import os
8dirname = os.path.basename(os.getcwd())
9
10fontsize = 14
11
12df = pd.read_csv("main.out", delimiter = ", ")
13colnames = list(df.columns.values)
14
15
18
19ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
20ax = plt.subplot()
21
22for colname in colnames[1:]:
23 plt.plot( df[colnames[0]].values
24 , df[colname].values
25 , linewidth = 2
26 )
27
28plt.xticks(fontsize = fontsize)
29plt.yticks(fontsize = fontsize)
30ax.set_xlabel(colnames[0], fontsize = fontsize)
31ax.set_ylabel("Runtime [ seconds ]", fontsize = fontsize)
32ax.set_title(" vs. ".join(colnames[1:])+"\nLower is better.", fontsize = fontsize)
33ax.set_xscale("log")
34ax.set_yscale("log")
35plt.minorticks_on()
36plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
37ax.tick_params(axis = "y", which = "minor")
38ax.tick_params(axis = "x", which = "minor")
39ax.legend ( colnames[1:]
40 #, loc='center left'
41 #, bbox_to_anchor=(1, 0.5)
42 , fontsize = fontsize
43 )
44
45plt.tight_layout()
46plt.savefig("benchmark." + dirname + ".runtime.png")
47
48
51
52ax = plt.figure(figsize = 1.25 * np.array([6.4,4.6]), dpi = 200)
53ax = plt.subplot()
54
55plt.plot( df[colnames[0]].values
56 , np.ones(len(df[colnames[0]].values))
57 , linestyle = "--"
58 #, color = "black"
59 , linewidth = 2
60 )
61for colname in colnames[2:]:
62 plt.plot( df[colnames[0]].values
63 , df[colname].values / df[colnames[1]].values
64 , linewidth = 2
65 )
66
67plt.xticks(fontsize = fontsize)
68plt.yticks(fontsize = fontsize)
69ax.set_xlabel(colnames[0], fontsize = fontsize)
70ax.set_ylabel("Runtime compared to {}".format(colnames[1]), fontsize = fontsize)
71ax.set_title("Runtime Ratio Comparison. Lower means faster.\nLower than 1 means faster than {}().".format(colnames[1]), fontsize = fontsize)
72ax.set_xscale("log")
73#ax.set_yscale("log")
74plt.minorticks_on()
75plt.grid(visible = True, which = "both", axis = "both", color = "0.85", linestyle = "-")
76ax.tick_params(axis = "y", which = "minor")
77ax.tick_params(axis = "x", which = "minor")
78ax.legend ( colnames[1:]
79 #, bbox_to_anchor = (1, 0.5)
80 #, loc = "center left"
81 , fontsize = fontsize
82 )
83
84plt.tight_layout()
85plt.savefig("benchmark." + dirname + ".runtime.ratio.png")

Visualization of the benchmark output

Benchmark moral
  1. The procedures under the generic interface setMatMulAdd are enhancements to the original BLAS routines.
    However, they are currently not optimized for cache-efficient data access on various hardware.
    Similarly, setMatMulAdd does not utilize blocking methods to improve cache access.
  2. For practical routine daily usages, setMatMulAdd offers a nice generic matrix multiplication interface.
    However, the current implementations are suboptimal to hardware-tuned BLAS libraries such as OpenBLAS and MKL.
    Such performance differences will be most striking for large matrices where cache-efficiency dominates the performance bottleneck.
Test:
test_pm_matrixMulAdd
Todo:
Critical Priority: The following BLAS Band-matrix routines must be added to this module:
  1. SGBMV, DGBMV, CGBMV, and ZGBMV (Matrix-Vector Product for a General Band Matrix, Its Transpose, or Its Conjugate Transpose).
  2. SSBMV, DSBMV, CHBMV, and ZHBMV (Matrix-Vector Product for a Real Symmetric or Complex Hermitian Band Matrix).
  3. STBMV, DTBMV, CTBMV, and ZTBMV (Matrix-Vector Product for a Triangular Band Matrix, Its Transpose, or Its Conjugate Transpose).


Final Remarks


If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

  1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
  2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Author:
Amir Shahmoradi, Friday 1:54 AM, April 21, 2017, Institute for Computational Engineering and Sciences (ICES), The University of Texas, Austin, TX

Variable Documentation

◆ MODULE_NAME

character(*, SK), parameter pm_matrixMulAdd::MODULE_NAME = "@pm_matrixMulAdd"

Definition at line 118 of file pm_matrixMulAdd.F90.