R vs Rcpp vs Armadillo 中矩阵 rowSums() 与 colSums() 的效率 [英] Efficiency of matrix rowSums() vs. colSums() in R vs Rcpp vs Armadillo
问题描述
来自 R 编程，我正在使用 Rcpp 扩展为 C/C++ 形式的编译代码.作为循环交换效果的实践练习(一般来说只是 C/C++)，我实现了与 R 的 rowSums() 和 colSums() 等价的矩阵函数使用 Rcpp(我知道这些作为 Rcpp 糖和犰狳存在 - 这只是一个练习).
Coming from R programming, I'm in the process of expanding to compiled code in the form of C/C++ with Rcpp. As a hands on exercise on the effect of loop interchange (and just C/C++ in general), I implemented equivalents to R's rowSums() and colSums() functions for matrices with Rcpp (I know these exist as Rcpp sugar and in Armadillo -- this was just an exercise).
我有 rowSums() 和 colSums() 的 C++ 实现以及 Rcpp Sugar 和 arma::sum() 版本="https://gist.github.com/mikmart/66bf16f0bd329ec468aade6bf81fee96" rel="noreferrer">这个matsums.cpp 文件.我的只是像这样的简单循环:
I have my C++ implementation of rowSums() and colSums() along with Rcpp sugar and arma::sum() versions in this matsums.cpp file. Mine are just simple loops like this:
NumericVector Cpp_colSums(const NumericMatrix& x) {
int nr = x.nrow(), nc = x.ncol();
NumericVector ans(nc);
for (int j = 0; j < nc; j++) {
double sum = 0.0;
for (int i = 0; i < nr; i++) {
sum += x(i, j);
}
ans[j] = sum;
}
return ans;
}
NumericVector Cpp_rowSums(const NumericMatrix& x) {
int nr = x.nrow(), nc = x.ncol();
NumericVector ans(nr);
for (int j = 0; j < nc; j++) {
for (int i = 0; i < nr; i++) {
ans[i] += x(i, j);
}
}
return ans;
}
(R 矩阵按列存储，因此外循环中的列应该是更有效的方法.这就是我最初测试的内容.)
在对这些进行基准测试时，我遇到了一些出乎意料的问题:行总和和列总和之间存在明显的性能差异(请参阅下面的基准):
While running benchmarks on these, I ran into something I wasn't expecting: there was a clear performance difference between row sums and col sums (see benchmarks below):
- 使用内置的 R 函数，
colSums()的速度大约是rowSums()的两倍. - 使用我自己的 Rcpp 和糖版本，情况正好相反:
rowSums()的速度大约是colSums()的两倍. - 最后，添加 Armadillo 实现，操作大致相同(col sum 可能会快一点，因为我也希望它们在 R 中).
- Using the builtin R functions,
colSums()is about twice as fast asrowSums(). - With my own Rcpp and the sugar version, this is reversed: it is
rowSums()that is about twice as fast ascolSums(). - And finally, adding the Armadillo implementations, the operations are roughly equal (col sum maybe a bit faster, as I would have expected them to be in R, too).
所以我的主要问题是:为什么 Cpp_rowSums() 比 Cpp_colSums() 快得多?
So my primary question is: why is Cpp_rowSums() significantly faster than Cpp_colSums()?
作为次要兴趣，我也很好奇为什么在 R 实现中颠倒了相同的差异.(我浏览了C 源代码，但无法真正分辨出显着差异.)(第三，犰狳如何获得相同的性能?)
As a secondary interest, I'm also curious why the same difference is reversed in the R implementations. (I skimmed through the C source, but could not really make out the significant differences.) (And third, how come Armadillo gets equal performance?)
我在 10,000 x 10,000 对称矩阵上测试了这两个函数的所有 4 个实现:
I tested all 4 implementations of both functions on a 10,000 x 10,000 symmetric matrix:
Rcpp::sourceCpp("matsums.cpp")
set.seed(92136)
n <- 1e4 # build n x n test matrix
x <- matrix(rnorm(n), 1, n)
x <- crossprod(x, x) # symmetric
bench::mark(
rowSums(x),
colSums(x),
Cpp_rowSums(x),
Cpp_colSums(x),
Sugar_rowSums(x),
Sugar_colSums(x),
Arma_rowSums(x),
Arma_colSums(x)
)[, 1:7]
#> # A tibble: 8 x 7
#> expression min mean median max `itr/sec` mem_alloc
#> <chr> <bch:tm> <bch:tm> <bch:tm> <bch:tm> <dbl> <bch:byt>
#> 1 rowSums(x) 192.2ms 207.9ms 194.6ms 236.9ms 4.81 78.2KB
#> 2 colSums(x) 93.4ms 97.2ms 96.5ms 101.3ms 10.3 78.2KB
#> 3 Cpp_rowSums(x) 73.5ms 76.3ms 76ms 80.4ms 13.1 80.7KB
#> 4 Cpp_colSums(x) 126.5ms 127.6ms 126.8ms 130.3ms 7.84 80.7KB
#> 5 Sugar_rowSums(x) 73.9ms 75.6ms 74.3ms 79.4ms 13.2 80.7KB
#> 6 Sugar_colSums(x) 124.2ms 125.8ms 125.6ms 127.9ms 7.95 80.7KB
#> 7 Arma_rowSums(x) 73.2ms 74.7ms 73.9ms 79.3ms 13.4 80.7KB
#> 8 Arma_colSums(x) 62.8ms 64.4ms 63.7ms 69.6ms 15.5 80.7KB
(同样，您可以找到 C++ 源文件 matsums.cpp 此处.)
(Again, you can find the C++ source file matsums.cpp here.)
平台:
> sessioninfo::platform_info()
setting value
version R version 3.5.1 (2018-07-02)
os Windows >= 8 x64
system x86_64, mingw32
ui RStudio
language (EN)
collate English_United States.1252
tz Europe/Helsinki
date 2018-08-09
更新
进一步调查，我也使用R的传统C接口编写了相同的函数:来源是这里.我使用R CMD SHLIB编译函数，并再次测试:行总和比列总和更快的相同现象持续存在(基准).然后我还查看了与objdump的反汇编，但在我看来(我对 asm 的理解非常有限)编译器并没有真正优化主循环体(行, cols) 是否离 C 代码更远?
Update
Investigating further, I also wrote the same functions using R's traditional C interface: the source is here. I compiled the functions with R CMD SHLIB, and tested again: the same phenomenon of row sums being faster than col sums persisted (benchmarks). I then also looked at the disassembly with objdump, but it seems to me (with my very limited understanding of asm) that the compiler doesn't really optimize the main loop bodies (rows, cols) any further from the C code, either?
推荐答案
首先，让我展示一下我的笔记本电脑上的计时统计信息.我使用 5000 x 5000 矩阵足以进行基准测试，microbenchmark 包用于 100 次评估.
First, let me show the timing statistics on my laptop. I use a 5000 x 5000 matrix which is sufficient for benchmarking, and microbenchmark package is used for 100 evaluations.
Unit: milliseconds
expr min lq mean median uq max
colSums(x) 71.40671 71.64510 71.80394 71.72543 71.80773 75.07696
Cpp_colSums(x) 71.29413 71.42409 71.65525 71.48933 71.56241 77.53056
Sugar_colSums(x) 73.05281 73.19658 73.38979 73.25619 73.31406 76.93369
Arma_colSums(x) 39.08791 39.34789 39.57979 39.43080 39.60657 41.70158
rowSums(x) 177.33477 187.37805 187.57976 187.49469 187.73155 194.32120
Cpp_rowSums(x) 54.00498 54.37984 54.70358 54.49165 54.73224 64.16104
Sugar_rowSums(x) 54.17001 54.38420 54.73654 54.56275 54.75695 61.80466
Arma_rowSums(x) 49.54407 49.77677 50.13739 49.90375 50.06791 58.29755
R 核心中的 C 代码并不总是比我们自己编写的好.Cpp_rowSums 比 rowSums 显示的要快.我觉得自己没有能力解释为什么 R 的版本比它应该的慢.我将重点关注:我们如何进一步优化我们自己的 colSums 和 rowSums 以击败犰狳.请注意，我编写 C，使用 R 的旧 C 接口并使用 R CMD SHLIB 进行编译.
C code in R core is not always better than what we can write ourselves. That Cpp_rowSums is faster than rowSums shows this. I don't feel myself competent to explain why R's version is slower than it should be. I will focuse on: how we can further optimize our own colSums and rowSums to beat Armadillo. Note that I write C, use R's old C interface and do compilation with R CMD SHLIB.
如果我们有一个远大于 CPU 缓存容量的 nxn 矩阵，colSums 从 RAM 加载 nxn 数据到缓存，但 rowSums 加载两倍，即 2 xnxn.
If we have an n x n matrix that is much larger than the capacity of a CPU cache, colSums loads n x n data from RAM to cache, but rowSums loads as twice as many, i.e., 2 x n x n.
考虑保存总和的结果向量:这个 length-n 向量从 RAM 加载到缓存中的次数是多少?对于colSums，它只加载一次，但对于rowSums，它被加载n 次.每次向其中添加矩阵列时，它都会加载到缓存中，但由于太大而被逐出.
Think about the resulting vector that holds the sum: how many times this length-n vector is loaded into cache from RAM? For colSums, it is loaded only once, but for rowSums, it is loaded n times. Each time you add a matrix column to it, it is loaded into cache but then evicted since it is too big.
对于大n:
colSums导致n x n + n数据从 RAM 加载到缓存；rowSums导致n x n + n x n数据从 RAM 加载到缓存.
colSumscausesn x n + ndata load from RAM to cache;rowSumscausesn x n + n x ndata load from RAM to cache.
换句话说，rowSums 理论上内存效率较低，而且可能会更慢.
In other words, rowSums is in theory less memory efficient, and is likely to be slower.
由于 RAM 和缓存之间的数据流很容易优化，唯一的改进是循环展开.将内部循环(求和循环)展开深度为 2 就足够了，我们将看到 2 倍的提升.
Since the data flow between RAM and cache is readily optimal, the only improvement is loop unrolling. Unrolling the inner loop (the summation loop) by a depth of 2 is sufficient and we will see a 2x boost.
循环展开工作是因为它启用了 CPU 的指令管道.如果我们每次迭代只做一个加法，流水线是不可能的；通过两个添加，这个指令级并行开始工作.我们也可以将循环展开深度为 4，但我的经验是深度 2 展开足以从循环展开中获得大部分收益.
Loop unrolling works as it enables CPU's instruction pipeline. If we just do one addition per iteration, no pipelining is possible; with two additions this instruction-level parallelism starts to work. We can also unroll the loop by a depth of 4, but my experience is that a depth-2 unrolling is sufficient to gain most of the benefit from loop unrolling.
优化数据流是第一步.我们需要先做缓存阻塞，以将数据传输从 2 x n x n 减少到 n x n.
Optimization of data flow is the first step. We need to first do cache blocking to reduce the data transfer from 2 x n x n down to n x n.
将这个nxn矩阵切分成若干行块:每块是2040 xn(最后一个块可能更小)，然后应用普通的rowSums 一块一块的.对于每个块，累加器向量的长度为 2040，大约是 32KB CPU 缓存可以容纳的长度的一半.对于添加到此累加器向量的矩阵列，另一半被反转.这样，累加器向量可以保存在缓存中，直到处理完该块中的所有矩阵列.因此，累加器向量只加载到缓存中一次，因此整体内存性能与 colSums 一样好.
Chop this n x n matrix into a number of row chunks: each being 2040 x n (the last chunk may be smaller), then apply the ordinary rowSums chunk by chunk. For each chunk, the accumulator vector has length-2040, about half of what a 32KB CPU cache can hold. The other half is reversed for a matrix column added to this accumulator vector. In this way, the accumulator vector can be hold in the cache until all matrix columns in this chunk are processed. As a result, the accumulator vector is only loaded into cache once, hence the overall memory performance is as good as that for colSums.
现在我们可以进一步对每个块中的 rowSums 应用循环展开.将外循环和内循环展开深度为 2，我们将看到提升.一旦外循环展开，块大小应该减少到 1360，因为现在我们需要缓存空间来保存每次外循环迭代的三个长度为 1360 的向量.
Now we can further apply loop unrolling for the rowSums in each chunk. Unroll both the outer loop and inner loop by a depth of 2, we will see a boost. Once the outer loop is unrolled, the chunk size should be reduced to 1360, as now we need space in the cache to hold three length-1360 vectors per outer loop iteration.
编写带有循环展开的代码可能是一项令人讨厌的工作，因为我们现在需要为一个函数编写多个不同的版本.
Writing code with loop unrolling can be a nasty job as we now need to write several different versions for a function.
对于colSums，我们需要两个版本:
For colSums, we need two versions:
colSums_1x1:内循环和外循环都以深度1展开，即这是一个没有循环展开的版本；colSums_2x1:不展开外循环，而展开深度为 2 的内循环.
colSums_1x1: both inner and outer loops are unrolled with depth 1, i.e., this is a version without loop unrolling;colSums_2x1: no outer loop unrolling, while inner loop is unrolled with depth 2.
对于rowSums，我们最多可以有四个版本，rowSums_sxt，其中s = 1 or 2 是内循环的展开深度，t = 1 or 2 是外环的展开深度.
For rowSums we can have up to four versions, rowSums_sxt, where s = 1 or 2 is the unrolling depth for inner loop and t = 1 or 2 is the unrolling depth for outer loop.
如果我们一个一个地编写每个版本，代码编写可能会非常乏味.经过多年或对此感到沮丧后，我开发了一个自动代码/版本生成"；使用内联模板函数和宏的技巧.
Code writing can be very tedious if we write each version one by one. After many years or frustration on this I developed an "automatic code / version generation" trick using inlined template functions and macros.
#include <stdlib.h>
#include <Rinternals.h>
static inline void colSums_template_sx1 (size_t s,
double *A, size_t LDA,
size_t nr, size_t nc,
double *sum) {
size_t nrc = nr % s, i;
double *A_end = A + LDA * nc, a0, a1;
for (; A < A_end; A += LDA) {
a0 = 0.0; a1 = 0.0; // accumulator register variables
if (nrc > 0) a0 = A[0]; // is there a "fractional loop"?
for (i = nrc; i < nr; i += s) { // main loop of depth-s
a0 += A[i]; // 1st iteration
if (s > 1) a1 += A[i + 1]; // 2nd iteration
}
if (s > 1) a0 += a1; // combine two accumulators
*sum++ = a0; // write-back
}
}
#define macro_define_colSums(s, colSums_sx1) \
SEXP colSums_sx1 (SEXP matA) { \
double *A = REAL(matA); \
size_t nrow_A = (size_t)nrows(matA); \
size_t ncol_A = (size_t)ncols(matA); \
SEXP result = PROTECT(allocVector(REALSXP, ncols(matA))); \
double *sum = REAL(result); \
colSums_template_sx1(s, A, nrow_A, nrow_A, ncol_A, sum); \
UNPROTECT(1); \
return result; \
}
macro_define_colSums(1, colSums_1x1)
macro_define_colSums(2, colSums_2x1)
模板函数计算(在 R 语法中)sum <- colSums(A[1:nr, 1:nc]) 矩阵 A 与 LDA(A 的前导维度)行.参数 s 是对内循环展开的版本控制.模板函数乍一看很可怕，因为它包含许多 if.但是，它被声明为static inline.如果通过将已知常量 1 或 2 传递给 s 来调用它，则优化编译器能够在编译时评估这些 if，消除无法访问的代码并删除代码".设置但未使用"变量(已初始化、修改但未写回 RAM 的寄存器变量).
The template function computes (in R-syntax) sum <- colSums(A[1:nr, 1:nc]) for a matrix A with LDA (leading dimension of A) rows. The parameter s is a version control on inner loop unrolling. The template function looks horrible at first glance as it contains many if. However, it is declared static inline. If it is called by passing in known constant 1 or 2 to s, an optimizing compiler is able to evaluate those if at compile-time, eliminate unreachable code and drop "set-but-not-used" variables (registers variables that are initialized, modified but not written back to RAM).
宏用于函数声明.接受一个常量 s 和一个函数名，它会生成一个具有所需循环展开版本的函数.
The macro is used for function declaration. Accepting a constant s and a function name, it generates a function with desired loop unrolling version.
以下是rowSums.
static inline void rowSums_template_sxt (size_t s, size_t t,
double *A, size_t LDA,
size_t nr, size_t nc,
double *sum) {
size_t ncr = nc % t, nrr = nr % s, i;
double *A_end = A + LDA * nc, *B;
double a0, a1;
for (i = 0; i < nr; i++) sum[i] = 0.0; // necessary initialization
if (ncr > 0) { // is there a "fractional loop" for the outer loop?
if (nrr > 0) sum[0] += A[0]; // is there a "fractional loop" for the inner loop?
for (i = nrr; i < nr; i += s) { // main inner loop with depth-s
sum[i] += A[i];
if (s > 1) sum[i + 1] += A[i + 1];
}
A += LDA;
}
for (; A < A_end; A += t * LDA) { // main outer loop with depth-t
if (t > 1) B = A + LDA;
if (nrr > 0) { // is there a "fractional loop" for the inner loop?
a0 = A[0]; if (t > 1) a0 += A[LDA];
sum[0] += a0;
}
for(i = nrr; i < nr; i += s) { // main inner loop with depth-s
a0 = A[i]; if (t > 1) a0 += B[i];
sum[i] += a0;
if (s > 1) {
a1 = A[i + 1]; if (t > 1) a1 += B[i + 1];
sum[i + 1] += a1;
}
}
}
}
#define macro_define_rowSums(s, t, rowSums_sxt) \
SEXP rowSums_sxt (SEXP matA, SEXP chunk_size) { \
double *A = REAL(matA); \
size_t nrow_A = (size_t)nrows(matA); \
size_t ncol_A = (size_t)ncols(matA); \
SEXP result = PROTECT(allocVector(REALSXP, nrows(matA))); \
double *sum = REAL(result); \
size_t block_size = (size_t)asInteger(chunk_size); \
size_t i, block_size_i; \
if (block_size > nrow_A) block_size = nrow_A; \
for (i = 0; i < nrow_A; i += block_size_i) { \
block_size_i = nrow_A - i; if (block_size_i > block_size) block_size_i = block_size; \
rowSums_template_sxt(s, t, A, nrow_A, block_size_i, ncol_A, sum); \
A += block_size_i; sum += block_size_i; \
} \
UNPROTECT(1); \
return result; \
}
macro_define_rowSums(1, 1, rowSums_1x1)
macro_define_rowSums(1, 2, rowSums_1x2)
macro_define_rowSums(2, 1, rowSums_2x1)
macro_define_rowSums(2, 2, rowSums_2x2)
注意，模板函数现在接受s和t，宏定义的函数应用了行组块.
Note that the template function now accepts s and t, and the function to be defined by the macro has applied row chunking.
尽管我在代码中留下了一些注释，但代码可能仍然不容易理解，但我不能花更多时间来更详细地解释.
Even though I've left some comments along the code, the code is probably still not easy to follow, but I can't take more time to explain in greater details.
要使用它们，请将它们复制并粘贴到名为matSums.c"的 C 文件中；并用 R CMD SHLIB -c matSums.c 编译它.
To use them, copy and paste them into a C file called "matSums.c" and compile it with R CMD SHLIB -c matSums.c.
对于 R 端，在matSums.R"中定义以下函数.
For the R side, define the following functions in "matSums.R".
colSums_zheyuan <- function (A, s) {
dyn.load("matSums.so")
if (s == 1) result <- .Call("colSums_1x1", A)
if (s == 2) result <- .Call("colSums_2x1", A)
dyn.unload("matSums.so")
result
}
rowSums_zheyuan <- function (A, chunk.size, s, t) {
dyn.load("matSums.so")
if (s == 1 && t == 1) result <- .Call("rowSums_1x1", A, as.integer(chunk.size))
if (s == 2 && t == 1) result <- .Call("rowSums_2x1", A, as.integer(chunk.size))
if (s == 1 && t == 2) result <- .Call("rowSums_1x2", A, as.integer(chunk.size))
if (s == 2 && t == 2) result <- .Call("rowSums_2x2", A, as.integer(chunk.size))
dyn.unload("matSums.so")
result
}
现在让我们进行一个基准测试，再次使用 5000 x 5000 矩阵.
Now let's have a benchmark, again with a 5000 x 5000 matrix.
A <- matrix(0, 5000, 5000)
library(microbenchmark)
source("matSums.R")
microbenchmark("col0" = colSums(A),
"col1" = colSums_zheyuan(A, 1),
"col2" = colSums_zheyuan(A, 2),
"row0" = rowSums(A),
"row1" = rowSums_zheyuan(A, nrow(A), 1, 1),
"row2" = rowSums_zheyuan(A, 2040, 1, 1),
"row3" = rowSums_zheyuan(A, 1360, 1, 2),
"row4" = rowSums_zheyuan(A, 1360, 2, 2))
在我的笔记本电脑上我得到:
On my laptop I get:
Unit: milliseconds
expr min lq mean median uq max neval
col0 65.33908 71.67229 71.87273 71.80829 71.89444 111.84177 100
col1 67.16655 71.84840 72.01871 71.94065 72.05975 77.84291 100
col2 35.05374 38.98260 39.33618 39.09121 39.17615 53.52847 100
row0 159.48096 187.44225 185.53748 187.53091 187.67592 202.84827 100
row1 49.65853 54.78769 54.78313 54.92278 55.08600 60.27789 100
row2 49.42403 54.56469 55.00518 54.74746 55.06866 60.31065 100
row3 37.43314 41.57365 41.58784 41.68814 41.81774 47.12690 100
row4 34.73295 37.20092 38.51019 37.30809 37.44097 99.28327 100
注意循环展开如何加快 colSums 和 rowSums.通过全面优化(col2"和row4")，我们击败了犰狳(参见本答案开头的时序表).
Note how loop unrolling speeds up both colSums and rowSums. And with full optimization ("col2" and "row4"), we beat Armadillo (see the timing table at the beginning of this answer).
在这种情况下，行组块策略并没有明显产生好处.让我们尝试一个包含数百万行的矩阵.
The row chunking strategy does not clearly yield benefit in this case. Let's try a matrix with millions of rows.
A <- matrix(0, 1e+7, 20)
microbenchmark("row1" = rowSums_zheyuan(A, nrow(A), 1, 1),
"row2" = rowSums_zheyuan(A, 2040, 1, 1),
"row3" = rowSums_zheyuan(A, 1360, 1, 2),
"row4" = rowSums_zheyuan(A, 1360, 2, 2))
我明白
Unit: milliseconds
expr min lq mean median uq max neval
row1 604.7202 607.0256 617.1687 607.8580 609.1728 720.1790 100
row2 514.7488 515.9874 528.9795 516.5193 521.4870 636.0051 100
row3 412.1884 413.8688 421.0790 414.8640 419.0537 525.7852 100
row4 377.7918 379.1052 390.4230 379.9344 386.4379 476.9614 100
在这种情况下，我们观察到缓存阻塞带来的收益.
In this case we observe the gains from cache blocking.
基本上这个答案已经解决了所有问题，除了以下问题:
Basically this answer has addressed all the issues, except for the following:
- 为什么 R 的
rowSums效率低于应有的效率. - 为什么没有任何优化，
rowSums(row1")比colSums(col1")快.
- why R's
rowSumsis less efficient than it should be. - why without any optimization,
rowSums("row1") is faster thancolSums("col1").
再说一次，我无法解释第一个，实际上我不在乎，因为我们可以轻松编写一个比 R 的内置版本更快的版本.
Again, I cannot explain the first and actually I don't care that since we can easily write a version that is faster than R's built-in version.
第二个绝对值得追求.我在我们的讨论室中复制了我的评论以作记录.
The 2nd is definitely worth pursuing. I copy in my comments in our discussion room for a record.
这个问题归结为:为什么将单个向量相加比逐元素相加两个向量慢?"
This issue is down to this: "why adding up a single vector is slower than adding two vectors element-wise?"
我不时看到类似的现象.我第一次遇到这种奇怪的行为是几年前，当我编写自己的矩阵乘法时.我发现 DAXPY 比 DDOT 快.
I see similar phenomenon from time to time. The first time I encountered this strange behavior was when I, a few years ago, coded my own matrix-matrix multiplication. I found that DAXPY is faster than DDOT.
DAXPY 这样做:y += a * x，其中 x 和 y 是向量，a是一个标量；DDOT 这样做:a += x * y.
DAXPY does this: y += a * x, where x and y are vectors and a is a scalar; DDOT does this: a += x * y.
鉴于比 DDOT 是一个归约操作，我希望它比 DAXPY 更快.但是不，DAXPY 更快.
Given than DDOT is a reduction operation I expect that it is faster than DAXPY. But no, DAXPY is faster.
但是，一旦我在矩阵乘法的三重循环嵌套中展开循环，DDOT 就比 DAXPY 快得多.
However, as soon as I unroll the loop in the triple loop-nest of the matrix-multiplication, DDOT is much faster than DAXPY.
您的问题也发生了非常相似的情况.归约操作:a = x[1] + x[2] + ... + x[n] 比按元素加法慢:y[i] += x[我].但是一旦循环展开，后者的优势就失去了.
A very similar thing happens to your issue. A reduction operation: a = x[1] + x[2] + ... + x[n] is slower than element-wise add: y[i] += x[i]. But once loop unrolling is done, the advantage of the latter is lost.
我不确定以下解释是否属实，因为我没有证据.
I am not sure whether the following explanation is true as I have no evidence.
归约操作有一个依赖链，所以计算是严格串行的；另一方面，逐元素操作没有依赖链，因此 CPU 可以更好地使用它.
The reduction operation has a dependency chain so the computation is strictly serial; on the other hand, element-wise operation has no dependency chain, so that CPU may do better with it.
一旦我们展开循环，每次迭代都有更多的算术要做，并且 CPU 可以在这两种情况下进行流水线操作.然后可以观察到归约操作的真正优势.
As soon as we unroll the loop, each iteration has more arithmetics to do and CPU can do pipelining in both cases. The true advantage of the reduction operation can then be observed.
回复 Jaap 关于使用 rowSums2 和 colSums2来自 matrixStats
仍然使用上面的 5000 x 5000 示例.
In reply to Jaap on using rowSums2 and colSums2 from matrixStats
Still using the 5000 x 5000 example above.
A <- matrix(0, 5000, 5000)
library(microbenchmark)
source("matSums.R")
library(matrixStats) ## NEW
microbenchmark("col0" = base::colSums(A),
"col*" = matrixStats::colSums2(A), ## NEW
"col1" = colSums_zheyuan(A, 1),
"col2" = colSums_zheyuan(A, 2),
"row0" = base::rowSums(A),
"row*" = matrixStats::rowSums2(A), ## NEW
"row1" = rowSums_zheyuan(A, nrow(A), 1, 1),
"row2" = rowSums_zheyuan(A, 2040, 1, 1),
"row3" = rowSums_zheyuan(A, 1360, 1, 2),
"row4" = rowSums_zheyuan(A, 1360, 2, 2))
Unit: milliseconds
expr min lq mean median uq max neval
col0 71.53841 71.72628 72.13527 71.81793 71.90575 78.39645 100
col* 75.60527 75.87255 76.30752 75.98990 76.18090 87.07599 100
col1 71.67098 71.86180 72.06846 71.93872 72.03739 77.87816 100
col2 38.88565 39.03980 39.57232 39.08045 39.16790 51.39561 100
row0 187.44744 187.58121 188.98930 187.67168 187.86314 206.37662 100
row* 158.08639 158.26528 159.01561 158.34864 158.62187 174.05457 100
row1 54.62389 54.81724 54.97211 54.92394 55.04690 56.33462 100
row2 54.15409 54.44208 54.78769 54.59162 54.76073 60.92176 100
row3 41.43393 41.63886 42.57511 41.73538 41.81844 111.94846 100
row4 37.07175 37.25258 37.45033 37.34456 37.47387 43.14157 100
我没有看到 rowSums2 和 colSums2 的性能优势.
I don't see performance advantage of rowSums2 and colSums2.
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