Optimization for custom small blas multiplication with dynamic
template parameters in C level.
- unroll for loops
- matrix access more cache coherent
- platform independant
Briefly, this commit brings 1~50% performance improvments for
most cases in small_blas_gem(m/v)_benchmark, but a small drop
for corner cases with small dimensions especially 1,2,3. Here
we list the results partially, which show decrease percentage
of executing time, compared to unoptimized version.
Platform: desktop PC (i7-7700 CPU MP8@3.60GHz + ubuntu 17.10)
(Lenovo Research Device+ Lab, <yangfan34@lenovo.com>)
Benchmark Time CPU
-----------------------------------------------------------
BM_MatrixMatrixMultiplyDynamic/2/2/2 -0.1082 -0.1083
BM_MatrixMatrixMultiplyDynamic/2/2/15 -0.1270 -0.1270
BM_MatrixMatrixMultiplyDynamic/2/4/2 -0.1433 -0.1433
BM_MatrixMatrixMultiplyDynamic/2/4/15 -0.2069 -0.2068
BM_MatrixMatrixMultiplyDynamic/2/6/2 -0.1446 -0.1446
BM_MatrixMatrixMultiplyDynamic/2/6/15 -0.2156 -0.2156
BM_MatrixMatrixMultiplyDynamic/2/8/2 -0.1788 -0.1788
BM_MatrixMatrixMultiplyDynamic/2/8/15 -0.3316 -0.3316
BM_MatrixMatrixMultiplyDynamic/2/10/2 -0.2025 -0.2025
BM_MatrixMatrixMultiplyDynamic/2/10/15 -0.3444 -0.3444
BM_MatrixMatrixMultiplyDynamic/2/12/2 -0.0515 -0.0515
BM_MatrixMatrixMultiplyDynamic/2/12/15 -0.3733 -0.3733
BM_MatrixMatrixMultiplyDynamic/2/15/2 -0.2784 -0.2784
BM_MatrixMatrixMultiplyDynamic/2/15/15 -0.3704 -0.3704
BM_MatrixMatrixMultiplyDynamic/4/2/2 -0.1839 -0.1839
BM_MatrixMatrixMultiplyDynamic/4/2/15 -0.1922 -0.1922
BM_MatrixMatrixMultiplyDynamic/4/4/2 -0.2248 -0.2248
BM_MatrixMatrixMultiplyDynamic/4/4/15 -0.3132 -0.3132
BM_MatrixMatrixMultiplyDynamic/4/6/2 -0.2311 -0.2311
BM_MatrixMatrixMultiplyDynamic/4/6/15 -0.3239 -0.3239
BM_MatrixMatrixMultiplyDynamic/4/8/2 -0.0574 -0.0574
BM_MatrixMatrixMultiplyDynamic/4/8/15 -0.4173 -0.4173
BM_MatrixMatrixMultiplyDynamic/4/10/2 -0.2861 -0.2861
BM_MatrixMatrixMultiplyDynamic/4/10/15 -0.4065 -0.4064
BM_MatrixMatrixMultiplyDynamic/4/12/2 -0.2976 -0.2975
BM_MatrixMatrixMultiplyDynamic/4/12/15 -0.4218 -0.4218
BM_MatrixMatrixMultiplyDynamic/4/15/2 -0.3116 -0.3116
BM_MatrixMatrixMultiplyDynamic/4/15/15 -0.4242 -0.4241
BM_MatrixMatrixMultiplyDynamic/8/12/2 -0.3675 -0.3674
BM_MatrixMatrixMultiplyDynamic/8/12/4 -0.5055 -0.5055
BM_MatrixMatrixMultiplyDynamic/8/12/6 -0.4302 -0.4302
BM_MatrixMatrixMultiplyDynamic/8/12/8 -0.4854 -0.4854
BM_MatrixMatrixMultiplyDynamic/8/12/10 -0.4882 -0.4882
BM_MatrixMatrixMultiplyDynamic/8/12/12 -0.5209 -0.5209
BM_MatrixMatrixMultiplyDynamic/8/12/15 -0.4558 -0.4558
BM_MatrixMatrixMultiplyDynamic/8/15/2 -0.2319 -0.2319
BM_MatrixMatrixMultiplyDynamic/8/15/4 -0.5105 -0.5105
BM_MatrixMatrixMultiplyDynamic/8/15/6 -0.4477 -0.4477
BM_MatrixMatrixMultiplyDynamic/8/15/8 -0.5479 -0.5479
BM_MatrixMatrixMultiplyDynamic/8/15/10 -0.4843 -0.4843
BM_MatrixMatrixMultiplyDynamic/8/15/12 -0.5212 -0.5212
BM_MatrixMatrixMultiplyDynamic/8/15/15 -0.4459 -0.4459
BM_MatrixVectorMultiply/1/1 +0.0978 +0.0978
BM_MatrixVectorMultiply/1/2 +0.0551 +0.0551
BM_MatrixVectorMultiply/1/3 -0.0019 -0.0020
BM_MatrixVectorMultiply/1/4 +0.0563 +0.0562
BM_MatrixVectorMultiply/1/6 +0.1379 +0.1379
BM_MatrixVectorMultiply/1/7 +0.1090 +0.1090
BM_MatrixVectorMultiply/1/12 +0.0901 +0.0901
BM_MatrixVectorMultiply/1/16 +0.0493 +0.0493
BM_MatrixVectorMultiply/1/20 +0.2255 +0.2255
BM_MatrixVectorMultiply/2/1 +0.1261 +0.1261
BM_MatrixVectorMultiply/2/2 +0.2328 +0.2328
BM_MatrixVectorMultiply/2/3 +0.1404 +0.1403
BM_MatrixVectorMultiply/2/4 +0.0257 +0.0256
BM_MatrixVectorMultiply/2/6 -0.1691 -0.1691
BM_MatrixVectorMultiply/2/7 -0.2619 -0.2619
BM_MatrixVectorMultiply/2/12 -0.4261 -0.4261
BM_MatrixVectorMultiply/2/16 -0.5387 -0.5387
BM_MatrixVectorMultiply/2/20 -0.6171 -0.6171
BM_MatrixVectorMultiply/3/1 +0.1664 +0.1664
BM_MatrixVectorMultiply/3/2 +0.0848 +0.0848
BM_MatrixVectorMultiply/3/3 -0.0044 -0.0044
BM_MatrixVectorMultiply/3/4 -0.0683 -0.0684
BM_MatrixVectorMultiply/3/6 -0.1652 -0.1652
BM_MatrixVectorMultiply/3/7 -0.1633 -0.1633
BM_MatrixVectorMultiply/3/12 -0.1921 -0.1921
BM_MatrixVectorMultiply/3/16 -0.3659 -0.3659
BM_MatrixVectorMultiply/3/20 -0.4137 -0.4137
BM_MatrixVectorMultiply/4/1 -0.0577 -0.0577
BM_MatrixVectorMultiply/4/2 -0.1337 -0.1338
BM_MatrixVectorMultiply/4/3 -0.1443 -0.1443
BM_MatrixVectorMultiply/4/4 +0.0013 +0.0013
BM_MatrixVectorMultiply/4/6 -0.1071 -0.1071
BM_MatrixVectorMultiply/4/7 -0.1396 -0.1397
BM_MatrixVectorMultiply/4/12 -0.2792 -0.2792
BM_MatrixVectorMultiply/4/16 -0.4485 -0.4486
BM_MatrixVectorMultiply/4/20 -0.3588 -0.3588
Change-Id: I64a8cf11391e3d06341a2b8764cd1b4f1b8a23f1
diff --git a/internal/ceres/small_blas.h b/internal/ceres/small_blas.h
index 264ac53..34b4ec7 100644
--- a/internal/ceres/small_blas.h
+++ b/internal/ceres/small_blas.h
@@ -38,6 +38,7 @@
#include "ceres/internal/port.h"
#include "ceres/internal/eigen.h"
#include "glog/logging.h"
+#include "small_blas_generic.h"
namespace ceres {
namespace internal {
@@ -89,6 +90,26 @@
B, num_row_b, num_col_b, \
C, start_row_c, start_col_c, row_stride_c, col_stride_c);
+#define CERES_GEMM_STORE_SINGLE(p, index, value) \
+ if (kOperation > 0) { \
+ p[index] += value; \
+ } else if (kOperation < 0) { \
+ p[index] -= value; \
+ } else { \
+ p[index] = value; \
+ }
+
+#define CERES_GEMM_STORE_PAIR(p, index, v1, v2) \
+ if (kOperation > 0) { \
+ p[index] += v1; \
+ p[index + 1] += v2; \
+ } else if (kOperation < 0) { \
+ p[index] -= v1; \
+ p[index + 1] -= v2; \
+ } else { \
+ p[index] = v1; \
+ p[index + 1] = v2; \
+ }
// For the matrix-matrix functions below, there are three variants for
// each functionality. Foo, FooNaive and FooEigen. Foo is the one to
@@ -160,24 +181,64 @@
const int NUM_COL_C = NUM_COL_B;
DCHECK_LE(start_row_c + NUM_ROW_C, row_stride_c);
DCHECK_LE(start_col_c + NUM_COL_C, col_stride_c);
+ const int span = 4;
- for (int row = 0; row < NUM_ROW_C; ++row) {
- for (int col = 0; col < NUM_COL_C; ++col) {
+ // Calculate the remainder part first.
+
+ // Process the last odd column if present.
+ if (NUM_COL_C & 1) {
+ int col = NUM_COL_C - 1;
+ const double* pa = &A[0];
+ for (int row = 0; row < NUM_ROW_C; ++row, pa += NUM_COL_A) {
+ const double* pb = &B[col];
double tmp = 0.0;
- for (int k = 0; k < NUM_COL_A; ++k) {
- tmp += A[row * NUM_COL_A + k] * B[k * NUM_COL_B + col];
+ for (int k = 0; k < NUM_COL_A; ++k, pb += NUM_COL_B) {
+ tmp += pa[k] * pb[0];
}
const int index = (row + start_row_c) * col_stride_c + start_col_c + col;
- if (kOperation > 0) {
- C[index] += tmp;
- } else if (kOperation < 0) {
- C[index] -= tmp;
- } else {
- C[index] = tmp;
- }
+ CERES_GEMM_STORE_SINGLE(C, index, tmp);
+ }
+
+ // Return directly for efficiency of extremely small matrix multiply.
+ if (NUM_COL_C == 1) {
+ return;
}
}
+
+ // Process the couple columns in remainder if present.
+ if (NUM_COL_C & 2) {
+ int col = NUM_COL_C & (int)(~(span - 1)) ;
+ const double* pa = &A[0];
+ for (int row = 0; row < NUM_ROW_C; ++row, pa += NUM_COL_A) {
+ const double* pb = &B[col];
+ double tmp1 = 0.0, tmp2 = 0.0;
+ for (int k = 0; k < NUM_COL_A; ++k, pb += NUM_COL_B) {
+ double av = pa[k];
+ tmp1 += av * pb[0];
+ tmp2 += av * pb[1];
+ }
+
+ const int index = (row + start_row_c) * col_stride_c + start_col_c + col;
+ CERES_GEMM_STORE_PAIR(C, index, tmp1, tmp2);
+ }
+
+ // Return directly for efficiency of extremely small matrix multiply.
+ if (NUM_COL_C < span) {
+ return;
+ }
+ }
+
+ // Calculate the main part with multiples of 4.
+ int col_m = NUM_COL_C & (int)(~(span - 1));
+ for (int col = 0; col < col_m; col += span) {
+ for (int row = 0; row < NUM_ROW_C; ++row) {
+ const int index = (row + start_row_c) * col_stride_c + start_col_c + col;
+ MMM_mat1x4(NUM_COL_A, &A[row * NUM_COL_A],
+ &B[col], NUM_COL_B, &C[index], kOperation);
+ }
+ }
+
}
CERES_GEMM_BEGIN(MatrixMatrixMultiply) {
@@ -220,24 +281,68 @@
const int NUM_COL_C = NUM_COL_B;
DCHECK_LE(start_row_c + NUM_ROW_C, row_stride_c);
DCHECK_LE(start_col_c + NUM_COL_C, col_stride_c);
+ const int span = 4;
- for (int row = 0; row < NUM_ROW_C; ++row) {
- for (int col = 0; col < NUM_COL_C; ++col) {
+ // Process the remainder part first.
+
+ // Process the last odd column if present.
+ if (NUM_COL_C & 1) {
+ int col = NUM_COL_C - 1;
+ for (int row = 0; row < NUM_ROW_C; ++row) {
+ const double* pa = &A[row];
+ const double* pb = &B[col];
double tmp = 0.0;
for (int k = 0; k < NUM_ROW_A; ++k) {
- tmp += A[k * NUM_COL_A + row] * B[k * NUM_COL_B + col];
+ tmp += pa[0] * pb[0];
+ pa += NUM_COL_A;
+ pb += NUM_COL_B;
}
const int index = (row + start_row_c) * col_stride_c + start_col_c + col;
- if (kOperation > 0) {
- C[index]+= tmp;
- } else if (kOperation < 0) {
- C[index]-= tmp;
- } else {
- C[index]= tmp;
- }
+ CERES_GEMM_STORE_SINGLE(C, index, tmp);
+ }
+
+ // Return directly for efficiency of extremely small matrix multiply.
+ if (NUM_COL_C == 1) {
+ return;
}
}
+
+ // Process the couple columns in remainder if present.
+ if (NUM_COL_C & 2) {
+ int col = NUM_COL_C & (int)(~(span - 1)) ;
+ for (int row = 0; row < NUM_ROW_C; ++row) {
+ const double* pa = &A[row];
+ const double* pb = &B[col];
+ double tmp1 = 0.0, tmp2 = 0.0;
+ for (int k = 0; k < NUM_ROW_A; ++k) {
+ double av = *pa;
+ tmp1 += av * pb[0];
+ tmp2 += av * pb[1];
+ pa += NUM_COL_A;
+ pb += NUM_COL_B;
+ }
+
+ const int index = (row + start_row_c) * col_stride_c + start_col_c + col;
+ CERES_GEMM_STORE_PAIR(C, index, tmp1, tmp2);
+ }
+
+ // Return directly for efficiency of extremely small matrix multiply.
+ if (NUM_COL_C < span) {
+ return;
+ }
+ }
+
+ // Process the main part with multiples of 4.
+ int col_m = NUM_COL_C & (int)(~(span - 1));
+ for (int col = 0; col < col_m; col += span) {
+ for (int row = 0; row < NUM_ROW_C; ++row) {
+ const int index = (row + start_row_c) * col_stride_c + start_col_c + col;
+ MTM_mat1x4(NUM_ROW_A, &A[row], NUM_COL_A,
+ &B[col], NUM_COL_B, &C[index], kOperation);
+ }
+ }
+
}
CERES_GEMM_BEGIN(MatrixTransposeMatrixMultiply) {
@@ -301,21 +406,54 @@
const int NUM_ROW_A = (kRowA != Eigen::Dynamic ? kRowA : num_row_a);
const int NUM_COL_A = (kColA != Eigen::Dynamic ? kColA : num_col_a);
+ const int span = 4;
- for (int row = 0; row < NUM_ROW_A; ++row) {
+ // Calculate the remainder part first.
+
+ // Process the last odd row if present.
+ if (NUM_ROW_A & 1) {
+ int row = NUM_ROW_A - 1;
+ const double* pa = &A[row * NUM_COL_A];
+ const double* pb = &b[0];
double tmp = 0.0;
for (int col = 0; col < NUM_COL_A; ++col) {
- tmp += A[row * NUM_COL_A + col] * b[col];
+ tmp += (*pa++) * (*pb++);
}
+ CERES_GEMM_STORE_SINGLE(c, row, tmp);
- if (kOperation > 0) {
- c[row] += tmp;
- } else if (kOperation < 0) {
- c[row] -= tmp;
- } else {
- c[row] = tmp;
+ // Return directly for efficiency of extremely small matrix multiply.
+ if (NUM_ROW_A == 1) {
+ return;
}
}
+
+ // Process the couple rows in remainder if present.
+ if (NUM_ROW_A & 2) {
+ int row = NUM_ROW_A & (int)(~(span - 1));
+ const double* pa1 = &A[row * NUM_COL_A];
+ const double* pa2 = pa1 + NUM_COL_A;
+ const double* pb = &b[0];
+ double tmp1 = 0.0, tmp2 = 0.0;
+ for (int col = 0; col < NUM_ROW_A; ++col) {
+ double bv = *pb++;
+ tmp1 += *(pa1++) * bv;
+ tmp2 += *(pa2++) * bv;
+ }
+ CERES_GEMM_STORE_PAIR(c, row, tmp1, tmp2);
+
+ // Return directly for efficiency of extremely small matrix multiply.
+ if (NUM_ROW_A < span) {
+ return;
+ }
+ }
+
+ // Calculate the main part with multiples of 4.
+ int row_m = NUM_ROW_A & (int)(~(span - 1));
+ for (int row = 0; row < row_m; row += span) {
+ MVM_mat4x1(NUM_COL_A, &A[row * NUM_COL_A], NUM_COL_A,
+ &b[0], &c[row], kOperation);
+ }
+
#endif // CERES_NO_CUSTOM_BLAS
}
@@ -352,21 +490,55 @@
const int NUM_ROW_A = (kRowA != Eigen::Dynamic ? kRowA : num_row_a);
const int NUM_COL_A = (kColA != Eigen::Dynamic ? kColA : num_col_a);
+ const int span = 4;
- for (int row = 0; row < NUM_COL_A; ++row) {
+ // Calculate the remainder part first.
+
+ // Process the last odd column if present.
+ if (NUM_COL_A & 1) {
+ int row = NUM_COL_A - 1;
+ const double* pa = &A[row];
+ const double* pb = &b[0];
double tmp = 0.0;
for (int col = 0; col < NUM_ROW_A; ++col) {
- tmp += A[col * NUM_COL_A + row] * b[col];
+ tmp += *pa * (*pb++);
+ pa += NUM_COL_A;
}
+ CERES_GEMM_STORE_SINGLE(c, row, tmp);
- if (kOperation > 0) {
- c[row] += tmp;
- } else if (kOperation < 0) {
- c[row] -= tmp;
- } else {
- c[row] = tmp;
+ // Return directly for efficiency of extremely small matrix multiply.
+ if (NUM_COL_A == 1) {
+ return;
}
}
+
+ // Process the couple columns in remainder if present.
+ if (NUM_COL_A & 2) {
+ int row = NUM_COL_A & (int)(~(span - 1));
+ const double* pa = &A[row];
+ const double* pb = &b[0];
+ double tmp1 = 0.0, tmp2 = 0.0;
+ for (int col = 0; col < NUM_ROW_A; ++col) {
+ double bv = *pb++;
+ tmp1 += *(pa ) * bv;
+ tmp2 += *(pa + 1) * bv;
+ pa += NUM_COL_A;
+ }
+ CERES_GEMM_STORE_PAIR(c, row, tmp1, tmp2);
+
+ // Return directly for efficiency of extremely small matrix multiply.
+ if (NUM_COL_A < span) {
+ return;
+ }
+ }
+
+ // Calculate the main part with multiples of 4.
+ int row_m = NUM_COL_A & (int)(~(span - 1));
+ for (int row = 0; row < row_m; row += span) {
+ MTV_mat4x1(NUM_ROW_A, &A[row], NUM_COL_A,
+ &b[0], &c[row], kOperation);
+ }
+
#endif // CERES_NO_CUSTOM_BLAS
}
@@ -374,6 +546,8 @@
#undef CERES_GEMM_EIGEN_HEADER
#undef CERES_GEMM_NAIVE_HEADER
#undef CERES_CALL_GEMM
+#undef CERES_GEMM_STORE_SINGLE
+#undef CERES_GEMM_STORE_PAIR
} // namespace internal
} // namespace ceres
diff --git a/internal/ceres/small_blas_generic.h b/internal/ceres/small_blas_generic.h
new file mode 100644
index 0000000..978c5d5
--- /dev/null
+++ b/internal/ceres/small_blas_generic.h
@@ -0,0 +1,315 @@
+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2018 Google Inc. All rights reserved.
+// http://ceres-solver.org/
+//
+// Redistribution and use in source and binary forms, with or without
+// modification, are permitted provided that the following conditions are met:
+//
+// * Redistributions of source code must retain the above copyright notice,
+// this list of conditions and the following disclaimer.
+// * Redistributions in binary form must reproduce the above copyright notice,
+// this list of conditions and the following disclaimer in the documentation
+// and/or other materials provided with the distribution.
+// * Neither the name of Google Inc. nor the names of its contributors may be
+// used to endorse or promote products derived from this software without
+// specific prior written permission.
+//
+// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
+// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
+// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
+// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
+// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
+// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
+// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
+// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
+// POSSIBILITY OF SUCH DAMAGE.
+//
+// Author: yangfan34@lenovo.com (Lenovo Research Device+ Lab - Shanghai)
+//
+// Optimization for simple blas functions used in the Schur Eliminator.
+// These are fairly basic implementations which already yield a significant
+// speedup in the eliminator performance.
+
+#ifndef CERES_INTERNAL_SMALL_BLAS_GENERIC_H_
+#define CERES_INTERNAL_SMALL_BLAS_GENERIC_H_
+
+namespace ceres {
+namespace internal {
+
+// The following macros are used to share code
+#define CERES_GEMM_OPT_NAIVE_HEADER \
+ double c0 = 0.0; \
+ double c1 = 0.0; \
+ double c2 = 0.0; \
+ double c3 = 0.0; \
+ const double* pa = a; \
+ const double* pb = b; \
+ const int span = 4; \
+ int col_r = col_a & (span - 1); \
+ int col_m = col_a - col_r;
+
+#define CERES_GEMM_OPT_STORE_MAT1X4 \
+ if (kOperation > 0) { \
+ *c++ += c0; \
+ *c++ += c1; \
+ *c++ += c2; \
+ *c++ += c3; \
+ } else if (kOperation < 0) { \
+ *c++ -= c0; \
+ *c++ -= c1; \
+ *c++ -= c2; \
+ *c++ -= c3; \
+ } else { \
+ *c++ = c0; \
+ *c++ = c1; \
+ *c++ = c2; \
+ *c++ = c3; \
+ }
+
+// Matrix-Matrix Multiplication
+// Figure out 1x4 of Matrix C in one batch
+//
+// c op a * B;
+// where op can be +=, -=, or =, indicated by kOperation.
+//
+// Matrix C Matrix A Matrix B
+//
+// C0, C1, C2, C3 op A0, A1, A2, A3, ... * B0, B1, B2, B3
+// B4, B5, B6, B7
+// B8, B9, Ba, Bb
+// Bc, Bd, Be, Bf
+// . , . , . , .
+// . , . , . , .
+// . , . , . , .
+//
+// unroll for loops
+// utilize the data resided in cache
+// NOTE: col_a means the columns of A
+static inline void MMM_mat1x4(const int col_a,
+ const double* a,
+ const double* b,
+ const int col_stride_b,
+ double* c,
+ const int kOperation) {
+ CERES_GEMM_OPT_NAIVE_HEADER
+ double av = 0.0;
+ int bi = 0;
+
+#define CERES_GEMM_OPT_MMM_MAT1X4_MUL \
+ av = pa[k]; \
+ pb = b + bi; \
+ c0 += av * *pb++; \
+ c1 += av * *pb++; \
+ c2 += av * *pb++; \
+ c3 += av * *pb++; \
+ bi += col_stride_b; \
+ k++;
+
+ for (int k = 0; k < col_m;) {
+ CERES_GEMM_OPT_MMM_MAT1X4_MUL
+ CERES_GEMM_OPT_MMM_MAT1X4_MUL
+ CERES_GEMM_OPT_MMM_MAT1X4_MUL
+ CERES_GEMM_OPT_MMM_MAT1X4_MUL
+ }
+
+ for (int k = col_m; k < col_a;) {
+ CERES_GEMM_OPT_MMM_MAT1X4_MUL
+ }
+
+ CERES_GEMM_OPT_STORE_MAT1X4
+
+#undef CERES_GEMM_OPT_MMM_MAT1X4_MUL
+}
+
+// Matrix Transpose-Matrix multiplication
+// Figure out 1x4 of Matrix C in one batch
+//
+// c op a' * B;
+// where op can be +=, -=, or = indicated by kOperation.
+//
+// Matrix A
+//
+// A0
+// A1
+// A2
+// A3
+// .
+// .
+// .
+//
+// Matrix C Matrix A' Matrix B
+//
+// C0, C1, C2, C3 op A0, A1, A2, A3, ... * B0, B1, B2, B3
+// B4, B5, B6, B7
+// B8, B9, Ba, Bb
+// Bc, Bd, Be, Bf
+// . , . , . , .
+// . , . , . , .
+// . , . , . , .
+//
+// unroll for loops
+// utilize the data resided in cache
+// NOTE: col_a means the columns of A'
+static inline void MTM_mat1x4(const int col_a,
+ const double* a,
+ const int col_stride_a,
+ const double* b,
+ const int col_stride_b,
+ double* c,
+ const int kOperation) {
+ CERES_GEMM_OPT_NAIVE_HEADER
+ double av = 0.0;
+ int ai = 0;
+ int bi = 0;
+
+#define CERES_GEMM_OPT_MTM_MAT1X4_MUL \
+ av = pa[ai]; \
+ pb = b + bi; \
+ c0 += av * *pb++; \
+ c1 += av * *pb++; \
+ c2 += av * *pb++; \
+ c3 += av * *pb++; \
+ ai += col_stride_a; \
+ bi += col_stride_b;
+
+ for (int k = 0; k < col_m; k += span) {
+ CERES_GEMM_OPT_MTM_MAT1X4_MUL
+ CERES_GEMM_OPT_MTM_MAT1X4_MUL
+ CERES_GEMM_OPT_MTM_MAT1X4_MUL
+ CERES_GEMM_OPT_MTM_MAT1X4_MUL
+ }
+
+ for (int k = col_m; k < col_a; k++) {
+ CERES_GEMM_OPT_MTM_MAT1X4_MUL
+ }
+
+ CERES_GEMM_OPT_STORE_MAT1X4
+
+#undef CERES_GEMM_OPT_MTM_MAT1X4_MUL
+}
+
+// Matrix-Vector Multiplication
+// Figure out 4x1 of vector c in one batch
+//
+// c op A * b;
+// where op can be +=, -=, or =, indicated by kOperation.
+//
+// Vector c Matrix A Vector b
+//
+// C0 op A0, A1, A2, A3, ... * B0
+// C1 A4, A5, A6, A7, ... B1
+// C2 A8, A9, Aa, Ab, ... B2
+// C3 Ac, Ad, Ae, Af, ... B3
+// .
+// .
+// .
+//
+// unroll for loops
+// utilize the data resided in cache
+// NOTE: col_a means the columns of A
+static inline void MVM_mat4x1(const int col_a,
+ const double* a,
+ const int col_stride_a,
+ const double* b,
+ double* c,
+ const int kOperation) {
+ CERES_GEMM_OPT_NAIVE_HEADER
+ double bv = 0.0;
+
+#define CERES_GEMM_OPT_MVM_MAT4X1_MUL \
+ bv = *pb; \
+ c0 += *(pa ) * bv; \
+ c1 += *(pa + col_stride_a ) * bv; \
+ c2 += *(pa + col_stride_a * 2) * bv; \
+ c3 += *(pa + col_stride_a * 3) * bv; \
+ pa++; \
+ pb++;
+
+ for (int k = 0; k < col_m; k += span) {
+ CERES_GEMM_OPT_MVM_MAT4X1_MUL
+ CERES_GEMM_OPT_MVM_MAT4X1_MUL
+ CERES_GEMM_OPT_MVM_MAT4X1_MUL
+ CERES_GEMM_OPT_MVM_MAT4X1_MUL
+ }
+
+ for (int k = col_m; k < col_a; k++) {
+ CERES_GEMM_OPT_MVM_MAT4X1_MUL
+ }
+
+ CERES_GEMM_OPT_STORE_MAT1X4
+
+#undef CERES_GEMM_OPT_MVM_MAT4X1_MUL
+}
+
+// Matrix Transpose-Vector multiplication
+// Figure out 4x1 of vector c in one batch
+//
+// c op A' * b;
+// where op can be +=, -=, or =, indicated by kOperation.
+//
+// Matrix A
+//
+// A0, A4, A8, Ac
+// A1, A5, A9, Ad
+// A2, A6, Aa, Ae
+// A3, A7, Ab, Af
+// . , . , . , .
+// . , . , . , .
+// . , . , . , .
+//
+// Vector c Matrix A' Vector b
+//
+// C0 op A0, A1, A2, A3, ... * B0
+// C1 A4, A5, A6, A7, ... B1
+// C2 A8, A9, Aa, Ab, ... B2
+// C3 Ac, Ad, Ae, Af, ... B3
+// .
+// .
+// .
+//
+// unroll for loops
+// utilize the data resided in cache
+// NOTE: col_a means the columns of A'
+static inline void MTV_mat4x1(const int col_a,
+ const double* a,
+ const int col_stride_a,
+ const double* b,
+ double* c,
+ const int kOperation) {
+ CERES_GEMM_OPT_NAIVE_HEADER
+ double bv = 0.0;
+
+#define CERES_GEMM_OPT_MTV_MAT4X1_MUL \
+ bv = *pb; \
+ c0 += *(pa ) * bv; \
+ c1 += *(pa + 1) * bv; \
+ c2 += *(pa + 2) * bv; \
+ c3 += *(pa + 3) * bv; \
+ pa += col_stride_a; \
+ pb++;
+
+ for (int k = 0; k < col_m; k += span) {
+ CERES_GEMM_OPT_MTV_MAT4X1_MUL
+ CERES_GEMM_OPT_MTV_MAT4X1_MUL
+ CERES_GEMM_OPT_MTV_MAT4X1_MUL
+ CERES_GEMM_OPT_MTV_MAT4X1_MUL
+ }
+
+ for (int k = col_m; k < col_a; k++) {
+ CERES_GEMM_OPT_MTV_MAT4X1_MUL
+ }
+
+ CERES_GEMM_OPT_STORE_MAT1X4
+
+#undef CERES_GEMM_OPT_MTV_MAT4X1_MUL
+}
+
+#undef CERES_GEMM_OPT_NAIVE_HEADER
+#undef CERES_GEMM_OPT_STORE_MAT1X4
+
+} // namespace internal
+} // namespace ceres
+
+#endif // CERES_INTERNAL_SMALL_BLAS_GENERIC_H_
