From ce9f4591847c6bd2dc6ebaa30fc5d72714e0280c Mon Sep 17 00:00:00 2001 From: Nunigan Date: Tue, 10 Aug 2021 06:37:20 +0200 Subject: new measurements --- buch/papers/multiplikation/code/MM | Bin 26848 -> 0 bytes buch/papers/multiplikation/code/MM.c | 19 +- buch/papers/multiplikation/code/MM.py | 77 ++++---- buch/papers/multiplikation/code/c_matrix.h | 204 ++++++++++++++------- buch/papers/multiplikation/code/c_meas_4096.pdf | Bin 17448 -> 22360 bytes buch/papers/multiplikation/code/ci.txt | 0 buch/papers/multiplikation/code/helper_class.py | 3 +- buch/papers/multiplikation/code/meas/MM.txt | 118 +++++++++++- buch/papers/multiplikation/code/meas/MM_dc.txt | 118 +++++++++++- buch/papers/multiplikation/code/meas/blas.txt | 114 +++++++++++- buch/papers/multiplikation/code/meas/ci/MM.txt | 11 ++ buch/papers/multiplikation/code/meas/ci/Wino.txt | 11 ++ buch/papers/multiplikation/code/meas/ci/blas.txt | 11 ++ buch/papers/multiplikation/code/meas/ci/dc.txt | 11 ++ .../multiplikation/code/meas/ci/strassen.txt | 11 ++ .../multiplikation/code/meas/old/8196/MM.txt | 1 + .../multiplikation/code/meas/old/8196/MM_dc.txt | 1 + .../multiplikation/code/meas/old/8196/blas.txt | 1 + .../multiplikation/code/meas/old/8196/strassen.txt | 1 + .../multiplikation/code/meas/old/8196/winograd.txt | 1 + buch/papers/multiplikation/code/meas/old/MM.txt | 12 ++ buch/papers/multiplikation/code/meas/old/MM_dc.txt | 12 ++ buch/papers/multiplikation/code/meas/old/blas.txt | 12 ++ .../multiplikation/code/meas/old/strassen.txt | 12 ++ .../multiplikation/code/meas/old/winograd.txt | 12 ++ buch/papers/multiplikation/code/meas/strassen.txt | 122 ++++++++++-- buch/papers/multiplikation/code/meas/winograd.txt | 116 +++++++++++- buch/papers/multiplikation/code/meas_4096.pdf | Bin 12952 -> 17369 bytes buch/papers/multiplikation/code/meas_4096.txt | 6 + buch/papers/multiplikation/images/algo_tab.pdf | Bin 0 -> 34251 bytes buch/papers/multiplikation/images/algo_tab.tex | 122 ++++++++++++ buch/papers/multiplikation/images/meas_c.pdf | Bin 23161 -> 23552 bytes buch/papers/multiplikation/images/meas_c.tex | 9 +- buch/papers/multiplikation/loesungsmethoden.tex | 48 ++--- buch/papers/multiplikation/problemstellung.tex | 145 ++++++++------- 35 files changed, 1096 insertions(+), 245 deletions(-) delete mode 100755 buch/papers/multiplikation/code/MM create mode 100644 buch/papers/multiplikation/code/ci.txt create mode 100644 buch/papers/multiplikation/code/meas/ci/MM.txt create mode 100644 buch/papers/multiplikation/code/meas/ci/Wino.txt create mode 100644 buch/papers/multiplikation/code/meas/ci/blas.txt create mode 100644 buch/papers/multiplikation/code/meas/ci/dc.txt create mode 100644 buch/papers/multiplikation/code/meas/ci/strassen.txt create mode 100644 buch/papers/multiplikation/code/meas/old/8196/MM.txt create mode 100644 buch/papers/multiplikation/code/meas/old/8196/MM_dc.txt create mode 100644 buch/papers/multiplikation/code/meas/old/8196/blas.txt create mode 100644 buch/papers/multiplikation/code/meas/old/8196/strassen.txt create mode 100644 buch/papers/multiplikation/code/meas/old/8196/winograd.txt create mode 100644 buch/papers/multiplikation/code/meas/old/MM.txt create mode 100644 buch/papers/multiplikation/code/meas/old/MM_dc.txt create mode 100644 buch/papers/multiplikation/code/meas/old/blas.txt create mode 100644 buch/papers/multiplikation/code/meas/old/strassen.txt create mode 100644 buch/papers/multiplikation/code/meas/old/winograd.txt create mode 100644 buch/papers/multiplikation/images/algo_tab.pdf create mode 100644 buch/papers/multiplikation/images/algo_tab.tex (limited to 'buch/papers/multiplikation') diff --git a/buch/papers/multiplikation/code/MM b/buch/papers/multiplikation/code/MM deleted file mode 100755 index d52dda4..0000000 Binary files a/buch/papers/multiplikation/code/MM and /dev/null differ diff --git a/buch/papers/multiplikation/code/MM.c b/buch/papers/multiplikation/code/MM.c index a897d4f..2588262 100755 --- a/buch/papers/multiplikation/code/MM.c +++ b/buch/papers/multiplikation/code/MM.c @@ -28,11 +28,12 @@ int main() { // omp_set_num_threads(4); // run_algo(openMP_MM, "openMP_MM",0); run_algo(MM_dc, "MM_dc",0); + run_algo(strassen, "strassen",0); run_algo(MM, "MM", 0); - run_algo(winograd, "winograd", 0); - run_algo_cblas(0); + run_algo(winograd, "winograd", 0); + run_algo_cblas(0); return 0; } @@ -414,12 +415,12 @@ void run_algo(void (*algo)(), char alog_name[], int print) for(int i=0; i const int A0[][2] = { - {75,47}, - {-41,-24} + {60,-84}, + {-66,-1} }; const int B0[][2] = { - {-53,-95}, - {-93,30} + {-45,87}, + {-38,-73} }; const double dB0[][2] = { - {-53,-95}, - {-93,30} + {-45,87}, + {-38,-73} }; const double dA0[][2] = { - {75,47}, - {-41,-24} + {60,-84}, + {-66,-1} }; const int A1[][4] = { - {47,11,-66,8}, - {36,98,39,82}, - {-32,12,40,-79}, - {61,-20,-85,-98} + {-72,-19,-91,62}, + {-36,-74,-44,-47}, + {-39,-31,50,-93}, + {-81,2,-17,-86} }; const int B1[][4] = { - {37,75,-53,9}, - {37,-33,-67,38}, - {70,39,-93,43}, - {43,41,23,-4} + {-66,39,-23,52}, + {-88,-13,13,-13}, + {-45,-70,28,-20}, + {96,5,88,96} }; const double dB1[][4] = { - {37,75,-53,9}, - {37,-33,-67,38}, - {70,39,-93,43}, - {43,41,23,-4} + {-66,39,-23,52}, + {-88,-13,13,-13}, + {-45,-70,28,-20}, + {96,5,88,96} }; const double dA1[][4] = { - {47,11,-66,8}, - {36,98,39,82}, - {-32,12,40,-79}, - {61,-20,-85,-98} + {-72,-19,-91,62}, + {-36,-74,-44,-47}, + {-39,-31,50,-93}, + {-81,2,-17,-86} }; const int A2[][8] = { - {-54,-87,87,69,52,-21,-86,55}, - {19,-75,-61,-50,-55,-23,66,-92}, - {-73,-67,-36,19,84,-11,24,46}, - {-98,62,-76,57,-100,6,-23,-51}, - {62,46,1,-64,42,-9,85,-12}, - {35,-59,-17,-47,78,86,-50,74}, - {-15,45,33,-59,-9,-81,49,96}, - {-57,22,-43,7,-30,-45,-5,13} + {-36,-2,-58,-32,34,-89,49,-55}, + {-68,-73,52,-3,-51,-37,-31,70}, + {73,-90,-21,-79,-15,96,-99,12}, + {68,-25,38,-73,-60,35,-99,72}, + {-43,-87,48,-84,-100,37,80,53}, + {-27,88,-5,-82,-57,-27,20,10}, + {-91,-47,54,-90,-99,-76,50,-18}, + {69,-36,76,5,-67,-38,-95,91} }; const int B2[][8] = { - {-71,-82,-80,-78,83,-97,48,-24}, - {15,75,15,-60,-63,-53,1,-50}, - {-84,63,67,-2,78,93,-13,95}, - {61,-26,-88,56,56,27,26,1}, - {2,54,21,36,9,-41,53,53}, - {85,-11,42,-51,-6,3,27,97}, - {10,-2,90,-76,-75,0,8,-37}, - {10,-64,47,-69,66,-50,89,-66} + {-84,22,-13,-66,-42,51,66,0}, + {37,-65,66,-85,-10,-23,77,5}, + {1,41,-79,0,63,-37,-10,29}, + {72,66,-99,92,-28,65,25,-40}, + {69,-49,65,-18,64,-97,-47,30}, + {36,86,66,-12,-17,89,1,-37}, + {-100,11,27,23,-75,-23,96,-9}, + {68,90,-87,-99,-70,-28,98,-76} }; const double dB2[][8] = { - {-71,-82,-80,-78,83,-97,48,-24}, - {15,75,15,-60,-63,-53,1,-50}, - {-84,63,67,-2,78,93,-13,95}, - {61,-26,-88,56,56,27,26,1}, - {2,54,21,36,9,-41,53,53}, - {85,-11,42,-51,-6,3,27,97}, - {10,-2,90,-76,-75,0,8,-37}, - {10,-64,47,-69,66,-50,89,-66} + {-84,22,-13,-66,-42,51,66,0}, + {37,-65,66,-85,-10,-23,77,5}, + {1,41,-79,0,63,-37,-10,29}, + {72,66,-99,92,-28,65,25,-40}, + {69,-49,65,-18,64,-97,-47,30}, + {36,86,66,-12,-17,89,1,-37}, + {-100,11,27,23,-75,-23,96,-9}, + {68,90,-87,-99,-70,-28,98,-76} }; const double dA2[][8] = { - {-54,-87,87,69,52,-21,-86,55}, - {19,-75,-61,-50,-55,-23,66,-92}, - {-73,-67,-36,19,84,-11,24,46}, - {-98,62,-76,57,-100,6,-23,-51}, - {62,46,1,-64,42,-9,85,-12}, - {35,-59,-17,-47,78,86,-50,74}, - {-15,45,33,-59,-9,-81,49,96}, - {-57,22,-43,7,-30,-45,-5,13} - }; -const int *Ap[3] = {(int*) A0,(int*) A1,(int*) A2}; -const int *Bp[3] = {(int*) B0,(int*) B1,(int*) B2}; -const double *dAp[3] = {(double*) dA0,(double*) dA1,(double*) dA2}; -const double *dBp[3] = {(double*) dB0,(double*) dB1,(double*) dB2}; -int n[3] = {2,4,8}; -int n_arrays = 3; + {-36,-2,-58,-32,34,-89,49,-55}, + {-68,-73,52,-3,-51,-37,-31,70}, + {73,-90,-21,-79,-15,96,-99,12}, + {68,-25,38,-73,-60,35,-99,72}, + {-43,-87,48,-84,-100,37,80,53}, + {-27,88,-5,-82,-57,-27,20,10}, + {-91,-47,54,-90,-99,-76,50,-18}, + {69,-36,76,5,-67,-38,-95,91} + }; +const int A3[][16] = + { + {-24,65,21,19,94,70,-90,-81,53,-41,-23,-1,58,-80,-54,59}, + {-42,76,-19,98,29,-56,92,14,45,11,82,83,48,-13,81,66}, + {43,-57,-67,95,5,72,11,0,-47,55,-24,36,84,54,-31,-54}, + {-39,-40,19,97,-82,-56,27,95,81,-21,-50,-74,-35,-87,-28,-26}, + {-74,-98,79,92,-24,-48,99,94,55,-83,70,98,-24,18,-67,14}, + {20,76,11,-23,-56,21,0,42,64,86,-74,44,93,-76,-30,97}, + {13,20,-73,-11,-30,80,53,-8,60,21,17,-42,82,-72,-6,-80}, + {36,-93,-64,-21,20,-85,15,24,99,81,-52,64,71,-56,52,63}, + {32,9,-2,-85,17,62,-98,-35,75,-58,-44,-20,-47,89,-95,52}, + {93,-43,86,68,-6,-25,90,57,60,-10,65,-97,43,46,-60,-41}, + {43,-33,0,50,-100,26,-60,95,39,-70,-61,-81,9,-23,-99,-4}, + {20,61,15,43,-96,93,-55,38,-29,-1,-10,26,-87,18,64,6}, + {-98,-84,51,16,-14,86,52,59,44,-39,-2,10,82,-66,54,19}, + {89,-49,-37,-6,-53,40,-11,46,-51,-56,86,34,11,13,-20,-49}, + {-90,14,28,-45,-25,-56,-51,-61,28,-8,51,91,95,-10,-85,58}, + {8,-44,88,-71,-27,11,89,37,86,-78,-44,-56,-87,0,-42,-61} + }; +const int B3[][16] = + { + {62,-30,62,92,29,-93,-95,44,-33,-88,-29,9,-88,-42,-90,-70}, + {60,37,-44,-93,-87,6,-53,2,-29,53,-49,59,6,83,-15,50}, + {-19,85,-49,-14,84,-4,12,88,-83,-81,-24,-16,-12,-42,-63,-71}, + {-42,-78,-58,-61,-29,67,-28,-46,64,7,6,-13,88,-42,95,-24}, + {-90,-56,8,-30,-89,70,37,-29,24,-8,-10,-2,-25,-63,-95,-91}, + {10,-81,42,-28,-13,-68,-72,-20,-22,5,-79,-50,-88,62,57,69}, + {-67,24,-71,-43,11,48,33,-93,-82,-65,-4,5,-15,25,-54,-45}, + {-49,19,-29,90,-97,-87,78,-39,-75,-85,-79,-35,54,3,-73,7}, + {-7,39,70,-42,32,-100,56,4,-24,-57,38,-49,-50,-44,79,-42}, + {37,-65,-55,22,-97,-42,-76,95,97,-27,38,11,0,-81,-23,35}, + {26,-70,10,-29,47,-70,-52,29,-13,-18,5,34,18,32,87,91}, + {-84,41,-19,96,-51,-19,81,75,81,92,2,-40,-42,-69,-10,-61}, + {-30,98,71,-51,91,-59,58,86,86,-22,-84,7,66,-55,-52,23}, + {-71,-44,-9,90,26,18,26,-10,-85,64,-47,3,72,81,74,-8}, + {52,-59,-91,22,8,-63,84,9,-11,-54,-78,-71,-98,42,96,57}, + {18,-39,34,-50,-62,-96,-2,-78,52,94,-33,2,-19,-9,-86,-75} + }; +const double dB3[][16] = + { + {62,-30,62,92,29,-93,-95,44,-33,-88,-29,9,-88,-42,-90,-70}, + {60,37,-44,-93,-87,6,-53,2,-29,53,-49,59,6,83,-15,50}, + {-19,85,-49,-14,84,-4,12,88,-83,-81,-24,-16,-12,-42,-63,-71}, + {-42,-78,-58,-61,-29,67,-28,-46,64,7,6,-13,88,-42,95,-24}, + {-90,-56,8,-30,-89,70,37,-29,24,-8,-10,-2,-25,-63,-95,-91}, + {10,-81,42,-28,-13,-68,-72,-20,-22,5,-79,-50,-88,62,57,69}, + {-67,24,-71,-43,11,48,33,-93,-82,-65,-4,5,-15,25,-54,-45}, + {-49,19,-29,90,-97,-87,78,-39,-75,-85,-79,-35,54,3,-73,7}, + {-7,39,70,-42,32,-100,56,4,-24,-57,38,-49,-50,-44,79,-42}, + {37,-65,-55,22,-97,-42,-76,95,97,-27,38,11,0,-81,-23,35}, + {26,-70,10,-29,47,-70,-52,29,-13,-18,5,34,18,32,87,91}, + {-84,41,-19,96,-51,-19,81,75,81,92,2,-40,-42,-69,-10,-61}, + {-30,98,71,-51,91,-59,58,86,86,-22,-84,7,66,-55,-52,23}, + {-71,-44,-9,90,26,18,26,-10,-85,64,-47,3,72,81,74,-8}, + {52,-59,-91,22,8,-63,84,9,-11,-54,-78,-71,-98,42,96,57}, + {18,-39,34,-50,-62,-96,-2,-78,52,94,-33,2,-19,-9,-86,-75} + }; +const double dA3[][16] = + { + {-24,65,21,19,94,70,-90,-81,53,-41,-23,-1,58,-80,-54,59}, + {-42,76,-19,98,29,-56,92,14,45,11,82,83,48,-13,81,66}, + {43,-57,-67,95,5,72,11,0,-47,55,-24,36,84,54,-31,-54}, + {-39,-40,19,97,-82,-56,27,95,81,-21,-50,-74,-35,-87,-28,-26}, + {-74,-98,79,92,-24,-48,99,94,55,-83,70,98,-24,18,-67,14}, + {20,76,11,-23,-56,21,0,42,64,86,-74,44,93,-76,-30,97}, + {13,20,-73,-11,-30,80,53,-8,60,21,17,-42,82,-72,-6,-80}, + {36,-93,-64,-21,20,-85,15,24,99,81,-52,64,71,-56,52,63}, + {32,9,-2,-85,17,62,-98,-35,75,-58,-44,-20,-47,89,-95,52}, + {93,-43,86,68,-6,-25,90,57,60,-10,65,-97,43,46,-60,-41}, + {43,-33,0,50,-100,26,-60,95,39,-70,-61,-81,9,-23,-99,-4}, + {20,61,15,43,-96,93,-55,38,-29,-1,-10,26,-87,18,64,6}, + {-98,-84,51,16,-14,86,52,59,44,-39,-2,10,82,-66,54,19}, + {89,-49,-37,-6,-53,40,-11,46,-51,-56,86,34,11,13,-20,-49}, + {-90,14,28,-45,-25,-56,-51,-61,28,-8,51,91,95,-10,-85,58}, + {8,-44,88,-71,-27,11,89,37,86,-78,-44,-56,-87,0,-42,-61} + }; +const int *Ap[4] = {(int*) A0,(int*) A1,(int*) A2,(int*) A3}; +const int *Bp[4] = {(int*) B0,(int*) B1,(int*) B2,(int*) B3}; +const double *dAp[4] = {(double*) dA0,(double*) dA1,(double*) dA2,(double*) dA3}; +const double *dBp[4] = {(double*) dB0,(double*) dB1,(double*) dB2,(double*) dB3}; +int n[4] = {2,4,8,16}; +int n_arrays = 4; diff --git a/buch/papers/multiplikation/code/c_meas_4096.pdf b/buch/papers/multiplikation/code/c_meas_4096.pdf index 5236afb..b42082f 100644 Binary files a/buch/papers/multiplikation/code/c_meas_4096.pdf and b/buch/papers/multiplikation/code/c_meas_4096.pdf differ diff --git a/buch/papers/multiplikation/code/ci.txt b/buch/papers/multiplikation/code/ci.txt new file mode 100644 index 0000000..e69de29 diff --git a/buch/papers/multiplikation/code/helper_class.py b/buch/papers/multiplikation/code/helper_class.py index 485fa76..ad67909 100755 --- a/buch/papers/multiplikation/code/helper_class.py +++ b/buch/papers/multiplikation/code/helper_class.py @@ -101,5 +101,6 @@ if __name__ == '__main__': helper = Helper() # n = np.arange(2,10) - n = np.logspace(1,3,3,base=2,dtype=(np.int)) + n = np.logspace(1,4,4,base=2,dtype=(np.int)) + # n=[8192] C = helper.write_c_matrix(n) diff --git a/buch/papers/multiplikation/code/meas/MM.txt b/buch/papers/multiplikation/code/meas/MM.txt index e296dd7..7bffb6e 100644 --- a/buch/papers/multiplikation/code/meas/MM.txt +++ b/buch/papers/multiplikation/code/meas/MM.txt @@ -1,12 +1,110 @@ -0.000001,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 0.000001,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000001,4 +0.000001,4 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000001,8 0.000001,8 -0.000010,16 -0.000081,32 -0.000654,64 -0.005556,128 -0.054253,256 -0.487317,512 -4.162845,1024 -125.909034,2048 -1111.312696,4096 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000021,16 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+129.295723,2048 +129.402601,2048 +129.300820,2048 diff --git a/buch/papers/multiplikation/code/meas/MM_dc.txt b/buch/papers/multiplikation/code/meas/MM_dc.txt index f6be928..b78b925 100644 --- a/buch/papers/multiplikation/code/meas/MM_dc.txt +++ b/buch/papers/multiplikation/code/meas/MM_dc.txt @@ -1,12 +1,110 @@ 0.000003,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 0.000002,4 -0.000010,8 -0.000068,16 -0.000594,32 -0.004264,64 -0.036289,128 -0.324645,256 -2.612010,512 -19.928951,1024 -159.333884,2048 -1147.106865,4096 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000001,4 +0.000008,8 +0.000008,8 +0.000008,8 +0.000008,8 +0.000007,8 +0.000007,8 +0.000007,8 +0.000007,8 +0.000018,8 +0.000008,8 +0.000075,16 +0.000063,16 +0.000088,16 +0.000062,16 +0.000086,16 +0.000092,16 +0.000081,16 +0.000080,16 +0.000070,16 +0.000085,16 +0.000581,32 +0.000659,32 +0.000584,32 +0.000714,32 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a/buch/papers/multiplikation/code/meas/blas.txt b/buch/papers/multiplikation/code/meas/blas.txt index 92a61b9..9414d8f 100644 --- a/buch/papers/multiplikation/code/meas/blas.txt +++ b/buch/papers/multiplikation/code/meas/blas.txt @@ -1,12 +1,110 @@ 0.000001,2 -0.000001,4 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 0.000001,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 +0.000000,8 0.000003,16 -0.000022,32 -0.000179,64 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000012,16 +0.000003,16 +0.000003,16 +0.000003,16 +0.000021,32 +0.000019,32 +0.000030,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000020,32 +0.000180,64 +0.000192,64 +0.000163,64 +0.000153,64 +0.000153,64 +0.000197,64 +0.000163,64 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index 0000000..b5989fb --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/8196/blas.txt @@ -0,0 +1 @@ +478.429957,8192 diff --git a/buch/papers/multiplikation/code/meas/old/8196/strassen.txt b/buch/papers/multiplikation/code/meas/old/8196/strassen.txt new file mode 100644 index 0000000..ca06e97 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/8196/strassen.txt @@ -0,0 +1 @@ +3014.235467,8192 diff --git a/buch/papers/multiplikation/code/meas/old/8196/winograd.txt b/buch/papers/multiplikation/code/meas/old/8196/winograd.txt new file mode 100644 index 0000000..2a529c4 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/8196/winograd.txt @@ -0,0 +1 @@ +10071.512655,8192 diff --git a/buch/papers/multiplikation/code/meas/old/MM.txt b/buch/papers/multiplikation/code/meas/old/MM.txt new file mode 100644 index 0000000..e296dd7 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/MM.txt @@ -0,0 +1,12 @@ +0.000001,2 +0.000001,4 +0.000001,8 +0.000010,16 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a/buch/papers/multiplikation/code/meas/old/strassen.txt b/buch/papers/multiplikation/code/meas/old/strassen.txt new file mode 100644 index 0000000..fdfbf2b --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/strassen.txt @@ -0,0 +1,12 @@ +0.000001,2 +0.000003,4 +0.000010,8 +0.000066,16 +0.000470,32 +0.003368,64 +0.024232,128 +0.172000,256 +1.209262,512 +8.457472,1024 +59.267256,2048 +414.648901,4096 diff --git a/buch/papers/multiplikation/code/meas/old/winograd.txt b/buch/papers/multiplikation/code/meas/old/winograd.txt new file mode 100644 index 0000000..d185906 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/old/winograd.txt @@ -0,0 +1,12 @@ +0.000001,2 +0.000001,4 +0.000002,8 +0.000011,16 +0.000100,32 +0.000654,64 +0.005229,128 +0.057440,256 +0.517850,512 +4.539413,1024 +130.627663,2048 +1179.261048,4096 diff --git a/buch/papers/multiplikation/code/meas/strassen.txt b/buch/papers/multiplikation/code/meas/strassen.txt index fdfbf2b..d6e040e 100644 --- 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b/buch/papers/multiplikation/code/meas/winograd.txt @@ -1,12 +1,110 @@ 0.000001,2 -0.000001,4 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,2 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 +0.000000,4 0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000002,8 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000011,16 +0.000021,16 +0.000011,16 0.000011,16 -0.000100,32 -0.000654,64 -0.005229,128 -0.057440,256 -0.517850,512 -4.539413,1024 -130.627663,2048 -1179.261048,4096 +0.000092,32 +0.000092,32 +0.000081,32 +0.000081,32 +0.000081,32 +0.000081,32 +0.000088,32 +0.000079,32 +0.000079,32 +0.000079,32 +0.000670,64 +0.000739,64 +0.000609,64 +0.000609,64 +0.000700,64 +0.000648,64 +0.000626,64 +0.000626,64 +0.000626,64 +0.000626,64 +0.005321,128 +0.005286,128 +0.005180,128 +0.005223,128 +0.005249,128 +0.005299,128 +0.005205,128 +0.005268,128 +0.005464,128 +0.005378,128 +0.053123,256 +0.052325,256 +0.052729,256 +0.052930,256 +0.052207,256 +0.053178,256 +0.052122,256 +0.052681,256 +0.052965,256 +0.052486,256 +0.527028,512 +0.525201,512 +0.521822,512 +0.525147,512 +0.525241,512 +0.527725,512 +0.526321,512 +0.526479,512 +0.524020,512 +0.520768,512 +4.732299,1024 +4.617253,1024 +4.647425,1024 +4.519233,1024 +4.917471,1024 +4.564929,1024 +4.870771,1024 +4.555407,1024 +4.727473,1024 +4.559349,1024 +136.409028,2048 +136.390557,2048 +136.541672,2048 +136.598491,2048 +137.720790,2048 +136.825926,2048 +136.367686,2048 +136.650627,2048 +136.642195,2048 +136.622805,2048 diff --git a/buch/papers/multiplikation/code/meas_4096.pdf b/buch/papers/multiplikation/code/meas_4096.pdf index e889d17..9e8fcea 100644 Binary files a/buch/papers/multiplikation/code/meas_4096.pdf and b/buch/papers/multiplikation/code/meas_4096.pdf differ diff --git a/buch/papers/multiplikation/code/meas_4096.txt b/buch/papers/multiplikation/code/meas_4096.txt index e69de29..cae1bc6 100644 --- a/buch/papers/multiplikation/code/meas_4096.txt +++ b/buch/papers/multiplikation/code/meas_4096.txt @@ -0,0 +1,6 @@ +2.048000000000000000e+03 4.096000000000000000e+03 +6.154183513402938843e+03 4.681333474493026733e+04 +7.375929301261901855e+03 5.846600176072120667e+04 +3.860573610544204712e+03 2.290433094644546509e+04 +4.884613198995590210e+03 4.359707747149467468e+04 +2.157390117645263672e-01 1.491588830947875977e+00 diff --git a/buch/papers/multiplikation/images/algo_tab.pdf b/buch/papers/multiplikation/images/algo_tab.pdf new file mode 100644 index 0000000..7f2bb4f Binary files /dev/null and b/buch/papers/multiplikation/images/algo_tab.pdf differ diff --git a/buch/papers/multiplikation/images/algo_tab.tex b/buch/papers/multiplikation/images/algo_tab.tex new file mode 100644 index 0000000..50ce392 --- /dev/null +++ b/buch/papers/multiplikation/images/algo_tab.tex @@ -0,0 +1,122 @@ +\documentclass{article} +\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{times} +\usepackage{geometry} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{algorithm} +\usepackage{algpseudocode} +\usepackage{mathrsfs} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{lipsum} +\usepackage{amscd} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{textcomp} +\usepackage{pgfplots} +\usepackage{txfonts} +\usepackage[all]{xy} +\usepackage{paralist} +\usepackage[colorlinks=true]{hyperref} +\usepackage{array} +\usepackage{tikz} +\usepackage{slashed} +\usepackage{pdfpages} +\usepackage{multicol} +\usepackage{cite} +\usepackage{url} +\usepackage{amsmath,amsfonts,amssymb} +\usepackage{tikz} +\usetikzlibrary{arrows,matrix,positioning} +\usetikzlibrary{overlay-beamer-styles} +\usetikzlibrary{matrix.skeleton} +\usetikzlibrary{automata,positioning} +\usetikzlibrary{decorations.text} +\usepackage{listings} +\usepackage{multirow} +\usepackage{color} + +\begin{document} + + + +\begin{table}[t] + \begin{tabular}{ll} + \begin{minipage}{0.4\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \label{multiplikation:alg:b1} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{B1}{$a, b$} + \State \textbf{return} $a+b$ + \EndFunction + \State + \State + \end{algorithmic} + \end{algorithm} + \end{minipage} + & + \begin{minipage}{0.4\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \label{multiplikation:alg:b2} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{B2}{$a, b$} + \State $ x \gets a+b $ + \State $ y \gets a \cdot b $ + \State \textbf{return} $x+y$ + \EndFunction + \end{algorithmic} +\end{algorithm} + + \end{minipage} + \end{tabular} +\end{table} + +\begin{table} + \begin{tabular}[t]{ll} + \begin{minipage}{0.4\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \label{multiplikation:alg:linear} + \Function{L}{$\mathbf{a}, \mathbf{b}$,n} + \State $ sum \gets 0$ + \For{$i = 0,1,2 \dots,n$} + \State $ sum \gets sum + A[i] \cdot B[i] $ + \EndFor + + \State \textbf{return} $sum$ + + \EndFunction + \State + \State + \end{algorithmic} + \end{algorithm} + \end{minipage} + & + \begin{minipage}{0.4\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \label{multiplikation:alg:q1} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{Q}{$\mathbf{A}, \mathbf{B}$,n} + \State $ sum \gets 0$ + \For{$i = 0,1,2 \dots,n$} + \For{$j = 0,1,2 \dots,n$} + \State $ sum \gets sum + A[i] \cdot B[j] $ + \EndFor + \EndFor + \State \textbf{return} $sum$ + \EndFunction + \end{algorithmic} + \end{algorithm} + \end{minipage} + \end{tabular} +\end{table} + +dhdfh +\end{document} diff --git a/buch/papers/multiplikation/images/meas_c.pdf b/buch/papers/multiplikation/images/meas_c.pdf index 3a4cfd8..e6af618 100644 Binary files a/buch/papers/multiplikation/images/meas_c.pdf and b/buch/papers/multiplikation/images/meas_c.pdf differ diff --git a/buch/papers/multiplikation/images/meas_c.tex b/buch/papers/multiplikation/images/meas_c.tex index 818a7e6..647a322 100644 --- a/buch/papers/multiplikation/images/meas_c.tex +++ b/buch/papers/multiplikation/images/meas_c.tex @@ -43,8 +43,8 @@ \begin{tikzpicture} \begin{axis}[ xmode=log, ymode=log, -xmin=60, xmax=5000, -ymin=1e-4, ymax=2e3, +xmin=60, xmax=10000, +ymin=1e-4, ymax=2e4, grid=both, major grid style={black!50}, xlabel = data Input ($n$), @@ -70,6 +70,7 @@ width=12cm, height=8cm, (1024,4.539413) (2048,130.627663) (4096,1179.261048) +(8192,10071.512655) }; \addlegendentry{Strassen} \addplot [ color=black, @@ -86,6 +87,7 @@ width=12cm, height=8cm, (1024,8.457472 ) (2048,59.267256) (4096,414.648901) +(8192,3014.235467) }; \addlegendentry{MM div and conq} @@ -103,6 +105,7 @@ width=12cm, height=8cm, (1024,19.928951 ) (2048,159.333884 ) (4096,1147.106865) +(8192,9606.402522) }; \addlegendentry{MM} @@ -120,6 +123,7 @@ width=12cm, height=8cm, (1024,4.162845 ) (2048,125.909034 ) (4096,1111.312696) +(8192,9376.173434) }; \addlegendentry{BLAS} \addplot[ color=blue, @@ -136,6 +140,7 @@ width=12cm, height=8cm, (1024,0.704748 ) (2048,6.845095 ) (4096,55.845038) +(8192,478.429957) }; \end{axis} \end{tikzpicture} diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex index a7612e1..464085d 100755 --- a/buch/papers/multiplikation/loesungsmethoden.tex +++ b/buch/papers/multiplikation/loesungsmethoden.tex @@ -39,13 +39,13 @@ Die \texttt{for i} Schleife iteriert \"uber alle Zeilen der $\mathbf{A}$ Matrix, \end{algorithmic} \end{algorithm} -Die Laufzeit dieser Struktur mit drei \texttt{For} Schleifen ist $\mathcal{O}\left(n^3\right)$ +Die Laufzeit dieser Struktur mit drei \texttt{For} Schleifen ist $\mathcal{O} (n^3)$ \subsubsection{Divide and Conquer Methode} F\"ur gewisse Algorithmen f\"uhren \textit{Divide and Conquer} Ans\"atze \cite{multiplikation:DAC} zu markant besseren Laufzeiten. Die Grundidee ist, dass ein Problem in mehrere, meist simplere und kleinere Teilprobleme aufgeteilt wird. -Das bekannteste Beispiel ist wohl die \textit{Fast Fourier Transform} wobei die Laufzeit von $\mathcal{O}\left(n^2\right)$ zu $\mathcal{O}(n \log n)$ verbessert werden kann. +Das bekannteste Beispiel ist wohl die \textit{Fast Fourier Transform} wobei die Laufzeit von $\mathcal{O} (n^2)$ zu $\mathcal{O}(n \log n)$ verbessert werden kann. Die Matrizenmultiplikation kann ebenfalls mit solch einem Ansatz berechnet werden. Zur vereinfachten Veranschaulichung kann die Situation mit $\mathbf{A}$ und $\mathbf{B}$ der Gr\"osse $2^n \times 2^n$ verwendet werden. @@ -68,7 +68,7 @@ Das Matrizen Produkt \end{bmatrix}, \end{equation} \begin{equation} -\mathbf{C}_{ij} = \sum_{k=1}2n \mathbf{A}_{ik} \mathbf{B}_{kj} +\mathbf{C}_{ij} = \sum_{k=1}^{2n} \mathbf{A}_{ik} \mathbf{B}_{kj} \label{multiplikation:eq:MM_block} \end{equation} ist identisch zu der Gleichung \eqref{multiplikation:eq:MM}, f\"ur die Multiplikation der Untermatrize $\mathbf{A}_{ik}$ und $\mathbf{B}_{kj}$ wird die Matrizenmultiplikation verwendet. @@ -109,7 +109,7 @@ Die Laufzeit dieser rekursiven Funktion kann mit dem \textit{Master Theorem} \ci Ohne auf dieses vertieft einzugehen, bestimmt die Anzahl rekursiver Aufrufe $\mathcal{T} $ der Funktion die Laufzeit. In diesem Fall wird die Funktion pro Durchlauf acht mal rekursiv aufgerufen, dies f\"uhrt \begin{equation} \label{multiplikation:eq:laufzeitdac} - \mathcal{T}(n) = 8 \cdot \mathcal{T}\left (\frac{n}{2}\right ) + n^2 = \mathcal{O}(n^{\log_2 8}) = \mathcal{O}\left (n^{3} \right ) + \mathcal{T}(n) = 8 \cdot \mathcal{T} \left(\frac{n}{2}\right ) + n^2 = \mathcal{O}(n^{\log_2 8}) = \mathcal{O} (n^{3} ) \end{equation} zu einer kubischen Laufzeit. Die Addition zweier Matrizen $\mathbf{A} + \mathbf{B} = \mathbf{C}$ hat eine Laufzeit von $\mathcal{O}(n^{2})$ und kann neben dem dominierendem Anteil von $\mathcal{O}(n^{3})$ ignoriert werden. @@ -202,7 +202,7 @@ Die Funktion wird sieben mal rekursiv aufgerufen. Dies f\"uhrt nach dem \textit{Master Theorem} zu einer Laufzeit von \begin{equation} \label{multiplikation:eq:laufzeitstrassen} \mathcal{T}(n) = -7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 = \mathcal{O}\left(n^{\log_2 7}\right ) = \mathcal{O}\left(n^{2.8074} \right ) +7 \cdot \mathcal{T}\left(\frac{n}{2}\right) + n^2 = \mathcal{O}(n^{\log_2 7} ) = \mathcal{O}(n^{2.8074} ) \end{equation} und ist somit schneller als die Standardmethode. Man beachte, dass die Anzahl von Additionen und Subtraktionen gr\"osser und die Anzahl der Multiplikationen kleiner wurde. @@ -267,7 +267,7 @@ sein, damit man etwas einspart. Die Implementation kann Algorithmus \ref{multiplikation:alg:winograd} entnommen werden. Falls $m=n=p$ werden $\frac{n^3}/{2}$ Multiplikationen benötigt. Im Abschnitt \ref{muliplikation:sec:bigo} wurde bereits erläutert: falls $n \rightarrow \infty$ können Konstanten vernachlässigt werden und - somit entsteht für diesen Algorithmus wieder die Ursprüngliche Laufzeit von $\mathcal{O}\left(n^3 \right)$. + somit entsteht für diesen Algorithmus wieder die Ursprüngliche Laufzeit von $\mathcal{O}(n^3 )$. \begin{algorithm}\footnotesize\caption{Winograds Matrizenmultiplikation} \setlength{\lineskip}{7pt} \label{multiplikation:alg:winograd} @@ -336,33 +336,33 @@ Die meisten Numerischen Bibliotheken von High-Level Skriptsprachen wie \texttt{M \item Level 2 \begin{itemize} \item Operationen der Art: $\mathbf{y} \leftarrow \alpha \mathbf{A}\mathbf{x}+\beta \mathbf{y}$ - \item Dieses Level hat $\mathcal{O}\left(n^2\right)$ Charakteristik + \item Dieses Level hat $\mathcal{O}(n^2)$ Charakteristik \end{itemize} \item Level 3 \begin{itemize} \item Operationen der Art: $\mathbf{C} \leftarrow \alpha \mathbf{A}\mathbf{B}+\beta\mathbf{C}$ - \item Dieses Level hat $\mathcal{O}\left(n^3\right)$ Charakteristik + \item Dieses Level hat $\mathcal{O}(n^3)$ Charakteristik \end{itemize} \end{itemize} Die \textit{BLAS} sind auf die modernen Computer Prozessoren optimiert und k\"onnen dank einer ausgeklügelter Verwendung der Speicherarchitektur zu erheblichen Leistungsoptimierungen f\"uhren. -\subsubsection{General Matrix Multiplication (GEMM)} - -Die \textit{Double-GEMM} \cite{multiplikation:DGEMM} ist definiert als: - -\textit{DGEMM performs one of the matrix-matrix operations} -$$ - C := \alpha \cdot op( A )\cdot op( B ) + \beta \cdot C, - $$ - \textit{where op( X ) is one of} -$$ -op( X ) = X \quad \text{ or } \quad op( X ) = X^T, -$$ - \textit{alpha and beta are scalars, and A, B and C are matrices, with op( A ) - an m by k matrix, op( B ) a k by n matrix and C an m by n matrix. - } +%\subsubsection{General Matrix Multiplication (GEMM)} +% +%Die \textit{Double-GEMM} \cite{multiplikation:DGEMM} ist definiert als: +% +%\textit{DGEMM performs one of the matrix-matrix operations} +%$$ +% C := \alpha \cdot op( A )\cdot op( B ) + \beta \cdot C, +% $$ +% \textit{where op( X ) is one of} +%$$ +%op( X ) = X \quad \text{ or } \quad op( X ) = X^T, +%$$ +% \textit{alpha and beta are scalars, and A, B and C are matrices, with op( A ) +% an m by k matrix, op( B ) a k by n matrix and C an m by n matrix. +% } %Die Implementation von $\alpha\mathbf{A}\mathbf{B} + \beta \mathbf{C} = \mathbf{C}$, wobei $\alpha = 1.0$ und $\beta = 0.0$ in der \texttt{C}-Version von \textit{BLAS}, ist als %\begin{lstlisting}[style=multiplikationC] @@ -379,7 +379,7 @@ $$ Folgende Algorithmen wurden jeweils in \texttt{C} und \texttt{Python} implementiert. \begin{itemize} \item Standard Matrizenmultiplikation - \item \textit{Devide and Conquer} Matrizenmultiplikation + \item \textit{Divide and Conquer} Matrizenmultiplikation \item Strassens Matrizenmultiplikation \item Winograds Matrizenmultiplikation \item \texttt{BLAS} Matrizenmultiplikation in \texttt{C} diff --git a/buch/papers/multiplikation/problemstellung.tex b/buch/papers/multiplikation/problemstellung.tex index e53b0de..c8ba274 100755 --- a/buch/papers/multiplikation/problemstellung.tex +++ b/buch/papers/multiplikation/problemstellung.tex @@ -14,87 +14,102 @@ Gezielt wird auf Algorithmen eingegangen, welche das Problem schneller als der S Die Big $\mathcal{O}$ Notation beschreibt die Laufzeitkomplexit\"at eines Algorithmus in Abhängigkeit zur Inputgrösse \cite{multiplikation:bigo}. $f(x) \in \mathcal{O}(g(x))$ besagt, dass die Funktion $f$ nicht wesentlich schneller w\"achst als $g$ wenn $x \rightarrow \infty$. % Es gibt eine Konstante $K$ derart, dass $f(x) \le K g(x)$ für $x\to\infty$ -Als Beispiel: benötigt eine Funktion $g$ $\mathcal{O}\left(n^2 \right)$ Multiplikationen, so wächst $f$ mit $\mathcal{O}\left(n+ n^2 \right)$ nicht wesentlich schneller falls $x\to\infty$. +Als Beispiel: benötigt eine Funktion $g$ $\mathcal{O} (n^2 )$ Multiplikationen, so wächst $f$ mit $\mathcal{O} (n+ n^2 )$ nicht wesentlich schneller falls $x\to\infty$. Vereinfacht werden f\"ur Algorithmen die folgende Notation verwendet: \begin{itemize} \item $f \in \mathcal{O}(1) \rightarrow f$ ist beschr\"ankt \item $f \in \mathcal{O}(n) \rightarrow f$ w\"achst linear - \item $f \in \mathcal{O}\left (n^2 \right ) \rightarrow f$ w\"achst quadratisch + \item $f \in \mathcal{O} (n^2 ) \rightarrow f$ w\"achst quadratisch \item $f \in \mathcal{O}(\log n) \rightarrow f$ w\"achst logarithmisch \item $f \in \mathcal{O}(n \log n) \rightarrow f$ hat super-lineares Wachstum - \item $f \in \mathcal{O}\left (e^n \right ) \rightarrow f$ w\"achst exponentiell + \item $f \in \mathcal{O} (e^n ) \rightarrow f$ w\"achst exponentiell \item usw. \end{itemize} In der Abbildung \ref{multiplikation:fig:bigo} k\"onnen die verschiedenen Laufzeiten miteinander verglichen werden. Bei einer logarithmischen Darstellung werden Polynome der Form $f(x) = x^k$ als Gerade und Exponentialfunktionen der Form $f(x) = a^x$ als nach oben gekr\"ummte Kurven dargestellt. -Sch\"on zu erkennen ist, dass Logarithmische Kurven beschr\"ankt sind. + \subsubsection{Beispiel Algorithmen} Es folgen einige Beispiele von Algorithmen welche zu einer bestimmten Zeitkomplexit\"atsklasse zugeteilt werden k\"onnen. -\begin{minipage}{0.4\textwidth} - \begin{algorithm}[H]\footnotesize\caption{} - \label{multiplikation:alg:b1} - \setlength{\lineskip}{7pt} - \begin{algorithmic} - \Function{B1}{$a, b$} - \State \textbf{return} $a+b$ - \EndFunction - \end{algorithmic} - \end{algorithm} - - \begin{algorithm}[H]\footnotesize\caption{} - \setlength{\lineskip}{7pt} - \begin{algorithmic} - \label{multiplikation:alg:linear} - \Function{L}{$\mathbf{a}, \mathbf{b}$,n} - \State $ sum \gets 0$ - \For{$i = 0,1,2 \dots,n$} - \State $ sum \gets sum + A[i] \cdot B[i] $ - \EndFor - - \State \textbf{return} $sum$ - - \EndFunction - \end{algorithmic} - \end{algorithm} -\end{minipage} -\hspace{2cm} -\begin{minipage}{0.4\textwidth} - - \begin{algorithm}[H]\footnotesize\caption{} - \label{multiplikation:alg:b2} - \setlength{\lineskip}{7pt} - \begin{algorithmic} - \Function{B2}{$a, b$} - \State $ x \gets a+b $ - \State $ y \gets a \cdot b $ - \State \textbf{return} $x+y$ - \EndFunction - \end{algorithmic} - \end{algorithm} - - - \begin{algorithm}[H]\footnotesize\caption{} - \label{multiplikation:alg:q1} - \setlength{\lineskip}{7pt} - \begin{algorithmic} - \Function{Q}{$\mathbf{A}, \mathbf{B}$,n} - \State $ sum \gets 0$ - \For{$i = 0,1,2 \dots,n$} - \For{$j = 0,1,2 \dots,n$} - \State $ sum \gets sum + A[i] \cdot B[j] $ - \EndFor - \EndFor - \State \textbf{return} $sum$ - \EndFunction - \end{algorithmic} - \end{algorithm} - -\end{minipage} + +\begin{table}[t] + \begin{tabular}{ll} + \begin{minipage}{0.48\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \label{multiplikation:alg:b1} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{B1}{$a, b$} + \State \textbf{return} $a+b$ + \EndFunction + \State + \State + \end{algorithmic} + \end{algorithm} + \end{minipage} + & + \begin{minipage}{0.48\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \label{multiplikation:alg:b2} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{B2}{$a, b$} + \State $ x \gets a+b $ + \State $ y \gets a \cdot b $ + \State \textbf{return} $x+y$ + \EndFunction + \end{algorithmic} + \end{algorithm} + + \end{minipage} + \end{tabular} +\end{table} + +\begin{table} + \begin{tabular}[t]{ll} + \begin{minipage}{0.48\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \label{multiplikation:alg:linear} + \Function{L}{$\mathbf{a}, \mathbf{b}$,n} + \State $ sum \gets 0$ + \For{$i = 0,1,2 \dots,n$} + \State $ sum \gets sum + A[i] \cdot B[i] $ + \EndFor + + \State \textbf{return} $sum$ + + \EndFunction + \State + \State + \end{algorithmic} + \end{algorithm} + \end{minipage} + & + \begin{minipage}{0.48\textwidth} + \begin{algorithm}[H]\footnotesize\caption{} + \label{multiplikation:alg:q1} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{Q}{$\mathbf{A}, \mathbf{B}$,n} + \State $ sum \gets 0$ + \For{$i = 0,1,2 \dots,n$} + \For{$j = 0,1,2 \dots,n$} + \State $ sum \gets sum + A[i] \cdot B[j] $ + \EndFor + \EndFor + \State \textbf{return} $sum$ + \EndFunction + \end{algorithmic} + \end{algorithm} + \end{minipage} + \end{tabular} +\end{table} \paragraph{Beschr\"ankter Algorithmus} @@ -111,7 +126,7 @@ Die \texttt{for}-Schleife wird $n$-mal durchlaufen und f\"uhrt deshalb zu $\math \paragraph{Quadratischer Algorithmus} Der Algorithmus \ref{multiplikation:alg:q1} hat ein quadratisches Verhalten. -Die beiden \texttt{for}-Schleifen werden jeweils $n$-mal durchlaufen und f\"uhrt deshalb zu $\mathcal{O}\left(n^2\right)$. +Die beiden \texttt{for}-Schleifen werden jeweils $n$-mal durchlaufen und f\"uhrt deshalb zu $\mathcal{O} (n^2 )$. \begin{figure} -- cgit v1.2.1 From 3c59b60807e1d1238bf591e238a42574327246ca Mon Sep 17 00:00:00 2001 From: Nunigan Date: Tue, 10 Aug 2021 07:29:49 +0200 Subject: update plots --- buch/papers/multiplikation/code/MM.py | 22 ++-- buch/papers/multiplikation/code/c_meas_4096.pdf | Bin 22360 -> 22207 bytes buch/papers/multiplikation/code/helper_class.py | 4 +- buch/papers/multiplikation/code/meas/ci/MM.txt | 11 -- buch/papers/multiplikation/code/meas/ci/Wino.txt | 11 -- buch/papers/multiplikation/code/meas/ci/blas.txt | 11 -- buch/papers/multiplikation/code/meas/ci/dc.txt | 11 -- .../multiplikation/code/meas/ci/strassen.txt | 11 -- buch/papers/multiplikation/code/meas_4096.pdf | Bin 17369 -> 18300 bytes buch/papers/multiplikation/images/meas_c.pdf | Bin 23552 -> 24028 bytes buch/papers/multiplikation/images/meas_c.tex | 115 +++++++++++---------- buch/papers/multiplikation/images/meas_python.pdf | Bin 21700 -> 26004 bytes buch/papers/multiplikation/images/meas_python.tex | 53 ++++++---- buch/papers/multiplikation/images/x.pdf | Bin 0 -> 23603 bytes 14 files changed, 105 insertions(+), 144 deletions(-) create mode 100644 buch/papers/multiplikation/images/x.pdf (limited to 'buch/papers/multiplikation') diff --git a/buch/papers/multiplikation/code/MM.py b/buch/papers/multiplikation/code/MM.py index 8a6824a..8057850 100644 --- a/buch/papers/multiplikation/code/MM.py +++ b/buch/papers/multiplikation/code/MM.py @@ -291,19 +291,21 @@ def mean_confidence_interval(data, confidence=0.95): n = len(a) m, se = np.mean(a), scipy.stats.sem(a) h = se * scipy.stats.t.ppf((1 + confidence) / 2., n-1) - return m, m-h, m+h + return m, h # test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if __name__ == '__main__': - A = plot_c_res(10, 4096) - name = ['MM', 'Wino', 'blas', 'strassen', 'dc'] - for i in range(5): - ci_inner = [] - for j in range(11): - ci_inner.append(mean_confidence_interval(A[i][j*10:(j+1)*10])) - np.savetxt('meas/ci/' + name[i]+'.txt',ci_inner) - - # arr = plot(1024) + # A = plot_c_res(10, 4096) + # name = ['MM', 'Wino', 'blas', 'strassen', 'dc'] + # for i in range(5): + # ci_inner = [] + # print(name[i]) + # for j in range(11): + # m,h=mean_confidence_interval(A[i][j*10:(j+1)*10]) + # print("({},{})".format(2**(j+1),m)) + # np.savetxt('meas/ci/' + name[i]+'.txt',ci_inner) + + arr = plot(4096) # n = np.logspace(1,12,12,base=2,dtype=(np.int)) # n=[2048,4096] # n = np.arange(1,50,2) diff --git a/buch/papers/multiplikation/code/c_meas_4096.pdf b/buch/papers/multiplikation/code/c_meas_4096.pdf index b42082f..f637ae4 100644 Binary files a/buch/papers/multiplikation/code/c_meas_4096.pdf and b/buch/papers/multiplikation/code/c_meas_4096.pdf differ diff --git a/buch/papers/multiplikation/code/helper_class.py b/buch/papers/multiplikation/code/helper_class.py index ad67909..3b74f67 100755 --- a/buch/papers/multiplikation/code/helper_class.py +++ b/buch/papers/multiplikation/code/helper_class.py @@ -101,6 +101,6 @@ if __name__ == '__main__': helper = Helper() # n = np.arange(2,10) - n = np.logspace(1,4,4,base=2,dtype=(np.int)) + n = np.logspace(1,11,11,base=2,dtype=(np.int)) # n=[8192] - C = helper.write_c_matrix(n) + # C = helper.write_c_matrix(n) diff --git a/buch/papers/multiplikation/code/meas/ci/MM.txt b/buch/papers/multiplikation/code/meas/ci/MM.txt index e4ad1ba..e69de29 100644 --- a/buch/papers/multiplikation/code/meas/ci/MM.txt +++ b/buch/papers/multiplikation/code/meas/ci/MM.txt @@ -1,11 +0,0 @@ -0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 -2.999999999999999864e-07 -4.555021440490437016e-08 6.455502144049043430e-07 -1.800000000000000130e-06 1.498379044967867836e-06 2.101620955032132425e-06 -1.199999999999999861e-05 9.737842837259007037e-06 1.426215716274099018e-05 -8.930000000000000221e-05 7.942767152586658090e-05 9.917232847413342352e-05 -6.922999999999999684e-04 6.611729768299406300e-04 7.234270231700593067e-04 -5.684200000000000363e-03 5.587928563282692010e-03 5.780471436717308717e-03 -5.177150000000000502e-02 5.161257221154376407e-02 5.193042778845624596e-02 -5.062468000000001078e-01 5.001729723042721565e-01 5.123206276957280592e-01 -4.504808599999999608e+00 4.404751183933223402e+00 4.604866016066775813e+00 -1.292894618999999921e+02 1.292188312556721144e+02 1.293600925443278697e+02 diff --git a/buch/papers/multiplikation/code/meas/ci/Wino.txt b/buch/papers/multiplikation/code/meas/ci/Wino.txt index 4ec0106..e69de29 100644 --- a/buch/papers/multiplikation/code/meas/ci/Wino.txt +++ b/buch/papers/multiplikation/code/meas/ci/Wino.txt @@ -1,11 +0,0 @@ -9.999999999999999547e-08 -1.262157162740991459e-07 3.262157162740991104e-07 -0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 -2.000000000000000333e-06 1.999999999999999909e-06 2.000000000000000757e-06 -1.199999999999999861e-05 9.737842837259007037e-06 1.426215716274099018e-05 -8.329999999999999189e-05 7.952898408510092581e-05 8.707101591489905797e-05 -6.478999999999999733e-04 6.173195729945008762e-04 6.784804270054990705e-04 -5.287299999999999986e-03 5.226513788941518357e-03 5.348086211058481615e-03 -5.267459999999999504e-02 5.240389179019239174e-02 5.294530820980759833e-02 -5.249752000000000862e-01 5.233835466989910090e-01 5.265668533010091634e-01 -4.671160999999999675e+00 4.572509907501117965e+00 4.769812092498881384e+00 -1.366769777000000090e+02 1.363957928284978891e+02 1.369581625715021289e+02 diff --git a/buch/papers/multiplikation/code/meas/ci/blas.txt b/buch/papers/multiplikation/code/meas/ci/blas.txt index 5d7ff7b..e69de29 100644 --- a/buch/papers/multiplikation/code/meas/ci/blas.txt +++ b/buch/papers/multiplikation/code/meas/ci/blas.txt @@ -1,11 +0,0 @@ -9.999999999999999547e-08 -1.262157162740991459e-07 3.262157162740991104e-07 -0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 -9.999999999999999547e-08 -1.262157162740991459e-07 3.262157162740991104e-07 -3.899999999999999929e-06 1.864058553533107683e-06 5.935941446466892176e-06 -2.100000000000000223e-05 1.871284586667546976e-05 2.328715413332453469e-05 -1.858000000000000168e-04 1.595988766828141249e-04 2.120011233171859087e-04 -1.264900000000000009e-03 1.221091632895032926e-03 1.308708367104967091e-03 -9.648900000000000185e-03 9.575266909835235610e-03 9.722533090164764760e-03 -7.737650000000000083e-02 7.445101996220353235e-02 8.030198003779646931e-02 -7.643868000000000329e-01 7.545731380187049586e-01 7.742004619812951072e-01 -7.632099399999999534e+00 7.613379481172315444e+00 7.650819318827683624e+00 diff --git a/buch/papers/multiplikation/code/meas/ci/dc.txt b/buch/papers/multiplikation/code/meas/ci/dc.txt index df268a9..e69de29 100644 --- a/buch/papers/multiplikation/code/meas/ci/dc.txt +++ b/buch/papers/multiplikation/code/meas/ci/dc.txt @@ -1,11 +0,0 @@ -2.999999999999999864e-07 -3.786471488222973584e-07 9.786471488222972253e-07 -1.100000000000000056e-06 8.737842837259009412e-07 1.326215716274099171e-06 -8.600000000000000670e-06 6.210712693650135778e-06 1.098928730634986641e-05 -7.819999999999998990e-05 7.075203863371232107e-05 8.564796136628765873e-05 -5.940000000000001269e-04 5.439534118129448707e-04 6.440465881870553830e-04 -4.433900000000000167e-03 4.349138038034851966e-03 4.518661961965148369e-03 -3.484430000000000166e-02 3.435947773230259988e-02 3.532912226769740344e-02 -2.948473000000000344e-01 2.887830472415335303e-01 3.009115527584665384e-01 -2.222850699999999957e+00 2.193855611791002858e+00 2.251845788208997057e+00 -1.765923450000000372e+01 1.762601016688562439e+01 1.769245883311438305e+01 -1.416103936000000090e+02 1.414816028568733657e+02 1.417391843431266523e+02 diff --git a/buch/papers/multiplikation/code/meas/ci/strassen.txt b/buch/papers/multiplikation/code/meas/ci/strassen.txt index 983fed9..e69de29 100644 --- a/buch/papers/multiplikation/code/meas/ci/strassen.txt +++ b/buch/papers/multiplikation/code/meas/ci/strassen.txt @@ -1,11 +0,0 @@ -0.000000000000000000e+00 0.000000000000000000e+00 0.000000000000000000e+00 -2.099999999999999799e-06 1.572163328693768390e-06 2.627836671306231420e-06 -1.130000000000000023e-05 7.484018077564905836e-06 1.511598192243509547e-05 -7.069999999999999733e-05 6.500652995675561090e-05 7.639347004324438376e-05 -5.040999999999999474e-04 4.766428619697257881e-04 5.315571380302741610e-04 -3.595999999999999804e-03 3.528938496002300557e-03 3.663061503997699052e-03 -2.544810000000000128e-02 2.513634544137222787e-02 2.575985455862777468e-02 -1.781816999999999984e-01 1.773043765864557864e-01 1.790590234135442105e-01 -1.255500000000000060e+00 1.247847949398645628e+00 1.263152050601354492e+00 -8.830237099999999728e+00 8.790366960647805428e+00 8.870107239352194028e+00 -6.190186909999999898e+01 6.183048085945843297e+01 6.197325734054156499e+01 diff --git a/buch/papers/multiplikation/code/meas_4096.pdf b/buch/papers/multiplikation/code/meas_4096.pdf index 9e8fcea..ecf2cff 100644 Binary files a/buch/papers/multiplikation/code/meas_4096.pdf and b/buch/papers/multiplikation/code/meas_4096.pdf differ diff --git a/buch/papers/multiplikation/images/meas_c.pdf b/buch/papers/multiplikation/images/meas_c.pdf index e6af618..faf347e 100644 Binary files a/buch/papers/multiplikation/images/meas_c.pdf and b/buch/papers/multiplikation/images/meas_c.pdf differ diff --git a/buch/papers/multiplikation/images/meas_c.tex b/buch/papers/multiplikation/images/meas_c.tex index 647a322..fe2bd2f 100644 --- a/buch/papers/multiplikation/images/meas_c.tex +++ b/buch/papers/multiplikation/images/meas_c.tex @@ -43,8 +43,8 @@ \begin{tikzpicture} \begin{axis}[ xmode=log, ymode=log, -xmin=60, xmax=10000, -ymin=1e-4, ymax=2e4, +xmin=30, xmax=10000, +ymin=1e-5, ymax=2e4, grid=both, major grid style={black!50}, xlabel = data Input ($n$), @@ -57,35 +57,36 @@ width=12cm, height=8cm, ] \addlegendentry{Winograd} \addplot[ color=purple, + error bars/.cd, y dir=both, y explicit, ] coordinates { -% (2, 0.000001) -% (4, 0.000001) -% (8, 0.000002) -% (16, 0.000011) -% (32, 0.000100) -(64, 0.000654) -(128, 0.005229) -(256, 0.057440) -(512, 0.517850) -(1024,4.539413) -(2048,130.627663) +%(2,1e-07) +%(4,5e-07) +%(8,2.0000000000000003e-06) +%(16,1.1999999999999999e-05) +(32,8.329999999999999e-05) +(64,0.0006479) +(128,0.0052873) +(256,0.052674599999999995) +(512,0.5249752000000001) +(1024,4.671161) +(2048,136.6769777) (4096,1179.261048) (8192,10071.512655) }; \addlegendentry{Strassen} \addplot [ color=black, ]coordinates { - % (2,0.000001 ) - % (4,0.000003 ) - % (8,0.000010 ) - % (16,0.000066 ) - % (32,0.000470 ) - (64,0.003368 ) - (128,0.024232 ) - (256,0.172000 ) - (512,1.209262 ) -(1024,8.457472 ) -(2048,59.267256) +%(2,1e-07) +%(4,2.1e-06) +%(8,1.13e-05) +%(16,7.07e-05) +(32,0.0005041) +(64,0.003596) +(128,0.0254481) +(256,0.1781817) +(512,1.2555) +(1024,8.8302371) +(2048,61.9018691) (4096,414.648901) (8192,3014.235467) }; @@ -93,17 +94,17 @@ width=12cm, height=8cm, \addlegendentry{MM div and conq} \addplot[ color=green, ] coordinates { - % (2,0.000003 ) - % (4,0.000002 ) - % (8,0.000010 ) - % (16,0.000068 ) - % (32,0.000594 ) - (64,0.004264 ) - (128,0.036289 ) - (256,0.324645 ) - (512,2.612010 ) -(1024,19.928951 ) -(2048,159.333884 ) +%(2,3e-07) +%(4,1.1e-06) +%(8,8.6e-06) +%(16,7.819999999999999e-05) +(32,0.0005940000000000001) +(64,0.0044339) +(128,0.0348443) +(256,0.29484730000000003) +(512,2.2228507) +(1024,17.659234500000004) +(2048,141.6103936) (4096,1147.106865) (8192,9606.402522) }; @@ -111,34 +112,34 @@ width=12cm, height=8cm, \addlegendentry{MM} \addplot [ color=red, ]coordinates { - % (2,0.000001 ) - % (4,0.000001 ) - % (8,0.000001 ) - % (16,0.000010 ) - % (32,0.000081 ) - (64,0.000654 ) - (128,0.005556 ) - (256,0.054253 ) - (512,0.487317 ) -(1024,4.162845 ) -(2048,125.909034 ) +%(2,0.0) +%(4,3e-07) +%(8,1.8000000000000001e-06) +%(16,1.1999999999999999e-05) +(32,8.93e-05) +(64,0.0006923) +(128,0.0056842) +(256,0.051771500000000005) +(512,0.5062468000000001) +(1024,4.5048086) +(2048,129.2894619) (4096,1111.312696) (8192,9376.173434) }; \addlegendentry{BLAS} \addplot[ color=blue, ] coordinates { - % (2,0.000001 ) - % (4,0.000001 ) - % (8,0.000001 ) - % (16,0.000003 ) - % (32,0.000022 ) - (64,0.000179 ) - (128,0.001278 ) - (256,0.010165 ) - (512,0.074739 ) -(1024,0.704748 ) -(2048,6.845095 ) +%(2,1e-07) +%(4,0.0) +%(8,1e-07) +%(16,3.9e-06) +(32,2.1000000000000002e-05) +(64,0.00018580000000000002) +(128,0.0012649) +(256,0.0096489) +(512,0.0773765) +(1024,0.7643868) +(2048,7.6320993999999995) (4096,55.845038) (8192,478.429957) }; diff --git a/buch/papers/multiplikation/images/meas_python.pdf b/buch/papers/multiplikation/images/meas_python.pdf index cea2232..cea4f4b 100644 Binary files a/buch/papers/multiplikation/images/meas_python.pdf and b/buch/papers/multiplikation/images/meas_python.pdf differ diff --git a/buch/papers/multiplikation/images/meas_python.tex b/buch/papers/multiplikation/images/meas_python.tex index ee4db43..c8892be 100644 --- a/buch/papers/multiplikation/images/meas_python.tex +++ b/buch/papers/multiplikation/images/meas_python.tex @@ -43,8 +43,8 @@ \begin{tikzpicture} \begin{axis}[ xmode=log, ymode=log, -xmin=30, xmax=1050, -ymin=0.01, ymax=900, +xmin=30, xmax=4100, +ymin=0.00001, ymax=60000, grid=both, major grid style={black!50}, xlabel = data input ($n$), @@ -68,7 +68,8 @@ width=12cm, height=8cm, (256, 8.29899 ) (512, 68.3699 ) (1024,537.374 ) - +(2046,4884.61) +(4096,43597.1) }; \addlegendentry{Strassen} \addplot [ color=black, @@ -79,10 +80,12 @@ width=12cm, height=8cm, % (16,0.00475407 ) (32,0.0485256 ) (64,0.220414 ) - (128,1.44718 2 ) - (256,9.93866 0 ) - (512,63.961 2 ) -(1024,461.494 2 ) + (128,1.44718 ) + (256,9.93866 ) + (512,63.961 ) +(1024,461.494 ) +(2046,3860.57) +(4096,22904.3) }; \addlegendentry{MM div and conq} @@ -98,6 +101,8 @@ width=12cm, height=8cm, (256,13.27 ) (512,105.397 ) (1024,847.321 ) +(2046,7375.93) +(4096,58466) }; \addlegendentry{MM} @@ -113,25 +118,33 @@ width=12cm, height=8cm, (256, 11.0062 ) (512, 85.4768) (1024,750.757 ) +(2046,6154.18) +(4096,46813.3) }; -% \addlegendentry{NumPy} -% \addplot[ color=blue, -% ] coordinates { + \addlegendentry{NumPy} + \addplot[ color=blue, + ] coordinates { % (2,1.83582e-05 ) % (4,7.86781e-06) % (8,1.00136e-05) % (16,5.4121e-05 ) -% (32,4.26769e-05) -% (64,0.000118494) -% (128,0.000244141 ) -% (256,0.000695705 ) -% (512,0.00221705 ) -% (1024,0.0188088 ) -% }; + (32,4.26769e-05) + (64,0.000118494) + (128,0.000244141 ) + (256,0.000695705 ) + (512,0.00221705 ) + (1024,0.0188088 ) +(2046,0.215739) +(4096,1.49159) + }; + \addplot [ + domain= 1:5000, + samples=100, + color=yellow, + ] + {(x-1000)^3}; + \addlegendentry{$\mathcal{O}\left(n^3\right)$} \end{axis} \end{tikzpicture} \end{document} - - - diff --git a/buch/papers/multiplikation/images/x.pdf b/buch/papers/multiplikation/images/x.pdf new file mode 100644 index 0000000..da4956f Binary files /dev/null and b/buch/papers/multiplikation/images/x.pdf differ -- cgit v1.2.1 From 6bab37ba5c4d1875f3c99f338a554537219013f6 Mon Sep 17 00:00:00 2001 From: Nunigan Date: Wed, 11 Aug 2021 21:22:29 +0200 Subject: update multiplikation --- buch/papers/multiplikation/images/meas_python.pdf | Bin 26004 -> 22384 bytes buch/papers/multiplikation/images/meas_python.tex | 44 ++++++++++------------ buch/papers/multiplikation/images/x.pdf | Bin 23603 -> 0 bytes buch/papers/multiplikation/loesungsmethoden.tex | 30 ++++++++++----- 4 files changed, 39 insertions(+), 35 deletions(-) delete mode 100644 buch/papers/multiplikation/images/x.pdf (limited to 'buch/papers/multiplikation') diff --git a/buch/papers/multiplikation/images/meas_python.pdf b/buch/papers/multiplikation/images/meas_python.pdf index cea4f4b..ab3b14b 100644 Binary files a/buch/papers/multiplikation/images/meas_python.pdf and b/buch/papers/multiplikation/images/meas_python.pdf differ diff --git a/buch/papers/multiplikation/images/meas_python.tex b/buch/papers/multiplikation/images/meas_python.tex index c8892be..d942f46 100644 --- a/buch/papers/multiplikation/images/meas_python.tex +++ b/buch/papers/multiplikation/images/meas_python.tex @@ -43,8 +43,8 @@ \begin{tikzpicture} \begin{axis}[ xmode=log, ymode=log, -xmin=30, xmax=4100, -ymin=0.00001, ymax=60000, +xmin=30, xmax=4200, +ymin=0.01, ymax=70000, grid=both, major grid style={black!50}, xlabel = data input ($n$), @@ -121,29 +121,23 @@ width=12cm, height=8cm, (2046,6154.18) (4096,46813.3) }; - \addlegendentry{NumPy} - \addplot[ color=blue, - ] coordinates { -% (2,1.83582e-05 ) -% (4,7.86781e-06) -% (8,1.00136e-05) -% (16,5.4121e-05 ) - (32,4.26769e-05) - (64,0.000118494) - (128,0.000244141 ) - (256,0.000695705 ) - (512,0.00221705 ) - (1024,0.0188088 ) -(2046,0.215739) -(4096,1.49159) - }; - \addplot [ - domain= 1:5000, - samples=100, - color=yellow, - ] - {(x-1000)^3}; - \addlegendentry{$\mathcal{O}\left(n^3\right)$} +% \addlegendentry{NumPy} +% \addplot[ color=blue, +% ] coordinates { +% % (2,1.83582e-05 ) +% % (4,7.86781e-06) +% % (8,1.00136e-05) +% % (16,5.4121e-05 ) +% (32,4.26769e-05) +% (64,0.000118494) +% (128,0.000244141 ) +% (256,0.000695705 ) +% (512,0.00221705 ) +% (1024,0.0188088 ) +% (2046,0.215739) +% (4096,1.49159) +% }; + \end{axis} \end{tikzpicture} diff --git a/buch/papers/multiplikation/images/x.pdf b/buch/papers/multiplikation/images/x.pdf deleted file mode 100644 index da4956f..0000000 Binary files a/buch/papers/multiplikation/images/x.pdf and /dev/null differ diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex index 464085d..be8c2d4 100755 --- a/buch/papers/multiplikation/loesungsmethoden.tex +++ b/buch/papers/multiplikation/loesungsmethoden.tex @@ -389,6 +389,14 @@ Folgende Algorithmen wurden jeweils in \texttt{C} und \texttt{Python} implementi Der Code kann im zum Buch gehörigem \textit{GitHub} \footnote{\url{https://github.com/AndreasFMueller/SeminarMatrizen.git}} Repository gefunden werden. Anzumerken ist, dass die Matrizenmultiplikation von \texttt{NumPy} als einzige Implementation Multiprocessing und Multithreading verwendet, dies f\"uhrt zu den tiefen Messzeiten. In Abbildung \ref{multiplikation:fig:python} und Abbildung \ref{multiplikation:fig:c_meas_4096} sind de Messresultate grafisch dargestellt. Die selben Messresultate sind tabellarisch in Tabelle \ref{multiplikation:tab:messung_Python} und Tabelle \ref{multiplikation:tab:messung_C} ersichtlich. + +In der Messung mit der Programmiersprache \texttt{C}, kann ein typischer Cache-Effekt beobachtet wer- +den. Bei den Algorithmen von Winograd und der Standardmethode hat bei einer Gr\"osse von +n = 2048 wohl eine Zeile der Matrix nicht an einer Cache Speicherstelle platzt. Diese beiden Al- +Algorithmen sind die Einzigen, welche \texttt{for}-Schleifen über die ganze Breite der Matrizen verwenden. +Dies führt dazu, dass ganze Zeilen zwischengespeichert werden müssen. Bei den anderen Algorith- +men ist dies nicht der Fall. + Die Hardwareinformationen des verwendeten Computers sind in der Tabelle \ref{multiplikation:tab:pc_config} aufgelistet. @@ -400,14 +408,15 @@ Die Hardwareinformationen des verwendeten Computers sind in der Tabelle \ref{mul \textbf{n} & \textbf{MM (\textit{s})} & \textbf{MM DC (\textit{s})} & \textbf{Strassen (\textit{s})} & \textbf{Winograd (\textit{s})} & \textbf{BLAS (\textit{s})} \\ \hline \multicolumn{6}{c}{} \\ - \textbf{32} & 0.000081 &0.000594 & 0.00047& 0.00010 & 0.000022 \\ - \textbf{64} & 0.00065 & 0.0042& 0.0033& 0.00065& 0.00017 \\ - \textbf{128} & 0.0055 & 0.036& 0.024& 0.0052 & 0.0012 \\ - \textbf{256} & 0.054 & 0.32 & 0.17 & 0.057& 0.010 \\ - \textbf{512} & 0.48 & 2.61 & 1.20 & 0.51 & 0.074\\ - \textbf{1024} & 4.16 & 19.92& 8.45 & 4.53 & 0.704 \\ - \textbf{2048} & 125.90 & 159.33& 59.26 & 130.62 & 6.84 \\ - \textbf{4096} & 1111.31 & 1147.10& 414.64 & 1179.26 & 55.84\\ + \textbf{32} & 0.000089 & 0.000594 & 0.0005 & 0.00008 & 0.000021 \\ + \textbf{64} & 0.00069 & 0.0044 & 0.0036 & 0.00064 & 0.00018 \\ + \textbf{128} & 0.0057 & 0.035 & 0.025 & 0.0052 & 0.0012 \\ + \textbf{256} & 0.052 & 0.29 & 0.178 & 0.053 & 0.0096 \\ + \textbf{512} & 0.51 & 2.22 & 1.25 & 0.55 & 0.077 \\ + \textbf{1024} & 4.50 & 17.65 & 8.83 & 4.67 & 0.764 \\ + \textbf{2048} & 129.28 & 141.61 & 61.901 & 136.67 & 7.63 \\ + \textbf{4096} & 1111.31 & 1147.10 & 414.64 & 1179.26 & 55.84 \\ + \textbf{8192} & 9376.17 & 9606.40 & 3014.23 & 10071.51& 478.42 \\ \multicolumn{6}{c}{} \\ \hline \hline @@ -427,13 +436,14 @@ Die Hardwareinformationen des verwendeten Computers sind in der Tabelle \ref{mul \textbf{n} & \textbf{MM (\textit{s})} & \textbf{MM DC (\textit{s})} & \textbf{Strassen (\textit{s})} & \textbf{Winograd (\textit{s})} & \textbf{\texttt{NumPy}(\textit{s})} \\ \hline \multicolumn{6}{c}{} \\ - \textbf{32} & 0.0240 &0.0271 & 0.04852& 0.01871 & 4.26e-05 \\ + \textbf{32} & 0.0240 &0.0271 & 0.04852& 0.01871 & 0.0000426 \\ \textbf{64} & 0.186 & 0.265& 0.2204& 0.1530& 0.000118 \\ \textbf{128} & 1.563 & 1.777& 1.447& 1.1947 & 0.000244 \\ \textbf{256} & 11.006 & 13.27 & 9.938 & 8.298& 0.000695 \\ \textbf{512} & 85.476 & 105.397 & 63.961 & 68.36 & 0.00221\\ \textbf{1024} & 750.757 & 847.321& 461.494 & 537.374 & 0.0188 \\ - \textbf{4096} & - & - & - & - & 1.633 \\ + \textbf{2048} & 6154.18 & 7375.93& 3860.57 & 4884.61 & 0.215 \\ + \textbf{4096} & 46813.3 & 58466 & 22904.3 & 43597.1 & 1.49 \\ \multicolumn{6}{c}{} \\ \hline \hline -- cgit v1.2.1 From 2b6637fb99a4aaebadc739b323b0ae440eb805e7 Mon Sep 17 00:00:00 2001 From: Nunigan Date: Wed, 11 Aug 2021 21:31:37 +0200 Subject: typo --- buch/papers/multiplikation/loesungsmethoden.tex | 12 ++++++------ 1 file changed, 6 insertions(+), 6 deletions(-) (limited to 'buch/papers/multiplikation') diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex index be8c2d4..0760719 100755 --- a/buch/papers/multiplikation/loesungsmethoden.tex +++ b/buch/papers/multiplikation/loesungsmethoden.tex @@ -265,7 +265,7 @@ N=2n, \quad T = n^2 \\ \end{equation} sein, damit man etwas einspart. Die Implementation kann Algorithmus \ref{multiplikation:alg:winograd} entnommen werden. -Falls $m=n=p$ werden $\frac{n^3}/{2}$ Multiplikationen benötigt. +Falls $m=n=p$ werden $\frac{n^3}{2}$ Multiplikationen benötigt. Im Abschnitt \ref{muliplikation:sec:bigo} wurde bereits erläutert: falls $n \rightarrow \infty$ können Konstanten vernachlässigt werden und somit entsteht für diesen Algorithmus wieder die Ursprüngliche Laufzeit von $\mathcal{O}(n^3 )$. \begin{algorithm}\footnotesize\caption{Winograds Matrizenmultiplikation} @@ -391,11 +391,11 @@ Anzumerken ist, dass die Matrizenmultiplikation von \texttt{NumPy} als einzige I In Abbildung \ref{multiplikation:fig:python} und Abbildung \ref{multiplikation:fig:c_meas_4096} sind de Messresultate grafisch dargestellt. Die selben Messresultate sind tabellarisch in Tabelle \ref{multiplikation:tab:messung_Python} und Tabelle \ref{multiplikation:tab:messung_C} ersichtlich. In der Messung mit der Programmiersprache \texttt{C}, kann ein typischer Cache-Effekt beobachtet wer- -den. Bei den Algorithmen von Winograd und der Standardmethode hat bei einer Gr\"osse von -n = 2048 wohl eine Zeile der Matrix nicht an einer Cache Speicherstelle platzt. Diese beiden Al- -Algorithmen sind die Einzigen, welche \texttt{for}-Schleifen über die ganze Breite der Matrizen verwenden. -Dies führt dazu, dass ganze Zeilen zwischengespeichert werden müssen. Bei den anderen Algorith- -men ist dies nicht der Fall. +den. +Bei den Algorithmen von Winograd und der Standardmethode hat bei einer Matrizengrösse von $n = 2048$ wohl eine Zeile der Matrize nicht an einer Cache Speicherstelle platzt. +Diese beiden Algorithmen sind die Einzigen, welche \texttt{for}-Schleifen über die ganze Breite der Matrizen verwenden. +Dies führt dazu, dass ganze Zeilen zwischengespeichert werden müssen. +Bei den anderen Algorithmen ist dies nicht der Fall. Die Hardwareinformationen des verwendeten Computers sind in der Tabelle \ref{multiplikation:tab:pc_config} aufgelistet. -- cgit v1.2.1