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') 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') 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 75f4f6666396ffbc403d04e6b533b69d38fc065b Mon Sep 17 00:00:00 2001 From: Reto Fritsche Date: Tue, 10 Aug 2021 23:12:40 +0200 Subject: found some typos, removed them, some other small changes --- buch/papers/mceliece/aufbau.tex | 11 ++++++----- buch/papers/mceliece/fazit.tex | 6 +++--- buch/papers/mceliece/funktionsweise.tex | 4 ++-- buch/papers/mceliece/references.bib | 8 ++++++++ 4 files changed, 19 insertions(+), 10 deletions(-) (limited to 'buch/papers') diff --git a/buch/papers/mceliece/aufbau.tex b/buch/papers/mceliece/aufbau.tex index 521488d..200cb7b 100644 --- a/buch/papers/mceliece/aufbau.tex +++ b/buch/papers/mceliece/aufbau.tex @@ -10,7 +10,8 @@ Das McEliece-Kryptosystem besteht aus folgenden Elementen: \subsection{Datenvektor $d_k$ \label{mceliece:subsection:d_k}} In diesem Vektor der Länge $k$ sind die zu verschlüsselnden Daten enthalten. -Beispielsweise + +Beispiel: \[d_4= \begin{pmatrix} 1\\ @@ -25,7 +26,7 @@ Beispielsweise $S_k$ ist eine Binäre Zufallsmatrix der Grösse $k \times k$. Auch muss diese Matrix in $\mathbb{F}_2$ invertierbar sein. Für kleine Matrizen kann durchaus jedes Matrizenelement zufällig generiert werden, -wobei danach mithilfe des Gauss-Algorythmusses deren Inverse bestimmt werden kann. +wobei danach mithilfe des Gauss-Algorithmus deren Inverse bestimmt werden kann. Da eine solche Matrix möglicherweise singulär ist, muss in diesem Fall eine neue Zufallsmatrix erzeugt werden. Für grössere Matrizen existieren bessere Methoden, auf welche hier nicht weiter eingegangen wird \cite{mceliece:GenerationRandMatrix}. @@ -53,9 +54,9 @@ Beispiel: \label{mceliece:subsection:g_nk}} Das wichtigste Element des McEliece-Systems ist ein fehlerkorrigierender Code, der in der Lage ist, $t$ Fehler zu korrigieren. -Im Zusammenhang mit McEliece werden dabei meist Goppa-Codes verwendet, -es können prinzipiell auch andere Codes wie beispielsweise Reed-Solomin verwendet werden, -jedoch besitzen einige Codes Schwachstellen \cite{mceliece:lorenz}. +Im Zusammenhang mit McEliece werden dabei meist binäre Goppa-Codes \cite{mceliece:goppa} verwendet, +es können prinzipiell auch andere Codes wie beispielsweise Reed-Solomon verwendet werden, +jedoch besitzen einige (unter anderem auch Reed-Solomon) Codes Schwachstellen \cite{mceliece:lorenz}. Das Codieren mit diesem linearen Code kann mithilfe dessen Generatormatrix $G_{n,k}$ erfolgen. Da es sich um einen fehlerkorrigierenden Code handelt, wird das Codewort länger als das Datenwort, diff --git a/buch/papers/mceliece/fazit.tex b/buch/papers/mceliece/fazit.tex index d618993..186708b 100644 --- a/buch/papers/mceliece/fazit.tex +++ b/buch/papers/mceliece/fazit.tex @@ -35,12 +35,12 @@ Grosse unterschiede zwischen den beiden Kryptosystemen gibt es jedoch bei der Si Der Kern der RSA-Verschlüsselung beruht auf dem Problem, eine grosse Zahl in ihre beiden Primfaktoren zu zerlegen. Bei genügend grossen Zahlen ist diese Zerlegung auch mit den heute besten verfügbaren Computern kaum innerhalb vernünftiger Zeit zu lösen. Weiter ist aber bekannt, -dass mithilfe des sogenannten Shor-Algorithmuses \cite{mceliece:shor} und einem Quantencomputer auch diese Zerlegung zügig realisiert werden könnte, +dass mithilfe des sogenannten Shor-Algorithmus \cite{mceliece:shor} und einem Quantencomputer auch diese Zerlegung zügig realisiert werden könnte, was zur Folge hätte, dass die Verschlüsselung von RSA unwirksam würde. Zurzeit sind die Quantencomputer jedoch noch bei weitem nicht in der Lage, grosse Zahlen mithilfe dieses Algorithmuses zu zerlegen. -Das McEliece-System hingegen beruht auf dem Problem des ``Syndrome decoding'' (Korrektur von Bitfehlern eines Codewortes, das mit dem entsprechenden Linearcode codiert wurde). +Das McEliece-System hingegen beruht auf dem Problem des ``Syndrome decoding'' (Korrektur von Bitfehlern eines Codewortes, das mit einem entsprechenden Linearcode codiert wurde). Für das ``Syndrome decoding'' sind bis heute keine Methoden bekannt, -welche nennenswerte Vorteile gegenüber dem durchprobieren (brute-force) bringen, +welche nennenswerte Vorteile gegenüber dem Durchprobieren (brute-force) bringen, auch nicht mithilfe eines Quantencomputers. \begin{center} \begin{tabular}{c|c|c} diff --git a/buch/papers/mceliece/funktionsweise.tex b/buch/papers/mceliece/funktionsweise.tex index 93bb1c7..7c69b13 100644 --- a/buch/papers/mceliece/funktionsweise.tex +++ b/buch/papers/mceliece/funktionsweise.tex @@ -18,7 +18,7 @@ Dazu erstellt sie die einzelnen Matrizen $S_k$, $G_{n,k}$ und $P_n$. Diese drei einzelnen Matrizen bilden den privaten Schlüssel von Alice und sollen geheim bleiben. Der öffentliche Schlüssel $K_{n,k}$ hingegen berechnet sich -aus der Multiplikation der privaten Matrizen\ref{mceliece:subsection:k_nk} +aus der Multiplikation der privaten Matrizen (Abschnitt \ref{mceliece:subsection:k_nk}) und wird anschliessend Bob zugestellt. \subsection{Verschlüsselung @@ -61,7 +61,7 @@ können nun die Bitfehler, verursacht durch den Fehlervektor $e'_n$, entfernt werden. Da es sich bei diesem Schritt nicht um eine einfache Matrixmultiplikation handelt, wird die Operation durch eine Funktion dargestellt. -Wie dieser Decoder genau aufgebaut ist ist, +Wie dieser Decoder genau aufgebaut ist, hängt vom verwendeten Linearcode ab. \begin{align*} c_{k}'\,&=\text{Linear-Code-Decoder($c''_n$)}\\ diff --git a/buch/papers/mceliece/references.bib b/buch/papers/mceliece/references.bib index 52aa166..0388ff4 100644 --- a/buch/papers/mceliece/references.bib +++ b/buch/papers/mceliece/references.bib @@ -37,4 +37,12 @@ year = {2021}, month = {8}, day = {9} +} + +@online{mceliece:goppa, + title = {Binary Goppa code}, + url = {https://en.m.wikipedia.org/wiki/Binary_Goppa_code}, + year = {2021}, + month = {8}, + day = {10} } \ No newline at end of file -- 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') 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') 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