1
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If an operation in Mathematica (say, computing the determinant of a $88\times88$ parametric matrix) takes much time to be computed on a personal computer (core i7), say a week, then, is there any hope that using a supercomputer get the result faster? Is this running time related to Mathematica's limitation or the computer we are running Mathematica on it?

My matrix $M$ is sparse and ArrayRules[SparseArray[M]] is as follows. Can someone please make comment on how I can ask Mathematica to compute its determinant faster?

sp:={{1, 1} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {1, 
   2} -> -E^(((I x)/2)) + E^((I x)/2) x, {1, 3} -> 1 - x, {1, 4} -> 
  1 + x, {2, 3} -> -1 - x, {2, 4} -> -1 + x, {2, 5} -> 
  1 + x, {2, 6} -> 
  1 - x, {3, 5} -> -1 + x, {3, 6} -> -1 - x, {3, 7} -> 
  1 + x, {3, 8} -> 1 - x, {4, 1} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {4, 2} -> 
  E^((I x)/2) + E^((I x)/2) x, {4, 7} -> -1 + x, {4, 8} -> -1 - x, {5,
    9} -> -E^(((I x)/2)) + E^((I x)/2) x, {5, 10} -> -E^(-((I x)/2)) -
    E^(-((I x)/2)) x, {5, 11} -> 1 + x, {5, 12} -> 
  1 - x, {6, 11} -> -1 + x, {6, 12} -> -1 - x, {6, 13} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {6, 14} -> 
  E^((I x)/2) + E^((I x)/2) x, {7, 13} -> -E^(-((I x)/2)) - 
   E^(-((I x)/2)) x, {7, 14} -> -E^(((I x)/2)) + E^((I x)/2) x, {7, 
   15} -> 1 - x, {7, 16} -> 1 + x, {8, 9} -> 
  E^((I x)/2) + E^((I x)/2) x, {8, 10} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {8, 15} -> -1 - x, {8, 
   16} -> -1 + x, {9, 17} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {9, 
   18} -> -E^(((I x)/2)) + E^((I x)/2) x, {9, 19} -> 1 - x, {9, 20} ->
   1 + x, {10, 19} -> -1 - x, {10, 20} -> -1 + x, {10, 21} -> 
  E^((I x)/2) + E^((I x)/2) x, {10, 22} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {11, 21} -> -E^(((I x)/2)) + 
   E^((I x)/2) x, {11, 22} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {11,
    23} -> 1 + x, {11, 24} -> 1 - x, {12, 17} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {12, 18} -> 
  E^((I x)/2) + E^((I x)/2) x, {12, 23} -> -1 + x, {12, 24} -> -1 - 
   x, {13, 25} -> -E^(((I x)/2)) + E^((I x)/2) x, {13, 
   26} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {13, 27} -> 
  1 + x, {13, 28} -> 
  1 - x, {14, 27} -> -1 + x, {14, 28} -> -1 - x, {14, 29} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {14, 30} -> 
  E^((I x)/2) + E^((I x)/2) x, {15, 29} -> -E^(-((I x)/2)) - 
   E^(-((I x)/2)) x, {15, 30} -> -E^(((I x)/2)) + E^((I x)/2) x, {15, 
   31} -> 1 - x, {15, 32} -> 1 + x, {16, 25} -> 
  E^((I x)/2) + E^((I x)/2) x, {16, 26} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {16, 31} -> -1 - x, {16, 
   32} -> -1 + x, {17, 33} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {17,
    34} -> -E^(((I x)/2)) + E^((I x)/2) x, {17, 35} -> 
  1 - x, {17, 36} -> 
  1 + x, {18, 35} -> -1 - x, {18, 36} -> -1 + x, {18, 37} -> 
  E^((I x)/2) + E^((I x)/2) x, {18, 38} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {19, 37} -> -E^(((I x)/2)) + 
   E^((I x)/2) x, {19, 38} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {19,
    39} -> 1 + x, {19, 40} -> 1 - x, {20, 33} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {20, 34} -> 
  E^((I x)/2) + E^((I x)/2) x, {20, 39} -> -1 + x, {20, 40} -> -1 - 
   x, {21, 41} -> -E^(((I x)/2)) + E^((I x)/2) x, {21, 
   42} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {21, 43} -> 
  1 + x, {21, 44} -> 
  1 - x, {22, 43} -> -1 + x, {22, 44} -> -1 - x, {22, 45} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {22, 46} -> 
  E^((I x)/2) + E^((I x)/2) x, {23, 45} -> -E^(-((I x)/2)) - 
   E^(-((I x)/2)) x, {23, 46} -> -E^(((I x)/2)) + E^((I x)/2) x, {23, 
   47} -> 1 - x, {23, 48} -> 1 + x, {24, 41} -> 
  E^((I x)/2) + E^((I x)/2) x, {24, 42} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {24, 47} -> -1 - x, {24, 
   48} -> -1 + x, {25, 49} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {25,
    50} -> -E^(((I x)/2)) + E^((I x)/2) x, {25, 51} -> 
  1 - x, {25, 52} -> 
  1 + x, {26, 51} -> -1 - x, {26, 52} -> -1 + x, {26, 53} -> 
  E^((I x)/2) + E^((I x)/2) x, {26, 54} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {27, 53} -> -E^(((I x)/2)) + 
   E^((I x)/2) x, {27, 54} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {27,
    55} -> 1 + x, {27, 56} -> 1 - x, {28, 49} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {28, 50} -> 
  E^((I x)/2) + E^((I x)/2) x, {28, 55} -> -1 + x, {28, 56} -> -1 - 
   x, {29, 65} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {29, 
   66} -> -E^(((I x)/2)) + E^((I x)/2) x, {29, 67} -> 
  1 - x, {29, 68} -> 
  1 + x, {30, 67} -> -1 - x, {30, 68} -> -1 + x, {30, 69} -> 
  E^((I x)/2) + E^((I x)/2) x, {30, 70} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {31, 69} -> -E^(((I x)/2)) + 
   E^((I x)/2) x, {31, 70} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {31,
    71} -> 1 + x, {31, 72} -> 1 - x, {32, 65} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {32, 66} -> 
  E^((I x)/2) + E^((I x)/2) x, {32, 71} -> -1 + x, {32, 72} -> -1 - 
   x, {33, 81} -> -1 - x, {33, 82} -> -1 + x, {33, 83} -> 
  1 - x, {33, 84} -> 
  1 + x, {34, 83} -> -1 - x, {34, 84} -> -1 + x, {34, 85} -> 
  E^((I x)/2) + E^((I x)/2) x, {34, 86} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {35, 85} -> -E^(((I x)/2)) + 
   E^((I x)/2) x, {35, 86} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {35,
    87} -> 1 + x, {35, 88} -> 1 - x, {36, 81} -> 1 - x, {36, 82} -> 
  1 + x, {36, 87} -> -1 + x, {36, 88} -> -1 - x, {37, 
   57} -> -E^(((I x)/2)) + E^((I x)/2) x, {37, 
   58} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {37, 59} -> 
  1 + x, {37, 60} -> 
  1 - x, {38, 59} -> -1 + x, {38, 60} -> -1 - x, {38, 61} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {38, 62} -> 
  E^((I x)/2) + E^((I x)/2) x, {39, 61} -> -E^(-((I x)/2)) - 
   E^(-((I x)/2)) x, {39, 62} -> -E^(((I x)/2)) + E^((I x)/2) x, {39, 
   63} -> 1 - x, {39, 64} -> 1 + x, {40, 57} -> 
  E^((I x)/2) + E^((I x)/2) x, {40, 58} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {40, 63} -> -1 - x, {40, 
   64} -> -1 + x, {41, 73} -> -E^(((I x)/2)) + E^((I x)/2) x, {41, 
   74} -> -E^(-((I x)/2)) - E^(-((I x)/2)) x, {41, 75} -> 
  1 + x, {41, 76} -> 
  1 - x, {42, 75} -> -1 + x, {42, 76} -> -1 - x, {42, 77} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {42, 78} -> 
  E^((I x)/2) + E^((I x)/2) x, {43, 77} -> -E^(-((I x)/2)) - 
   E^(-((I x)/2)) x, {43, 78} -> -E^(((I x)/2)) + E^((I x)/2) x, {43, 
   79} -> 1 - x, {43, 80} -> 1 + x, {44, 73} -> 
  E^((I x)/2) + E^((I x)/2) x, {44, 74} -> 
  E^(-((I x)/2)) - E^(-((I x)/2)) x, {44, 79} -> -1 - x, {44, 
   80} -> -1 + x, {45, 65} -> 1, {45, 66} -> 
  1, {45, 73} -> -1, {45, 74} -> -1, {46, 65} -> 
  I x, {46, 66} -> -I x, {46, 73} -> -I x, {46, 74} -> 
  I x, {47, 49} -> 1, {47, 50} -> 
  1, {47, 57} -> -1, {47, 58} -> -1, {48, 49} -> 
  I x, {48, 50} -> -I x, {48, 57} -> -I x, {48, 58} -> 
  I x, {49, 61} -> -1, {49, 62} -> -1, {49, 69} -> 1, {49, 70} -> 
  1, {50, 61} -> -I x, {50, 62} -> I x, {50, 69} -> 
  I x, {50, 70} -> -I x, {51, 1} -> 1, {51, 2} -> 
  1, {51, 9} -> -1, {51, 10} -> -1, {52, 1} -> 
  I x, {52, 2} -> -I x, {52, 9} -> -I x, {52, 10} -> 
  I x, {53, 13} -> -1, {53, 14} -> -1, {53, 21} -> 1, {53, 22} -> 
  1, {54, 13} -> -I x, {54, 14} -> I x, {54, 21} -> 
  I x, {54, 22} -> -I x, {55, 17} -> 1, {55, 18} -> 
  1, {55, 25} -> -1, {55, 26} -> -1, {56, 17} -> 
  I x, {56, 18} -> -I x, {56, 25} -> -I x, {56, 26} -> 
  I x, {57, 33} -> 1, {57, 34} -> 
  1, {57, 41} -> -1, {57, 42} -> -1, {58, 33} -> 
  I x, {58, 34} -> -I x, {58, 41} -> -I x, {58, 42} -> 
  I x, {59, 45} -> 1, {59, 46} -> 
  1, {59, 53} -> -1, {59, 54} -> -1, {60, 45} -> 
  I x, {60, 46} -> -I x, {60, 53} -> -I x, {60, 54} -> 
  I x, {61, 29} -> -1, {61, 30} -> -1, {61, 37} -> 1, {61, 38} -> 
  1, {62, 29} -> -I x, {62, 30} -> I x, {62, 37} -> 
  I x, {62, 38} -> -I x, {63, 77} -> 1, {63, 78} -> 
  1, {63, 85} -> -1, {63, 86} -> -1, {64, 77} -> 
  I x, {64, 78} -> -I x, {64, 85} -> -I x, {64, 86} -> 
  I x, {65, 5} -> -E^(I t - (I x)/2), {65, 6} -> -E^(
   I t + (I x)/2), {65, 81} -> E^((I x)/2), {65, 82} -> 
  E^(-((I x)/2)), {66, 5} -> -I E^(I t - (I x)/2) x, {66, 6} -> 
  I E^(I t + (I x)/2) x, {66, 81} -> 
  I E^((I x)/2) x, {66, 82} -> -I E^(-((I x)/2)) x, {67, 67} -> 
  E^((9 I d \[Pi])/11 + (I x)/2), {67, 68} -> 
  E^((9 I d \[Pi])/11 - (I x)/2), {67, 71} -> -E^(
   I b - (9 I d \[Pi])/11 - (I x)/2), {67, 72} -> -E^(
   I b - (9 I d \[Pi])/11 + (I x)/2), {68, 67} -> 
  I E^((9 I d \[Pi])/11 + (I x)/2) x, {68, 
   68} -> -I E^((9 I d \[Pi])/11 - (I x)/2) x, {68, 71} -> -I E^(
   I b - (9 I d \[Pi])/11 - (I x)/2) x, {68, 72} -> 
  I E^(I b - (9 I d \[Pi])/11 + (I x)/2) x, {69, 83} -> E^(
  I d \[Pi] + (I x)/2), {69, 84} -> E^(
  I d \[Pi] - (I x)/2), {69, 87} -> -E^(
   I b - I d \[Pi] - (I x)/2), {69, 88} -> -E^(
   I b - I d \[Pi] + (I x)/2), {70, 83} -> 
  I E^(I d \[Pi] + (I x)/2) x, {70, 84} -> -I E^(
   I d \[Pi] - (I x)/2) x, {70, 87} -> -I E^(
   I b - I d \[Pi] - (I x)/2) x, {70, 88} -> 
  I E^(I b - I d \[Pi] + (I x)/2) x, {71, 3} -> 
  E^((I d \[Pi])/11 + (I x)/2), {71, 4} -> 
  E^((I d \[Pi])/11 - (I x)/2), {71, 7} -> -E^(
   I b - (I d \[Pi])/11 - (I x)/2), {71, 8} -> -E^(
   I b - (I d \[Pi])/11 + (I x)/2), {72, 3} -> 
  I E^((I d \[Pi])/11 + (I x)/2) x, {72, 
   4} -> -I E^((I d \[Pi])/11 - (I x)/2) x, {72, 7} -> -I E^(
   I b - (I d \[Pi])/11 - (I x)/2) x, {72, 8} -> 
  I E^(I b - (I d \[Pi])/11 + (I x)/2) x, {73, 11} -> -E^(
   I b - (2 I d \[Pi])/11 - (I x)/2), {73, 12} -> -E^(
   I b - (2 I d \[Pi])/11 + (I x)/2), {73, 15} -> 
  E^((2 I d \[Pi])/11 + (I x)/2), {73, 16} -> 
  E^((2 I d \[Pi])/11 - (I x)/2), {74, 11} -> -I E^(
   I b - (2 I d \[Pi])/11 - (I x)/2) x, {74, 12} -> 
  I E^(I b - (2 I d \[Pi])/11 + (I x)/2) x, {74, 15} -> 
  I E^((2 I d \[Pi])/11 + (I x)/2) x, {74, 
   16} -> -I E^((2 I d \[Pi])/11 - (I x)/2) x, {75, 19} -> 
  E^((3 I d \[Pi])/11 + (I x)/2), {75, 20} -> 
  E^((3 I d \[Pi])/11 - (I x)/2), {75, 23} -> -E^(
   I b - (3 I d \[Pi])/11 - (I x)/2), {75, 24} -> -E^(
   I b - (3 I d \[Pi])/11 + (I x)/2), {76, 19} -> 
  I E^((3 I d \[Pi])/11 + (I x)/2) x, {76, 
   20} -> -I E^((3 I d \[Pi])/11 - (I x)/2) x, {76, 23} -> -I E^(
   I b - (3 I d \[Pi])/11 - (I x)/2) x, {76, 24} -> 
  I E^(I b - (3 I d \[Pi])/11 + (I x)/2) x, {77, 27} -> -E^(
   I b - (4 I d \[Pi])/11 - (I x)/2), {77, 28} -> -E^(
   I b - (4 I d \[Pi])/11 + (I x)/2), {77, 31} -> 
  E^((4 I d \[Pi])/11 + (I x)/2), {77, 32} -> 
  E^((4 I d \[Pi])/11 - (I x)/2), {78, 27} -> -I E^(
   I b - (4 I d \[Pi])/11 - (I x)/2) x, {78, 28} -> 
  I E^(I b - (4 I d \[Pi])/11 + (I x)/2) x, {78, 31} -> 
  I E^((4 I d \[Pi])/11 + (I x)/2) x, {78, 
   32} -> -I E^((4 I d \[Pi])/11 - (I x)/2) x, {79, 35} -> 
  E^((5 I d \[Pi])/11 + (I x)/2), {79, 36} -> 
  E^((5 I d \[Pi])/11 - (I x)/2), {79, 39} -> -E^(
   I b - (5 I d \[Pi])/11 - (I x)/2), {79, 40} -> -E^(
   I b - (5 I d \[Pi])/11 + (I x)/2), {80, 35} -> 
  I E^((5 I d \[Pi])/11 + (I x)/2) x, {80, 
   36} -> -I E^((5 I d \[Pi])/11 - (I x)/2) x, {80, 39} -> -I E^(
   I b - (5 I d \[Pi])/11 - (I x)/2) x, {80, 40} -> 
  I E^(I b - (5 I d \[Pi])/11 + (I x)/2) x, {81, 43} -> -E^(
   I b - (6 I d \[Pi])/11 - (I x)/2), {81, 44} -> -E^(
   I b - (6 I d \[Pi])/11 + (I x)/2), {81, 47} -> 
  E^((6 I d \[Pi])/11 + (I x)/2), {81, 48} -> 
  E^((6 I d \[Pi])/11 - (I x)/2), {82, 43} -> -I E^(
   I b - (6 I d \[Pi])/11 - (I x)/2) x, {82, 44} -> 
  I E^(I b - (6 I d \[Pi])/11 + (I x)/2) x, {82, 47} -> 
  I E^((6 I d \[Pi])/11 + (I x)/2) x, {82, 
   48} -> -I E^((6 I d \[Pi])/11 - (I x)/2) x, {83, 51} -> 
  E^((7 I d \[Pi])/11 + (I x)/2), {83, 52} -> 
  E^((7 I d \[Pi])/11 - (I x)/2), {83, 55} -> -E^(
   I b - (7 I d \[Pi])/11 - (I x)/2), {83, 56} -> -E^(
   I b - (7 I d \[Pi])/11 + (I x)/2), {84, 51} -> 
  I E^((7 I d \[Pi])/11 + (I x)/2) x, {84, 
   52} -> -I E^((7 I d \[Pi])/11 - (I x)/2) x, {84, 55} -> -I E^(
   I b - (7 I d \[Pi])/11 - (I x)/2) x, {84, 56} -> 
  I E^(I b - (7 I d \[Pi])/11 + (I x)/2) x, {85, 59} -> -E^(
   I b - (8 I d \[Pi])/11 - (I x)/2), {85, 60} -> -E^(
   I b - (8 I d \[Pi])/11 + (I x)/2), {85, 63} -> 
  E^((8 I d \[Pi])/11 + (I x)/2), {85, 64} -> 
  E^((8 I d \[Pi])/11 - (I x)/2), {86, 59} -> -I E^(
   I b - (8 I d \[Pi])/11 - (I x)/2) x, {86, 60} -> 
  I E^(I b - (8 I d \[Pi])/11 + (I x)/2) x, {86, 63} -> 
  I E^((8 I d \[Pi])/11 + (I x)/2) x, {86, 
   64} -> -I E^((8 I d \[Pi])/11 - (I x)/2) x, {87, 75} -> -E^(
   I b - (10 I d \[Pi])/11 - (I x)/2), {87, 76} -> -E^(
   I b - (10 I d \[Pi])/11 + (I x)/2), {87, 79} -> 
  E^((10 I d \[Pi])/11 + (I x)/2), {87, 80} -> 
  E^((10 I d \[Pi])/11 - (I x)/2), {88, 75} -> -I E^(
   I b - (10 I d \[Pi])/11 - (I x)/2) x, {88, 76} -> 
  I E^(I b - (10 I d \[Pi])/11 + (I x)/2) x, {88, 79} -> 
  I E^((10 I d \[Pi])/11 + (I x)/2) x, {88, 
   80} -> -I E^((10 I d \[Pi])/11 - (I x)/2) x, {_, _} -> 0}
Improve this question
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1
  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Kuba
    Commented Oct 24, 2022 at 6:52

1 Answer 1

Reset to default
3
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It's not so much about a supercomputer but about using the right tools. As pointed out in the comments, a SparseMatrix should be significantly faster:

s=SparseArray[yourMatrix,{88,88}];
Det[s]
Improve this answer
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11
  • $\begingroup$ Thanks, then I should run this new command and see how long it will take to compute it. $\endgroup$
    – charmin
    Commented Oct 19, 2022 at 15:03
  • $\begingroup$ it gives error that the matrix is not square. $\endgroup$
    – charmin
    Commented Oct 19, 2022 at 15:13
  • 1
    $\begingroup$ @charmin, provide SparseArray the actual dimensions of your matrix: s = SparseArray[matrix, {88, 88}] $\endgroup$
    – Domen
    Commented Oct 19, 2022 at 15:40
  • 1
    $\begingroup$ You complained that Mathematica gives error that the matrix is not square. My comment is meant to solve this problem by explicitly making the matrix be square. $\endgroup$
    – Domen
    Commented Oct 19, 2022 at 15:59
  • 1
    $\begingroup$ @charmin can you upload the whole matrix somewhere? Maybe at pastebin $\endgroup$
    – infinitezero
    Commented Oct 20, 2022 at 10:05

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