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}