# Solve huge symbolic system of linear equations

I have a system of 76 symbolic linear equations (i.e. some coefficients are symbolic) with a sparse coefficient matrix. However, neither Solve[] nor LinearSolve[] can solve this system in a reasonable amount of time (I have not received an answer in an hour).

On the other hand, Maple gives me the correct answer in a few seconds.

The 76 linear equations $equations in 76 $coefficients are (also can be found here.):

$equations = Uncompress["1:eJztXUuPHVcRdsbxOCCEEIsIKRISWURZzu3nDHt2WaCIP2BQLFkYInDy570IzNx+1a3qOqe+73S3HeVmYcczp/q8qurUu/709++/ff3Ns2fP3n36+Mc3b9798Prm6V8vHv/4y39+fPX29e+nX/717Y/vll/+7c2/vhv/+dnjH9+++uHN9/9+9fbN/x7/e3P+8fPHP15VNQ7SwyDNHQ5SvX5+ASJ+\ 1/Xvnv7xXauGeF99LrZ7hvyn/ngAsg9DfvL4DwFZP7CQzR0NWbH7bO8xyFtztjUO2WOQ9mwDkOcTWuaczpaArDBIe7b160891B6HfNe+e/n0s3+8evvqv3dvHv/3mab7L5/hdL8s5aSJ+DxgvI5T/nPqTKomuAIB0sIgdZSNCJDAwtRe6i4P8tPlaTb4LM0DDNLiJ9Z2MHqfRmI8wZBVE4Y0uMBC1nc\ 0ZHy1BkHYORt6zuaBhWzps23j+/xJcxf6KWhYtjzhEA5Z009BHV+txr4uDPn0kwtew87ZxB9LjQn02bbzPtVz8/RyDV8D3zDJqeCN8MxtQekNNqKJu8H5Ac1rF/xRy7wt2sOpQu7v5rwS/BIqYtfDXOxxnUBpz0xZZyW53xFUsEiMYaFIgAQkHK2OBcQVBRKWPYQ4Tys8Ez8lVKUTqyo1ccVOK1n0nC\ 24z01eufhbpU/odMBbpZUses75rbpSaVrpjlOphoxjr4NJBH3T5heASjUkuE+KSnmJ0uGAhCkElCgpKvUkylUq/XWCSoWJo8ZtmC1Owa0WUM960mjDrNZtmHJIkx/SOkPeL0Mesl/pnYm8y5CHgiGAPJvUNX6VuMaVb7+8RJGT5pzSjhuQU7VmGbWRS4tmHuTmkgrrwCxa6AkYhF5c7qW+nwyn8J4iR\ nxrKh0u+z6Bwvd5/AzQykgI+jWTQ3pnyHuzFjNk2VJ/coYsqN3X+SHOcgMqUNwoY9CYNs/RWnNNz1nzJsF7WjWOGxMdnwChX4LCNeVv0ZgQN88pw14dn1OzN948d0+LJ7Qwtfh41EOygdmqJmS7OBk5BLiH2aqOygubUHW7vpFLiW+kKf35C4e385aYqewQoWh5L9LyllRV9hWovIfCriXxro0+0NRE\ jbeW5SuT47HLmqu+KEKa0z0uf2kxRwryAXFOO+uierUACYhzWk3BfZUtLmcyLsGA1dzcGS1z0PTe0HJOQ0srLe1MbGk5p8A9F9+nljniL6qWOeKvuHr/m7ico+mVdiYC2qnGBFq2KnCV7elhwhUNgAAd0t1DeGpwnx/PD5qkjSJlasrbP4yNImAQ1k+neAdb7UyLaNiLsDJptWVKeO/JVmKII1tFDgw\ jLHluqWv8uoiwIuKM0dtwCaiZ2e+owOHmpUBQlxa58aCuiFnJKmmjjArbvxr88B8Ze1hw0vzZkN+yku4uQX552gICWnnRzDEf7BrnpdW/0wHRWgYlafZPi4MNfSuAIGnQ8wBx0DEp7hrnpU1QtOcIiNbSkHTkFGCC0gIzfSuA8Kqx734/ya0mSDfOLhxGs4ssrZ0ZEZBSTpI2e+Fiy0AKr6JC6BLV9O\ BbmWrvoTPbSZnWujv6LWTeiGFTNOtkWIqcMh8d9nnRRXd4LH+nBTGhZnRRiWvhY31gBYrF91As4ah64CA4hnU0Q+riDiIF2dMyU087pXranNXT5qyOjtvq4iGaGtHic6rXuWfCQi/OloB0XCBbvM49cdU0MfQ7vs49HhlZQCeOm7A0lhp6loe3FxdKgEgCM9kB3m7PC5x9MG8SZy8tZtpBI3/Xr375V\ 9kvn5/2qKXl/OMzRNSLs0CYyLJFgKo63n25DGkdYS6wGcJON+xpAlzLjhQjAsmR+TDAD3dTi9X1lA/Ncy/T+n8TQ5q8YbZxvpK/tvh965NJmWV/iWntcdu4MoQCae0aMp7WruR6IK1dQQJp7RoyntbupF7j0sEhYVZOyYBd7U0aMi7RqlsJpLXPPGrnZCiCiMORsBdEjIKYCNj8sV5rVUQRj2B7R9Sq\ KDQi27PdtVaFZwq+Uun16T3q6XU44K5Pr9Zt6cTDQ55ex+lXlAzVQErNMC9BW8ZzgLnIPRVGZG/kdSXP0S4jVOL+B3MobIRK+5C6xpvYNdqsp+V39X0JgpxwPSdcDuD845szGiYU5fy1VXkUqp2blUPyKXdNPFnOnAlrGWlS6PFCXqLjprm0eQ9/17Mb5HLsbG8rMbIETkNHFwm7DBQbd4ZIMJcqf6e\ u53JBQddzaUPoU/wHdG7K42LNLEneUmZmCUe/CdmPeGxwY06bsGx6UYp5eeTE6xsNDdnSUiI9Z0trni24Wj4m3Z7t/iWj7NkeGpwzr9Y3s0ysbuuoArWUSFSBcrV2ASLWPvMoqxCz4J6zLlEJscdTnnvCf017+Ds6WrCL++k1JO027OhAfCAewYnYwD38HU2oXZytaCKh3aMdrfD2O2aIdsRG6NCTbs\ uN6G9Hq+RQdOJQmOGYZVEFYZ1dgGixOjILe38tHYWHKt52znxUwR/l2Y/e40k7ntW74W+Trnpe1RmkHvUb6QBNhVS6Wu0yxPXtLkM8rWNFw04Uxzh5eca3ZrlMzrNV2FI5z3f5Id5XhLjU5b9SkFwtDHzepq1+mVhLk19Lm7+ANl+dpM0nnbf5SindfEeK7XxqScmkzyyLOXnZ+LdmyPrxCsILZLevW\ VO0tHv5Yk/v77rEOPOZPep5Y3loG5Q1CmSEmXQNeGFMze59yhlpHwh+yExOfyLN0+SZxRXrQwsG8dlafMoHn5VGCx4FBYPoWykoNUDPWVAwgHfhxU0sTz/hMsQ09vH1vI9oRKELI9C3UlDeYMsMMZO0hZMun1C6ZQy6ZgvRYJ8t+IGbIQZqi/rZgB7hVT0mMgmx62EumpUxJC6nzKuKfyiigm5UK00a\ VsRkhRs2onnwUuTF92RsrMvnIjZWa7Ia/j40g4s3TPE5UbwNiS4OxOd+AbZW4xyg7Z50QSLA1urYsUtyv7Z/dCK21u3cB3sajfFaRQU+Cc/W+hu5h/GlJYqt4dWGbZzUjaawpK0vX9/QjZMSX/ECbpYhaJwUZSJ2zLUEJB8IG373UwKXMJ2d9BqkuTFfV7LJx0K1Hg6IIZ5VTQyZJtohcITpbQAlaw0\ 7IEASZlGvZHteZLr2NclC0p0hDulronUusAaNPVuCD5X3NfEDR5pwfk6eyZ2VpUTB9yrvjaryKY11PtjXS2GVQ6oUjyvLcmCC46IV1wQnxUFaPMbqGvmWm/OQyDfeRKpWC0S+8SbSwpqq9mz3yEX6WVOpOieASjU+xCnGwSRCtKHnBKjUwSQ8DbjABQJSDEWlfKk73jXgcMDtyqwKzO4dapWnhVc4jV\ CyiRmGQSK93wgXuGbMXVxPurk83Uj7au2veJhlrIlboZ9gSrAmIlJ2KD4sqJ82p/LFUoHa+QbhDnCiW/Qj5yxoZctHDNJnCzTFVlQGNMUulPMsDhFyHt+IkC+zyvcIoOdEG41aHCpRyNVjtoH1PNIU24CwZLRpU2xN3IH20AaklAeZV2ytuxDusIxkwyoHt8mGNaeeMs5H4obBPFa5G9ZrT98PkDq7L\ FZOuXcX7khGlAbBMyGNHz0PEslH0g5+2orb0UUiOtrY09FlNHpajexpNRLwkesTopMcAB+5ovQtsorgW9nR2POzplKN93R3aoBK9Zy0sQegUk1rtEkWoFKN96CvhaJSPSdt7NkiHoXlgOtUehEBFyc9vS8iAg436IQJPBHOFsD/h5nQlagszhQ3wzARcHT0RUeLbwURcLRu0tHGgYIa5nQl8o5W8Q/h\ G5rMaKV5zxrmHa4xFqC0ozFuEgGHp93y2L7cpVrmSgRcfUwf1ERxay8GRw7x6v1Ioc4ZYo4lGQEX16DNodCGS5ofFjTDnPjEVqXXU6WfHrYIqTfiHA5SzZiy/uRLbhhNCxRPPe6kiTRFsbVIYBDcHFRQvoRvvMD3nuAbsdDR0oBAUWguoEQRjWh0GD/QFKXQXLCTQKHv7dBuPTsmpEWaohiQUgpLCxT\ wHiJNUbQ/6X58wIwxXYgCXtE+czNJUSAvlrj+aTGE90/T3legl4u2ydMJNPFeLvmOK8NSPkyB0uVrbjTqMsTzt6zU3SCqQ4KcV2xqAnRjg08Tbmdv62Xstj5UvWFB9PESkOqg0TwSuZ7sSbfhky6svx6QCxUIU38dJ8pr/XUf8uNufaJWe0j9dQWJdkmwZ7t1/fVfMJVqfLj2MsnsE+2SQFGpwwF37W\ XClztxOODWvUxeSCrV1p+E1Wfy6Ghj1mxu2/cRj0QVa1rH3d8NzlHaqCohSPWaH5mDBPfJB3Las901P1I/4kfkR2oBqdvvEf9ZU6nGB1p4BahU0xod5QJQqcYHek40i5miUocDErRGG5cLgqbpyJo0lV6YHka7xWj9qxpH0r0spTZJCsPjvl7G89TsHUaK2zkOClDDu2VcA9Ryq0XDSO3Z7uoXUuzik\ EATfULlfqHzYq5UmsOka7C3S9/8amkx+CMP9nY4YEE4WFE4iSZHEU7SrH74i8SH8weFZfOMgkNi+QHWsDjaz7uK8p8FImBLNAlIKAQUSHuGiHK4BYKoMUo7soHqpGqVdIIUE2Iy3BYLyMS6DnfHAvKlW2nRBSj6+vQTSWlELIy8ST5hDQZkxDN5kzigEzlaGHozcjh0MUws2eUVb7AL9Wn8EmkGMOHb\ 6nv3VWIX0h8QMFYt9DGA4LGP1UPcNGYSxOHHujnFTWTGVpEHealAiB5LgT29ULOkWikHymB4Bdhu5hmmQqOJehtTJHaqA4zXX0dMlO871Od7/UyBU3jl2Coe/qTuraLfMaBUhmOxI8pW8JB8Qy96n0CdVxWCWsXLNGlORnsePoSfpKAaJB/sQO+z3VJS0HSIN1sAiN5hFxuLPCMC4suhs34qL+vnsuD\ FWbQwtTzNZ1JPSSA00i3UKSqCeo+AXQvTNs4ud6M+bFl/wS+wAbdXxTwvOl4LHObmPKTAoRboDytPZc/20N5H+QbcU+vKfATvb5890w5Lo36tKU1Tlb92ZjST4uHSmyvvC2E+3zfSlfcXJu3K+2IiT96/NctNrGWR9xVf+uziVNdfL3Fb3sHcmiGJXbvvjnhUAu1K87qS++7Yktf5d2cF/+ox6s1k+U\ pHeT9T0UQTw9+JnF0vT3YjfVMMSfYBHYbkL2tK1EmhaBPCv4GvmnYPAi/yaUNuM9xb85UUAnpfsX0lUtiV7907t47drFuGlm3l7/I14CvvhMUQz2KynE2d75YxdQFeryT/58Rm81aliMNJmYisw0lIUQEJ8OknmMPpk0uIgMi4LPkMEXA4KYjAPl5eQgQcTgoiYWpzi+wFcEsMySNxnZ8o0A6hyZOC2\ 0tcDPGYOdZ3YZMONW5fc0EHeXnD7Wv+3tx0qml5vq22y8oDgkKAlUOyhNvX3OJL6s3IP5VuX3NOrtnmcUoo7t5y3xusw43NgCvacWDifiHaLca0ARz4OAvIVEkYuDoLSJvPAFe0E1SAe5SZgg7yJnH/LlNUWd4k7v5k0gblTe7hxIWCuUGaWyfWPRzq+CXSDMDpDTlYTS6Vc93O3nwlJZWfPHZuPpLX\ RgojwrQiKMK7KtwlHMn81J6AgGSrXGV11KorQIjmF9EAAZkiiHv5ibZLtHeyYoI0xlti56zpOWu+jQTfk5yub9bw0Xq0dxJNc7f3iXub6/icmmz5lg50CyM0td7e5x4O05rIW6abW9fOq7aFC7vGC/4UUPVHVl8bz6WIlLXTINf0JQ/yI6+vzacvOUknRE3vI+prqxP62KrgfzRUek1fQun7iF4Vmr7\ BSH2KSh0OuGvxSSh9KVkObVn46WEWJocHevz3HLlcT2ZS9Y3J1pLTbVPR3wE9aU6jxvXYRPmU2dkc+KySnZtoJUgBElCPX1zebaRwtQ4QwfX2SBNEHWNuXO3CI5L3zrje+G2iQcSQpHlmGJK3/Lve+PdmRwGDvMVtUh3gmzM2fPR3ce1vIvqbtjsUtFhk7PkX90nYAOLvjm49GLc7aD5Gy5VALXeNCb\ REWtDucM8ugTjRAwTokO4exowG79vI84PGuZHVdocmfsl8JxnbnS97HPD0u65zEaOTjw9zXed2RykXcd7TH4+CisqIo2xYnWZ2Nezobt7ZMLEWjSZHXm4dXxJUKEXYPIhO1MFT+SI+DlP8DDfGRWtpyTQenNzXaxaMAj++gn2yG021U/xF5jNUeP8MLT8AjbMN1tC8mM7MA3wlBl1o2YzPI4zL2j9p7\ sJKWBXvE+ILGNO+koKiqny9RTqQ5ZBKjRpyy/Q7rQsSpUX4Uhb7lE6YXlYY23le64jEyfDu5GfPAgsUZ3CGCIjPeg42SocONysOiyrql5G3v6f6ckXFtZt50Yn+C25Y9DLEjXnOqxaRjWJcT2xqAlxLARMjAhlgX2N3pVMhJ6txQFg0ryLOU6BKTZeqCdG3Z7KA4ybYcPHclCi+HFVEFDfvIryCOYkP\ 14mMSL6SMZWy9PIdh+i3D3hstgufog2ggGDvCGVHBkEBgr0jlBGQccFeMSJAsDfWA1aELOh5QBtdC3qa8CWa6WJphxR3dpTDPeThiGC/mdViU8HeyNp7GlEcLp0xEZ9FHsiSFJXvtQjs22xP+SIkbpLdMsTLfQuL9QeWUCwu9bdVJ19T70z+br2Vb0oxHCST80pNZoBI+83LNSfPpfB+niCZgSgOKzV\ iQrw9uj9Fwy0ECF7ZJRJ9oUBMZZeArnIt25KZEyjbol0vdFD3IWVbtAwFKt72bAvKtlypNIRJRCfFOMVoSBrvASp1MImgb5qnFASmMEmYKJVqfQXkDBSVai2ysBL8qlgpftXvUH0tXI5WSJl47HXEUatBArWTcAZx7f+UhaS70RUYF47o/6Rv5UOYCOZ9utXXpuoZZZ2q8+QVaUWvWYVOg5WKUqKxuN\ dY3jsvuUjsduVasyc9V17aqie4qTkWYHJE59pElTIvdDqyEMKbNK4ne9JtHz3plYpup7FaYFU5zqLLvNKBtqs8Vnb3WXOKG/0thuRr83T5cix9vjSMG0O+VkFc4dRnK6dqSpIJ6d6zTdyaIUwxIVuSbKcaPwIBpysoqqEgSD3SZcBU9Q6SugDB3XCEp5JwN4bFOQGCNxyKuBJNkW7cEMAX0qYt7ED1/\ M0ccgXOMdqVB1TPd6qt4w45oHq+plBaIvwQ/UsLpFC+CTm9z02r55uC9jjp8klTO4bvRarnG5BS7rVurPi8aCORVGtFfB2emhnpR6nTgaMiuliYfoSFGNBDDeZGIQ2+446Oke6YOnZS9CXwqqMj1zsal3t+tUxFuouz/UjbW2ryosM35vzwHTh3p/WzCEgpMezBuTucc/PY3jlvKR+vPDzRRLI4FLE8\ zkKLhDRjKUg1dYMN/g/TGBw4"];$coefficients = Uncompress["1:eJwtzLtNQ1EQRdEnoBE6sOd3TQ+O6AACJCKCR//C4q5k9iRnvX7+vH/dj+M4Xx7n/n3+nk+P5+Oyc92Jndypnd6ZnbVz23k7n//nF71qaGpp6+jSm/KCF7zgBS94wQte8IIXvOQlL3nJS17ykpe85CWveMUrXvGKV7ziFa94xWte85rXvOY1r3nNa17zhje84Q1veMMb3vCGN7zFW7zFW7zFW/0H\ Ks1gug=="];


and I try the following code to solve:

$result = Solve[$equations,$coefficients]  or $matrix = CoefficientArrays[$equations,$coefficients];
$result = LinearSolve[$matrix[[2]], \$matrix[[1]]]


Is there any way to solve this system with Mathematica in a reasonable amount of time?

• If possible, please try to post code directly in your question to make it permanent and give viewers direct feedback what awaits them. It is also good practice to provide all relevant code to show what you already tried. – Yves Klett Jul 2 '13 at 14:52
• Can you say anything about the values of the constants scalar0, k, etc. Adding some assumptions might make a difference. – Corey Kelly Jul 2 '13 at 17:12
• Can you at least be certain that the determinant is non-zero? i.e. 1 + e5 scalar0 != 0, -1 + 2 e5 scalar0 + 6 e6 scalar0 != 0, etc. Mathematica might require this as an assumption. – Corey Kelly Jul 3 '13 at 11:01
• It seems that this example is sensitive to some heuristic internal settings that involve use of Together and related. In the version currently under development it gives a result, with leaf count around 100 M, in under a half minute on my desk top. In version 9, same machine, it gives a result around 2/3 that size, in about 45 minutes. – Daniel Lichtblau Jul 3 '13 at 14:58
• I tweaked some of the heuristics and it gives considerable improvement. Not sure if this will survive to the next release-- depends on whether in so doing I broke other things of importance. – Daniel Lichtblau Jul 5 '13 at 15:45