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linear algebra - What is most efficient way to convert system of equations to collection of functions?


I have the type of system M.x = b, where M is a known matrix and b is a known vector. M contains many parameters, call the entire parameter set 'a', so M => M[a].


I want to be able to efficiently evaluate x[a], i.e. x[a] is now a collection of functions with variables/parameters 'a'. How to do this in an optimal way?


The system can become very large, so that (symbolic) evaluation of LinearSolve[M,b] will take a long time. Also note that x can have many elements, and one could be interested in just evaluating x[[i]] in which case it is redundant to evaluate all the other elements of x.


EDIT


the system can e.g. be defined by:


b = {1, 0, 0, 0, 0, 0, 0, 0, 0, 0}

M = {{1, 0, 0, 0, 1, 0, 0, 0, 1}, {-c21 - c31, 0, -I omge, 0, c12, 0, 

  I Conjugate[omge], 0, c13}, {0, 
  I (-dge + dse) - c1/4 - c12/2 - c2/4 - c21/2 - c31/2 - c32/
   2, -I omse, 0, 0, 0, 0, I Conjugate[omge], 
  0}, {-I Conjugate[omge], -I Conjugate[omse], -I dge - c1/4 - c13/2 -
    c21/2 - c23/2 - c3/4 - c31/2, 0, 0, 0, 0, 0, 
  I Conjugate[omge]}, {0, 0, 
  0, -I (-dge + dse) - c1/4 - c12/2 - c2/4 - c21/2 - c31/2 - c32/2, 
  0, -I omge, I Conjugate[omse], 0, 0}, {c21, 0, 0, 
  0, -c12 - c32, -I omse, 0, I Conjugate[omse], c23}, {0, 0, 
  0, -I Conjugate[omge], -I Conjugate[omse], -I dge - I (-dge + dse) -

    c12/2 - c13/2 - c2/4 - c23/2 - c3/4 - c32/2, 0, 0, 
  I Conjugate[omse]}, {I omge, 0, 0, I omse, 0, 0, 
  I dge - c1/4 - c13/2 - c21/2 - c23/2 - c3/4 - c31/2, 
  0, -I omge}, {0, I omge, 0, 0, I omse, 0, 0, 
  I dge + I (-dge + dse) - c12/2 - c13/2 - c2/4 - c23/2 - c3/4 - c32/
   2, -I omse}, {c31, 0, I omge, 0, c32, 
  I omse, -I Conjugate[omge], -I Conjugate[omse], -c13 - c23}}

Answer



Exploring the above matrix M we get Dimensions[M]=={10,9}.


Also MatrixRank[M]==9 and MatrixRank[M[[1 ;; 9]]]==9 so we transform the system to a square system:



M = M[[1 ;; 9]];
b = b[[1 ;; 9]];
det = Det[M];

then simply calculating determinants we obtain the xi


solvefor[i_] := Module[{B},
  B = M; B[[All, i]] = b;
  Det[B]/det]

example : solvefor[1] solves for x1



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