Skip to main content

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



Comments

Post a Comment

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more