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

Popular posts from this blog

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...

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...

What is and isn't a valid variable specification for Manipulate?

I have an expression whose terms have arguments (representing subscripts), like this: myExpr = A[0] + V[1,T] I would like to put it inside a Manipulate to see its value as I move around the parameters. (The goal is eventually to plot it wrt one of the variables inside.) However, Mathematica complains when I set V[1,T] as a manipulated variable: Manipulate[Evaluate[myExpr], {A[0], 0, 1}, {V[1, T], 0, 1}] (*Manipulate::vsform: Manipulate argument {V[1,T],0,1} does not have the correct form for a variable specification. >> *) As a workaround, if I get rid of the symbol T inside the argument, it works fine: Manipulate[ Evaluate[myExpr /. T -> 15], {A[0], 0, 1}, {V[1, 15], 0, 1}] Why this behavior? Can anyone point me to the documentation that says what counts as a valid variable? And is there a way to get Manpiulate to accept an expression with a symbolic argument as a variable? Investigations I've done so far: I tried using variableQ from this answer , but it says V[1...