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Mathematica Algebra Calculus Wolfram Language

Using Solve[] to solve symbolic system of equations

Brandon Breitling

Brandon Breitling, POET LLC

Posted 10 years ago

I've been able to solve cubic spline functions given values of the function using the code below. F1[t_] = k1 + k2t + k3t^2 + k4t^3; F2[t_] = k5 + k6t + k7t^2 + k8t^3; F3[t_] = k9 + k10t + k11t^2 + k12t^3; F4[t_] = k13 + k14t + k15t^2 + k16t^3; F5[t_] = k17 + k18t + k19t^2 + k20*t^3; Solve[{F1[0] == 0, F1[5] == 3, F2[5] == 3, F1'[5] == F2'[5], F1''[5] == F2''[5], F2[15] == 11, F3[15] == 11, F2'[15] == F3'[15], F2''[15] == F3''[15], F3[30] == 16, F4[30] == 16, F3'[30] == F4'[30], F3''[30] == F4''[30], F4[45] == 19, F5[45] == 19, F4'[45] == F5'[45], F4''[45] == F5''[45], F5[72] == 20, F1'[0] == 0, F1''[0] == 0}] What I would like to be able to do is to solve these symbolically like: Solve[{F1[t0] == Fi1, F1[t1] == Fi2, F2[t1] == Fi2, F1'[t1] == F2'[t1], F1''[t1] == F2''[t1], F2[t2] == Fi3, F3[t2] == Fi3, F2'[t2] == F3'[t2], F2''[t2] == F3''[t2], F3[t3] == Fi4, F4[t3] == Fi4, F3'[t3] == F4'[t3], F3''[t3] == F4''[t3], F4[t4] == Fi5, F5[t4] == Fi5, F4'[t4] == F5'[t4], F4''[t4] == F5''[t4], F5[t5] == Fi6, F1'[t0] == 0, F1''[t0] == 0}] for just the k values i.e. I don't want it to think of the Fi's and t's as unknowns but rather as values that I can then plug in.

I've been able to solve cubic spline functions given values of the function using the code below.

F1[t_] = k1 + k2*t + k3*t^2 + k4*t^3;
F2[t_] = k5 + k6*t + k7*t^2 + k8*t^3;
F3[t_] = k9 + k10*t + k11*t^2 + k12*t^3;
F4[t_] = k13 + k14*t + k15*t^2 + k16*t^3;
F5[t_] = k17 + k18*t + k19*t^2 + k20*t^3;

Solve[{F1[0] == 0, F1[5] == 3, F2[5] == 3, F1'[5] == F2'[5], 
  F1''[5] == F2''[5],
  F2[15] == 11, F3[15] == 11, F2'[15] == F3'[15], F2''[15] == F3''[15],
  F3[30] == 16, F4[30] == 16, F3'[30] == F4'[30], F3''[30] == F4''[30],
  F4[45] == 19, F5[45] == 19, F4'[45] == F5'[45], F4''[45] == F5''[45],
  F5[72] == 20, F1'[0] == 0, F1''[0] == 0}]

What I would like to be able to do is to solve these symbolically like:

Solve[{F1[t0] == Fi1, F1[t1] == Fi2, F2[t1] == Fi2, 
  F1'[t1] == F2'[t1], F1''[t1] == F2''[t1],
  F2[t2] == Fi3, F3[t2] == Fi3, F2'[t2] == F3'[t2], 
  F2''[t2] == F3''[t2],
  F3[t3] == Fi4, F4[t3] == Fi4, F3'[t3] == F4'[t3], 
  F3''[t3] == F4''[t3],
  F4[t4] == Fi5, F5[t4] == Fi5, F4'[t4] == F5'[t4], 
  F4''[t4] == F5''[t4],
  F5[t5] == Fi6, F1'[t0] == 0, F1''[t0] == 0}]

for just the k values i.e. I don't want it to think of the Fi's and t's as unknowns but rather as values that I can then plug in.

POSTED BY: Brandon Breitling

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Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 10 years ago

Maybe something like this? fparams = {Fi1, Fi2, Fi3, Fi4, Fi5, Fi6}; tparams = {t0, t1, t2, t3, t4, t5}; Random example: fmat = RandomReal[{-10, 10}, {3, 6}]; tmat = RandomReal[{-10, 10}, {3, 6}]; subsF = Map[Thread[fparams -> #] &, fmat]; subsT = Map[Thread[tparams -> #] &, tmat]; soln /. subsF[[1]] /. subsT[[1]] (* Out[224]= {{k1 -> 9.94179828016, k10 -> 126.921208535, k11 -> 14.0653575745, k12 -> 0.543238518835, k13 -> 86.9409886567, k14 -> -8.1197838958, k15 -> -5.8283104079, k16 -> -0.433645163347, k17 -> -34.2801747093, k18 -> 107.627824592, k19 -> -42.6687115907, k2 -> 0.352538732018, k20 -> 3.47490257264, k3 -> 0.045099495612, k4 -> 0.00192315937676, k5 -> 9.96105988746, k6 -> 0.418727726288, k7 -> 0.120914957506, k8 -> 0.0308704908054, k9 -> 392.500018004}} *)

Maybe something like this?

fparams = {Fi1, Fi2, Fi3, Fi4, Fi5, Fi6};
tparams = {t0, t1, t2, t3, t4, t5};

Random example:

fmat = RandomReal[{-10, 10}, {3, 6}];
tmat = RandomReal[{-10, 10}, {3, 6}];

subsF = Map[Thread[fparams -> #] &, fmat];
subsT = Map[Thread[tparams -> #] &, tmat];

soln /. subsF[[1]] /. subsT[[1]]

(* Out[224]= {{k1 -> 9.94179828016, k10 -> 126.921208535, 
  k11 -> 14.0653575745, k12 -> 0.543238518835, k13 -> 86.9409886567, 
  k14 -> -8.1197838958, k15 -> -5.8283104079, k16 -> -0.433645163347, 
  k17 -> -34.2801747093, k18 -> 107.627824592, k19 -> -42.6687115907, 
  k2 -> 0.352538732018, k20 -> 3.47490257264, k3 -> 0.045099495612, 
  k4 -> 0.00192315937676, k5 -> 9.96105988746, k6 -> 0.418727726288, 
  k7 -> 0.120914957506, k8 -> 0.0308704908054, k9 -> 392.500018004}} *)

POSTED BY: Daniel Lichtblau

John Doty

John Doty, Noqsi Aerospace Ltd

Posted 10 years ago

Loops in Mathematica are clumsy and hard to debug. Write and debug a function that does the job for a single vector in your matrix. Then use Map to apply it to each vector in your matrix in turn.

POSTED BY: John Doty

Brandon Breitling

Brandon Breitling, POET LLC

Posted 10 years ago

Thanks Daniel, That worked exactly. I don't suppose you and the wider community could help me if I had a matrix of t0 through t5's and Fi1 through Fi6's? Basically if the fit we have done so far is for a single experiment, and we have maybe a hundred experiments of data and we want to fit the data for each run. Otherwise I would just have to plug in actual values and record the results each time. I don't know if there is a way to do this, perhaps a loop? I'm not too familiar with loops in Mathematica.

POSTED BY: Brandon Breitling

Daniel Lichtblau

Daniel Lichtblau, Wolfram Research

Posted 10 years ago

Specify explicitly the variables for which you wish to solve. The rest are then effectively treated as parameters. F1[t_] = k1 + k2t + k3t^2 + k4t^3; F2[t_] = k5 + k6t + k7t^2 + k8t^3; F3[t_] = k9 + k10t + k11t^2 + k12t^3; F4[t_] = k13 + k14t + k15t^2 + k16t^3; F5[t_] = k17 + k18t + k19t^2 + k20*t^3;! vars = Complement[Variables[{F1[t], F2[t], F3[t], F4[t], F5[t]}], {t}] eqns = {F1[t0] == Fi1, F1[t1] == Fi2, F2[t1] == Fi2, F1'[t1] == F2'[t1], F1''[t1] == F2''[t1], F2[t2] == Fi3, F3[t2] == Fi3, F2'[t2] == F3'[t2], F2''[t2] == F3''[t2], F3[t3] == Fi4, F4[t3] == Fi4, F3'[t3] == F4'[t3], F3''[t3] == F4''[t3], F4[t4] == Fi5, F5[t4] == Fi5, F4'[t4] == F5'[t4], F4''[t4] == F5''[t4], F5[t5] == Fi6, F1'[t0] == 0, F1''[t0] == 0}; soln = Solve[eqns, vars]; The solution is complicated, unsurprisingly.

Specify explicitly the variables for which you wish to solve. The rest are then effectively treated as parameters.

F1[t_] = k1 + k2*t + k3*t^2 + k4*t^3;
F2[t_] = k5 + k6*t + k7*t^2 + k8*t^3;
F3[t_] = k9 + k10*t + k11*t^2 + k12*t^3;
F4[t_] = k13 + k14*t + k15*t^2 + k16*t^3;
F5[t_] = k17 + k18*t + k19*t^2 + k20*t^3;!
vars = Complement[Variables[{F1[t], F2[t], F3[t], F4[t], F5[t]}], {t}]
eqns = {F1[t0] == Fi1, F1[t1] == Fi2, F2[t1] == Fi2, 
   F1'[t1] == F2'[t1], F1''[t1] == F2''[t1], F2[t2] == Fi3, 
   F3[t2] == Fi3, F2'[t2] == F3'[t2], F2''[t2] == F3''[t2], 
   F3[t3] == Fi4, F4[t3] == Fi4, F3'[t3] == F4'[t3], 
   F3''[t3] == F4''[t3], F4[t4] == Fi5, F5[t4] == Fi5, 
   F4'[t4] == F5'[t4], F4''[t4] == F5''[t4], F5[t5] == Fi6, 
   F1'[t0] == 0, F1''[t0] == 0};
soln = Solve[eqns, vars];

The solution is complicated, unsurprisingly.

POSTED BY: Daniel Lichtblau

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