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Added Gaussian Elimimnation in C++
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‎contents/gaussian_elimination/code/c++/gaussian_elimination.cpp‎

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#include <iostream>
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#include <algorithm>
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#include <stdexcept>
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#include <vector>
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using namespace std;
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class EquationSolver
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{
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private:
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// subtract row from another row after multiplcation
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void subRow(vector<long double> &toBeSubtracted, const vector<long double> &toSubtract,
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long double mult, int start)
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{
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for (int i = start; i < toBeSubtracted.size(); i++)
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toBeSubtracted[i] -= mult * toSubtract[i];
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}
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// A start variable is given since if we know all values are 0, we don't need
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// to subtract.
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// swap two rows
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void swapRow(vector<long double> &eqn1, vector<long double> &eqn2, int start)
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{
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for (int i = start; i < eqn1.size(); i++)
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swap(eqn1[i], eqn2[i]);
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}
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// A start variable is given since if we know all values are 0, we don't need
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// to swap.
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void gaussianElimination(vector<vector<long double>> &eqns, int varc)
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{
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// 'eqns' is the matrix, 'varc' is no. of vars
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for (int i = 0; i < varc - 1; i++)
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{
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if (eqns[i][i] == 0)
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{
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for (int j = i + 1; j < varc; j++)
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{
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if (eqns[j][i] != 0)
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{
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swapRow(eqns[j], eqns[i], i);
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break;
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}
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}
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if (eqns[i][i] == 0)
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throw runtime_error("Singular matrix");
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// Cannot solve since coefficient of one variable is always zero
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}
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for (int j = i + 1; j < varc; j++)
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if (eqns[j][i])
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subRow(eqns[j], eqns[i], (eqns[j][i] / eqns[i][i]), i);
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}
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if (eqns[varc - 1][varc - 1] == 0)
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{
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if (eqns[varc - 1][varc] == 0)
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throw runtime_error("Singular matrix");
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// Cannot solve since final equation is '0*xn = 0'
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else
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throw runtime_error("No solutions");
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// Cannot solve since final equation is '0*xn = c'
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}
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}
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void gaussJordan(vector<vector<long double>> &eqns, int varc)
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{
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// 'eqns' is the (Row-echelon) matrix, 'varc' is no. of vars
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for (int i = varc - 1; i >= 0; i--)
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{
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eqns[i][varc] /= eqns[i][i];
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eqns[i][i] = 1;
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for (int j = i - 1; j >= 0; j--)
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{
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eqns[j][varc] -= eqns[j][i] * eqns[i][varc];
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eqns[j][i] = 0;
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}
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}
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}
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vector<long double> backSubs(vector<vector<long double>> &eqns, int varc)
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{
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// 'eqns' is matrix, 'varc' is no. of variables
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vector<long double> ans(varc);
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for (int i = varc - 1; i >= 0; i--)
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{
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long double sum = 0;
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for (int j = i + 1; j < varc; j++)
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sum += eqns[i][j] * ans[j];
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ans[i] = (eqns[i][varc] - sum) / eqns[i][i];
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}
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return ans;
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}
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public:
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vector<long double> solveByGaussJordan(vector< vector<long double> > eqns, int varc)
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{
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this->gaussianElimination(eqns, varc);
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this->gaussJordan(eqns, varc);
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vector<long double> ans(varc);
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for (int i = 0; i < varc; i++)
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ans[i] = eqns[i][varc];
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return ans;
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}
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vector<long double> solveByBacksubs(vector< vector<long double> > eqns, int varc)
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{
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this->gaussianElimination(eqns, varc);
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vector<long double> ans = this->backSubs(eqns, varc);
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return ans;
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}
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};
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int main() {
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int varc = 3;
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vector< vector<long double> > equations { { 2 , 3, 4, 6 },
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{ 1 , 2, 3, 4 },
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{ 3 ,-4, 0, 10 }};
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EquationSolver solver;
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vector<long double> ans;
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try
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{
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ans = solver.solveByGaussJordan(equations, varc);
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}
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catch(runtime_error e)
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{
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cout << "Error found: " << e.what() << endl;
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return -1;
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}
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cout << "The solution is," << endl
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<< "x == " << ans[0] << endl
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<< "y == " << ans[1] << endl
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<< "z == " << ans[2] << endl;
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}

‎contents/gaussian_elimination/gaussian_elimination.md‎

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@@ -232,7 +232,7 @@ $$
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$$
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There are plenty of different strategies you could use to do this, and no one strategy is better than the rest.
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One method is to subtract a multiple of the top row from subsequent rows below it such that all values beneath the pivot value are zero.
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One method is to subtract a multiple of the top row from subsequent rows below it such that all values beneath the pivot value are zero.
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This process might be easier if you swap some rows around first and can be performed for each pivot.
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After you get a row echelon matrix, the next step is to find the reduced row echelon form. In other words, we do the following:
@@ -314,6 +314,9 @@ In code, this process might look like this:
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{% sample lang="c" %}
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[import:5-13, lang:"c_cpp"](code/c/gaussian_elimination.c)
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[import:19-34, lang:"c_cpp"](code/c/gaussian_elimination.c)
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{% sample lang="cpp" %}
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[import:22-29, lang:"c_cpp"](code/c++/gaussian_elimination.cpp)
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[import:37-51, lang:"c_cpp"](code/c++/gaussian_elimination.cpp)
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{% sample lang="hs" %}
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[import:10-17, lang:"haskell"](code/haskell/gaussianElimination.hs)
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[import:44-46, lang:"haskell"](code/haskell/gaussianElimination.hs)
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\left[
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\begin{array}{ccc|c}
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3 & -4 & 0 & 10 \\
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0 & \mathbf{\frac{10}{3}} & \mathbf{3} & \mathbf{\frac{2}{3}}
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0 & \mathbf{\frac{10}{3}} & \mathbf{3} & \mathbf{\frac{2}{3}}
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\\
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2 & 3 & 4 & 6
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\end{array}
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[import:32-40, lang:"java"](code/java/GaussianElimination.java)
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{% sample lang="c" %}
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[import:36-41, lang:"c_cpp"](code/c/gaussian_elimination.c)
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{% sample lang="cpp" %}
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[import:11-19, lang:"c_cpp"](code/c++/gaussian_elimination.cpp)
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{% sample lang="hs" %}
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[import:19-33, lang:"haskell"](code/haskell/gaussianElimination.hs)
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[import:42-42, lang:"haskell"](code/haskell/gaussianElimination.hs)
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[import:1-45, lang:"julia"](code/julia/gaussian_elimination.jl)
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{% sample lang="c" %}
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[import:15-48, lang:"c_cpp"](code/c/gaussian_elimination.c)
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{% sample lang="cpp" %}
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[import:11-67, lang:"c_cpp"](code/c++/gaussian_elimination.cpp)
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{% sample lang="rs" %}
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[import:41-78, lang:"rust"](code/rust/gaussian_elimination.rs)
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{% sample lang="hs" %}
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[import:67-93, lang:"julia"](code/julia/gaussian_elimination.jl)
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{% sample lang="c" %}
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[import:64-82, lang:"c_cpp"](code/c/gaussian_elimination.c)
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{% sample lang="cpp" %}
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[import:70-83, lang:"c_cpp"](code/c++/gaussian_elimination.cpp)
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{% sample lang="rs" %}
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This code does not exist yet in rust, so here's Julia code (sorry for the inconvenience)
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[import:67-93, lang:"julia"](code/julia/gaussian_elimination.jl)
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[import:47-64, lang:"julia"](code/julia/gaussian_elimination.jl)
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{% sample lang="c" %}
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[import:50-62, lang:"c_cpp"](code/c/gaussian_elimination.c)
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{% sample lang="cpp" %}
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[import:86-99, lang:"c_cpp"](code/c++/gaussian_elimination.cpp)
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{% sample lang="rs" %}
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[import:79-94, lang:"rust"](code/rust/gaussian_elimination.rs)
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{% sample lang="hs" %}
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## Visual Representation
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We have thus far used Gaussian elimination as a method to solve a system of equations; however, there is often a much easier way to find a similar solution simply by plotting each row in our matrix.
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We have thus far used Gaussian elimination as a method to solve a system of equations; however, there is often a much easier way to find a similar solution simply by plotting each row in our matrix.
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For the case of 2 equations and 2 unknowns, we would plot the two lines corresponding to each equation and the $$(x, y)$$ location of their point of intersection would be the solution for $$x$$ and $$y$$.
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Similarly, for the case of 3 equations and 3 unknowns, we would plot 3 planes and the $$(x, y, z)$$ location of their point of intersection would be the solution for $$x$$, $$y$$, and $$z$$.
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[import, lang:"julia"](code/julia/gaussian_elimination.jl)
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{% sample lang="c" %}
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[import, lang:"c_cpp"](code/c/gaussian_elimination.c)
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{% sample lang="cpp" %}
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[import, lang:"c_cpp"](code/c++/gaussian_elimination.cpp)
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{% sample lang="rs" %}
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[import, lang:"rust"](code/rust/gaussian_elimination.rs)
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{% sample lang="hs" %}

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