MSolver is designed as an educational tool, aiming to show how to solve systems of linear equations by Gauss Elimination or to compute matrix determinants (with a help of Laplace Expansion). Tool may be adapted to show solutions to the problem in various contexts.
To install it, clone or download the repository and compile files m_solver.cpp and determinant.cpp. Compilation requires C++11 standard or newer.
Below we describe basic usages of MSolver with examples. Please note, that the actual output is more verbose and presents subsequent operations required to archieve described results. In the examples, only the final result of computation is presented.
m_solver.cpp contains an implementation of Gaussian Elimination algorithm, explainig the steps executed along the computation. Input specification is as follows:
- number of equations (E);
- number of variables (V);
- number of augmented columns (A);
- coefficient matrix (dim E x (V + A));
Number format is either an integer or a rational of the form p/q.
Augmented columns may contain polynomial symbolic expressions, using single-lettered variable names. Exponent is specified via ^ symbol. For example, a correct token could be -1/2+2p^2-3/2pq.
A single token cannot contain whitespaces to avoid ambiguity.
Example usages:
To solve a linear system:
2x + 3y - z = 4
x + y + 2z = 1
-x + 3z = 1
A correct input would be:
3
3
1
2 3 -1 4
1 1 2 1
-1 0 3 1
Producing result of
1 0 0 -1
0 1 0 2
0 0 1 0
Meaning, that (x, y, z) we search for is (-1, 2, 0).
To compute rank of a matrix:
| 2 1 -1 0 -2 |
M = | 4 3 -1 2 -4 |
| 1 1 0 1 -1 |
| 0 1 1 2 0 |
A correct input would be:
4
5
0
2 1 -1 0 -2
4 3 -1 2 -4
1 1 0 1 -1
0 1 1 2 0
Producing result of
1 1 0 1 -1
0 1 1 2 0
0 0 0 0 0
0 0 0 0 0
Meaning, that the rank of the matrix is 2.
Compute the inverse of a matrix:
| 0.25 -0.25 -0.5 |
M = | 0.25 0.75 1.5 |
| -0.25 -0.75 -2.5 |
An input of a form would be apropriate:
3
3
3
1/4 -1/4 -1/2 1 0 0
1/4 3/4 3/2 0 1 0
-1/4 -3/4 -5/2 0 0 1
Producing result of a form:
1 0 0 3 1 0
0 1 0 -1 3 2
0 0 1 0 -1 -1
To represent vectors (1, 1, 1, 0), (1, 2, 0, -1) and (2, 0.5, 0.5, -0.5) in a basis consisting of (2, 1, -1, 0), (1, 1, 0, 1), (0, 0, 1, -1) and (1, -1, 1, -1), we can use an input of a from:
4
4
3
2 1 0 1 1 1 2
1 1 0 -1 1 2 1/2
-1 0 1 1 1 0 1/2
0 1 -1 -1 0 -1 -1/2
yelding a result equal to
1 0 0 0 0 1 1/2
0 1 0 0 1 0 1/2
0 0 1 0 1 2 1/2
0 0 0 1 0 -1 1/2
meaning that the vectors' coordinates are (respectively) [0, 1, 1, 0], [1, 0, 2, -1] and [0.5, 0.5, 0.5, 0.5].
Given a linear map f in a matrix form, compute Ker f and Im f, where
| 1 0 -3 -1 |
M_f = | 2 -1 -4 -1 |
|-1 -1 5 2 |
This can be solved by an input:
3
4
2
1 0 -3 -1 0 x
2 -1 -4 -1 0 y
-1 -1 5 2 0 z
Producing an output:
1 0 -3 -1 0 x
0 1 -2 -1 0 2x-y
0 0 0 0 0 3x-y+z
Since last row in the variable part is zeros and rank(M_f) = 2, dim Im f = rank(M_f) = 2 and dim Ker f = 4 - dim Im f = 2. If last two variables are parameters, say p and q, Ker f is of the form (3p + q, 2p + q, p, q), thus spanned by (3, 2, 1, 0) and (1, 1, 0, 1). Im f is constrained by 0 = 3x-y+z, meaning it's of a form (x, y, y - 3x), thus spanned by (1, 0, -3) and (0, 1, 1).
determinant.cpp contains an implementation of mixture of Gaussian Elimination, Laplace Expansion and Sarus's Method. It supports polynomial elements of a form described in the previous section.
Input specification is as follows:
- the determinant's dimension (N);
- matrix of polynomials - determinant entries (dim N x N);
To compute characteristic polynomial of the matrix M, we compute det (M - tI), so for matrix:
| 1 0 0 |
M = | 1 4 10 |
| -0.5 -1.5 -4 |
We can input (please remember that polynomial tokens must not contain any whitespaces)
3
1-t 0 0
1 4-t 10
-1/2 -3/2 -4-t
Yelding result (the characteristic polynomial of M) in a form Wyznacznik D = -1+t+t^2-t^3 which can be decomposed using different methods (not implemented here) to a form -(t-1)^2 (t + 1), which means that the eigenvalues of the matrix M are 1 (with corresponding eigenspace dimension of 2) and -1.