# Matrix Algorithms

### Matrix Algorithms

This tool offers algorithms for matrices, in particular, one can compute a QR decomposition of a matrix, compute its eigenvalues, its inverse matrix, and solve the corresponding homogeneous or heterogeneous linear equation system. For a given matrix A, the homogeneous linear equation system is A*x=0 where the heterogenous one is A'*x=b where A=(A'|b).

 matrix 3/4 5/6 4/3 8/7 3 3 1/2 6 2 1 1 2 4 3 1 2 action DecomposeQR DecomposeQRC EigenvaluesShift EigenvaluesDoubleShift Inverse GaussJordan LinearEquationSystem

### Roots of Polynomials

With the tool below, one can compute the roots of a polynomial. The tool computes the companion matrix of the polynomial, i.e., a matrix whose characteristic polynomial is the given polynomial (after normalizing its leading coefficient to 1). The roots are then computed as eigenvalues of that matrix. The input are the coefficients of the polynomial; i.e., x^4-10x^3+35x^2-50x+24 in the case below:

 polynomial 1 -10 35 -50 24 action EigenvaluesShift EigenvaluesDoubleShift

### Linear Inequalities

This tool implements algorithms for checking satisfiability of a given set of linear inequalities: Given linear inequalities over rational numbers with lower and upper bounds (that can be also -oo and +oo), it will check whether there is a solution of rational numbers. To check satisfiability, one can choose between the Simplex algorithm (which first computes a solution on the rationals and then narrows it to an integer solution) and the Fourier-Motzkin variable elimination method.

 matrix -oo <= 5 1 <= 5 -oo <= 2 1 <= 4 -oo <= 1/2 1 <= 2 -oo <= 1/5 1 <= 1 1 <= 1/2 1 <= +oo 3/2 <= 1 1 <= +oo compute rational solution using the Simplex algorithm compute integer solution using the Simplex algorithm compute rational solution using Fourier-Motzkin elimination