Numerical Analysis Questions Medium
Gaussian elimination is a widely used method in numerical linear algebra for solving systems of linear equations. It is an algorithm that transforms a system of linear equations into an equivalent system that is easier to solve.
The process begins by representing the system of equations as an augmented matrix, where the coefficients of the variables are arranged in a rectangular array along with the constants on the right-hand side. The goal is to transform this matrix into an upper triangular form, where all the elements below the main diagonal are zero.
The algorithm proceeds by performing a series of elementary row operations on the augmented matrix. These operations include multiplying a row by a nonzero scalar, adding or subtracting one row from another, and swapping rows. The objective is to eliminate the coefficients below the main diagonal by subtracting appropriate multiples of one row from another.
By applying these row operations systematically, the augmented matrix is transformed into an upper triangular form. This process is known as forward elimination. Once the upper triangular form is obtained, the system of equations can be easily solved by back substitution, starting from the last equation and working upwards.
Gaussian elimination is a powerful method because it guarantees a unique solution if one exists. It is also computationally efficient, with a time complexity of O(n^3), where n is the number of variables or equations. However, it may encounter numerical stability issues when dealing with ill-conditioned systems or round-off errors.
Overall, Gaussian elimination is a fundamental technique in numerical linear algebra that allows us to solve systems of linear equations efficiently and accurately.