Final Answer:
Applying Gaussian elimination with partial pivoting and back substitution to solve Ax = b provides the exact solution, x-hat (^), for the nearby problem.
Step-by-step explanation:
Gaussian elimination with partial pivoting is a robust method used to solve systems of linear equations, represented as Ax = b. The process involves transforming the augmented matrix [A | b] through row operations to an upper triangular form. Partial pivoting helps avoid division by very small numbers and reduces numerical errors by interchanging rows to place the largest pivot element in a column at the diagonal position.
After this elimination process, back substitution is applied to solve for x in the upper triangular matrix by starting from the last equation and working upwards. However, it's important to note that while Gaussian elimination with partial pivoting is highly accurate in solving linear systems, it is not immune to numerical errors when dealing with matrices that are ill-conditioned or close to singularity.
In such cases, the computed solution x-hat (^) may slightly deviate from the exact solution due to the limitations of floating-point arithmetic and numerical precision. These deviations occur due to rounding errors or inaccuracies introduced during the computation process, particularly in divisions or operations involving very small or large numbers.
Nevertheless, in general scenarios where the matrix is well-conditioned and the operations are executed accurately, Gaussian elimination with partial pivoting followed by back substitution yields a highly accurate and reliable solution for the nearby problem Ax = b. It remains a fundamental and widely used technique in numerical linear algebra for solving systems of linear equations.