Numerical linear systems

Objective. Use elimination while controlling stability.

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Learn: the key idea

Numerical linear systems are solved on computers using Gaussian elimination, reducing Ax=b to upper-triangular form via row operations, then back-substituting. Partial pivoting swaps rows to put the largest available pivot on the diagonal at each step, preventing division by very small numbers which would amplify rounding error. For the system 0.0001x + y = 1 and x + y = 2, eliminating without pivoting divides by 0.0001 and destroys accuracy, while swapping rows first keeps the pivot large and stable. Stability, not just correctness in exact arithmetic, is the key concern in numerical linear algebra.

Worked example

Use Gaussian elimination to solve the numerical linear system x + y = 5 and 2x + 3y = 13.

1 $From equation 1: x = 5 - y$
2 $Substitute into equation 2: 2(5-y)+3y=13$
3 $Simplify: 10 + y = 13, so y = 3$
4 $Back-substitute: x = 5 - 3 = 2$

Answer: x = 2, y = 3

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Common mistake. Performing Gaussian elimination without pivoting when a diagonal entry is very small; dividing by a tiny pivot massively amplifies rounding error, even though the algorithm is mathematically correct in exact arithmetic.

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